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Home , Adriaan Fokker, Arthur Berger (composer), Benjamin Boretz, Ben Johnston (composer), Carlton Gamer, Charles Wuorinen

American S ociety o-f University Composers

Proceedings of the Second Annual Conference / April 1967 AMERICAN SOCIETY OF UNIVERSITY COMPOSERS

Proceedings of the Second Annual Conference, April, 1967

Copyright © 1969 The American Society of University Composers, Inc. clo Department of Music, Dodge Hall C<1lumbia University New York, N. Y. 10027 Subscription Information: Proceedings of The American Society of University Composers are published annually by the American Society of University Com posers, Inc. Subscriptions are available at $5.00 per year (domestic) and $6.00 per year (foreign). Back issue rate: $5.00 per volume. Subscription requests should be addressed to:

Proceedings Editor American Society of University Composers Department of Music, Columbia University New York, New York 10027, U.S.A. AMERICAN SOCIETY OF UNIVERSITY COMPOSERS

Proceedings of the Second Annual Conference, April, 1967

H eld at Washington University, St. Louis, Missouri in cooperation with the Department of Music, Washington University

The Society gratefully acknowledges the support of

THE A LICE M. DITSON F UND in assisting the publication of this volume. Founding Committee

BENJAMIN BORETZ, DONALD MARTINO,

J. K. RANDALL, CLAUDIO SPIES, HENRY WEINBERG,

PETER WESTERGAARD, CHARLES WUORINEN

National Council

ALLEN BRINGS, BARNEY CHILDS,

RANDOLPH E. COLEMAN (Chairman), DAVID EPSTEIN,

CARLTON GAMER, LOTHAR KLEIN,

DONALD MACINNIS, CLIFFORD TAYLOR

Executive Committee

ELAINE BARKIN, RICHMOND BROWNE, CHARLES DODGE,

WILLIAM HIBBARD, HUBERT S. HOWE, JR.,

BEN JOHNSTON, JOEL MANDELBAUM,

RAOUL PLESKOW, HARVEY SOLLBERGER

Editor of Proceedings

HUBERT S. HOWE, JR. CONTENTS I Proceedings of the 1967 Conference

Part I: Sung Language

9 PETER WESTERGAARD Sung Language

Part II: Performance Pr.oblems

39 MELVIN STRAUSS Performance Problems in Contemporary Music

42 JoEL CHADABE Performing Problems

45 FRED COULTER Simplification of Complex Musical Structures

48 CHARLES WUORINEN Comment on Fred Coulter

Part III: Professional Colloquium

51 PAUL A. PISK Arnold Schoenberg as Teacher

54 ROY TRAVIS Directed Motion in Webern's Piano Variations, Op. 27/II

61 JOHN ROGERS Toward a System of Rotational Arrays Part IV: Microtonal Music in America

77 LEIGH GERDINE Introductory Remarks

79 PETER YA TES Micro tones

89 BEN JOHNSTON Three Attacks on a Problem

99 LEJAREN A. HILLER, JR. Electronic Synthesis of Microtonal Music

107 JOEL MANDELBAUM The Isolation of the Microtonal Composer

113 CARL TON GAMER Deep Scales and Difference Sets in Equal-Tempered Systems

123 Other Presentations

125 Membership List, February, 1969

The music examples in this issue are reprinted by kind permission of the copyright owners as follows:

The Proceedings are printed by LIPSIUS PRESS, 460 Amsterdam Avenue, New York, N. Y.10024, U.S.A. PART I

SUNG LANGUAGE

PETER WESTERGAARD 9

SUNG LANGUAGE

I WOULD like to talk about some of the problems that come up when composers write vocal music with a text.

The problems that I have in mind are not the aesthetic ones caused by mixing up two different communications systems; ·they are the compositional ones caused by the fact that both systems depend on the articulation and perception of sequences of differentiated sounds.

That is not to say ·that both systems use the same differentiations or that the same differentiations, should they exist in both systems, would serve the same structural functions. But there is no avoiding the fact that in vocal music with a text some of the differentiations are bound to be the same for both systems and some of the struc tural functions of each will either join forces or collide.

Now while I assume that you all know all about the differentia tions among sounds and their structural functions in music, I won't 10 assume the same for language. So at the risk of appearing to treat my colleagues like a bunch of freshmen, I will devote the first half of my talk to an informal resume of some of the findings of linguistics that seem to me to be relevant to composing.

1.1 SPEECH SOUNDS

1. SPOKEN LANGUAGE

Consider for a moment your capabilities as a sound generator. If you push with your diaphragm or squeeze with the mus.cles between your ribs or just plain let your chest sag, you can force air out of your lungs, up your windpipe, through your open vocal chords, and (if you've closed off your nasal passages with your soft palate) out through your mouth. As the air moves out through the relatively wide aperture, the friction will produce a wide band of white noise of rather weak intensity [h] .1 To get a stronger signal you must limit the aperture and increase the pressure. By placing the tip of your tongue against your lower front teeth and the edges of your tongue against your back upper teeth and then almost closing your jaws you can c-reate a much smaller aperture at the front of the mouth. Air passing through this aperture will be vibrating at around 3900 c/ s and up [s]. By gradually widening the aperture and moving it back into your mouth, you can lower the bottom limit of the band to around 2000 c/s [s+SJ.2 By limiting the aperture to a thin space between the tip of your tongue and your upper front teeth you can get another kind of band

1 Square brackets around a symbol mean the sound indicated by the sym bol. One of the advantages of giving this paper orally was that I could simply make the necessary sound at the appropriate place in the sentence. Anyone who reads the paper will of course have to make the sound for himself. For those not familiar with I.P.A., here are some of the symbols used in this paper that are not self-explanatory together with examples of words that use the sound they represent. [f1 ship [f IP] [8] thin [81n] [3] vision [vi;•ml [17] sing [s111] [x] loch [bxl [:J] log [l:J g] [t] bet [btt] [re] bat [bretl

2 + means a continuous change from the first sound to the second. [8]; by putting your lower lip against your upper front teeth, yet an- 11 other [f]. Since the envelope of these sounds is controlled by the breathing apparatus, the rise time tends to be pretty long - the bigger the aperture, the longer; steady states are possible for as long as you've got air. (The bigger the aperture, the more air you'll have to use.)

To get a fast attack on a high narrow band, put the front of your tongue against the ridge of your hard palate (just above your upper front teeth) so it closes off the air; build up the pressure, and then release it in a sudden puff [t]. To get a slightly wider, considerably lower band in a slightly slower attack, close off the air with your lips instead [p]. To get a still wider and lower band in a still slower attack close off the air with the very back of your tongue against the soft part of the roof of your mouth [k].

Now none of the sounds described so far has had a pitch. To get a pitch you have to close your vocal cords to a thin chink. The moving air will then set the cords in periodic vibration and itself be set into periodic vibrations. Pitches can be added to any of the foregoing noises. Because of the common envelope they are easily associated into a single complex sound [t], [d]; [p], [b]; [k], [g]; [s], [z]; [SJ, [J]; ff], [v]; [e], [8]. In each case the rise time is a little longer for the complex (or voiced) sound than it is for the straight noise (or unvoiced) sound.

To produce a pitch without a noticeable noise component you must cut down the friction by sending the periodically vibrating air through a wider aperture. But even with a wider aperture you can still shape the passage and channel the vibrating air past soft tissues or bony walls, thereby damping some partials and boosting others. Jamming the edges of your tongue against your upper molars will channel the vibrating air between the soft groove of your tongue and the hard roof of your mouth [r]; putting the tip of your tongue against the ridge just above your upper front teeth will channel the vibrating air past your even harder back upper teeth and the edges of your tongue, to produce yet another timbre [l]. To get an essen tially different kind of timbre you can open the valve to your nose and close your mouth sending the vibrating air through the soft tissues of the nasal passages: if you close off the whole resonating chamber of your mouth by closing the valve at the back of the mouth you get one timbre [>;]; if you close off the front half of your mouth (including your teeth) by putting your tongue against the ridge above your upper front teeth, you get another [n]; or, if you close off the mouth at the lips only, yet another [m]. 12 Now you can make any of these sounds as soft as you want to, but you can't make all of them equally loud. Try as you may you will never be able to get [h] or [8] anywhere near the maximum loudness of [s] or [ S]. Of the voiced sounds described so far the ones with the biggest dynamic range were the last five [r], [1], [m], · [n], and [1]. But there is another class of sound with a much greater dynamic range. If you really want to make a loud sound you've got to send the vibrating air out by more of the hard resonating cham bers of the mouth [;,].

You can vary the timbre of these sounds by changing the posi tion of your tongue and lips. Starting with the tongue forward and high you'll get a timbre with the strongest partials in the range of 270 c/ s [i]. By gradually lowering your tongue (but keeping it for ward) you can gradually raise the area of the strongest partials, or formant, and make it even stronger [i ->- re]. In the course of doing this you also are bringing into play a higher formant. For (i] it was around 2290 c/ s but very weak. As your tongue gets lower, this formant gets lower and its partials get stronger [re]. Starting at [c] with your tongue forward and about halfway down, you can move it back, gradually rounding your lips as weil [e ->- ;,] or [e ->- o]. This keeps the lower formant in more or less the same position but makes it stronger. More important, it strengthens the second formant and takes it from around 1840 c/ s at [e] down to around 840 c/ s at [i]. Now moving the tongue up (but keeping it back) and increas ing the lip rounding you can again affect the lower formant, bringing its frequency down to around 300 c/ s [;, ->- u]. The upper form ant's frequency moves around a bit and becomes somewhat weaker but at [u] it is still much stronger and much lower than at [i]. To make it disappear, move the tongue back up front and unround your lips [u + i] . (See Ex. 1)

Now you can of course produce any of these different sounds in any sequence, but the mechanics of the apparatus favor certain combinations over others. This is particularly marked in rapid sequences. You can do the six sounds [t, k, t, p, k, t] in about 2/ 3 of a second if you're a flutist, but anyone who speaks English can do the six sounds [s, t, r, 1, t, SJ in about half that time ("stritch"). Furthermore, when you do, you create, instead of a pattern of six items, a single, unified, although complex, envelope: the [s] , [t] and [r] become transient stages in a compound attack to the nucleus [1] sound, which is suddenly terminated in the compound decay [t S]. The [st] is easy to make fast and unified because both sounds are made with the same part of the tongue in the same part of the mouth; the [t] just stops the [s]. Meanwhile the sides of the tongue Example 1. Vowel Formants 13

HIGH

3310(-28) 2670 (-43) [i] 2790 (-24) [u] 950 (-19) 310(-4) 370 (-3) 3070 (-27) 2680 (-34) (I) 2480 (-23) [U] 1160 (-12) E-< z 430 (-3) 470 (-1) t:O 0 >(") ~ 2990 (-24) 2780 (-27) 2710(-34) ~ ~ [e] 2330 (-17) [A) 1400 (-10) [ ::i J 920 (-7) 610(-2) 760 ( -1) 590 (0) 2850 (-22) 2810(-28) [re] 2050(-12) [a] 1220 (-5) 850 (-1) 850 (-1)

LOW

Formant frequencies (c/s) and amplitudes (db) relative to that of the lowest formant of [::i] Women's voices, averaged. The frequencies run higher for children 3730 for [i] and lower 3220 370 for men 3010 for [i]. 2290 270

Figures from G. E. Peterson and H. L. Barney, "Control Meth ods used in a Study of Vowels," J.A.S.A. 24 (1952) 807, re printed in E. G. Richardson, Technical Aspects of Sound, Elsevier, 1953, p. 215. have gotten into position for [r] which is released instantaneously by the unstopping of the [t]. The transition of [r] to [1] is so grad ual that the second sound barely gets a chance to sound on its own before being cut off by the rapid decay enforced by [t S] but because it's easier to make louder than any other sound it seems to function as the nucleus of the whole envelope. The same factors that make it possible to articulate "stritch" very rapidly also make it possible to articulate "stritch" very slowly without losing the sense of a uni fied envelope. l 14 1.2 PHONEM ES

Now consider what happens when we think of these complex envelopes not as potential musical materials but as syllables in our own language. Take the attacks in "Sam" and "sham" or the decays in "mass" and "mash." While objectively as composers we might think of [s] and [SJ as points on a continuum of available sounds ranging from [s] to [x], i.e. [s + x], subjectively as speakers and listeners of English we conceive of them as discrete. We might say 1/3 v 3 [ s ::em] or [s::em] and still be understood to say someone's name 2/3v but if we say [ s ;:em) we run the risk of being heard to say "sham" or being taken for drunk.

Furthermore if we were to examine each word in English that has [s] in it and substitute a [ S ], we would either get a different word or a nonsense utterance. Thus we can say that [s] and [ S] are "significantly different" segments of the syllables of English or SEGMENTAL PHONEM ES.

Now listen to the differences in intensity and duration in the following: [p' p P pl].4 Such differences may be considerable in themselves, but when the sounds are used in the words "Paul," "appall," "apple," and "lap" they do not seem particularly sig nificant. The main reason for this is that the differences in the sound are consistent with the position of the sound in the envelope. We can check this out by trying other sets of words with p sounds at the same points: "pin" "opine" "open" "nip." We also find the same effect with other stops: "call," "recall," "lacquer," "lack." Further more, we find that when we try putting the end sound of "lap" at the beginning of "Paul," we can't; when we try putting the middle

1/3v s [ s l means a sound between the [s] of Sam and the [fl of sham but 2/3v closer to the [s]. Similarly [ s l means a sound between [s] and [f] but closer to [fl.

4 These are just the different p sounds isolated from the words "Paul" "appall," "apple," ad "lap." [p'] is characterized by the puff of air rushing out as the lips are suddenly opened. [pl] is characterized by the pop of the lips closing off the air stream. [p] and [P] combine these two noises in varying strengths.

------sound of "appall" or "apple" at the beginning of "Paul" or the end is of "lap" we can, but it doesn't make much difference; and finally when we put the beginning sound of "Paul" at the end of "lap" it just sounds like the same thing, said a little more emphatically. In short, what we have here are systematically distributed, but non significant varieties of a sound in our language. [p' p P pl] are called the ALLOPHONES of the phoneme / p/. We can consider a group of sounds as allophones of one phoneme if

1) they are phonetically similar

2) they are in complementary distribution (that is, in those situations where one sound normally appears, the other sounds in the group normally don't)

3) they follow the same kinds of patterns as other groups of sounds do under similar circumstances.

By careful analysis of the whole range of possible English syl lables, linguists have arrived at a relatively small number of seg mental phonemes. (Ex. 2.) Linguists are pretty much in agreement about most· of the items on this list although the slashed / 1/ is still somewhat controversial and some linguists don't think [t S] needs its own phoneme. I think you'll find that the consonants are pretty well self-explanatory but the vowels are not. For one thing, you'll notice that the only vowels included are the ones popularly known as the "short" vowels. Now the duration of vowels does vary, but these variations are systematic rather than significant. Take the [I] in "hit," "hid," "hiss," and "his." Obviously it gets longer. But so does the [A] in "but," "bud," "bus," and "buzz." These varia tions in duration are of course just allophones of the vowel in ques tion and are determined by the envelope characteristics of the following consonant: short for unvoiced stops, a bit longer for voiced stops or nasals, longer yet for unvoiced spirants, and longest for voiced spirants.

What are popularly called "long" vowels do in fact last longer, but this is because they involve a glide from one vowel timbre to another. The exact path of a glide is hard to map, but at least you can usually say where it starts and where it ends. English vowel glides usually start at a vowel timbre that can be established as an allophone of one of the "short" vowels, but they often end higher or in between the established allophones. Compare the words "lie," "lay," and "lea." (Ex. 3.) The glide in "lie" begins at the vowel of "cot" and 16 Example 2. Segmental Phonemes of English

(unvoiced) p t k f e s s c consonants: (voiced) b d g v 0 z 3 j mn iJ lr\yhwl (stops) (spirants) (com- (semi- (pre-vocalic plex) vowels) post-vocalic off-glides) vowels: -t- i u (as in "just," (as in "gist") the adverb) (as in "put")

;J e 0 (as in "just," (as in "bet") the adjective) (as in "coat")

a: a 0 (as in "hat") (as in "cot") (as in "Jog" )

In this notation, / cl corresponds to l tf!, and / j/ to l d3! . / y/, /hi, and / w/ are explained in the text. Symbols adapted from: George L. Trager and Henry Lee Smith, Jr. An Outline of English Structure, American Council of Learned Societies, 1957. The model of the English language used in this paper is essentially the one developed by Trager and Smith.

Example 3. /YI and / w/ as Off-Glides

Front Central Back

High ~1~1 ..c: 0 ::sOil 0 Middle fl 'O

Low ·a

(the dotted lines indicate dialect variations) goes up to a point between the vowels of "bet" and "gist." The glide 17 in "lay" begins at the vowel of "bet" and goes up to the vowel of "gist." The glide in "lea" begins at the vowel of "gist" and goes up to the /i/ of French or Italian - that is, to a higher position than any English simple vowel. It should be easy to see how these glide des tinations can be considered as allophones of a single phoneme: they are all front, all pretty high (certainly all higher than the starting point), and all follow the simple rule: the higher the starting point the higher the destination (and, incidentally, the faster it is reached). Furthermore, when we compare these destinations to the vowel-like sounds used as initial consonants in "yip," "yet," or "yacht," we find the same vowel-timbres but in reverse order. Since these sounds are not used in anything else that sounds like a final consonant we can conclude that they too are allophones of the single phoneme /y/. A similar process leads to a back lip-rounded phoneme / w I which can function either as a glide destination "now," "dough," "do," or as an initial consonant "wagon," "woe," "woo," and to a central, unrounded phoneme /h/ which can function either as a glide destination "burn," "barn," "born," or, with a little [h], as an initial consonant "hut," "hot," "hone."

These segmental phonemes I have been talking about are the minimal timbral units to be significantly differentiated in English. But other kinds of differentiations are both available and significant.

Consider the contrast in the loudness of otherwise similar syllables in the following examples: "He constructs phonemes" : "Phonemes are constructs" "Come on, let's go" : That's just a come on" "I like Jane's dress" : "I like Jane's dress"

Or the effect of the slight pause between syllables in the second of the next two items: "nitrate" : "night-rate"

Or the syllable-segment paused on in the next two: "ice cream" : "I scream"

Or the length of pause in the next two pairs of items: "Come on up here" : "Come on, up here" "I told him I had to" : "I told him, I had to" 18 Or the pitch in the next pair:

23 1 231 3 12231 "I told him. I had to." : "I told him I had to."

Or the pitch on this pair:

4- 1 1 4- 1 I 1- 4 ~ 1-4 4 "I - like Jane's dress!" "I - like Jane's dress?"

Such differentiations become significant when applied to syllables or sequences of syllables but seem to act independently of the parti cular segmental phonemes involved. They are hence called the SUPRASEGMENTAL PHONEMES of the language. (Ex. 4.) Linguists are not in general agreement about them; still in dispute are such matters as the number of phonemes in each category, the range covered by each phoneme, and even the assumption that pitch phonemes can be handled in levels at all ..

Many of the difficulties stem from the fact that different kinds of differentiation may contribute to what seems to be the same phon eme: Stresses are normally differentiated by loudness but a change 2 3 ' ' in pitch often contributes and may even function by itself ("I like 2 1 ' ' Jane's dress."). A stress can be intensified without increasing the loudness by pausing on the syllable before ("I - like Jane's dress."). Terminal junctures are normally differentiated by length of pause but pitch direction preceding the pause is almost always a contribut ing factor (steady for the shortest (I), rising for the middle ( U) and falling for the longest ( #)), and under certain circumstances may actually be the dominant factor.

Example 4. Suprasegmental Phonemes of English

Stresses (from weakest to strongest) : \ A I Junctures (from shortest to longest): + II # Pitches (from lowest to highest) : 1 2 3 4

(sy mbols adapted from Trager and Smith, op. cit.) Are these then just allophones of the same phoneme? The first 19 criterion for allophones is that the sounds be phonetically similar. In fact it seems that the only reason for grouping these phenomena into a single phoneme would be that they serve a common function. But questions of function have no place at the phonemic level of structure. It seems to me simpler then to consider the phonemes as ranges of loudness, pause, and pitch, and handle the problem of their interaction at a level of linguistic structure where we can start talking about function - in short, at the next level : the morphemic.

1. 3 MORPHEMES

Take the item "constructs." We recognize it as a word because we have heard it and many structurally similar words many times before. More precisely we recognize a sequence of segmental phon emes which form a familiar base followed by another sequence (in this case, the simple phoneme I s/ ) forming a familiar affix that tells us that the item, if a noun, is plural; furthermore we recognize a familiar sequence of stresses - primary-tertiary - applied to these syllables, indicating that the item is the noun. Each type of sequence is familiar in that it is recurrent in the language; such recurrent se quences are the MORPHEMES of the language. The base and the affix are segmental morphemes; the stress sequence is a supraseg mental morpheme. With the exception of bases, most morphemes have a number of ALLOMORPHS. Allomorphs depend on the same kind of structural criteria as allophones except that the first, the one requiring similarity of sound, is missing. For example, here are the allomorphs of a segmental morpheme, the plural affix: -s after bases ending in / p t k f 8/ -;z (or -z) after bases ending in I s z J 3 c j/ -z elsewhere except in the following irregular instances: die: dice, house: houses, knife: knives, ox: oxen, child: children, sheep: sheep, etc. (The list is long but finite.)

Suprasegmental allomorphs will be clearer if we use another example.

Compare: "Come on, let's go" and "That's just a come-on." We have here two contrasting stress patterns, tertiary-primary indi cating a verbal use of the phrase, and primary-tertiary indicating a nominal use of the phrase. These patterns can be applied to other similarly constructed phrases with the same results: "come down" I "come-down," "come back"/ "come-back," "sit down" / "sit-down,'' 20 "get up"/"get-up,'' and hence can be considered morphemes. Possible allomorphs become evident when we compare the minimum different iation in loudness required to stress the first syllable of "come-on" in the following:

2 2 3 l 3 - gradually lower - 2 - l "That's just a come-on" I "That's just a come-on"

Where the syllable is higher than the surrounding ones it doesn't need to be that much louder. Where it's just another stage in a gradual descent, it does. But the syntax of "come-on" can be kept equally clear in both cases.

Next, consider the effect of the pause and the relative strength of the two principal stresses in these two items:

3 2 3- 1 1- 2 2 3- 1 \ {\ \ I ' ' ' Come on up here Come on I I up here

So long as we retain enough force on the "on,'' we can shorten the pause:

3 1 2 3- 1 \ ' . . Come on I up here or transfer it to the "on":

3 1-, 2 2 3-, 1 ' . Come on I up here and still retain the syntax of the second item. But if we make the "on" as short as the "come" or the "up," no amount of force will retain that syntax:

3 2 3-1 \ . , Come on + up-here

In this case, even if we change the pitch of "on" to that of the beginning of "here" : 3--1 21 Come on + up here

we still don't get a clear indication of the second syntax.

There are two points to be drawn: the first is that while syntax morphemes may have a normal form involving just stress or just pause phonemes, they almost always have allomorphs in which an other category is introduced, thereby changing the whole balance of the mechanism; the second is that there are limits to the extent to which one category can compensate for another - there's nothing like a pause to separate items; there's nothing like loudness to con centrate stress.

So far all my examples have shown pitch affecting syntax indi rectly by the way in which it can contribute (or not contribute) to stress. But pitch patterns can affect syntax directly as well. In both of the following, a long pause separates the two halves:

2 3 1 2 3 1 3 1 2 2 3 1 I told him. I had to. I told him. . . I had to.

In both examples the pause is long enough to separate one sentence from another, but the pitch contours make it clear that the first example consists of two simple sentences while the second consists of one complex sentence. The "I had ·to" was the same in both cases and could stand by itself:

2 3 I I had to.

It has the same pattern as the first "I told him.":

2 3 1 I told him.

When the second "I told him" is made to stand by itself it makes us expect a continuation:

3 1 2 I told him.

Furthermore, when we apply the same pitch contours to other word sequences we get either a syntax contrast of the same sort or some- 22 thing that doesn't sound quite like English.

We can use the same pitch contour on word sequences with more syllables as long as the principal stresses are lined up with the same part of the contour:

3 1 2 2 3 1 T explained to him it was necessary

1 3 1 2 3 1 I explained to him. It was necessary.

if I say "it was necessary to do it," the principal sentence stress (In "it was necessary" the lexical stress morpheme ,_,_ or ' - - - is retained because the principal sentence stress falls on this word; but has been shifted to "do" and so word stress is modified to · ---. )

The particular pitch given to the weak syllables doesn't seem to make much difference:

3 3 112 3 21 123 11 1 2 I explained to him. I explained to him. I explained to him.

so long as they don't break out of the contour.

3 4 1 1 2 I explained to him.

(In this last case it's not so much the syntax that's altered as the attitude of the speaker. I'll return to this function of pitch contours in a moment but first let me summarize from the previous examples: it should be evident that the pitch contours 3 1 2 and 2 3 1 with principal stresses on the middle element in both cases are syntax morphemes.)

Now compare the effect of stress position in the following :

2 (2) 2 2 3-1 ' I like Jane's dress I like Jane's dress.

The syntax is the same (both have a 2 3 1 contour) but in the second example the function of the position of the principal stress (and the top pitch) has been to communicate the speaker's attitude towards the separate items in the utterance - something that could not be communicated by syntax alone. This is called a shift morpheme in that it deperids on the shifting of normal syntactical stress patterns. Allomorphs due to pitch: 23

2 3- 1 ' ' I like Jane's dress or to pause:

3 3- 1 ' I like - Jane's dress can occur. (The latter example also carries an implication of hesita tion or deliberation.) The combination of both can produce complex shifts:

3-1 3- 1 ' I - like Jane's dress

However, when we invert the pitch contours of this last example something quite different about the speaker's attitude is communi cated:

1- 3 1- 3 I like Jane's dress?

Here the attitude of the speaker towards the whole utterance has been altered: he is presumably quoting someone else and questioning the meaning of what has been said. If the same pitch contour is in tensified by making the high's higher a further implication of attitude (toward Jane, her dress, or perhaps the someone who said the speaker liked it) is suggested:

l - 4 l --4 4 ' ' I like Jane's dress.

We're now in a position to summarize the principal functions of the different categories of sound differentiation in our language. The function of the minimal timbral limits - the segmental pho nemes - is to form the familiar sequences which, when given stress panerns, form WORDS. Words are put into sequence divided into a hierarchy of syntactical units by pauses of various sizes. The

~ ------~ 24 separate units are subjected to various stress and pitch patterns which indicate their syntactical function in the whole utterance. Certain alterations of these patterns can be used to indicate the attitude of the speaker towards what he's saying or even towards his listener.

Example 5. Summary of Functions of Phonemes at the Morphemic Level

segmental phonemes stress pause pitch

WORDS words Iw ords (lexical stress only) ( + juncture only) - SYNTAX SYNTAX syntax (function) (grouping) (function)

attitude attitude ATTITUDE (shift morpheme) (hesitation)

2. SUNG LANGUAGE

Now to the real problem at hand: What happens to language when it is sung instead of spoken and how can we control what hap pens compositionally? I will deal with the problem at the phonemic level of structure first and ask two questions: What are the differences between the sounds of sung words and spoken words? Can the differ ences be subjected to compositional controls to make them systema tically distributed - in short to make the sung sounds legitimate allophones of the phonemes of the spoken language? If they can, then one possibility would be to arrange these compositional allophones according to the morphemes of the language. Finally I will look into the possibility that there might also be purely musical patterns which could have an analogous structural effect on sung sounds as do the morphemes on spoken sounds. If there are, then a second possibility exists: compositional allomorphs.

2.1 SUNG SOUNDS

Compare the sounds of the following:

2 3 I I I like Jane's dress 25 J J Jane's dress.

Two gross differences are obvious: the sung version is higher and slower. I could of course sing it at approximately the same pitch level and tempo, but the higher, slower version makes certain fine differences easier to hear. The fine differences may be summarized as a freezing of the transients of each syllable at the vowel and on one pitch. Take the item:

; J II Jane's

The [d] and the [J] combine to form [j] and move into the [e] as in the spoken example except that the initial pitch of [j] is about the same while the pitch of [e] is higher so the pitch glide covers more territory in the same amount of time.

At the [e] however, the transition stops - frozen at the timbre [e] and the pitch G - and there it stays, until that moment when there'll be just time enough to get in the remaining three sounds of "Jane's" and the first two sounds of "dress" before the [e] of "dress" comes in, equally spaced with the beginnings of the vowels of the first three words. The remaining three sounds of "Jane's" are in speech-like transition as far as time is concerned, but the pitch of both off-glide and nasal stay at the pitch of the [e] while the [J] drops below. Thus the entire additional length of the sung version is to be found in the steady-state section of the vowel.

2.2 COMPOSITIONAL ALLOPHONES

2.21. SEGMENTAL ALLOPHONES

Can we then consider the various stages in the envelopes of our example allophones of the segmental phonemes of speech? When we check against the three criteria for allophones we find the following:

1) phonetic similarity: no great problem here; with the exception of the steady-state section of the vowel, all the separate 26 sounds have pretty much the same formant and envelope charac teristics as their cousins in speech. Obviously had I attempted to make a sharp attack on the second note or a gradual decay, this condition would not hold. The same would be true if a soprano attempted to sing "like" at a pitch above the lower for mant of the off-glide destination:

f f f l tike Jane's dress..

2) complementary distribution: Again, no problem. Those characteristics special to sung language don't have any other function in spoken language, and so are free to function here. Furthermore if in the midst of singing I briefly use some charac teristics of spoken language such as

~ J d J J II B [a..,i] like Jane's dress.

or even

[spoken] J J J Like Jane's dress.

the effect is simply emphatic - as in the possibility of using the /p/ of "Paul" at the end of "lap."

3) pattern congruity: Here is where the problems are. Note the inconsistencies. Some of the consonants glide in pitch and some don't. The four vowels are of four different lengths; three are long enough to be frozen at a steady state but the one in "like" glides off immediately as in speech. Now obviously to show pattern congruity we would have to have a great deal more data than this example. But even with this little we can at least indicate some of the constraints necessary to maintain pattern congruity.

For the pitches of consonants these constraints would include the following: off-glides following vowels, and semi-vowels following J vowels or off-glides either stay on the pitch of the preceding vowel 27 or glide to the pitch of the next vowel. Initial voiced consonants start in the pitch area of speech and glide to the pitch of the vowel. Final voiced stops and spirants glide from the pitch of the vowel back to the pitch area of speech.

Since we don't usually notate the pitch of consonants, we in a sense leave these constraints to the performer; normally, then, pattern congruity for the pitch of consonants and off-glides depends on the competence of the singer. On the other hand, once we start notating these sounds as separate elements - as some composers do nowa days - the responsibility for pattern congruity becomes ours and a new system of systematic constraints has to be set up.

For the durations of vowels these constraints could be sum marized as follows: the total duration of the sung vowel is a function (roughly exponential) of the duration of the spoken vowel. You'll remember that the durations of spoken vowels vary according to the consonant that follows: unvoiced stops produce the shortest vowel, then voiced stops and nasals, then unvoiced spirants, then voiced spirants, and finally - as may be the case with so-called "long vowels," the longest of all - no consonant.

If we consider the shortest vowel to have the value of one, applying an exponential function would ·simply mean that the shortest spoken vowel would have the same duration when sung, that a spoken vowel of middle duration would be somewhat longer when sung, and that a long spoken vowel would be much longer when sung.

Since we do usually notate the durations from the beginning of one vowel to the beginning of the next and since we often notate envefope characteristics as well, these constraints are our business; normally then, pattern-congruity for vowel duration depends on the competence of the composer. Now in our particular example it hap pens that we could notate the durations as four quarter notes without any articulation signs and get the same results from any competent singer: "I" legato, to "like" staccato, and no gap between "Jane's" and "dress" (the only articulation coming from the pitch and timbre changes on the [z] and the [dr]) . But this is true only because the systematically determined exaggerations of the spoken vowel lengths happen to be possible within the frame of steady quarter notes: Spoken, the vowel of "I" would be at the long end of the scale; sung it becomes much longer, lasting almost the whole quarter. Spoken, the vowel of "like" is at the short end of the scale; sung, it has the same duration; the extrn time is given to silence. Spoken, the vowel 28 of "Jane's" is in the middle of the scale; sung it becomes slightly lorrger, the remaining time being taken up by the consonant dusters [nz] and [dr]. If, on the other hand, the composer decides he has to have a dotted half, tenuto, for his high note, there is nothing the singer can do to prevent a non-systematic distortion.

f" j J57l I like Jane's dress.

2.22 SUPRASEGMENTAL ALLOPHONES

Given the kind of constraints we've been talking about, it should be possible for composer and singer to arrive at satisfactory allo phones for the segmental phonemes of the spoken languages. Can we do the same for the suprasegmental phonemes? Can we establish in a piece of music levels of loudness at peak of accent, duration between attacks, and pitch which could be considered allophones of the phonemes of stress, juncture, and pitch? Put this way the question seems to answer itself. It sounds as though all we have to do is think up symbols for four accent levels, choose four durations and four pitches that we'd like to hear over and over again, and then just sit back and let the phonemes of the text write the piece for us.

Now obviously such a piece would not only be pretty boring, but pretty silly as well. The problem here lies not in the mechanism of transfer from the sounds of speaking to the sounds of singing; that's easy to handle: Even if we take the phonetically least similiar category, pitch, and subject it to the three criteria for allophones, we would have no trouble showing that while the pitch glides during the spoken vowel and stays put during the sung vowel, phonemically the pitch of the spoken vowel is conceived of as the discrete pitch at the beginning of the vowel, and thus the steady pitch of the sung vowel is simply a systematic intensification of that phoneme.

The problem here lies in our assumptions about the mechanism at either end of the transfer. At the language end, we may very well conceive of the four phonemes in each category as four discrete levels but we must not forget that as sounds they are four ranges, not levels, and indeed they're not even bounded ranges but overlapping ones. (A loud tertiary stress may well be louder than the weakest secondary stress; a long middle terminal juncture may be longer than the shortest final juncture; a high 2 may be higher than the lowest 3. These seeming contradictions are possible because of the way in 29 which the three categories of suprasegmental phonemes interact.) At the music end, we may conceive of accent, duration, and pitch inde pendently, but there is no reason to suppose that they can perform their usual functions independently.

We must find if the suppposed allophones interact in the same way in sung language. If they do, it would seem that the allophones could also consist of overlapping ranges instead of discrete levels.

2 3 1 1 To go back to our example : "I like Jane's dress" and

j II tike Jane's dress.

In neither case is there any doubt about where the primary stress is. The sung "like" may be louder than the poken "like" but both have the same relation to their surroundings - · that is, neither needs to be that much louder than the syllables before and after because it's slightly higher in pitch. If we want the same pitches but we want primary stresses to fall on the first and third notes we must compen sate by giving them a considerably stronger accent:

> J J tlke Jane's dress.

2 3 ' But the same would have been true in the spoken version ("I like

Jane's dress) . On the other hand, if we want primary stresses to fall on the first and third notes but we don't want to have to go in for such strong accents, then we have to change the pitches:

j j I lt!TUt'~; dress. 2-1 2 3 1 30 Compare the spoken version: I like Jane's dress

In short, the effect of pitch on stress for sung language seems to follow much the same rules as for spoken language. Much the same results can be found when we experiment with pause and pitch, and with pause and stress.

Such experiments lead me to believe that we must consider the sung allophones of the suprasegmental phonemes of the language to be sets of overlapping ranges; in each case the particular point on that range would be determined, as in the spoken language, by the interaction of the three suprasegmental categories. In other words, the phonemes of the spoken language become compositional condi tions, instead of compositional determinants. It seems to me that we could at least begin to write music under such circumstances.

2.3 COMPOSITIONAL MORPHEMES

We're now ready to move to the morphemic level of structure.

2.31 SUNG ALLOPHONES IN THE MORPHEMIC SEQUENCES OF THE SPOKEN LANGUAGE.

Two questions remain to be answered. The first is easy. If the sung allophones are arranged in the sequences of the morphemes of the language, will the results be intelligible? Compare

J j $ J J j II totd him. I had to.

with

j j j J j told him I · had to.

or 31 FJ A 011 s I ex - plained to him. I t was-nee-es- sar-y. with

~n J Fl El A o j I ex.-ptained to him, it was nec-es-sar-y. or even

45J J •£9 ,El £iJ -£1 11 I ex - plained to him it was nec-~s-sar-y.

Or compare:

fy j j j j II I ti'k.e Jane's dress.

with

~~ j I tlke Jane's dress. or

> > j ~ ~ 'I® tlf

~ j= ~ .l ~jJ tike Jane's .dress? 32 or even

r like Jane's dress?

It seems to me that the change in meaning is clear in each case, and is the same change we found in our investigation of syntax and attitude morphemes in the spoken language. Indeed, I think we can even go so far as to say that while there may be some loss of intel ligibility for segmental morphemes in the sung version there is virtually none for the suprasegmental morphemes. The reasons for this seem to have to do with the phenomenon of freezing the syllable at the vowel. This does two things: it breaks up the sense of the syllable as a whole, and it destroys the balance in sound between vowels and consonants. In the spoken language it's the consonants that carry the bulk of information at the segmental level. Now the vowels may not carry much segmental information, but it's the vowels that get the stresses, that get paused on, and that get the significant p1tches. Obviously, exaggerating the vowels at the expense of the consonants is not going to impair the effect of the suprasegmental morphemes: on the contrary, it may even intensify or clarify it.

2.32 COMPOSITIONAL ALLOMORPHS

Now to the final question, that of compositional allomorphs. Obviously, vocal music, to have any interest as music, would have to include structures other than those determined by the processes I have outlined. What about these structures? Will they have an effect on the morphemes of the language? Are any of them closely enough related to the morphemes to intensify them? Are any of them strong enough to project what the morpheme projects without imitating the sound-structure of the morpheme?

Before we consider such structures it might be well to consider what kinds of sound differentiations are available to project them. It may seem that we've already used up all the available differentia tions in constructing our allophones. But, we have not yet dis cussed the possibility of fine gradations in pitch and duration, nor have we discussed ways in which sounds of the vocal line might form patterns with sounds of the accompaniment. Compare the followinl!: 33

d j Uf

Note that once 'a metrical frame is established, there is no need to go in for the extreme accents discussed earlier: the shift morpheme is just as clear. Rhythms within a metric frame are more than sequences of duration - they include a sense of stress which requires no extra accent to project it. Clearly, such rhythms can be used ,al) allomorphs of stress morphemes. ·

Or compare the following:

I like Jane's dress.

and

If\ I I I

~ I tik.e J'ane's dress. . -

In the first we would conclude that the stress is on "Jane's" because that is the only differentiated pitch and there is no difference in the loudness. In the second, our past experience with suspension patterns 34 makes us change our minds. In other words certain kinds of struc tures using finer differentiations of pitch than speech contours may imply stress sequences other than those implied by the contours.

Clearly such structures could be used as allomorphs.

Compare the following:

~ ~~ ~ ~ ~ ~ J s I totd him. I had to.

and

fyj J *J ,. J j J II I told liim. I had to.

If contour were the only consideration the syntax would be identical. In the second example, however, the syntactic state of nonfinality, which in speech would have been projected by a 3-1-2 pitch sequence with less of a pause is here projected by the incomplete tonal struc ture at the end of the first phrase (as contrasted with the completed structure at the end of the last phrase).

A final pair of examples:

f JG f==1 like Jane~:; dress?

and

F ~£¥ f II tike Jane's dr'ess?

The 1-4 contour in speech usually implies unexpectedness or surprise.· At least in tonal music the interval of an octave does not share that implication. It seems to me that there exist, or could be constructed, com~ 35 positional allomorphs for all of the suprasegmental morphemes of the language. The same cannot be said for the segmental morphemes simply because of the staggering number of them. We can of course intensify the patterns of segmental morphemes by imitating them in other characteristics. Or we can take that kind of highly patterned language in which morphemic patterns are already intensified by non-morphemic correspondences - as i the case in poetry - and intensify these correspondences in musical structures.

AFTERWORD

I have been concentrating on the compositional conditions for making the language element in vocal music as intelligible as possible. But before I stop I would like to point out that this kind of tight relationship between the structure of the spoken language and the structure of the vocal line can serve not only to make the language more intelligible, but to make the music more intelligible as well. Let me show you what I mean by describing three different kinds of situations.

Those of us who still write whole movements in 4/ 4 do so, not because we want to have a nice solid downbeat every measure but because we want a frame of reference against which we can place a variety of cross accents. Those of us who prefer to follow a bar of 4/ 4 with one of 2/ 4 plus a 16th and another of 3/ 4 minus a triplet 8th are after another kind of variety, but we don't necessarily want each measure to begin with the same kind of accent. Unfortunately, our standard notation for accents is inadequate to the variety demanded in both these cases. The stress morphemes of the spoken language - not just the lexical, but the syntax and shift morphemes as well - can be powerful indicators of just the kind of accent we want. (All we have to do is educate singers to read the sentences instead of just enunciating the words. )

A second situation: Many of us are interested in changing tim bres within a single instrument in order to get patterns of differen tiated timbres, or in order to clarify the patterns of pitch, or both. We try to get instrumentalists to remake their techniques to increase the numbers of differentiations available and the speed at which they can follow one another. But we have in the segmental phonemes a far greater variety of sounds than that of any instrument. Furthermore, because of their use in language we have the guarantees that our nota tion will be understood by the performer and that the differentiations will be understood by the listener. Now the patterns formed by these 36 phonemes in the word morphemes of ordinary language may not be tight enough to interest us as composers, but, as I've just suggested, there is a more highly patterned form of language available to us - that of poetry.

Finally, a third situation: Many of us are involved in setting up new kinds of pitch-structures, structures functioning at various levels, as in tonality. Just as in a tonal piece, at any point in the course of the piece we should be able to consider the structure to be complete or incomplete at any of the various levels. But in tonal music both the intervallic materials and the techniques for using the materials at the various levels of structure are common knowledge. In our music we change the intervallic materials and sometimes the techniques for every piece. We are asking a good deal of the listener when we expect him to grasp what is going on. We can however give him at least a toehold in the structures of our music if we just have the sense to align our structures with the structures of a system he's already intuitively familiar with - the spoken language. PART II 37

PERFORMANCE PROBLEMS

MELVIN STRAUSS 39

PERFORMANCE PROBLEMS IN CONTEMPORARY MUSIC

IT SEEMS ·appropriate that we begin the discussion of "Performance Problems in Contemporary Music" by examining our terms. Our per formance problems are a separate subject from our auffiihrungspraxis. And if we are to discuss the former of these subjects, a further edit ing seems necessary: I think that what we mean by "performance problems" is really "rehearsal problems."

Some of the issues encountered in the preparation of contem porary music are what might be called socio-professional. They are probably best solved by a conductor who is also an amateur labor leader, administrator, psychiatrist, stagehand and cheer leader. Such extra-musical considerations will be excluded from this discussion.

The primary and almost univernal rehearsal problem, which we cannot dismiss, is time : sufficient time in which to conquer the tech nical challenges and then to get on whh the business of making musical sense. In the rare environments where there is sufficient time, a whole array of rehearsal problems becomes easily apparent. There is only one aspect of this problem to which I care to speak; that is, the rehearsal problems of the conductor. 40 In the first place; a conductor is not always necessary, even when specified by the composer. The need for a conductor can often be eliminated in chamber works by the presence of a "coach." Suffi cient rehearsal time will frequently enable the players to achieve a high degree of ensemble that makes the conductor get "in the way". Almost invariably the presence of a conductor will rob the players of their initiative and their sense of musical responsibility. Composers too often demand the services of a conductor when one is not really necessary, and conductors too often refuse to relinquish their cher ished podium. There is little one can imagine that is sillier than the sight of three or four players being conducted. On the other hand, the experience of hearing three or four players failing to achieve an ensemble is painful. The inconspicuous approach is often the solu-. tion:. have the ~ 'conductor" sit between the players as a page turner and, in crucif\l moments, give the necessary cues.

In works of orchestral dimension, the conductor should not be embarrassed to play the simple role of a time-beater. If one can generalize, it might be said that, unlike some music of the past, it is now frequently desirable for the maestro to become physically unobtrusive. However, this should not be interpreted to mean that any shy, inexperienced soul can now conduct. It takes a person of considerable conducting skill and experience to play the unobtrusive role and to nevertheless control the ensemble and the shape of a work. The composer of a work i.s probably t·he only desirable sub stitute for an experienced conductor.

The main function of the conductor is one that is easily forgotten. He is in the best position to listen. If he has a concept of the desired sound (a large assumption), he can be of enormous assistance to one and all. Otherwise, he can be a serious detriment and a nuisance. The fact that a good deal of our music contains chance elements does not change the point. Improvisation, for example, must still be shaped and, to one extent or another, planned and remembered.

The ideas of the composer must be exteriorized. This is the responsibility of the performer. But the conductor is also a performer. Since he is freed of the burden of producing the sounds, he is therefore able to deal with larger concepts; namely, ideas. A dogged attention to technical detail is only appropriate when ·a concept of the whole is understood. The conductor who prefers to use the bulk of his time working on technical minutia should not be asked to conduct works involving new and difficult ideas. However, profound thoughts and understanding do not insure good performances. A conductor who is not .attracted to the communication aspects of performing should find other fields of empl9yment. Taken in perspective, then, it might be noticed that the definition 41 of "conductor" has changed in our time, especially with regard to the conducting of contemporary music. It should not be long before composers can stop worrying about the Romantic type conductor who has little or no interest in or understanding for their music. He will be replaced by a member of the new breed - a more humble, less egocentric figure who will be a master of unobtrusive- ness, a shrewd assessor of rehearsal problems and a champion of contemporary music. How he will deal with his Board of Directors is quite another (but not at all unrelated) problem. 42 JOEL CHADABE

PERFORMING PROBLEMS

PERFORMING PROBLEMS invariably derive from the specific musical composition which provides the occasion, whether that composition is a graph, a collection of lines on a single page, or a detailed score with conventional signs. The success of a performance is most sig nificantly judged (or should be) by the conviction and imagination that understanding provides (taking for granted technical capacity). One performs to make music breathe.

The basic question is the extent to which freedom in performance is desirable, and this question is wholly connected to the concept of music as a score of some type. A score can be a cue for improvisation, a verbal direction, or it can be extremely specific, yet it is still a score and performance freedom is thought of as a kind of rubato, in some cases a crazy kind of rubato that has a running relationship with tradition. Traditional musical notations imply more or less traditional rubatos, but a "rubato" in Earle Brown's December 1952, for example, still exists (at least mystically) even though the music compels it. It now seems to me that the basic change in rubato relationships with scores comes with a combination of composer performer functions, namely what is now often called "improvisation." Groups that do this take away the score; composing and 43 performing become the same thing. Although this is not what I intend to discuss (because "performing" implies a division of function, an exterior score), it does seem an irresistible next step from an extreme rubato interpretation of a score, and like scores it can produce both good and bad compositions. But the definitive quality of such music is the absence of forethought, because to think a composition through, in any way, before beginning the "performance," is a type of score and one may as well think it through well.

Outside of spontaneous composition we are always dealing with the problem of freedom in performance, what some people call "interpretation." I would like to make a small observation that in some ways applies to the subject.

It seems to me that there are different types of decisions made by a composer. Some are thematic decisions that define the essential character of a composition, and some are arbitrary decisions that can be changed without affecting anything essential. The material of the composition is thematic, whether notes, dynamics, timbre, a form, a process, or whatever the composer is working with. In John Cage's 4' 33" of Silence the form (defined by time durations) is thematic, as is the silence. (What better way of throwing complete emphasis on form than to specify that no specific material be chosen? The piece is an aesthetic statement in the same way that Rauschen berg's erased de Kooning drawing is.) From my point of view, the piece presents an interesting example of performance freedom, be cause here the performer must find a way (not specified by the composer) to perform the essential character, while he is instructed not to do anything about what is not essential.

Variations IV provides a more clear example of my ,idea. Here the process is thematic and given, but the performer selects the mate rial because it is not the specific material ·that defines the piece. The performer is "free" with whatever is not thematic. (In cases where the performer is directed to make choices, this very flexibility of result, this process, is thematic and so the game is still played accord ing to rules.)

. Franco Evangelisti's piece Aleatorio illustrates well the difference between thematic decisions and arbitrary ones because Evangelisti sets up the scheme of the piece (which defines its essential character) , but sometimes suggests a solution (putting it in parenthesis) with the explanation that it is merely a suggestion by the composer and need not necessarily be done that way. 44 What I am saying is that one difference between aesthetics in composition is a difference in the amount of information the com poser gives and what he leaves to the performer. But although a difference in degree (of given information) can become a difference in kind (of composition), it is always a difference in degree.

The Boulez Structures (Book I) for two pianos is highly organ ized and, not only is the performer not offered specific freedoms, there are discrete scales of degree in dynamics and attacks in the score that simply cannot be played on the piano with accuracy. There are so many definitive qualities that they are beyond the reach of any performer. Boulez is, of course, aware of this. It is not his point to offer performers an opportunity, but rather to state the piece in all its complexity in an abstract form (the score symbolizes a composition which can never be precisely performed) and the per former does his best in a context where expressive freedom is detri mental to the concept of the piece.

The composition which gives occasion to this discus·sion, my own Diversions for two pianos, was written with these ideas in mind. It is not a serial piece and so different things at different times seemed to me to define its essential character. I always wrote all the notes. But there are many elements of duration and rhythm that will vary from performance to performance, sometimes because the player must choose (although always within the clear context of the piece) and sometimes because one piano may fade out faster than ano ther. An ensemble situation is set up where what one player does influences what the other one does, and freedoms are specifically offered.

This whole attitude implies a somewhat different look at issues such as "freedom," "indeterminacy," etc. If Cage and Boulez are part of the same continuum, then Cage is simply freer than Boulez, both can be understood in terms of the other, and we are all more victims of our history than we would like to admit. But this excludes the all-or-nothing leap-to-faith idea that a composer either makes an object or allows complete (whatever that is) indeterminacy. Adher ence to any theory, whether of indeterminacy or determinacy, is not a criterion to judge the value of music. Theories are used; they do not use. The composers offers his piece, by whatever means defi,nes it, and leaves the rest up to the performer. Freedom is always defined, in life and in art. FRED CouL TER 45

SIMPLIFICATION OF COMPLEX MUSICAL STRUCTURES

SUMMARY: A new method of analysis is proposed to simplify for the lay performer scores which, due to the complexity of the notation, obscure their musical import and consequentl y undermine the possi bility of a secure performance.

PROBLEM: After a detailed study of the "format" technique employed by Stockhausen in the Zeitmasse, Gruppen, and Klavier.9tucke XI, many of the students involved wrote music using similar techniques. Consequently polyrhythmic structures of concurrent divisions of a whole note into 7, 11, and 19 equal parts were not unusual. To fur ther complicate matters, melodic material was presented in a Klang f arben manner, moving from one rhythmic system to another with each new pitch.

Rehearsal of this music was tedious, as the performers had not experienced music beyond Debussy and Bart6k in their studio experi ence. The majority of rehearsal time was spent counting out rhythms so that practically no time was left for interpretative work. Even after a great deal of practice, the performers felt insecure as to where their pitches fit into the total complex. 46 Consequently the music had to be notated in such a manner that the performer would feel secure with the material more readily.

PROCEDURE: Two elements had to be clarified: rhythm and melody.

Rhythmic clarification was a matter of simple mathematics. Each temporal division within a given area was considered a multiplicand. The product of all multiplicands was then considered the number of temporal units (TU) in that area. Working with equal divisions of the whole note into 3, 4, and 5 parts would yield 60 TUs (3 X 4 X 5) . The TU value of any given division was found by dividing the total TU by that factor; for example, a note valued at 1/ 5 of a whole note would equal 12 TUs (60/ 5). After transforming a section into a stream of TUs, the TU realization was then translated into a non polyrhythmic mensural notation. In Example 1, part A is the original rhythmic notation, part B the TU realization, and part C the final rhythmic product.

So that the performer could see how his part fit into the total musical complex, the combined rhythm of all parts was written above his part. Below his part was given the general pitch and dynamic shape of accentuated material, its instrumentation and rhythm. The latter additions to the score functioned as melodic material. Part D of Example 1 shows one part of the total score as it appears to the per former.

RESULTS: Rehearsal time could almost solely be devoted to problems of interpretation. During his independent preparation period, the per former was able to see how his part fit into the total complex. Need less to say, he was able to play with far greater rhythmic security. The final result was a far more successful performance with an economy of rehearsal time.

CONCLUSIONS AND IMPLICATIONS: This type of analysis permits per formance of music that is too frequently ignored because of the diffi culties outlined above, and it constitutes the first step in a program to acquaint audiences with recent serialized musical thought. While it undoubtedly requires considerable extra time from the composer, an unexpected benefit from this type of simplification was an increased interest, remarkable enthusiasm, and strong desire for further ex posure to contemporary music on the part of all musicians involved in this study. 47

A r------5 -- flute -+------i11-----~1 4 4

5 ~------' 5 clarinet --•--&------1.~----~1

l_:___--- 3 ______J

4 -----~ bassoon -----R------1 I -.- 5 ------1 B bsn fl clor _ ob ob I I I I I 111111 ti II I 111 11111111 111111 111111111 111 II 1111111 11111111111 I I I I bsn bsn '-' fl ob ( clor c fl ..... ob clar --- bsn .. ~ ...... D -=:::: -:::;::::;- "" ri.... ,,.. f ~ 1... clarinet ~~fi!#~d f"'-' :1.~... N fl. :·1 '"'' I I 48 CHARLES WuoRINEN

COMMENT ON FRED COULTER

WHILE I have no objection to the presentation of a commonly em ployed solution to rhythmical and counting difficulties found in many recent scores, I would take issues with Mr. Coulter's brief report in one important respect: I feel that the role of anyone who stands in a tutorial rela.tion to people learning to play contemporary music has an obligation to reveal to those stud,ents not merely the administrative solutions to various performance problems, but - perhaps more im portantly - to attempt to come to grips with the compositional im plications of such problems. Thus, in the case of the Stockhausen works Mr. Coulter has cited, one is tempted at the very least to raise the question of the relation of the notational practice to ·the composi tional content. In the case of Mr. Stockhausen, this has already been done on many occasions, and the general conclusion arising from such investigations is that the notation is designed (in the words of a contributor to Perspectives of New Music) to "impress rather than to inform." This being so, one would think it a part of the teacher-of rhythmic-simplification-method's primary responsibility io demon strate whether the notational "difficulties" are motivated by the need to express compositional relations that are uniquely representable in that and only ·that way, or whether, on the other hand, the intent is to impress, bamboozle, and obfuscate. PART III 49

PROFESSIONAL COLLOQUIUM

PAUL A. P1sK* 51

ARNOLD SCHOENBERG AS TEACHER

THROUGH his whole creative life Schoenberg gathered students around him. In Europe he taught mostly privately except for his activity at the Berlin Hochschule fiir Musik, but in this country he was affiliated with academic institutions of higher learning, his last position being at UCLA, so that he readily qualifies as a university composer.

He wrote several textbooks, the famous Harmony (1911) and later Structural Functions of Harmony (1954), also a Counter point text, published only posthumously ( 1964), and, last but not least, the Models for Beginners in Composition for use in his uni versity classes. All these books are practical. Schoenberg objected strenuously to being called a theorist and used the well-known com parison (in the Preface of his Harmony) that a cobbler, teaching his apprentices how to put leather together, should be called a

*The author was a personal student of Arnold Schoenberg in Vienna 1916-1919. 52 "master" (not a theorist) like a composer should who teaches stu dents to put chords together. His craft should not at all be theoretical. However, his books abound in aesthetic and philosophical ideas.

Schoenberg taught two types of students, elementary ones, single or in groups, in harmony, counterpoint or beginning composition, and advanced ones in analysis, technique of composition and orchestra tion. In his harmony presentation he was an adherent of Sechter's degree-method: connection of chords on all scale degrees by proper voice-leading without preferences. Functional elements, traditional cadences, etc. were taken up much later. Therefore the students acquired from the outset a facility to use more chords than the conventional ones. In his second book however, Schoenberg deals with the intricacies of romantic harmonic practices. The counter point book adheres to the organization in species, however not modal, but tonal with immediate application to practical use in writing music. It is not "historical" at all like Fux or Jeppesen.

The "models" concentrate on the principle to fit melody and harmony together, first in very simple units of 1 or 2 measures, later in bar groups or phrases. The directions are so detailed and systematic that the student cannot help but succeed even if he has no natural feeling for musical logic. The melodic elements (all tonal) are basically triadic or scale segments, the rhythm is developed from even notes to various groupings, small motives are constructed and their repetition and manipulation shown. The harmony unfolds slowly. First melodies based on one chord, then two (already func tional as e.g. I-IV). Later more than two chords are used, and those on all scale degrees are introduced. Harmonic rhythm is considered. Schoenberg shows how to build from small elements phrases, phrase groups, sentences and periods. The repetition and variation of thematic material is taken up, also various types of "accompaniment", and the book closes with examples of small binary and ternary forms. Elaboration and dissolution of material is shown. The book contains an excellent glossary in which basic terms for analysis are lucidly explained. This section is of special importance, as our terminology for musical form elements is not unified at all.

Schoenberg's instruction of advanced students in composition was given in two directions. First correction of actual student work, secondly analysis of masterpieces. The first consisted in minute examination of every detail ("Where does this come from?" "What use did you make of this motive or chord?") By this method the intellectual processes of composing were clarified and directed. Spe cial consideration was given to structural balance between sections, symmetry, variation technique and repetition. Different methods of connection of contrasting material were explored. This lead to the 53 analysis of the musical ideas, "character" of themes and their evaluation. Problems in unity or overall organization of the piece were detected and solutions attempted. For this purpose the second aspect, analysis, was brought in. Only materials which had immediate bearing on the student's pieces were used as examples. For contrapuntal devices (and others, of course) Bach, for architectural clarity, Mozart, and for manipulation of material (developmental techniques) Beethoven and Brahms. But very often less known compositions of other masters were analyzed if their content had relation to the student's work. For devices in romantic harmony, Wagner, Bruckner and Mahler were used, the latter especially as master of variation and transformation of musical ideas. The author does not recall one instance in which Schoenberg would have used his own music as examples for the general student. He might have done it in the later phases of collaboration with his special disciples, themselves already composers of note. But he was adamant in requiring from everybody who worked with him complete mastery of the traditional techniques before he permitted experimentation with (to them) new ideas (atonality). Very often performances of the music (at that time there were no recordings yet) were an important stimulus.

If the author may summarize what he learned from the personal instruction by Arnold Schoenberg, it would be in four directions. 1) Strictest intellectual self-evaluation and self-criticism. 2) Utmost sincerity of musical writing. No compromise in using elements foreign to the personality nor unmotivated emotionalism. 3) Feeling for style in music. 4) Aesthetic judgement of real artistic values versus ephemeral inferior ones. 54 RoY TRAVIS

DIRECTED MOri'ION IN WEBERN'S PIANO VARIATIONS, OP. 27 /II

THIS BRIEF report will be an illustrated paraphrase of part of an article of mine which appeared in Perspectives of New Music in the Spring-Summer issue of 1966. Although that article dealt with two brief piano pieces of Schoenberg and Webern, I shall confine myself to only one of these, namely, the middle movement of the Webern Op. 27 Piano Variations.

Peter Westergaard has already analyzed this movement rigor ously in an excellent article which appeared in Vol. 1, No. 2 of Perspectives of New Music. There are, however, several aspects of this piece which seem to me to merit further consideration:

1. the tendency of the tones to dispose themselves in fixed sys tems of registration, like so many stars around a galactic nucleus,

2. the meaning for such systems that the idea of degree progres sion might have, and

3. the possibility of hearing the complete piece in terms of a neo-Schenkerian or neo-Salzerian structural hierarchy. 1. CONCERNING FIXED SYSTEMS OF REGISTRATION : 55

First of all, I should like to point out a rather striking similarity ~\:)tween what Mr. Westergaard calls "the inversional symmetry around A" in this piece and a fixed system of registration which appears in measures 1-25 of the Webern Symphony, Op. 21, another context in which "inversionally related row-forms are canonically disposed." Here the "axis of symmetry" (to borrow George Perle's term) is not only A (in this case the A below middle C), but every single tone within the first 25 measures is confined to the system of registration illustrated in Example 1. That system consists of the axis tone A, the tone a perfect twelfth above A, the tone a perfect twelfth below A, and a succession of perfect fourths from the two extremes of the system into the octave E-flat, a tritone away from the axis, and the only tone which appears in more than one register (v. Example 1).

The question arises whether a similar system of registration is employed in the second movement of the Op. 27 Piano Variations. In Example 2 (second stave) I have eliminated the hand-crossings in order to reveal more clearly the actual sequence of auditory events, so that a possible system of registration might become more evident.

iii (C or~rigftt i9~'l,Uh ;v • rsol Ed. it; o,._. V i • "~") 56

~ , I ~~~~~~~~~~~~ M"1 Row- 'Po .. / -•' F» '"' I ~~· ~~~~~~~~~~~~§~~~~~~~~~§

What emerges is not a single system for the entire piece, but four slightly different sub-systems corresponding to each of the four pairs of row-forms (indicated at the top of Example 2). All four sub systems have in common not only the axis of symmetry A, but the three other dyads circled (F#-C, C#-F, and G#-Bb). What is more, two other dyads (Eb-D# and D-E, each encircled by a broken line) are shared by three of the four systems, so that the following single system of registration is heard as a ldnd of "common denominator" to the four sub-systems listed at the bottom of Example 2. There is 57 I-5 .. , ) 3_ __ , ( 'p.. ~ i__ , ;,.~~ ~~ i , 'h ; /~'· '- i O ,;_~ ' .... --: 1 '.~ ;,. J li'i , , ' " l.N:f__)/';, ·' r' '"- - /~ '8\t5 '-1, , .., /,i 1., ' ' ,, / '•\l/-'•H/ \;,;5 ' ' I' /, , ' "' 'ii-- ' .I'. b.._, 'I '·' /\,~ - - ·,. ,

9H 1~ i r ri t ~~" ~ llit ,,_, I

7 IC l f J,,r( _:~ 1f H~ +11 "' f ~ ~ n' ~L1 flf-1 ) ~! l? l', 7 • ,.. H i '- ___:_.--' 3 ;-;:: · - -·r •--;~ .,. "ff ~~H~ · -tr:!; ~ PL = = " r-.. /-;:-\ PL

'--"~ 111 • '<..:./ \l'.::_,I I Pio.nJI2-~ : jP5.od J- s• ~~ - ~------

thus a remarkable parallelism between the "common denominator" system of registration just illustrated and the system of measures 1-25 of the Symphony. Not only are the two axis-tones the same (except for the octave translation) but so are the extremes of the two systems (D and E), not to mention the octave duplication of E-fiat (here spelled Eb-D#), the tone which lies a tritone above and below the axis A of either system. (Compare the "fixed " system of the open ing ·25 ·measures of the Symphony with the "common denominator" system of the second movement of the Piano Variations.) 58 2. CONCERNING THE POSSIBLE MEANINGS OF DEGREE PROGRESSION WITHIN A FIXED SYSTEM OF REGISTRATION:

It is evident that a tension arises between the clear canonic state ment of a row-form together with its inversion, and the voice-cross ings necessitated by such "pre-ordained" systems of registration. It might be argued that the prevailing eighth-note lag between canonic "voices" (pointed out by Mr. Westergaard) is sufficient to distin guish them. However, the marked dynamic contrast between succes sive articulation patterns tends to emphasize a given dyad rather than its component canonic "voices." (If, for example, the left hand were to have been played forte and the right hand piano, it might have been easier to hear a principal form of the row crossing its inversion in the monochrome timbre of the piano.) Furthermore, the canonic "voices" are only partially clarified by the elaborate panto mime of crossing hands, because the hands exchange row-forms sev eral times in mid-row (e.g., in mm. 5, 8, 17, and 19). Therefore if discrete row forms in canonic relationship emerge at all in perform ance, they do so imperfectly, and in terms primarily of a visual rather than an auditory experience.

It is interesting to speculate on the possible meanings that the Salzerian concept of melodic progression could have within the fixed system and sub-systems of registration of this piece, in which the composer has legislated that all activity is to be confined to a par ticular group of dyads (or pairs of tones) of which each member and its counterpart are always to be equidistant from a common axis of symmetry. For example, there would seem to be three pos sible ways of interpreting progressions between tones in different re~~n: _

1. Melodic progression may occur freely from any chromatic degree to any other, regardless of respective position within the fixed "galaxy" of tones. The chief impediment to hearing such progressions in this piece would be the one already mentioned, namely, that since a given melodic progression is always presented together with its inversion, in purely auditory terms there is no way for the listener to know when a voice-crossing is intended. (It is this which makes it so difficult to hear the crossings of row-forms indicated visually at the top of Example 2.)

2. Melodic progression is to be understood only as motion from a given dyad to a directly adjacent dyad closer to or further away from the comon axis of symmetry. This definition seems to offer the advantage of immediate auditory intelligibility. Unfortunately, however, as far as I am able to determine, it is clearly applicable 59 only to measures 6-9 of this movement. (See bracketed context, Example 2.)

3. Melodic progression may occur from any degree to any other, regardless of octave registration, providing that it is not necessary to assume a ·voice-crossing or inversion of a dyad. The recognition of stepwise melodic progression in spite of octave transfers of reg ister is, of course, a commonplace of Schenkerian and Salzerian analysis. If the octave transfers of Example 3 were admitted as a registral "simplification" of the original music, it would be possible to explain the entire movement in terms of the structural hierarchy set forth in Examples 4 a, b, and c. It is, of course, true that much of the charm and vitality of the original is lost the moment one eliminates in this manner the tension between the fixed systems of registration described earlier, and the predominantly stepwise melodic progressions which have been made explicit in Examples 3 and 4 a, b, and c. Nevertheless, I submit that it is far more difficult to hear the serial labyrinth suggested by the diagram at the top of Example 2 than it is to hear the structural hierarchy I am about to discuss and illustrate next.

3. A SPECULATIVE EXPLANATION OF WEBERN'S OP. 27, SECOND MOVEMENT IN TERMS OF A QUASI-SCHENKERIAN STRUCTURAL HEIRARCHY:

Felix Salzer's Structural Hearing has already provided eloquent evidence that valuable insights into the music of Bart6k, Hindemith, and Stravinsky can be gained by a broader application of the con cepts of Heinrich Schenker. But there have been few attempts to explain the music of Schoenberg, Berg and Webern in these terms; the music of atonality and serialism does not readily yield to analysis from the standpoint of directed motion. Such an approach implies not only the possibility of recognizing clearly established origins and goals of motion, but the concomitant possibility of understanding the details of such motions as elaborations on various levels (fore ground, middleground, or background) of a primordial progression or structure which can be assumed to underlie the entire composi tion and to unfold through time some sort of tonic sonority, triadic or otherwise. (In an earlier article I have discussed several examples from Stravinsky and Bart6k involving dissonant tonic sonorities. See "Towards a New Concept of Tonality?" in the Journal of Music Theory, Vol. II, No. 2.)

The tonic sonority in this piece is the dyad G#-Bb, a diminished third or major second, depending on spelling, which appears at the 60 beginning and end of every statement of both form parts (Example 4a). The primordial progression or structure which underlies the entire composition is simply an alternation between the "tonic dyad" (G#-Bb) and the "polar dyad" a tritone away (D-E) (Example 4b). All of the other tones in the piece can readily be understood either in terms of neighboring motions around either of the two dyads, or in terms of passing motions from one dyad to another (Example 4c). This repeated progression between major seconds a tritone apart would at the same time explain the over-all impression of harmonic stasis and the apparently contradictory sense of harmonic thrust, which impelled Mr. Westergaard to recognize "Haydnesque wit" in the built-in luftpause immediately preceding the inevitable return to the final "tonic dyad" of measure 22 ( v. Example 4c). JoHN RoGERS 61

TOWARD A SYSTEM OF ROTATIONAL ARRAYS

ROTATIONAL MANIPULATIONS of pitch and/ or pitch class (pc) order ings are a well known feature of much twentieth-century music. These manipulations have been applied to linear, to vertical, and to linear vertical sets - that is, to "tunes," "harmonies,'' and "harmonized tunes." It is with this latter type of rotational structuring - hence forth called rotational arrays - that we will be concerned.

I. THE MODEL OR "TYPE l" ARRAY

Example Ia shows the hexachordal "generating set" and one "generated" array of Stravinsky's Variations. 1 This type array is our basic "model array" for all other types of rotational arrays, and is called a Type I array.

Note the following characteristics:

1. Only six distinct pc's occur.

2. Each pc occurs six times, once in each column and once in each row.

1 Spies, Claudio. "Stravinsky's Variations," Perspectives of N ew Music, F all-Winter, 1965, pages 62-74.

------62 Example Ia

(letter names) (n umbers)

D c A B E A# 0 10 7 9 2 8 c A B E A# D 10 7 9 2 8 0 A B E A# D c 7 9 2 8 0 10 B E A# D c A 9 2 8 0 10 7 E A# D c A B 2 8 0 10 7 9 A# D c A B E 8 0 10 7 9 2

3. The succession of pc's in row 1 is the same as that in column 1.

4. Each row is a cyclic permutation (rotation) of row 1, and each column is a cyclic permutation of column 1.

5. It therefore follows that the succession of pc's in column 2 is the same as that in row 2; and that an analogous relation holds between all respective rows and columns.

6. The "top right - bottom left" diagonals state only one pc each. The "middle" one of these diagonals states all six occur rences of that pc which is the final pc ·of the generating set. Diagonals which are complementary distances, mod 6, on each "side" of the "middle" diagonal also produce six occur rences of one pc.

7. Each of the six "top left - bottom right" diagonals states a "second order rotation" (every other pc, thus either 0, 7, 2, or 10, 9, 8) of the generating set.

Example lb shows the Type I array based on the complementary hexachord of the Variations.

Example lb

(l etter nam es) (numbers)

G# C# D# G F# F 6 11 5 4 3 C# D# G F# F G# 11 1 5 4 3 6 D# G F# F G# C# 1 5 4 3 6 11 G F# F G# C# D# 5 4 3 6 11 1 F# F G# C# D# G 4 3 6 11 5 F G# C# D# G F# 3 6 11 5 4 Example le combines these two "complementary" arrays (Ia and 63 lb). Note that 12 pc sets are formed by combining corresponding rows of the arrays. Example Id combines the complementary arrays in such a fashion as to produce 12 pc sets in corresponding columns.

Example le

0 10 7 9 2 8 + 6 11 1 5 4 3 Equals 12 pc's 10 7 9 2 8 0 + 11 1 5 4 3 6 Equals 12 pc's 7 9 2 8 0 10 + 1 5 4 3 6 11 Equals 12 pc's 9 2 8 0 10 7 + 5 4 3 6 11 1 Equals 12 pc's 2 8 0 10 7 9 + 4 3 6 11 1 5 Equals 12 pc's 8 0 10 7 9 2 + 3 6 11 1 5 4 Equals 12 pc's

Example Id

0 10 7 9 2 8 10 7 9 2 8 0 7 9 2 8 0 10 9 2 8 0 10 7 2 8 0 10 7 9 8 0 10 7 9 2 + + + + + + 6 11 1 5 4 3 11 1 5 4 3 6 1 5 4 3 6 11 5 4 3 6 11 1 4 3 6 11 1 5 3 6 11 1 5 4 12 pc's 12 pc's 12 pc's 12 pc's 12 pc's 12 pc's

All sets of distinct pc's of sizes 2 through 12 will generate Type I rotational arrays. Example le shows the ultimate in this type, that based on a 12 pc generating set. In this case, the 12 pc set is made up of the two complementary hexachords with which we have been working. The characteristics of this 12 X 12 Type I array are analogous to those listed earlier for the 6 X 6.

II. DUPLICATING ARRAYS

Example Ila shows two Type II duplicating arrays. Each is de rived from one of the two complementary hexachordal generating sets of Stravinsky's Variations. 64 Example le

0 IO 7 9 2 8 6 il 5 4 3-- 12 pc's IO 7 9 2 8 6 11 1 5 4 3 0-12 pc's 7 9 2 8 6 11 5 4 3 0 I0-12 pc's 9 2 8 6 11 5 4 3 0 IO 7-12 pc's 2 8 6 11 1 5 4 3 0 IO 7 9-12 pc's 8 6 11 5 4 3 0 IO 7 9 2-12 pc's 6 11 5 4 3 0 IO 7 9 2 8- 12 pc's 11 5 4 3 0 IO 7 9 2 8 6- 12 pc's 5 4 3 0 IO 7 9 2 8 6 11-12 pc's 5 4 3 0 IO 7 9 2 8 6 11 1-12 pc's 4 3 0 IO 7 9 2 8 6 11 5-12 pc's 3 0 10 7 9 2 8 6 11 5 4- 12 pc's 12 pc's 12 pc's 12 pc's 12 pc's 12 pc's 12 pc's 12 pc's 12pc's12 pc's 12 pc's 12 pc's 12 pc's

Example Ila

(first hexachord) (second hexachord) 0 IO 7 9 2 8 6 11 5 4 3 0 9 11 4 IO 2 6 8 0 11 IO l 0 2 7 1 5 3 6 10 9 8 11 4 0 5 11 3 1 IO 6 5 4 7 0 2 0 6 IO 8 5 7 6 5 8 3 7 0 4 2 11 6 6 9 2 4 8 7

Note the following characteristics (here with reference to the array generated by the first hexachord; the same general characteris tics, of course, apply to all Type II duplicating arrays) :

1. Each row is a transposition of the corresponding row of the Type I array. That is, each row is a transposition of _a rotation of the generating set.

2. The sequence of subtractions to obtain the five rows is the generating set itself (0, 10, 7, 9, 2, 8 - O; 10, 7, 9, 2, 8, 0 -10; 7, 9, 2, 8, 0, 10 - 7; 9, 2, 8, 0, 10, 7 - 9; 2, 8, 0, 10, 7, 9--2; 8, 0, 10, 7, 9, 2-8).

3. The rows therefore state a sequence of transpositions of the set in the pattern of the inverse of the generating set ( 0, 10, 7, 9, 2, 8; 0, 9, 11, 4, 10, 2; 0, 2, 7, J, 5, 3; 0, 5, 11, 3, 1, 10; 0, 6, 10, 8, 5, 7; 0, 4, 2, 11 , 1, 6).

4. The "top right-bottom left" diagonals are obtained by -sub tracting the pc's of the generating set from the duplicated pc found in each corresponding diagonal of the Type I array 65 (8, 8, 8, 8, 8, 8-0, 10, 7, 9, 2, 8; 2, 2, 2, 2, 2, 2-0, 10, 7, 9, 2, 8; 9, 9, 9, 9, 9, 9 - 0, 10, 7, 9, 2, 8; 7, 7, 7, 7, 7, 7 - 0, 10, 7, 9, 2, 8; 10, 10, 10, 10, 10, 10-0, 10, 7, 9, 2, 8; 0, 0, 0, 0, 0, 0-0, 10, 7, 9, 2, 8). Note that one definition of the invernion of a set is the subtraction of each pc in the set from some constant pc.

5. Therefore, the "top right - bottom left" diagonals state a sequence of transpositions of the inverse of the generating set in the pattern of the retrograde of the set itself (8, 10, 1, 11, 6, O; 2, 4, 7, 5, 0, 6; 9, 11, 2, 0, 7, 1; 7, 9, 0, 10, 5, 11; 10, 0, 3, 1, 8, 2; 0, 2, 5, 3, 10, 4). Note that if 1the order of read ing the diagonals is reversed, the diagonals would then s·tate a sequence of transpositions of the inverse of the generating set in the pattern of the set itself.

6. Characteris1vics 3 and 5 above combine to show that rows state generating sets in a sequence of transpositions which ·corresponds to the inverse of the original generating set while "top right-bottom left" diagonals state inversions of the gen erating set in a sequence of transpositions which corresponds to the retrograde of the generating set (or to the set itself).

7. The "top left - bottom right" diagonals are obtained by sub tracting the pc's of the genernting set from the sequence of second order rotations of the generating set found in the cor responding diagonals of ·the Type I array ( 0, 7, 2, 0, 7, 2 - 0, 10, 7, 9, 2, 8; etc.). Though in this particular case a set of six distinct pc's is produced, the characteristics of this diagonal are much harder to forecast, and it is of little sys tematic importance. (One should also note that the number of pc's in these diagonals will not necessarily agree with the number of pc's in the generating set. In Type I arrays, the number of pc's will agree only if 2 is relatively prime to the number of pc's in the generating set; in Type II arrays, the agreement will vary according to the interva11ic arrangement of the generating set and the characteristics of the Type I .array on which the Type II is based.)

8. There is total pc duplication in column 1.

9·. The pc's of columns 2 and 6 are related by inversion (10, 9, 2, 5, 6, 4 inverts into 2, 3, 10, 7, 6, 8) and those of columns 3 and 5 are similarly related (7, 11, 7, 11, 10, 2 inverts into 66 5, 1, 5, 1, 2, 10). Column 4 is inversionally degenerate m terms of its trichords (9, 4, 1 inverts into 3, 8, 11).

In order to get a clear understanding of points 8 and 9 above, we must define carefully the levels or orders of intervals found be tween the pc's of the generating set. Let us call first order intervals those obtained by the mod 12 subtraction of adjacent pc numbers, and let us continue our interval counting "around the horn" so that there are the same number of intervals as there are pc's in the gen erating set. Second order intervals may then be defined as those between "every other" pc; third order intervals as those between "every third" pc; etc., so that for a generating set of size n, there are n-1 order interval successions. Note that each interval of one order interval succession is the sum of all the first order intervals it contains - two in the case of second order intervals, three in the case of third, etc. Example Ilb shows the intervals of each order found in our hexachordal generating set.

Example Ilb

0 10 7 9 2 8 Generating Set

10 9 2 5 6 4 First Order Interval Succession 7 11 7 11 10 2 Second Order Interval Succession 9 4 3 8 11 Third Order Interval Succession 2 10 5 5 1 Fourth Order Interval Succession 8 2 3 10 7 6 Fifth Order Interval Succession

One notices that the above order interval number successions appearing in five rows are exactly the same as the pc number suc cessions appearing in columns 2 through 6 of our Type II duplicat ing array. Pitch class duplications in columns of a Type II array occur in those spots where there are interval duplications in any one order interval succession; pc duplications "between" columns of a Type II array occur in those spots where there are interval dupli cations between order interval successions.

If the pc's of the generating set are An, where n equals 1 through 6, then the succession of first order intervals is A2 - A 1 , A 3 - A 2 , A 4 -A3 , A 5 - A 4 , A 6 -A6, Ai -A6 • The fifth order interval s ucces~ion simply reverses the order of pc numbers in each in·terval (A6 - A 1 , A 1 - A 2 , etc.) and thus is necessarily the inversion of the first order interval succession. An analogous relation holds be tween second and fourth order successions, and between the two 3 element units of the third order succession. Obviously, the characteristics of the "middle column" (that 67 column whose number is n/2) will vary according to whether the generating set contains an odd or even number of elements. If the generating set contains an odd number of elements, there is no integral column number of n/ 2, and this column does not exist. (In a 7 X 7 array, for example, column 2 inverts into column 7, column 3 into 6, and column 4 into 5.) If the generating set contains an even number of elements, there is an integral column number of n/ 2 and this column consists at the maximum of 2 inversionally related sets of size n/ 2.

There are no particular necessary conditions for the production of a Type II duplicating array except the avoidance of pc duplica tion in the original generating set. Constructing a set with appropri ate interval characteristics will, of course, yield an array with desired duplication characteristics. Type II arrays may be based on generat ing sets of size 2-12. Example Ile shows a 12 X 12 array derived from the Type I 12 X 12 array shown in Example le.

Example Ile 0 10 7 9 2 8 6 11 1 5 4 3 0 9 11 4 10 8 1 3 7 6 5 2 0 2 7 1 11 4 6 10 9 8 5 3 0 5 11 9 2 4 8 7 6 3 1 10 0 6 4 9 11 3 2 1 10 8 5 7 0 10 3 5 9 8 7 4 2 11 1 6 0 5 7 11 10 9 e 4 1 3 8 2 0 2 6 5 4 1 11 8 10 3 9 7 0 4 3 2 11 9 6 8 1 7 5 10 0 11 10 7 5 2 4 9 3 1 6 8 0 11 8 6 3 5 10 4 2 7 9 1 0 9 7 4 6 11 5 3 8 10 2 1 1 pc 7 pc's 7 pc's 8 pc's 8 pc's 8 pc's 9 pc's 8 pc's 8 pc's 8 pc's 7 pc's 7 pc's The characteristics of the above array are the 12 X 12 analog of those listed for the 6 X 6 array found in Example Ila.

III. NON-DUPLICATING ARRAYS If there are no interval duplications within any of the order interval successions in the generating set, then there will be no pc duplications in corresponding columns of a Type II array. This type array will be called a Type II non-duplicating array. It is possible to convert our original hexachordal generating set to a set of this type by simply re-ordering the set to 0, 8, 10, 2, 9, 7 or 0, 8, 7, 9, 68 10, 2. Example Illa shows a Type II non-duplicating array based on the first of these two re-orderings.

Example Illa

0 8 10 2 9 7 0 2 6 1 11 4 0 4 11 9 2 10 0 7 5 10 6 8 0 10 3 11 1 5 0 5 1 3 7 2

The characteristics of Example IIla are analogous to those of Example Ila with the added characteristic that no pc's are duplicated in any one column except, of course, column 1. Example Illb shows a Type II non-duplicating array based on the "hexachordally combinatorial" inversion, and Example Ille shows a "row-wise" com bination of the arrays of Examples Illa and Illb.

Example I/lb

1 5 3 11 4 6 1 11 7 0 2 9 1 9 2 4 11 3 1 6 8 3 7 5 1 3 10 2 0 8 1 8 0 10 6 11

Example Ille

0 8 10 2 9 7 5 3 11 4 6 0 2 6 1 11 4 11 7 0 2 9 0 4 11 9 2 10 9 2 4 11 3 0 7 5 10 6 8 6 8 3 7 5 0 10 3 11 1 5 3 10 2 0 8 0 5 1 3 7 2 8 0 10 6 11

Type II non-duplicating arrays are possible with generating sets of sizes 2-7; they are impossible with generating sets of sizes 8-12. (The reasons for this will not be discussed here. See footnotes 2 and 3 for a full discussion of these reasons and for a fairly complete discussion of some compositional uses for Type II arrays of sizes 2-7.) Perhaps it should be mentioned that, because of the row-column identity of the original model (Type I) array, the following 90 degree rotation of a Type II array is systematically valid. The array so produced will be called a Type III array. Example IIId shows the 69 Type III transformation of the array succession of Example Ille.

Example llld

0 0 0 0 0 0 8 2 4 7 10 5 10 6 11 5 3 1 2 1 9 10 11 3 9 11 2 6 1 7 7 4 10 8 5 2 + + + + + + 1 1 1 1 1 1 5 11 9 6 3 8 3 7 2 8 10 0 11 0 4 3 2 10 4 2 11 7 0 6 6 9 3 5 8 11

IV. TYPE IV OR "SUMMATION" ARRAYS: DUPLICATING, NON-DUPLICATING, AND TOTAL NON-DUP LICATING

Consideration of the problem of producing 12 tone sets between corresponding rows of inversionally related Type lI arrays (or cor responding columns of I-related Type III arrays) led to the forma tion of summation arrays.2 Both Type II and III arrays are based on the ordered differences of the original generating set; Type IV arrays are based on "ordered sums." Consider the following array based on our familiar generating set.

Example IVa

0 8 10 2 9 7 4 6 10 5 3 8 8 0 7 5 10 6 4 11 9 2 10 0 6 4 9 5 7 11 2 7 3 5 9 4

1. Each row is a ,transposition of the corresponding row of the Type I array. That is, each row is a transposition of a cyclic permutation of the generating set. 2. The sequence of additions to obtain the six rows is the gen erating setitself (0, 8, 10, 2, 9, 7 + O; 8, 10, 2, 9, 7, 0 + 8; 10, 2, 9, 7, 0, 8. + 10; 2, 9, 7, 0, 8, 10 + 2; 9, 7, 0, 8, 10, 2 + 9; 7, 0, 8, 10, 2, 9 + 7).

2 Rogers, John. "Some Properties of Non-Duplicating Rotational Arrays," Perspectives of N ew Music, Fall-Winter, 1968, pages 80-1 02. 70 3. The rows therefore state a sequence of trarispos1tions of the set in the pattern of the set itself (0, 8, 10, 2, 9, 7; 4, 6, 10, 5, 3, 8; 8, 0, 7, 5, JO, 6; 4, 11, 9, 2, 10, O; 6, 4, 9, 5, 7, 11; 2, 7, 3, 5, 9, 4) .

4. The "top right-bottom left" diagonals are obtained by adding the pc's of the generating set to the duplicated pc found in each corresponding diagonal of the Type I array (7, 7, 7, 7, 7, 7 + 0, 8, 10, 2, 9, 7; 9, 9, 9, 9, 9, 9 + 0, 8, 10, 2, 9, 7; 2, 2, 2, 2, 2, 2 + 0, 8, 10, 2, 9, 7 ; 10, 10, 10, 10, 10, 10 + 0, 8, 10, 2, 9, 7; 8, 8, 8, 8, 8, 8 + 0, 8, 10, 2, 9, 7; 0, 0, 0, 0, 0, 0 + 0, 8, 10, 2, 9, 7) .

5. The above diagonals therefore state a sequence of transposi tions of the set in the pattern of the retrograde of the gen erating set (7, 3, 5, 9, 4, 2; 9, 5, 7, 11, 6, 4; 2, 10, 0, 4, 11 , 9; JO, 6, 8, 0, 7, 5; 8, 4, 6, 10, 5, 3; 0, 8, 10, 2, 9, 7) .

6. Row 1 equals "top right-bottom left" diagonal 6; row 2 equals diagonal 5; etc. to within rotation.

7. The "top left-bottom right" diagonals are obtained by adding the pc's of the generating set to the second order rotation of the generating set found in the corresponding diagonals of the Type I array. These diagonals do not and in general will not articulate meaningful variants of set statements.

8. The pc's in column 1 represent a segment of a whole tone scale. Since these numbers represent a doubling of the orig inal pc's, it is clear that column 1 is the whole tone or M2 equivalent of the original generating set. Pc duplications in column 1 will occur wherever there were numbers in the gen erating set that were not distinct, mod 6; that is, wherever there were tritone separated pitches.

9. The pc's of columns 2 and 6 are rotations of each other ( 8, 6, 0, 11 , 4, 7; 7, 8, 6, 0, 11, 4) , and those of columns 3 and 5 are similarly related (10, 10, 7, 9, 9, 3; 9, 3, 10, 10, 7, 9). Column 4 is composed of two 3 element units (2, 5, 5; 2, 5, 5) .

Let us call first order sums those obtained by the mod 12 addi tion of adjacent pc numbers, and let us continue our summing around the horn so that there are the same number of sums as there are pc's in the generating set. Second order sums may then be defined as those of "every other" pc; third order sums as those of "every third"

__J' pc; etc., so that for a generating set of size n, there are n-1 order 71 summation successions.

Example !Vb 0 8 10 2 9 7 Generating Set 8 6 0 11 4 7 First Order Summation Succession 10 10 7 9 9 3 Second Order Summation Succession 2 5 5 2 5 5 Third Order Summation Succession 9 3 10 10 7 9 Fourth Order Summation Succession 7 8 6 0 11 4 Fifth Order Summation Succession

One should note that the above order summation number suc cessions appearing in five rows are exactly the same as the pc num ber success,ions appearing in columns 2 through 6 of the Type IV array. Pc duplications in columns of a Type IV array occur in those spots where there are summation duplications in any one order sum mation succession; pc duplications between columns of a Type IV array occur in those spots where there are summation duplications between order summation successions.

Let the pc's of the generating set by represented by A 0 , where n equals 1 through 6. The succession of first order sums is A2 + A i ; Ag + A 2 ; A 4 + A 3 ; A 5 + A 4 ; A 6 + A 5 ; Ai + A 6, and the suc cession of fifth order sums is A 6 + A 1 ; A 1 + A 2 ; A 2 + A 3 ; A 3 + A 1 ; A 4 + A 5 ; A 5 + A 6 . The fifth order summation succession sim ply reverses the order of pc numbers in each sum. Since addition is commutative (does not depend on order), ,this means that the first and fifth order sums must be the same, to within rotation. An analo gous relation holds between second and fourth order successions, and between the two 3 element units on the third order succession.

Obviously, the characteristics of the "middle column" (that col umn whose number is n/ 2) will vary according to whether 2 is rela tively prime to the number of pc's in the generating set. If the gen erating set contains an odd number of elements, there is no integral column number of n/ 2, and this "redundant" column does not exist. (In a 7 X 7 array, for example, column 2 equals column 7 ; 3, 6; and 4, 5.) If the generating set contains an even number of elements, there is an integral column number of n/ 2 and this column consists of no more than 2 duplicated sets of size n/ 2. In certain cases, this column may contain only one duplicated pc.

Example IVc shows a "row-wise" combination of the arrays of Examples IVa and IIIb. Note that corresponding rows are related by inversion and sum to 12 pc's. 72 Example !Ve

Type IV Type II 0 8 10 2 9 7 1 5 3 11 4 6 4 6 10 5 3 8 11 7 0 2 9 8 0 7 5 10 6 9 2 4 11 3 4 11 9 2 10 0 6 8 3 7 5 6 4 9 5 7 11 3 10 2 0 8 2 7 3 5 9 4 8 0 10 6' 11

Type IV non-duplicating arrays of any size exist only in those cases in which the array is a first order rotation of a Type II array. Example IVd shows a combination of two 6 X 6 arrays. The Prime array is a Type II and the Inversional array, a Type IV. Note that total pc duplication occurs at order position 4 of the Type IV array, and thus it must be a rotation of a Type II array. If it is considered to be Type II, it is based on a rotation of a transposition of the prime set (0, 1, 9, 11, 10, 2 + 6 equals 6, 7, 3, 5, 4, 8 which becomes 5, 4, 8, 6, 7, 3 under first order rotation). In such cases as this, the ordered sums of the inverted set happen to equal the ordered differ ences of the prime set, to w1thin rotation.

Example /Vd

Prime Array (II) ln versiona l Array (IV) 0 1 9 11 10 2 5 4 8 6 7 3 0 8 10 9 11 3 7 5 6 2 4 0 2 1 5 3 4 11 9 10 6 8 7 0 11 3 2 10 7 8 4 6 ·s 9 0 4 2 3 11 1 9 5 7 6 10 8 0 10 11 7 9 8 3 2 6 4 5

With generating sets of sizes 5 and 3, it is possible to produce Type IV total non-duplicating arrays.

Example !Ve Example !Vf

0 1 3 0 1 2 4 5 2 4 1 2 3 5 6 6 3 4 4 6 7 2 3 8 9 4 5 6 10 5 6 7 9 In the 3 X 3 array, columns 2 and 3 are rotations of each other; 73 a similar relation holds between columns 2 and 5 and columns 3 and 4 of the 5 X 5 array. Such arrays are also possible with generating sets of size 2, but are impossible with generating sets of sizes 4 and 6 since these sizes must produce pc duplications in the "middle" column. Since, to avoid pc duplication in column 1, the pc's of the generating set must be distinct, mod 6, this means that such arrays are obviously impossible with generating sets whose size is seven or more (it is impossible for more than 6 pc's to be distinct, mod 6). Thus Type IV total non-duplicating arrays are possible only with generating sets of sizes 2, 3, and 5.

V. SOME SPECULATIONS

We have noted earlier that Type II non-duplicating arrays are impossible of generation with sets larger than 7. But generating sets of these sizes are possible if the modulus (corresponding to some equal tempered division of the octave) is changed. Prime numbered moduli are particularly versatile in the number of different size gen erating sets they allow.

Suppose we were to choose the often proposed "19-tone" division of the octave, a division which possesses certain well known relation ships to the 12-tone division. Though generating sets of sizes n ( 19) and n-2 (17) are still impossible, a generating set of size n-1 is pos sible.2 3 I am at present working on the complete categorization of all Type II generating sets in sys,tems of from 2-19 equal tempered divisions of the octave. The complete system as I now envision it will contain at least the following possibilities: ( 1) the possibility of finding common tones, intervals, and chords not only within any one tuning system, but between systems, and the use of these common elements as modulatory devices; (2) the possibility of contextually redefining the octave and projecting some equal tempered scheme onto this "new octave," and the possibility of using this as both a "thematic" and a modulatory device; (3) the possibility of both row wise and column-wise combinations of different but related arrays; and ( 4) the possibility of using the completely symmetrical nature of Type II and III arrays to produce completely symmetrical pitch structures which will expand and contract in relation to the size of the modulus.

3 Mitchell and Rogers. "A Problem in Mathematics and Music," American Mathematical Monthly, October, 1968. 74 Let me say in conclusion that if any of you are fascinated by the mathematics of this system, I will be happy to go into more detail about it,2 3 particularly if you might be interested in working on some problems I have been unable to solve. Also, one thing that is holding up my research in this area is simply the lack of adequate computer time. Thus if any of you have a lot of computer time available and would like to run some of these programs, I will be happy to send you FORTRAN versions of the search programs for successful Type II generating sets of sizes 2-18, mods 5-19. These programs can be run with only slight changes on any computer with a FORTRAN compiler. PART IV 75

MICROTONAL MUSIC IN AMERICA

LEIGH GERDINE 77

INTRODUCTORY REMARKS

MY FIRST duty is to apologize for appearing on a program designed for composers: I am not a composer; a number of years ago I decided that I could make a greater contribution as a good (or at least, con scientious) theorist than as a third-rate composer. But I have always been fascinated by both the theory and the fact of microtonal music. When I was new at Washington University, I cut up an old piano and re-tuned it to Yasser's scale, with results that were perplexing and disappointing, but stimulating.

Because I am a department head, I find my time taken up with a lot of routine paper-shuffiing. This makes research into microtonal music difficult, for me and I find myself devoting only distracted atten tion to it from time to time. My job with this panel, then, will be chiefly to keep the peace and to intrude as little as necessary upon the substantive discussion. I am serving as moderator faute de mieux.

It is my presumption that I am serving as moderator because I have been fascinated by the possibilities of microtones, and because I am an admirer of the pioneering work done by Professor Adriaan D. Fokker ("On the Expansion of the Musician's Realm of Har- 78 mony," in Acta Musicologica Fasc. II-IV, Volume XXXVIII, pages 197-202). Currently I have been having some preliminary conversa tions with administrative officers in my own university about the practicability of applying to the National Science Foundation for funds to build an electronic instrument which would make it possible to experiment with 31 notes or even 53 notes to the octave. The experiments must be made, and electronic mechanisms seem to me the simplest way to keep the material under control.

One further question occurs to me as worth mentioning: has the rapid development of elecronic music overleaped the need for experi menting with microtones? Not at all in my view. Harry Partch called his book Genesis of a Music. He is indicating in his title that other musics are indeed possible; that music is not only a goddess with a Viennese accent. In microtonal expeliments may lie the way to a much more significant, harmonically richer music. PETER YATES 79

MICROTONES

THE Harvard Dictionary and the Oxford Companion agree that a microtone is any interval smaller than a semitone; both proceed with dubious efficiency to discuss microtones as subdivisions of the equal-tempered scale and interval (quarter-, sixth-, eighth-, sixteenth tones) as exemplified in music by Charles Ives, Alois Haba, Bela Bart6k, Ernest Bloch, and others. None of these has become a developed system, although Ives and Alois Haba wrote whole quarter tone compositions. There is also vague reference to Greek and Hindu usage and to the distinction between G# and Ab in meantone tuning. This tells nothing practical about its past history, so we may as well start fresh. In this very broad, rapid survey I can do no more than indicate areas of microtonal usage in music past and present, leav ing to my colleagues the more specific details of ratios, cents, and certain compositional applications.

I would make the point, however, that musicians historically have tuned by tuning order from the instrument, as most piano tuners still do today, since they had no means, except the monochord, of transferring theoretical ratios to actual tunings. 80 Music is by nature tonal and microtonal, adapted to no fixed intervallic scale. If you improvise music by beating on a hollow log or the frame of a piano, you will find one central quality of thump which you return to; this sound to which you return is the tonal center of the composition, from which the other sounds you are able to produce deviate by unequal microtonal intervals. The sounds differ also by microtonal complexities of timbre. The ear will not deny a melody produced by a mode of unequal, microtonally variable in tervals which has no exact pitch or scale.

Jazz, drawing upon African and Caribbean drumming, has made much use of such distinctions and is now drawing on the microtonal drum melodies of India.

During the 1920s, Edgard Varese composed rich tapestries of instrumental and percussive sound from a large variety of instruments and sound-producers, using the principle that melodic and harmonic patterns must alter without reference to the octave, thus denying both tonal modulation and the tone-row. Influenced by this example, and by the growing interest of Henry Cowell and other composers in Oriental music, John Cage, Lou Harrison, Gerald Strang, and Cowell himself experimented with composition for percussion groups using sound-producers of no determinate tone. Six or a dozen Ford Model A brake drums, or bells or bars of indeterminate pitch, could be arranged to give an unequal-intervalled melodic mode or gramut of microtonally distinct pitches of similar timbre. By combining bat teries of such means one could produce a music of gamelan-like rhythms around reiterated melodic figures, as in Cage and Harrison's joint composition, Double-Music, or the elaborate melodic polyphony developed by Harrison, or the complex of sounds for their own sake on a numerical matrix of Cage's Construction in Metal.

Varese had shown by the use of such primitive sound-producers of sliding tone as the lion's roar or the hand-operated siren the audi ble interest of a fluctuating sound with no intervals and no tonal center. Indeterminate sliding tones are a commonplace of jazz. Cage carried over this principle to electronic sound with his Imaginary Landscape No. 1, put together of segments of the continuum of fundamental pitches, recorded as a continuous curve from the lowest to the highest audible limits on a record made by a telephone com pany to test its lines. Slides without interval are of course microtonal in effect. Charles Ives used sliding quartertones to give the extraordi nary sound of his song, Like A Sick Eagle, for solo voice and instru ments of flexible pitch. [Here a tape of this song, recently recorded under the direction of Gregg Smith, was played through the courtesy of John McClure of Columbia Records.] History does not record when musicians first discovered the inter- 81 vallic principle of music. It was presumably a vocal phenomenon, a rough grouping around the octave, 5th, and perhaps the 3rd, an interval with no determinate fixed pitches. Gilbert Chase has shown in America's Music that such approximate intervals are essential to the correct performance of American folk music from the 18th century to the present, in hymns, anthems, secular songs, Negro shouts, blues, jazz, the approximate pitches embellished by microtonal deviations and by slides: for example, the flatted third, which he calls "this ambivalent, this worried or slurred tone that constitutes the true 'blue note.' " And he continues: "The blue scale (diatonic, with microtonally flatted third and seventh) . . . has permeated large sections of American music both in the popular and in the fine-art idioms." (p. 453)

Such deviations from correct intervallic pitch are true of nearly all folk music, but musicians in practice have denied the esthetic implications of this fact by notating and performing such music in tones reproducing the fixed intervallic system of the keyboard. Musical attention is now turning from the fixed notes of the keyboard to explore the music which has been lost in the cracks between the keys. Recording has made us aware that our notation is entirely inadequate to writing out the microtonal deviations of music in other systems than our own. Charles Seeger has invented an electronic machine, the Melograph, to notate graphically such microtonal devia tions .in melodic inflection and in timbre. Similar equipment is in use in Sweden and elsewhere. Our ears, which are more musical than our notational systems, have learned to accept microtonally deviating melody even when accompanied on guitar or piano by impeccably correct basses, for example in songs by Leadbelly. Regularizing such melody to the correctness of pitched, harmonically related tones de bases it to the prettiness of the harmonized Negro spirituals.

Ear-determined intervals are usually acoustical approximations of the lowest intervals of the overtone series, as Pythagoras discov ered while exploring his belief that all relationships in nature can be stated geometrically as ratios of simple whole numbers. Western musical theory has always referred to these correct or just intervals, even when the same theorist makes music in a scale consisting of incorrect or tempered intervals, as we have been doing for 300 years. We may say, therefore, that music consists of correct tones correctly related (which is called just intonation) and of an enormous number of incorrect, microtonally altered tones, used either as embellish ments of the correct tone, which is sometimes not sounded (as in much Oriental music), or in scales or clusters of microtonal discord used as the basis of composition. 82 When several American composers, among them Arthur Farwell and Colin McPhee, began transcribing Negro, American Indian, Indo nesian, Indian, and other melodies in scales exotic to our own and using them in compositions notated to our 12-tone equal tempera ment, the resulting microtonal distortion regularized the music for our ears, while distorting it for listeners who understood the originals. Such imitation of melodic exotica precipitated a determination to fin d some way of using the original melodies correctly. Today American music contains exotic melodies of both types. The gradual change in ethnomusicology from documentary studies to actual performance with Oriental, Indonesian, Indian, or African instruments, following the example of the UCLA Institute of Ethnomusicology, should speed up the use of similar scales and instruments in our own music. We are also growing aware of the microtonal composition of si mi lar tones sounded by va1ious instruments (timbre, the basis of orches tration; or in more complex analysis the sound envelope, basis of electronic composition), of an instrument damped or undamped or an open string plucked at different points (the basis of harpsichord registration), or at different volume or amplification, or in relation to contiguous tones (the real significance of intonation in perform ance), or in space (the basis of acoustics) . The idea of a fixed scale of determinate intervals (though the overall pitch might vary, as concert pitch has continued changing until the present time) appeared with gradual recognition of the over tone series. It was discovered that a scale of just or correct intervals will be larger than a correct octave. This difference between a cor rect octave scale in just intonation and a correct octave is the root of all highly cultivated art music. There are two solutions: one, to retain the open-ended scale of just intervals, disregarding the correct octave, which will be widened (George Ives, the father of Charles, invented scales without octaves matching the tones of the overtone series); the other solution is to temper (or narrow) one or more of the intervals of the scale so that the octave will be correct. The scales of just intonation now in use employ varying ratios ·and numbers of tones, from the 12-tone sys tems of Lou Harrison and LaMonte Young to the 43-tone systems of Harry Partch and Eivind Groven, and beyond that, to the explora tory systems of Ben Johnston, which can add up to as many as 53 tones. [A recently issued demonstration record made by Eivind Groven can be obtained from Norsk Phonogram a/ s, Kirkeveien 64, Oslo 3, Norway.*]

* Since writing this article the author has received the pamphlet which should have come with the record. Groven's present organ "was completed in 1965 and is now installed in Valerengen Church in Oslo. It has 43 pitches to To temper a tuning means to tune some intervals more narrowly 8:3 than the correct or just relationship, with the result that other intervals are wider than the just relationship. Every tempered tuning, or temperament, therefore involves a pattern of microtonal discord within a correct octave.

Various systems have been conventionalized, each producing a distinct music. We are so accustomed to our own equal-tempered scale of more or less discordant intervals within a correct octave that until recently we have thought all other systems obsolete or barbaric. We are now coming to recognize that it is our own system which is obsolete, and we are far along in the process of abandoning it. By the end of this century musicians may be performing the music of Beethoven and Schoenberg in an intonational system as incorrect for that music as our system is incorrect for Mozart, the Bachs, Monte verdi, Bull, Palestrina, Vittoria, Machaut, Gregorian chant, and for all folk music. And I am sure that experts will soon be dismissing our equal-tempered tuning for reasons as impertinent to the facts as the reasons present-day musicologists give for dismissing meantone tuning. Even the scholar of tuning, J. Murray Barbour, found statis tical arguments to dismiss meantone as inapplicable to music of its own period. (Tuning and Temperament, A Historical Survey: Chap ter VIII, From Theory to Practice.) In his History of Tuning, re corded on tape, which Wesleyan University Press may soon issue as a text with the examples on records, the late Wesley Kuhnle had made a start towards solving such tuning deviations, having found what appears to be the correct tuning of the Hexachord by John Bull (a tempered Pythagorean) and the Toccata in F# minor by J. S. Bach (a meantone accord). This has been until now an almost entirely disregarded field of musical scholarship. It is hard to imagine how different our understanding of music would be, if we learned to perform and hear it as the composer or the habit of the culture intended. The esthetic differences would be greater than most of us are at present able to conceive.

The so-called Pythagorean scale of the medieval period seldom used all 12 tones; when tuned on a keyboard instrument Pythagorean gives a scale of unequal intervals with correct 5ths and predominantly very wide 3rds. This may have been the first keyboard tuning; extended to the full 12 tones, with some tempering at the extreme of the cycle (tempered Pythagorean), it seems to have persisted as every octave and 33 stops, and is provided with 2 manuals and a pedal-key board. The replacing mechanism (the computer) is operated by transistors ..." By the "replacing mechanism" ... ''.the requisite groups of notes are 'brought in' according to the demands of the harmonic progressions." 84 a principal tuning for English keyboard music through the lifetime of Purcell. During the high renaissance of vocal polyphony voices sang a correct just intonation with correct octave, altering certain enharmonic or mutable tones to preserve correct relationships in ver tical harmony.

When keyboard music rose to prominence with the beginning of the 17th century, just intonation would not serve. While the English composers seem to have retained their harmony built around the correct 5th, European composers preferred to build around a new scale having an extremely narrow 5th and a correct, just major third. This scale, called meantone, persisted for more than 200 years, longer than our equal temperament, from around 1600 until approximately 1800. Each of these scales, Pythagorean, just intonation, and mean tone, is made up of unequal intervals. In meantone this unequal inter vallic pattern gave to each key a distinctive harmonic coloring of microtonal intervallic differences. Meantone permitted no enhar monic tones: the extreme tone of the cycle, tuned from C, is either G# or Ab. In meantone modulation a note alters its tonal and har monic meaning by unequal degrees, while those of equal tempera ment do not. That is why for any one meantone tuning only 8 keys are acceptable; beyond these the proportion of extremely discordant or "Wolf" tones becomes too great to manage. There are, however, several ways of widening the number of acceptable keys by sharp or flat tuning, starting backwards from the last note of the cycle, which can be either G# or Ab, or by an accord or altered tuning like that which is needed for correct playing of Bach's F# minor Toccata. [Two taped examples were played, recorded by Wesley Kuhnle: the extreme discords which result from playing the second slow section of the Bach F# minor Toccata in regular meantone; and the extraordi nary tonal coloring produced by playing the same passage with the correct tuning accord.]

During the 17th century the Dutch physicist, Christiaan Huygens, extended the rneantone just 3rd to an equal-intervalled scale of 31 tones, which very nearly approximates just intonation with slight tempering of the 5th. This scale has been revived by the present-day Dutch physicist, Adriaan Fokker, who has built an organ with a 31-tone keyboard. The Dutch composer Henk Badings has written several works using this scale. The scale has several conveniences. It consists of equal instead of unequal intervals, each interval equal to the diesis between G# and Ab in just intonation, which is one fifth of a whole tone: notated G, G-half-sharp, G-sharp, A-fl at, A-half-flat, A. Charles Ives, Carl Ruggles, and Bela Bart6k extended the equal- 85 tempered scale by eliminating the key-signature and using both sharps and fiats to indicate microtonally distinct tones, which can be sung or played on flexibly pitched instruments, thus giving a scale of more than 12 notes. Ives notates both D# and Eb in a passage for the voice of his song, Like A Sick Eagle.

The sound of meantone, for music to be played in that tuning, is rich far beyond the capacity of equal temperament.The richness of meantone consists in the play between strictly concordant and severely discordant intervals; the sound is too rich to be acceptable in large vertical relationships on a harpsichord but sumptuous in arpeggiation. A modified meantone, called well-tempered and at tributed to J. S. Bach, is still the best practical tuning for a keyboard instrument. The subtly unequal intervals retain some of the modula tory coloration of meantone, yet permit playing with full chords in all keys. Well-tempered did not survive, probably because it is im practical for orchestral tuning. [A taped example was played, re corded by Wesley Kuhnle: Beethoven's Prelude through all the keys, Opus 39 No. 2, in a well-tempered tuning, the modulating sounding like a kaleidoscope of color-changes.]

Equal temperament divides the octave into equal parts, all inter vals except the octave being discordant. The loss of modulatory color change is made up by increase of harmonic flexibility, once one has learned to accept the increased increment of discord. The change to equal temperament was not proof of musical progress but a com promise which musicians for a long time bad refused. It is the most practical tuning for a large orchestra. Because it lacks richness and modulatory variety it invited more elaborate orchestration. The har monic flexibility led to chrnmaticism, atonality, polytonality, and diagrammatic serialism, which replace the sonority of correct intona tion with the brilliance of discord. Since brilliance instead of intona tion is now the issue, there has been a steady raising of official con cert pitch. But the more these effects are detached from correct intonation, the more the composer is reduced to a bare 12 tones, which for lack of an acoustically correct referrent he cannot effec tively extend in the manner of Bart6k, Ruggles, and Ives. It is in part for this reason that Schoenberg's composition remained ambi guously between the true emancipated dissonance of his theory and traditional harmony.

Nineteenth-century harmonic theory, a melange of rules derived from just intonation, meantone, Italianate consonance and Germanic dissonance, rationalized again to the different conditions of equal temperament, emphasized the notated composition rather than its 86 sound. The serialists, inheritors of this tradition, stress the importance of the score, with some aleatory concessions; their opponents dwell on the sound, across the entire field of microtonal distinctions, from just intonation to noise, with alteration of the sound envelope. It is this broad field of microtonal distinction which the music of Charles Ives is now bringing forcibly to notice: just intonation (in embedded octaves) , consonance (in simple triads), dissonance (which Schoen berg called "extended consonance") , discord (arbitrary tones with out harmonic relationship) , and noise.

All electronic or computer music is by implication of the medium microtonal, since it involves the entire field of sound instead of a 12-note system, in the same way that music in equal temperament is by implication atonal or polytonal and meantone implies the clash between consonance and dissonance. The full consequences follow by at least a half-century the implications of the medium : the con sonance-dissonance clash and modulatory color-change of meantone produced sonata-form; equal temperament equaled out to chroma ticism and atonality. Form in music responds to microtonal devia tions in the medium.

The human ear, as it grows more attuned to microtonal differ ence instead of the fixed pitches of the keyboard, grows also more aware of the need for a system of acoustically correct and coordinate tonal relationships. Thus there has occurred in this century a still spreading revival of interest in just intonation. Intonation is itself a source of ceaseless microtonal confusion, nobody except keyboard players being able to agree on the correct width of a major third. As a matter of fact the keyboard octave in equal temperament is often tuned wide, except in the middle octaves. Some string quartets play a slightly modified just intonation, others an unavoidably modi fied equal temperament - a correct equal temperament is difficult to achieve by ear. Schoenberg was explicit in requiring without devi ation an exact equal temperament, as the Kalisch Quartet professes to have played his music. Certainly the intonation of this famous Quartet, as preserved by its records of the four Schoenberg quartets, is extraordinarily satisfying. (Privately recorded for Alfred Newman, 1937; afterwards issued as long-play records by Alco.)

The only system of acoustically correct tonal relationships is just intonation, in one of several more or less practical scales. Lou Har rison prefers a 12-tone just intonation; when tempered to accommo date a correct uctave it may be called a just temperament. A scale of this type practically eliminates modulation. Harrison, combining the methods of polyphony with those of serialism, finds 12 tones quite sufficient for most purposes and can draw a satisfying melodi-

~------_ I ous complexity from a pentatonic scale. He is also at ease in the 87 microtonal usages of Oriental music. LaMonte Young works with a very precise, slowly undulating, sustained harmony, so exact in its pitches that the resulting difference tones sound in the harmony like instruments. Josef Yasser proposed a 19-tone just temperament, and some musicians are exploring the 22-tonc Hindu system. In Hindu music one does not use all 22 tones at one time but chooses the correct additional leading tones according to the melodic consistency of the raga. I have mentioned already the 31-tone temperament of Adriaan Fokker, which embodies a correct meantone and the correct leading tones for all occasions.

The most ample system is the 43-tone just intonation of Harry Partch, which provides a richness and extension of harmony possible with no other system in current use. The fine intervals of this scale permit the notation of idiomatically inflected speech. Partch also uses arbritrary tones, produced by a variety of primitive instruments, and noise. Partch writes for instruments adapted or invented and built by himself; there is only one complete set. His adapted keyboard is · good only for melodic playing. The keyboard of Fokker's 31-tone organ permits extensive playing in chords. I do not have the speci fications of Eivind Groven's 43-tone organ, which produces sound by means of finely tuned crystals.

I should mention finally the microtonal authority of noise (the indeterminate mingling of sounds), exemplified by the very rich tex tures of sonority in Cage's Fontana Mix. Noise is also a microtonal medium, much richer in potentiality than 12-tone discord, and mu sicians have scarcely begun learning how to use it. Noise should be considered in two aspects: the beating of severe dissonance and dis cord, which more easily becomes microtonal harmony when well mixed with simple concords and consonances, as in works by Ives; and extraharrnonic noise.

With the advent of the computer as a composing instrument, the entire field of sound has at last been brought within reach of musical governance, and we may expect not only a music of indeterminate sounds of microtonal complexity but elaborate scales in microtonal just intonation as well as innumerable scales drawing upon our entire knowledge of scale systems from all musical cultures, each enriching the others. Such music need not be confined to the computer; dis criminative instrumental techniques can produce a great variety of microtonal scales.

When Wesley Kuhnle was preparing his History of Tuning, he wondered that composers do not find pleasure tuning their instru- 88 ments to various systems and learning to distinguish the results. Careful tuning, in any system, he believed, is the best method of ear training. He hoped that the History of Tuning, apart from its historical interest, might awaken composers to the aesthetic significance and beauty of correctly realized tuning distinctions, so that these micro tonal distinctions might be clearly achieved in composition and per formance. Our host on this panel, Ben Johnston, confirms out of his own experience the importance of such microtonally discriminative ear training. I am w-ateful that he has assembled us here to talk about it.

BIBLIOGRAPHY

Barbour, J. Murray, Tuning and Temperament. East Lansing, Michigan State College Press, 1951. Chase, Gilber·t, America's Music, From the Pilgrim to the Present, revised sec ond edition. New York, McGraw-Hill, 1955-1966. Fokker, A. D., Just Intonation and the combination of harmonic diatonic me lodic groups. The Hague, Martinus Nijhoff, 1949. -----, Jan van Dijk, B. J. A. Pels, Recherches Musicales, Theoriques et Pratiques, extrait du Tome X, Archives du Musee Teyler. The Hague, Martinus Nijhoff, 1951. (Gives extensive information about the 31-tone organ at the Teyler Museum, with photographs.) LL S. Lloyd and Boyle, Hugh, Intervals, Scales and Temperaments. London, Macdonald & Co., 1963. Mandelbaum, Mayer Joel, Multiple Division of the Octave and the Tonal Re sources of 19-Tone Temperament. Ann Arbor, University Microfilms, Inc., 1964. (The most thorough and complete study of the history of intonation.) Partch, Harry, Genesis of a Music. Madison, The University of Wisconsin Press, 1949. Yates, Peter, An Amateur at the Keyboard. New York, Pantheon Books, 1964. (Appendix on Temperament and Tuning.) ---- - , Twentieth Century Music, Its Evolution from the End of the Har monic Era into the Present Era of Sound. New York, Pantheon Books, 1967. BEN JOHNSTON 89

TIIREE ATIACKS ON A PROBLEM

THROUGH PHILOSOPHY ScHoNBERG's SOLUTION to the impasse of music in his day was an heroic but tempo·rary expedient. The sickness he diagnosed was real; the therapy he devised was more than adequate for him and for his time, since he enjoyed great creative vitality. Between his right and his left hand, no collusion: his practice did not restrict itself to his theory.

A great composer and an independent thinker, Schonberg would have deplored the academy which has been erected upon the very part of his technique which he refused to teach to his students (regarding it, no doubt, as his own peculiar solution to composing). Schonberg did not wish to break with the past. He wanted very much to bridge the abyss he saw before him, not simply to leap it and start life all over again.

At the onset of this century the grand European tradition of music was sick with a mechanical over-ripeness. Its very means for creating the primary aesthetic illusion, that of organic unity, were 90 creaky. Ih an effort to conceal its wobbly wheels, the whole mech anism had been clothed more and more magnificently, so that finally the term "19th century" came to mean "pretentious."

Music, even if viewed purely as an experience of sensation, is an exercise in attention. It attracts and keeps attention; it arouses inter est. If it is to do this, it cannot forever reiterate the same sounds, the same patterns.

As an expression of man's inner life of feelings, music cannot afford to seek again and again the same responses, because these responses will deteriorate not only in intensity but also in significance. At first bringing challenge and discovery, a significant musical com position eventually breeds a contempt that masquerades as comfort and acceptance. A musical system is a set of logical symbolic relations. As any such system becomes thoroughly known, it gradually bankrupts itself of information to the point of exhaustion-to the point of contempt, one might say. We become contemptuous: of cliche, of platitude, of perfunctoriness, of insincerity. The symbols become exhausted: of meaning.

It was this depletion which Schonberg faced courageously. Music's pitch usages he found badly outworn. Schonberg's solution, based upon an extension and refinement of thematicism, largely replaced tonal devices in his music as a means to create an impression of organic unity. This technique later proved to be as applicable to any other measurable aspects of music as to pitch. A not altogether un predictable result of this has been that many composers, ti:red of tonal cliches, have either abandoned pitch, or, more accurately, have organized it as if it were noise.

In an article addressed to the general public (in Saturday Review of Literature), Aaron Copland describes the nearly exclusive pre occupation of a whole younger generation of composers with neg lected parameters of sound and performance. "They have changed the rules on us," he comments wryly. Well, why not? Who made the rules? Who, for that matter, made the game?

What is really behind the development to which Copland refers, is the bankruptcy of overused parameters, notably of pitch and metrics. And the problem of these aspects of music is not being solved, but evaded. The currently fashionable means for organizing sound: serial, aleatoric, statistical, are not more productive of musical order than earlier systems, but less so. However, these trends are definitely moves in a constructive 91 direction, since they represent new parts of a larger complete system than earlier musicians used. The main organizational techniques of the past also fall within this larger system.

Elswhere I have discussed this system, a heirarchy of kinds of aesthetic order, based upon a theory of scales of measurement devised by the Harvard psychophysicist S. S. Stevens. The kind of scalar order of which tonality and metricality are special cases is mathematically the most sophisticated of the kinds possible within this system, which also provides a number of less intricate modes of order. Other special cases of this most versatile of scalar order types include the pitch systems of classical Chinese music, of classical Islamic music, of classical Indian music.

Schonberg replaced a tired system with a less subtle but more vigorous one. Alternatively, others have tried to replace tonality with parallel alternative systems such as the Asiatic ones just mentioned. But when your own tradition outgrows its childhood, it is not a work able next step to abandon your own line of development for a parallel one. Nor can you permanently abandon overworked sides of your development to concentrate on potentially healthier ones.

Unfortunately, in tacitly accepting as an arbitrary "given" the twelve-tone equal-tempered scale, Schonberg committed music to the task of exhausting the remaining possibilities in a closed pitch system.

There are people who cannot stand to be confined in no matter how large an enclosure. They find the walls and are made miserable by them. They don't care how many bars you put in the cage - 12, 24, 48, 19, 22, 31 , 53 - they will hurt themselves on them. Or break them down.

When you reach a philosophical impasse, you need to get to a more basic idea. You need to raise your assumptions about your field to a higher level of abstraction. If you do this successfully, what used to be basics will turn out to be special cases of more general principles. Einstein's physics do not invalidate Newton's. It simply reveals Newton's laws to be special cases of more general ones. The discovery of DNA, a substance which controls the intra cellular synthesis of proteins and is thus basic to life itself, does not knock out all of biochemistry up to that breakthrough, but rather forces a reevaluation of many of its assumptions.

So let us revivify the problem of pitch as some composers have begun to revivify the problem of metrics. There is no need to rule out 92 chance, statistics, serialism. But let us also pursue, still more subtly, the techniques of rational proportion, which are capable of vastly more than has hitherto been asked of them. What can be grasped with equal alacrity by ear, by mathematics, and by intuitive feeling is the best material for art. And this intelligibility is not a mere matter of conditioning: some relations are naturally more easily understood than others. This makes them on the one hand primary, and on the other hand obvious. I am interested in a spectrum from the most obvious to the most subtle.

THROUGH THEORY The term "just intonation" connotes to most people either a very specialized concern with Renaissance and Medieval choral music or an equally specialized concern with pre-piano keyboard music. To performers with uncommon erudition it may connote (if somewhat foggily) an ideal of playing in tune by ear in a way that necessitates careful and minute adjustments of pitch not taken care of by the design and the manual performing techniques of their instruments. This ideal also brings out a bias against keyboard and other fixed pitch instruments which make these tuning adjustments impossible. But often the "in-tune" playing defended in the name of just intona tion is quite as literally irrational as the temperament its defenders profess to deplore. It is not that players cannot hear subtle pitch differences, rather it is that there is great confusion about what they intend when "playing in tune."

It is a rare performer who has any knowledge of tuning and tem perament. Available books on the subject confuse the avemge musi cian by translating the whole discussion into charts of numbers and other rudimentary but nonetheless mysterious mathematical symbols.

Most composers have little grasp of the subject and almost as little curiosity. This includes the majority of those who find mathe matics a convenient and fascinating tool in composition. References to tuning and temperament in writings by composers are almost parenthetical, and generally refer to the problems rather than present them clearly.

Concern with this field by musicologists is usually connected with ancient or medieval music theory, with Asiatic or North African music theory and performance, or with the tuning of early keyboard instruments. This leaves almost only mavericks like Yasser, who are attracted to speculate in what almost amounts to a branch of applied number theory.

~------Even acoustics, turning resolutely to the fi elds of hearing and 93 experimental psychology, leaves very much behind such latter-day Pythagoreans as Helmholtz.

In the last distillation of all, there remain a tiny handful of com posers, who are incurably fascinated with pitch relations, who are intrigued rather than frightened away by all the arithmetic, and who refuse to be bound to twelve-tone equal temperament, which after all was no more than an expedient necessitated by keyboard instruments intended to play tonal music.

This small group fringes off into a wider group of composers who have an occasional interest and concern with alternative tuning systems, but who find no more basic quarrel with 12-tone equal temperament than that many of them consider it a happenstance of historical tradition.

In any case, anyone who wants to tackle the problem of writing music in a new tuning system must either ( 1) rely mostly on fixed pitch instruments or (2) realize the music directly, without the inter vention of performers, or ( 3) reeducate the listening habits of performers.

Most of the earlier gropings in this direction were either entirely theoretical, or involved experiment with keyboard, fretted and pitched percussion instruments. Undoubtedly the most extensive work of this kind has been done by Harry Partch, who has literally designed and built an orchestra of such instruments for his own use. The training and coaching of performers, the maintenance and carrying about of his instruments, the persuading people that so unconventional a project is worth so much effort and expense, not to speak of the theoretical research, the design and building of instruments, and the composing itself - all these constitute a monumental life-work, mostly carried out, moreover, in conditions of poverty, public indifference, and rejection by his colleagues. Anyone interested in this field owes him not merely a debt, but an apology. Partch was determined to get this kind of music out of the limbo of theorizing, and he did it. Almost no one else has produced more than an occasional oddly tuned keyboard piece, such as Ives' Quarter-tone Impressions.

The second alternative, adopted by many electronic composers, can either be done by ear, as Varese insisted, or by computation and measurement, as Stockhausen made his Gesang der Junglinge. In designing such music by computation the danger lies precisely in the fact that almost anything conceivable can be executed, with sufficient ingenuity and patience. But since not everyone shares the faith that 94 whatever is mathematically intelligible will be audibly so, this can be a serious problem. In losing the performer, one removes an automatic check: whatever can be played can be understood by ear, but not so, whatever can be synthesized. If you operate as Varese did you are safe from that pitfall. But is the unaided ear really better than the ear plus the analytical intelligence?

It is the third alternative, refraining the performer's ear, which offers the most fruitful challenge, forbidding as it seems. If a per former can teach his ear to know its way around in a strange pitch system, then a listener can make sense of the result. It is far easier for performers to do this if the new system is based upon one with which they already are intimately familiar.

The earliest attempts to do this inserted additional melodic tones "between the cracks of the keys," providing a "hyperchromatic" melodic scale, and the possibility of "distorted" intervals adjacent to the more familiar ones. Until after the advent of electronic music none of these attempts involved a serial use of pitch. Haba's string quartets use the new pitch relations as a further complication of modulatory chromaticism inherited from the late nineteenth century. Ives' quartertone piano pieces and some of his songs point to a fresher approach to harmony derived from impressionism.

Impressionist harmony, especially that of Debussy and of later Scriabine, strongly implies a just intonation using higher partials of the overtone series, the very thing later to be developed systematically by Partch. Similarly much of Ives' harmony (not only in the expHcitly quartertone pass,ages) implies an extended just intonation tuning.

This use of harmonic intervals tuned just (by eliminating the roughness of beats) provides a better point of departure than any tempered scale for an expanded pitch system. Microtones are an interesting by-product of this kind of tuning, as the just intervals do not fit together in such neat circular patterns as do their tempered equivalents. Cyclic patterns (of chains of equal-sized intervals) do not return to octaves of their starting pitch, but instead to displaced octaves, microtonally distant from octave equivalents.

To make a just intonation pitch system, you select a small number of generative intervals which you can tune precisely, by ear. For a tridaic system the unison, the octave, the perfect fifth, and the major third will suffice. All other intervals can be obtained by combining these. Such a system yields scales more and more closely approxi mating equal-interval scales at five notes per octave, at seven, at twelve, at nineteen, 'at thirty-one, at fifty-three. Partch's system, a hexadic one, needs as generative intervals, the unison, the octave, 95 the perfect fifth, the major third, the "natural" seventh, and the "natural" eleventh. He linearizes a scale (by choice) at forty-three tones per octave.

Keyboard and other fixed pitch instruments are by their very design unsuited to just intonation. The best one can do with them is to select any convenient, desired number of tones from the system, and be content to use only these. This more or less drastically restricts movement within the infinite system of pitches provided by just in tonation. It is only with instruments of flexible pitch, with performers who adjust the tuning by ear as they play, that one can have full freedom of movement within such a wide-open pitch system. Partch's extensive reliance upon fixed-pitch instruments is mostly due to his introduction of unfamiliar generative intervals, "natural" sevenths and elevenths. In an expanded triadic system no such difficulty exists, since the generative intervals are known to every musician, and pre cision in tuning them by ear is mostly a matter of clearing up about wbat to listen for.

THROUGH PRACTICE I have been warned by just about everyone who knew I was writing this paper not to permit it to become just more words without music. Aesthetic philosophies and theories of music are known by their fruits: by the music written and played and heard with their aid.

For several years I have been writing music based on the posi tion which I have just presented. I certainly cannot claim that what I have written up to now exploits fully the philosophical and theo retical possibilities which I can already see. I am quite prepared to devote the rest of my creative life to this artistic effort. I would not wish for anyone to adopt my technique. There are as many composi tional aprpoaches to the musical problems with which I am con cerned, as there are composers. In speaking of my own music I am giving the example which I know best.

About all words can do to intensify the listening experience is to call attention to sounds which may especially interest us. Words do not usually do this, but lead us away from the experience of hearing sounds.

It is at present most unfashionable to speak about the "meaning" of a piece of music, about the experience of its intuitive, emotional, 96 symbolic content. Almost the only way to do this is to attempt a translation into poetry - and we have had a surfeit of nineteenth century-style program notes. We have the same difficulty in speaking of any powerful experience. We run a risk of detracting from it, so that many people prefer to say nothing.

With musical meaning, as with listening itself, about all words can do is to hint at the character, the color, the intensity which the experience of a piece of music may bring.

In the telling it is the remaining side of music, its intellectual machinery, which alone can receive justice. But I prefer to think of all intellection in the process of composing as a nozzle. Once, in answer to a question about technique, I replied that it is what I do with my mind while I am composing. To me this nozzle, this focus sing mechanism, is of great importance. Without it what comes out is weak and diffuse. I believe that the manner of the focussing should be relevant to, and symbolic of the inner meaning of the music. What makes Beethoven's Piano Sonata, Op. 109, worth studying, for instance, is the discovery that a structural, intellectual analysis of the work points to and deepens the direct intuition which you get when the sonata is brought to life in performance. It is, I think, dan gerous to discuss this aspect of a work when your listeners have not had this intuition of it.

For these reasons, more than from modesty, I shall keep my remarks on my String Quartet No. 2 brief and suggestive.

I wanted to write a piece in which the players would need to listen to each other carefully and to take mueh greater care than usual in locating the pitches. It would be a little like mountain climbing: the foothold of each note would be dependent upon mak ing precisely the right connection - the right interval - with some other player's note.

(I actually believe that all music not played on fixed-pitch instru ments should be played in this manner: it is keyboards, more than anything else, which create the illusion that "C" is some absolute pitch, that our scale is a fixed ladder of "steps.")

The intervals the players must tune in this way are precisely the consonant ones of triadic music, but the harmonic character of the music is almo~t nev,er triadic. There are three distinct kinds of interval texture in this piece. Where I used mostly consonant intervals and that type of dis- 97 sonance ,traditionally called "diatonic,'' I thought of the texture as "diatonic." The second movement is in this idiom. The term "diatonic" in a way implies even an occasional use of so-called "functional diatonicism." This music has, on the contrary, a harmonic idiom of rapid chromatic changes and microtonal cross relations, far closer in sound to Gesualdo than to Bach. These microtonal pitch relations describe a strict spiral pattern, ascending one octave of a fifty-three tone just intonation scale.

A second type of texture results from emphasizing dissonant in tervals produced with the aid of simple consonances, but predominat ing over them. When a large number of these dissonances are what, in traditional harmonic terminology, are called "chromatic intervals" (that is, augmented or diminished intervals) , I thought of the tex ture as "chromatic." This is the character of the opening movement. I composed this movement entirely of permutations of a single three note motif, interpreted with a great variety of tunings, and always combined into one of three strict permutations of a twelve-tone set. The starting tone of each successive set-form rises one more pitch in a complete fifty-three tone octave. In this movement, the rhythmic and durational relations are governed by a proportional system anal ogous to the just intonation system which governs the pitch relations.

The third kind of pitch texture, which dominates the final move ment, is created by melodic and harmonic use of microtonal intervals and microtonal alterations of larger intervals. These intervals also result from combinations of simple consonances, and occur mostly with these. In some places, however, the playern are told how to find these microtonal variants by melodic size. The middle section of this movement serially treats a 31-note scale. The movement also seri alizes durations and is a microtonally exact retrograde inversion of itself.

As you listen to the quartet, you become increasingly aware of microtonally altered intervals and of actual microtones. In the first movement these occur only in the widely leaping melodic lines of the instruments and never in the harmony. In the second movement they turn up in the harmony, in sharp contrast to the uncomplicated melodic lines, and to the harmonious consonances of the just intona tion. In the last movement such intervals are much of the time in the foreground of attention, set off more than ever by the clear con sonances they surmund. But in the middle section they eclipse all other types of intervals, in a frenzy of contrapuntal activity. 98 If I may venture a "subjective" adjective or two, I would describe the effect of the first movement as shifting and iridescent; the second, as clear and lyrical; the last, as intensely unsettling and expressionistic.

I hope I have conveyed the conviction I have that music should be heard and not seen. We have set ourselves to talk about it: that is already error enough. LEJAREN HILLER 99. ;

ELECTRONIC SYNTHESIS OF MICROTONAL MUSIC

l. INTRODUCTION It hardly needs pointing out that electronic media - tape, syn thesizers, computers, even specially designed electronic musical instru ments - offer a promising solution to many of the difficulties of realization of compositions not written for the ordinary chromatic scale. Nevertheless, it is also obvious that relatively little has been accomplished so far to assess the value of such media in this pll[' ticular application. There are relatively few tape compositions, exclu sive of concrete music, in which a precise articulation of the frequency domain is a dominant feature. Among early compositions, Stock hausen's Studie II and Moroi and Mayazumi's Variations Sur 7 can be cited, but these are rather exceptional. Most tape music is char acterized by a fairly diffuse or indifferent approach to detailed pitch articulation, to say nothing of detailed time articulation, an equally critical problem. With electronic music synthesizers or computer generated sound, the situation seems somewhat better, and substan tial numbers of experiments and actual compositions already exist which bear this out. 100 I am going to confine my remarks today primarily to processes and examples, so I shall not discuss reasons why one might or might not want to exploit the combination of electronics and unusual tunings and scales. Therefore, I shall review techniques for frequency control, measurement and transformation as found in the electronic music studio and in the computer laboratory. I shall then demonstrate how such processes work out in practice, illustrating my remarks with several short musical examples, pointing out advantages, disadvan tages and possible alternate methods. I shall then tum to computer methods as an alternate approach. I hope my remarks shall provide a few clues as to how we may proceed in the future.

II. ELECTRONIC TAPE METHODS Normally an oscillator provides a given frequency according to a setting on a rotating knob, a push button or other device which changes a circuit parameter in the oscillator circuit. Alternately, a given frequency can be generated indirectly by a fixed input voltage if the circuit is a voltage-controlled oscillator of the type that have recently come into considerable use in electronic tape music studios. In either case, it is seldom possible to trust the markings on the dials or buttons or keyboard inputs of such equipment unless it is high precision equipment that is extremely reliable or is self-adjusting by means of some sort of reliable feedback servomechanism. Most mark ings on frequency dials are not fine enough, the dials can slip out of adjustment, the oscillator can drift with room temperature and age, and so on. It is therefore obvious that one of the most important acquisitions of an electronic music studio is one or more pieces of equipment for precise frequency measurement. What are the avail able choices? They seem to be limited to the following: (a) The ear (b) Frequency meters ( c) Stroboscopic devices (d) Frequency counters ( e) Lissajous patterns and other special techniques.

The ear is obviously an excellent instrument if reliable. However, not all people, not even musicians, have similar hearing acuity. Quite apart from physiological differences in acuity of pitch perception, both relative and absolute, there arise questions of habit and experi ence and of difficulties associated with listening to entirely new types of pitch relationships. It is one thing to identify a B-flat or a tem pered fifth played on a piano; it is something quite different to iden tify correctly a 13: 11 frequency ratio. Perceiving more unusual pitch relationships can often be done too with practice, but one needs to 101 have heard accurately constructed prototypes. Again, the ear can fool you sometimes. I remember several years ago I constructed a short melodic line out of undistorted square waves. I did this by ear and was off by an octave throughout, the reason being, of course, that square waves lack even harmonics and therefore can generate a different subjective fundamental from the nominal one.

Frequency meters can be dismissed in a word. They lack suffi cient precision. On the other hand, stroboscopic instruments such as the familiar Stroboconn are a different matter. One problem, how ever, is that the Stroboconn measures deviations in cents from twelve note equal~tempered tuning with concert A normally set at 440 cps. These instruments are fine for normal tuning jobs, but become more and more clumsy to use the more one. departs from the ordinary chromatic scale. Even the computation of frequency ratios requires consultation of special charts. In spite of these limitations, strobo scopic instruments are useful, nonetheless, if one understands their particular manner of measurement.

Frequency counters are probably the single most useful measuring device. They are moderately priced, easy to use and accurate in proportion to the sampling time of the frequency being measured. They read directly in cps, which often eliminates much hand calcula tion. Their main deficiency, as far as I can judge, is in comparative measurements, for the ear can easily detect as beats slight mistunings which fall within the limits of precision of the usual frequency counter.

Liss·ajous patterns are useful for comparing one sinusoidal fre quency to another, provided the desired frequency ratio is fairly simple. These are easily generated on an oscilloscope by feeding one single to the vertical plates and the other to the horizontal plates of the oscilloscope. Within its limits, this kind of display has occasional utility.

Another major problem in the electronic music industry is that of frequency transposition. Frequency transposition is seldom heard in electronic music because it is hard to do as compared to writing the identical transformation into an ordinary instrumental score. It is not surprising, therefore, that this too has had an inhibiting effect on the exploitation of pitch relationships in the electronic medium.

The available techniques for pitch transposition seem to be limited to the following: 102 ( 1) Starting all over again from primary materials. This indeed does work and yields satisfactory results. However, it is frustrating and boring to have to redo essentially the same material over and over again. (2) Electronic transposition. With modulators or other non linear electronic circuits, it is possible to effect frequency transposi tions by side-band generation.

Once side ~bands have been formed, it is possible to remove the original frequency by selective filtering. The trouble with this process is that side-band generation yields sum and difference frequencies and not frequency ratio alterations. Consequently, the spectra of sounds so transposed are also transformed. For example, a sound with harmonic partials is converted to a sound with inharmonic partials. If this happens, one loses a sense of definite pitch because it is impossible to define an audible fundamental for a complex sound with strong inharmonic partials. ( 3) Variable speed tape recorders. An ordinary tape recorder equipped with a variable speed motor can effect transposition, but at the cost not only of concurrent time compression or expansion but also transformation of the timbre formant along with the fundamental. It is this latter condition which causes the quality of a sound to be altered as well as its pitch. A familiar example of this occurs when recorded piano sounds are transposed an octave. Both these effects are eliminated when a rotating playback head is substituted for a stationary one as in the Springer Zeitregler. The past two years we have had this latter device in our own studio and find it very useful for pitch transposition as well as time compression and expansion. One must remember, however, that the percentage of alteration one can employ with this device is limited by the amount of distortion one introduces. This in tum depends on the complexity of the sound. A sine tone can tolerate very little alteration, a noisy sound quite a bit. On the average, a major third seems to be about the safe limit for most sounds.

Ill. SOME EXAMPLES Given these possibilities and limitations, I thought it might be useful next t:o discuss several brief examples of electronic music written in unusual scale systems. These are taken from my Seven Electronic Studies of 1962 and 19631 . I should like to make a few

1 Tape recordings of the studies discussed in this talk were performed as illustrations. At the time of editing this talk for publication, it seems likely that these studies will be issued on a disk recording in the near future. comments on how the studies were actually made2 • 103

Study No. 2, called Proportions, deals with just intonations and precise intervallic relationships. It is also a study of harmonic pro gression. Call it a kind of tonality if you wish, but I think you will agree that it bears little relationship to conventional notions of tonal order. I set up the basic structural plan in terms of a more or less traditional tabulation of consonant to dissonant intervals as shown in Table I.

There are seventeen intervals listed in order from most distantly related to mostly closely related to an arbitrary fundamental. I then arranged a short piece of 17 sections, each section of which would be based on one of these pitch levels as related to a final pitch level of G set equal to 96 cps. Thus, the piece progresses in accordance to the plan shown in Table II.

It goes through a series of "keys" which become more and more simply related to the final tonic. Moreover, within each section, ver tical relationships are also restricted to intervals in just intonation, so the actual pitche·s change from section to section. In principle, this

Table Study No. 2: "Key Scheme" for the Seventeen Variations

Frequency Cents from Relative Variat ion Interva l Ratio Starting Point "Key", , Pitch Multiplier 1 Diminished 5th 64:45 609.777 Db 136.5 1.422 2 Augmented 4th 45:32 590.224 C# 135.0 1.406 3 Semitone Grave 16:15 111.731 G# 102.4 1.067 4 Minor 7th 16:9 999.091 F# 170.7 1.777 5 Major 7th 15:8 1088.269 Gb 180.0 1.875 6 Minor Tone 10:9 182.404 Ab 106.7 1.111 7 Major Tone 9:8 203.910 A 108.0 1.125 8 Minor 7th 9:5 1017.597 F 172.8 1.800 9 Minor 6th 8:5 813.687 Eb 153.6 1.600 10 Harmonic Minor 7th 7:4 968.826 Fb 168.5 1.750 11 Minor 3rd 6:5 315.641 Bb 115.2 1.200 12 Major 3rd 5:4 386.314 B 120.0 1.250 13 Major 6th 5:3 884.359 E 160.0 1.66•7 14 Perfect 4th 4:3 498.045 c 128.0 1.333 15 Perfect 5th 3:2 701.955 D 144.0 1.500 16 Octave 2:1 1200.000 G 192.0 2.000 17 Unison 1:1 0.000 G 96.0 1.000

2 A much more complete description of all seven studies is contained in L. A.· Hiller, Jr., "Seven Electronic Studies for Two-Channel Tape Recorder" (1963), Technical Report No. 6, University of Illinois Experimental Music Studio, Urbana, 1963. 104 Table II Study No. 2: Structural Plan of the Composition Variation "Key" Pattern A - Principal subject Db 2 Pattern B t Episode I: Chords with inharmonic 3 Pattern C - partials ICG# 4 Pattern DJ L F# 5 Pattern A' - Principal subject retrograde Gb 6 Pattern E } Episode II : Contrasting contrapuntal 7 Pattern E' materials in 3 simultaneously occurring 8 Pattern E" "keys" {t 9 Pattern A + A' - Principal subject against its retrograde Eb form 10 Pattern F } Episode Ill: Contrasting contrapuntal rFb 11 Pattern F' - materials in 3 simultaneously occurring i Bb 12 Pattern F" "keys" LB 13 Pattern A - Prin cipal subject retrograde E 14 Pattern D} Episode IV: Ch ord s with inharmonic re 15 Pattern C - partial s i D 16 Pattern B LG 17 Pattern A - Principal subject G

Table Ill Study No. 2: Chords with Square Root Overtone Structures

Chord 1 Chord 2 Chord 3 Chord 4 (Variation 2) (Variation 3) (Variation 15) (Variation 16) Partials l2it.1" on C# /iion G# /non D /2i1-T on G /r 135 102.4 144 192 ff 145 206 13 234 177.7 250 343 14 204.8 288 vrs- 301 229.5 323 430 /6 25 1.5 353 .;r 356.5 272 383 509 18 290 408 (9 405 307.2 422 576 /IO 324 456 !IT 446 636 (IT II3 48 8 694 fi4 IE" 523 743 /i6 (fl 556 791 IIT' ITS 590 836 would constitute a scale of 17 x 17 or 289 pitches per octave as a 105 basic repertory. In practice, I used many fewer pitches than this. One other detail that might be interesting to mention is the use in several of the variations of square root overtones to yield quite satisfactory clangorous tones. The composition of these tones is shown in Table III. As to composing method, all these pitches were generated in dependently, vertical structures were assembled by successive mon tage, and everything was spliced together by the classical techniques of tape editing. This obviously was slow and laborious work. Today, if I were to compare the same type of piece, I would save time with the Eltrn transposing playback transport. Even more than this, this piece, I believe, could be rather easily made by computer sound synthesis. In the next piece, Study No. 4, called Homage to Helmholtz, I was interes,ted in several other relationships among tones, notably the use of an additive scale rather than a logarithmic one. Although an additive scale is, in effect, an overtone series, it also has the virtue of corresponding to the ear's response in terms of judging conson ance and dissonance. Helmholtz observed that listeners seem to judge the range of about 32 cycles as the harshest dissonance between two tones regardless of pitch level. This corresponds to a large interval in the bass and a microtone in the high treble. So I constructed this piece on a scale with a constant frequency difference of 32 cps. up to the ratio of 81 : 80. You can hear an arpeggio of diads built on this difference at the opening of the study. Secondly, I built melodic lines on some frequency ratios one does not normally hear, namely on tones in the ratios of 9:11:12:13:14:17:18: 19. Thirdly, in the center of the piece you can hear some conventional seventh chords correctly tuned. My final example, Study No. 6, called Even-Tempered Scherzo, is made up of equal-tempered soales of 2 to 24 pitches to the octave. The center section of this study, which presents all these scales in succession, could easily have been made by computer synthesis, but whether the opening and closing sections could be as easily made I'm not so sure. In these sections of the study, the materials from all the scales are scrambled together.

IV. COMPUTER METHODS I think it is obvious that both the electronic music synthesizer and the electronic computer frequently offer a much more elegant 106 approach to pitch and time articulation than conventional tape tech niques. Computers particularly, offer the possibility of programming much of the compositional content of pieces such as I have just dis cussed as well as generating the actual sounds. Everything that applies to the generation of computer sounds in general applies equally to the generation of particular tunings and scales. In addition, however, several other points can be noted. For example, we have some pro grams now that link our compositional routines to our sound syn thesis. Some of them are designed to generate pitches in the ordinary chromatic scale with the pitches being assigned integer values from 1 upwards starting with C1 . A very simple little loop does the actual frequency computation for entry into the actual conversion routine. It is the obvious one of repeated multiplication by 1 2 V 2 until a con ditional transfer is activated (actually a TIX order in SCATRE lan guage). This 1 2 V 2 value is on one card which can simply be replaced by another card or alternately merely overwritten if a different equal tempered scale is desired. For example, I propose to do this in the near future in order to generate some prototypes for rehearsals to be used in preparing a quarter-tone string quartet I wrote several years ago. For this I simply need to insert the value of the 24 y 2 . I need hardly point out, by the way, that frequency transposition becomes a trivial task in the computer.

Along a somewhat different line, but also important, is the accu rate tracking of frequency by computer analysis. We are currently writing a program that will, we hope, plot frequency versus time of any monophonic line of music. This in turn can be used to evaluate intonations, tunings and scales actually used in performance. This problem is not quite so easy as it would seem, one complication being that the fundamental of a tone is often very weak compared to its upper partials. Accordingly, it may be obscured by inharmonicities of upper partials and other noise content. Nevertheless, we feel that the problem is amenable to computer processing and hope to have some thing to show on this subject within a few months. JOEL MANDELBAUM 107

THE ISOLATION OF THE MICROTONAL COMPOSER

THE INTEREST that the American Society of University Composers has shown in microtonal music today represents a welcome break in the clouds of isolation in which the majority of microtonal com posers have operated since a brief and, I suspect, muted limelight was put on their work in the 1920's.

Conventional wisdom on the subject today seems to be that "microtonalism" was an experiment of the 20's that proved a useless mutation and has found its way to the "dustbin of history." To some extent this conclusion seems outwardly to be borne out by events. The aesthetic populism of the 1930's and early 1940's which par tially eclipsed the Schoenberg school, all but totally eclipsed the quarter-tone composers such as Georgi Rimsky-Korsakov, Fr. Wies meyer, JOrg Mager, Silvestro Buglioni, Dominguez de Burrueta, Panach Ramos, Marina Scriabin, Yvette Grimaud, Hans Barth and Mildred Cooper as well. And when a new interest in "avant garde" techniques developed in the late 40's and early 50's, the Schoenberg school came fully into its own, but the quarter-tone composers made 108 no comeback whatsoever. Microtonal composition has continued (it never completely stopped, even in the 30's and 40's), and some of the composers have achieved a measure of renown: Haba in Czecho slovakia, Wyschnegradsky in France, Partch in America, and an international group of composers organized around Fokker in Hol land. But contact between these composers and the "mainstream avant garde" has been minimal; often non-existent.

The isolation has been taken by many to indicate that the musical community at large no longer finds anything worthy of note in the microtonal composers; while this may indeed be largely true, the reason, I would suggest, comes from a wide divergence of viewpoint as yet unresolved, rather than a finished dialectic which has discarded the microtonalists permanently as a used-up "antithesis." What was not clear in the 20's, but seems clearer now, is that most microtonal composers and theorist-advocates, however much they may disagree with each other on details, share in common an analysis of the early 20th-century crisis of musical materials which is fundamentally different from what has become the "mainstream" view.

A number of microtonalists, and Joseph Yasser in particular, have seized upon an article by Schoenberg in Modern Musicl which asserts that the dissonances which he and other composers used are simply the more outlying consonances of the overtone series. Many a Schoenberg disciple has refuted this view and correctly pointed out that it had little or no application to Schoenberg's own music. The harmonic series is no longer widely regarded as valid absolute basis for establishing a musical system. The newer "mainstream" conven tional wisdom on this subject, as expressed by Forte, is that " ... the tonic-dominant relationship is far from axiomatic. On the contrary, the overtone series as a natural basis for this relationship is open to question.... The pure overtone series as it exists in nature is un known to modern ears. Fine adjustments have been made to conform to equal temperament." 2 The majority of microtonalists, however, have continued right up to this day to accept the harmonic series as the generator of their musical systems. They have used microtones to capture the "fernerliegenden Konzonanten" of which Schoenberg wrote.

Taken in the broadest sense, the rnicrotonalists see the early 20th century crisis of tonality as a conflict between music based on the

l"Problems of Harmony." Modern Music XI, Nov. 1934, p. 167. 2Allen Forte, Contemporary Tone Structures. Bureau of Publications, Teachers College, Columbia University, New York, 1955, p. 11. "natural" heirarchy of the small-number ratios and the 12-tone equal- 109 tempered scale. Because the 12-tone scale admits within "tolerable" limits of error only the intervals of the first six numbers (the so-called "senario" of Renaissance theory), the saturation of harmonic possibilities within the senario - whose structural basis was limited to major and minor triads - produced a crisis. One of the two hypo- theses had to be abandoned: either the hypothesis of a harmonic heirarchy based on the number series, or the 12-tone equal-tempered scale which unequivocally limited the applicability of this heirarchy to the first six numbers.

While "mainstream" theory and practice has opted to keep 12- tone temperament and drop the hypothesis of a harmonic heirarchy, microtonal usage makes it possible to expand organically the effective range of the numerical heirarchy and makes unnecessary, for those who accept these theoretical premises, its abandonment. The micro tonalist can argue that the affective power of "tonality" is capable of infinite expansion and therefore need never be "exhausted." It is the 12-tone equal-tempered scale whose intersection with "tonality" is depressingly finite.

Because microtonality offers a "way out" for a composer with strong tonal predilections, it is quite natural that it has tended to attract tonally oriented composers whose characteristic sound-ideals seem startlingly "reactionary" and hence often "primitive" next to those of the "mainstream" composers. It is this "reactionary" out ward appearance based on this fundamental theoretical divergence which has isolated most of the microtonal composers from the pro fessional center of gravity far more than the complexity of his available tonal materials or their forbidding unfamiliarity.

In an age which respects many a composer who questions whether pitch specifications deserve a primary function among the properties of new music, the microtonalist's obsession with small pitch distinctions seems all the more irrelevant to many.

II. If it is the ideological isolation from the musical "mainstream" which has kept the microtonal composer obscure, an even greater cause for his real difficulties is his general isolation from other microtonal composers. The field of microtonal music is factionally divided almost beyond recognition. Perhaps this is innate in its very nature. There is only one way to divide the octave into twelve equal parts; there are, however, an infinite number of ways not to do so. 110 First, there is the basic question of whether to subject the octave to an equal temperament. The majority of theorists and composers seem certainly to have favored equal temperaments, but the contrary opin ion has been maintained at times most forcefully, especially by Partch. If, indeed, a theory of just ratios is called upon to support the abandonment of 12-tone equal temperament, Partch and others argue, why substitute one imperfect temperament for another?

However, it seems that no two advocates of "just intonation" can agree on which just intonation to use on fixed-pitch instruments. Partch's system, while demonstrably succe3sful for himself, has yet to be taken up by other composers, perhaps because he has made the only instruments which can play music tuned his most specific way. Ben Johnston varies his own "just intonation" from work to work, which is fine as long as fixed-pitch instruments are not used. Seem ingly unbridgeable are the gaps between composers whose just systems seek refinements of fifths and thirds by means of commatic alterations, such as Eivind Groven of Norway (and sometimes John ston), and composers such as Partch, whose emphasis is on obtaining intervals of the higher partials (in Partch's case through 11).

With advocates of equal temperament, the problem is slightly less acute. Composers with interests as diverse as Carlton Gamer, Henk Badings, and myself, have found 31-tone equal temperament satisfactory for our work. However, there are many different views as to the ideal number of equal tones per octave. Controversy exists between those favoring wide major thirds and those favoring small ones; between those willing to involve very small distinctions in order to gain greater accuracy with the natural consonances and those preferring simpler systems. Perhaps the greatest controversy of all is between those (so far the vast majority of microtonalists favoring temperament) who would favor temperaments closely duplicating the small-number ratios and those who favor temperaments which specifically avoid them, thereby offering a new language unfettered by involuntary allusions to the past. (Elliott Carter has verbally ex pressed himself as strongly in favor of the latter option.)

Perhaps most serious is the absence of a standard notational system whereby different microtonal composers could communicate in a manner easily interpreted by their peers. Despite their differences, microtonal composers usually seem to regard the works of their confreres with sympathy, but often they cannot decipher their scores.

The sympathy, despite doctrinal differences, is not surprising, for the very sensation of intervals outside of 12-tone equal temperament is a prominent fact of this music - more prominent now, perhaps, than it will seem later - and those who respond to it positively have 111 a strong common bond. Perhaps the time has come to use this bond to break down the isolation each school of microtonal composers (including the "schools" consisting of one composer) now has from the others. An intonational lingua franca is needed with a common nomenclature and notation. To those composers who would rebel at abandoning their own personal systems, such a lingua franca could be a point of departure from which to measure their own intervals (as 12-tone temperament is today, except that a new lingua franca would be closer to most of their systems than is 12-tone temperament, and hence the departures would be smaller and the notation more precise) .

As the tuning system making possible the union of the greatest number of diverse approaches, I wouid suggest 31-tone equal tem perament. (It was not the first microtonal tuning which attracted me, and a prominent reason I have come to prefer it for some of my own work is that others were already using it.) 31 -tone equal tempera ment provides excellent intonations for just major thirds ( 5: 4) and for natural sevenths (7 :4), thereby expanding the repertory of pure consonances beyond the traditional senario, and its one flaw, a slightly out-of-tune perfect fifth, can easily be adjusted whenever voices or instruments without fixed pitches are used.

The smallest unit in 31-tone equal temperament closely approxi mates the diesis (128 : 125) which is the difference between three just major thirds and an octave. It is therefore admirably suited to mark the difference between "G #", derived upwards from C by two major thirds (C-E-G#) , and Ab, derived downward from C by a major third. 3 It is useful to have these traditional names of tones, and even diatonic staff notation during any transitional period; the prac tice is no more outmoded for 31-tone music than for most recent 12-tone music - in fact, it is probably less so. Eventually, however, should 31-tone temperament or any other new temperament become widely used, a new staff system with an irregular pattern of horizontal lines unequal distances apart which repeats every octave might prove

BA conventional, diatonic nomenclature follows from this relationship, with G# and Ab not an identity, but rather consecutive tones in the system. Not so conventional is the alignment that the practice of just major thirds provides - Ab is higher than G#. For the remaining tones, semi-sharps <=F) , sesqui-sharps ( =#f:), and double sharps ( * ) suffice, with their enharmonic equivalents in fiats. The 5 steps from C to D are: C · ~ C:f = Dbb * cffe= JJbt- C=tf:I= = Db * C•* = D~ "' D 112 more efficient. Rather similar models of such a notation were inde pendently drawn up by A. D. Fokker in Holland and Erwin Wilson in Los Angeles.

Composers not wishing to play with the just intervals in 31-tone temperament might enjoy ways of using the primeness of the number 31. No matter what interval one cares to use, a complete cycle of 31 of them is required to return to the original tone. In 12-tone tempera ment the octave can be bisected or trisected. On the other hand the perfect fifth, which cannot be divided into equal parts in 12-tone temperament, can be bisected or trisected in 31-tone temperament.

Rather interesting possibilities for serial techniques in 31-tone temperament are suggested by Carlton Gamer in this same issue.

In recommending a lingua franca such as 31 -tone temperament, I am aware of the objections of such composers as Partch, to whom just intonation is just intonation and 31-tone temperament has essentially the same limitations as any other equal temperament, including 12-tone. Nevertheless, in spite of the richness of Partch's own music, and the magnificent consort of instruments he has built, I know of no other composers working in his tuning system and with his instruments. On the other hand, the two instances I know of where relatively crude 31-tone tempered organs were built (one designed by Fokker in Haarlem, and the other by David Rothenberg in New York, and built by Pels and Moog respectively), a cluster of composers has quickly formed around the instruments. Needless to say, a proliferation of 31-tone instruments, especially around our universities, would be desirable.

III.

Perhaps, in their quest for unity, it would not be unwise for microtonal composers to resort to the favorite device of our pro fession: a publication. Ivor Darreg, an indefatigable explorer of microtonal systems, has made such a suggestion, complete with the name such a publication might bear. He proposes that the generic name for all non-12-tone tunings be "Xenharmonic" ("xen" meaning "strange"). A publication would appear four times a year and would be called Xenharmonic Quarterly, the initials of which would be a fine addition to the librarians' catalogues and bibliographic footnotes. CARLTON GAMER 113

DEEP SCALES AND DIFFERENCE SETS IN EQUALTEMPERED SYSTEMS

THE 15TH CENTURY theorist and composer Ugolino of Orvieto, in his Declaratio musicae disciplinae, maintains that speculative music, or the study of the mathematical foundations of music, is as essential a discipline for the true musician as is composition or performance. Quoting from Aristotle's Physics, he describes the speculative method as follows:

"There is an inborn path in us from .tho se things which are more known and more certain . .. into the specifics. We are taught that in acquiring knowledge and in learning we ought to proceed from those things which are more known [to] us to those things which are less known to us, for this is the natural order of learn ing, so that through the cognition of the known we may come into the conception of the unknown, and thus from the facts known [to] us we may arrive at the facts of nature."1

1 .Seay, Albert: "Ugolino of Orvieto, Theorist and Composer," Musica D is ciplina, Vol. IX (1955), p. 148. 114 I should like to speak today, briefly and rather informally, about certain resources available to the composer who wishes to employ equal-tempered systems containing more than 12 tones per octave. Underlying my remarks are certain assumptions and beliefs: first, that an increasing number of composers will probably employ such systems in the near future, especially as instruments for the electronic genera tion of sound become more readily available; second, that a valid theoretical basis for the employment of such systems is therefore of crucial importance; and third, that such a basis is to be found in "those things which are more known to us," namely certain prop erties of the 12-tone equal-tempered system.

I am well aware that a number of writers on multiple division have shared these same assumptions and beliefs. Nevertheless, their theoretical formulations have differed in many respects from those I shall present because their analysis of the properties of the 12-tone equal-tempered system has been different. I can therefore offer these remarks only as a small contribution to an already large and growing theoretical literature2 in the hope that they may provide some new and useful insights.

In confining myself to equal-tempered systems I am still aware of the importance of just, or "proportional," systems. If I do not deal with these, it is because I prefer to extrapolate from one equal tempered system to another, as extrapolation from a tempered to a non-tempered system is extremely problematic.

Among equal-tempered systems I shall concentrate on the 19- and 31-tone systems, though the concepts I discuss are applicable to many other systems as well. The 19- and 31-tone systems are of particular importance in the literature. Furthermore, as 19 and 31 are prime numbers some significant differences as well as similarities can be noted between these systems and the 12-tone equal-tempered system.

Some of the reasons for the importance attached by theoretical writers to the 19- and 31-tone systems are disclosed by Tables I and II, where the frequency ratios of certain intervals in these systems are compared with those in the 12-tone system and with the just ratios through the 7th partial.3 (See Tables I and II.)

2 See Mandelbaum, Joel: Multiple Division of the Octave and the Tonal Resources of 19-tone Temperament, Doctoral Dissertation, Indiana University (1961), for a bibliography on multiple division.

3 Frequency ratios are from Novaro, Augusto: Sistema Natural de la Mu sica (Mexico, D. F., 1951). pp. 53-4. TABLES OF FREQUENCY RATIOS, D EEP SCALES, AND DIF F ERENCE SETS IN THE 12-, 19-, AND 31 - TONE EQUAL- T EMPERED SYSTEMS

I. Frequency Ratios (after Novaro~

Interval: Pl M2 M3 P4 P5 MB M7 PB ETS 12 Scale degree: 0 . 2. 4 5 7 9 11 12(0) Freq. ratio: 1.1225 1.2600 1.3348 1.4983 1.6818 1.8877 2 19 Scale degree: 0 . . 3 .. 6 8 .. 11 . . 14 17 19(0) Freq. ratio: 1 1.1157 1.2447 1.3389 1.4938 1.6665 1.8593 2 31 Seale degree : 0 .. .. 5 .. .. 10 . . 13 .. .. 18 .... 23 .. .. 28 . . 31(0) Freq. ratio: 1.1183 1.2505 1.3373 1.4956 1.6725 1.8702 2

II. Just and Tempered Frequency Ratios

4th partial 5th partial &th partial 7th partial ETS Just freq . ratios: 1 1.2500 1.5000 1.7500 12 Scale degree : 0 4 7 10 Freq. ratio: 1.2600 1.4983 1.7818 19 Scale degree: 0 6 11 15 Freq. ratio: 1 1.2447 1.4938 1.7285 31 Scale degree: 0 10 18 25 Freq. ratio: 1.2505 1.4956 1.7489

...... VI 116 The intervals chosen for comparison in Table I are those of the so-called "major scale collection" of the 12-tone system. Note the close correspondence of the frequency ratios of these intervals in all three systems. Whereas the "perfect fifth" of the 12-tone system approaches the just ratio 1.5 more closely than do the "fifths" in the other two systems - a situation which can be partially remedied in these other systems by "octave stretch" in tuning4 -the "major third" of the 31-tone system is exceptionally good and, while it is not shown in the table, the "minor third" of the 19-tone system, with a frequency ratio of 1.2001, is also outstanding. In Table II, note the close approximation of the 25th scale degree in the 31-tone system to the just ratio of the 7th partial.

I have listed the "major scale collection" not only as the basis of comparison between these systems and not only because of the particular intervals it contains, but more especially because of the nmltiplicity of occurrence of the intervals in this coilection.

Milton Babbitt has formulated the theorem that "given a collection of p;tches (pitch classes), the multiplicity of occurrence of any inter val (an interval being equated with its complement, since ordering is not involved) determines the number of common pitches between the original collection and the transposition by the interval. " 5 This theorem explains the heirarchical structure of the "circle of fifths" in the traditional major-minor key system.

By equating each of the intervals in the 12-tone system with its complement, these intervals can be paired into 6 non-zero interval classes, each of which occurs with unique multiplicity in the "major scale collection". The interval content of this collection can be repre sented by an interval vector, each successive term of which stands for a non-zero interval ciass, the magnitude of each term indicating the multiplicity of occurrence of that interval class in the collec tion. The interval vector representing the interval content of the "major scale collection'', with successive terms indicating the multi plicity of occurrence numbers of the non-zero interval classes from 1 to 6 is [2 5 4 3 6 1). (See Table III, row 2.) From this it can be

4 Regarding "octave stretch," see Ward, W. D .: "Subjective Musical Pi.tch," The Journal of the A coustical Society of America, Vol. 26, Number 3 (May, 1954); Kolinski, Mieczyslaw: "A New Equi-distant 12-Tone Temperament," Journal of the A merican Musicological Society, Vol. XU, Nos. 2-3 (Summer Fall, 1959); and Pikler, Andrew G.: "History of Experiments on the Musical Interval Sense," Journal of Music Theory, Vol. 10, Number 1 (Spring, 1966) .

5 Babbitt, Milton: "Past and Present Concepts of the Nature and Limits of Music," International Musicological Society: R eport of the Eighth Congress, Vol. I (New York, 1961), p. 402. Ill. Deep Scales

ETS Generator Scale Interval vector 12 5 (or 7l 024579 [1 4 3 2 5 OJ 02457911 [2 5 4 3 6 ll 19 8 (o r 11) 0 3 5 6 8 11 13 14 16 [2 4 7 0 6 5 1 8 3] 0 3 5 6 8 11 13 14 16 17 [3 5 8 1 7 6 2 9 41 31 13 (or 18) 0 2 3 5 8 10 13 15 16 18 20 21 23 26 28 [3 8 10 1 13 5 6 12 0 11 7 4 14 2 9] 0 2 3 5 8 10 13 15 16 18 20 21 23 26 28 29 [4 9 11 2 14 6 7 13 1 12 8 5 15 3 10]

IV. Combinatoriality

ETS and scale Set-forms generator 12-5 0 2 4 5 7 9 6 8 10 11 3

Po: 0 3 5 6 8 11 13 14 16 P4: 4 7 9 10 12 15 17 18 1 31-13 Po : 0 2 3 5 8 10 13 15 16 18 20 21 23 26 28 P9: 9 11 12 14 17 19 22 24 25 27 29 30 1 4 6

.,,_... --....) 118 seen that the interval class 1 (the so-called "semitone" and its com plement) occurs twice in the collection, the interval class 2 (the so-called "whole tone" and its complement) occurs five times, the interval class 3 occurs four times, and so on. Applying Babbitt's theorem, we find that transposition by 1 therefore entails two com mon pitches between the original collection and the transposition; transposition by 2, five common pitches; transposition by 3, four common pitches, and so on. (As for the interesting question of why transposition by 6 entails two common pitches rather than one, I refer you to Hubert Howe's article on pitch structures in Perspectives of New Music.) 6 The fact that each interval class occurs with unique multiplicity in the major scale collection thus enables maximum hier archization of the original collection and its transpositions on the basis of pitch intersection.

Besides the "major scale collection" 0 2 4 5 7 9 11 , another sub-collection of the total 12-tone chromatic with a similar capacity for heirarchization is the collection 0 2 4 5 7 9. (See Table III, row 1.) This collection, which might be termed the "Guidonian hexachord," once played a prominent role in Mediaeval music theory, though it was of course tuned differently. Recently, it has again assumed theoretical importance as one of the six "all-combinatorial source sets" in the "12-tone pitch-class system."7 As in the "major scale collection'', each interval class occurs in it with unique multi plicity, with this difference: that zero is the multiplicity of occurrence number of one of the interval classes, namely 6, the so-called "tritone." Transposition by 6 therefore entails no common pitches between the original collection and the transposition. This fact is of course the basis of the combinatorial properties of the collection. (See Table IV.)

Both of the collections I have mentioned can be generated by the interval 5 (the "perfect fourth") or its complement 7 (the "perfect fifth"), intervals which are prime to 12. Other intervals prime to 12 are 1 and 11 , which generate the sub-collections 0 1 2 3 4 5 6, the "chromatic heptachord", and 0 1 2 3 4 5, the "half chromatic scale" (not listed in Table III) , each of which has the capacity for maximum hierarchization which characterizes the "major scale collection" or the "Guidonian hexachord." Note that the "half chromatic scale," like the "Guidonian hexachord," is also an "all combinatorial source set."

6 Howe, Hubert S.: "Some Combinational Properties of Pitch Structures," Perspectives of New Music, Vol. 4, Number 1 (Fall-Winter, 1965), p. 52.

7 See Babbitt, Milton: "Some Aspects of Twelve-Tone Composition," The Score and /.M.A. Magazine (June, 1955). The four collections I have just described are representative of a 119 type of collection termed a deep scale by Terry Winograd, a former student at The Colorado College who has written a paper on the properties of such scales.s A deep scale in an n-tone equal-tempered system contains each interval class of the system with unique multiplicity, thus enabling maximum hierarchization of the scale and its transpositions on the basis of pitch intersection. The number of interval classes in an n-tone equal-tempered system is equal to n/ 2 where n is even (as in the 12-tone system) or n-1 I 2 where n is odd (as in the 19- and 31-tone systems). The number of tones in a deep scale in any n-tone equal-tempered system is equal either to the number of interval classes in the system or to that number plus 1. Thus, in the 12-tone system, the number of tones in a deep scale is 6 or 7; in the 19-tone system, 9 or 10; and in the 31 -tone system, 15 or 16.

In this connection, criticism must be made of proposals to estab lish a 12-tone scale as the "diatonic" scale in the 19-tone equal tempered system and a 19-tone scale as "diatonic" in the 31-tone system. Such scales will necessarily lack depth in transposition.

A deep scale in an n-tone equal-tempered system can be gener ated by any interval prime to n, or by the complement of such an interval. As Milton Babbitt has pointed out, in a system where n itself is prime, a deep scale can be generated by any interval in the system. 9 Thus 9 different kinds of deep scales are possible in the 19-tone system and 15 different kinds of deep scales in the 31-tone system. Table III lists deep scales in the 19- and 31-tone systems analogous to the "major scale collection" or the "Guidonian hexa chord" in the 12-tone system - namely, those deep scales generated by the interval representing the "perfect fourth" (or the "perfect fifth") : 8 (or 11) in the 19-tone system and 13 (or 18) in the 31-tone system. Following each deep scale is its interval vector. Deep scales can of course be generated by all other intervals in the 19- and 31-tone systems as well as by those employed as generators in Table III.

Just as both of the 6-tone deep scales in the 12-tone equal tempered system are "all-combinatorial source sets," so all 9 of the 9-tone deep scales in the 19-tone system and all 15 of the 15-tone

s Winograd, Terry: "An Analysis of the Properties of 'Deep Scales' in a T-Tone System." Unpublished.

9 Babbitt, Milton: "Twelve-Tone Rhythmic Structure and the Electronic Medium," Perspectives of New Music, Vol. 1, Number 1 (Fall, 1962), pp. 76-7. Babbitt's term for "deep scale" is "maximal sub-collection." 120 deep scales in the 31-tone system are "all-combinatorial source sets," with this qualification, that where n is odd, association of a deep scale (or set) with its complementary set will form an "aggregate minus-one." The combinatorial properties of the deep scales in Table III are illustrated in Table IV. To find the transposition which is combinatory with a given deep scale in Table III, find the zero term in its interval vector and transpose by the interval represented by that term. Inversional combina:toriality, though not shown in Table IV, is also a property of every deep scale which is combinatory in transposition.

I shall turn now to the concept of the difference set. If the in tegers 0, 1, 2, ... n-l representing the pitches (or pitch classes) in an n-tone equal-tempered system are regarded as a set of residue classes modulo n and each residue class is equated with its comple ment (mod n), then a difference set is a set of residues such that each non-zero residue (mod n ) can be expressed as the difference between two members of the set in exactly one way. In other words, a difference set contains each interval class in the system once and only once. If the interval content of a difference set is represented by an interval vector, each term in the interval vector will be a "1". Two independent difference sets exist in the 12-tone equal tempered system, namely the "all-interval tetrads" 01 4 6 and 0 1 3 7, as listed in Table V, and their inversions. Difference sets are not found in all equal-tempered systems, but only in those systems in which the total n.umber of interval classes contained is a "triangular number" - that is, a number belonging to the series 3, 6, 10, 15, 21, ... 1 0 For example, the 15 interval classes of the 31-tone equal tempered system can be contained in any one of five independent difference sets, the "all-interval hexads" listed in Table V, or in their inversions. Of particular interest is the difference set 0 10 18 25 27 30, derived by inversion from the third of these hexads. The first four tones of this set form a perfect "dominant seventh chord,'' as can be seen by reference to the frequency ratios given in Table II.

An extension of the idea of the difference set is the difference set complex, a set of sets such that each interval class in the · system is contained once and only once in some set in the complex. For exam ple, the 9 interval classes in the 19-tone equal-tempered system can all be contained in any one of four independent difference set com plexes of three triads each, as listed in Table V, or in their inversions. The 15 interval classes in the 31-tone system can all be contained in

10 And not necessarily in all of these systems. In systems containing 21 interval classes, for example, no difference sets are found. V. Difference Sets (after Winograd)

ETS Difference sets Difference set complexes 12 0 1 4 6 0 1 3 7 and their inversions

19 o 1 4 I 0 2 9 I 0 5 11 0 1 5 I 0 2 8 I 0 3 10 0 1 6 I 0 2 10 I 0 3 7 0 1 8 I 0 2 5 I 0 4 10 and their inversions

31 0 1 3 8 12 18 O 1 3 I 0 4 11 I 0 5 15 I 0 6 18 I 0 8 17 0 1 3 10 14 26 O 1 3 I 0 4 11 I 0 5 17 I 0 6 15 I 0 8 18 0 1 4 6 13 21 o 1 3 I 0 4 12 I o 5 14 I 0 6 16 I O 7 18 0 1 4 10 12 17 0 1 3 I 0 4 17 I 0 5 11 I 0 7 15 I 0 9 19 0 1 8 11 13 17 and 60 others and their inversions and their inversions

...... N ...... 122 any one of 64 independent difference set complexes of five triads each, as partially listed in Table V, or in their inversions. Terry Winograd in another article lists the difference sets and difference set complexes in a number of different equal-tempered systems.11

Time and space will not permit me to discuss the applications of the concepts of the deep scale or the difference set to musical com position. In general, whatever applications have been made or can be made of these concepts in the 12-tone equal-tempered system, whether combinational or permutational, tonal or serial, can also be made in the 19- and 31-tone systems and a number of other systems as wen.12

In closing, I hope that this brief presentation will at least have revealed to the composer some paths which lead "from those things which are more known and more certain into the more unknown and uncertain and from generalities into specifics."

11 Winograd, Terry: "The Application of Block Designs to Compositional Structure." Unpublished. 12 For some discussion of compositional applications see my article: "Some Combinational Resources of Equal-Tempered Systems," Journal of Music Theory, Vol. II, Number 1 (Spring, 1967). OTHER PRESENTATIONS 123

Mr. William Hibbard and Mr. Harvey Sollberger spoke on the panel concerning Performance Problems.

Mr. Ross Lee Finney, Mr. Lindsey Merrill, and Mr. H. Owen Reed participated in the Professional Colloquium.

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