A Research into the Accords of the Sho

by Tadao Osanai

Synopsis:

The sho笙serves as the standard of pitch ih the gagaku雅 楽*Japa-

Table 1. nese classical court-music, and produces 11 so- rts of peculiar accord called aitake fit. The

writer desired and endeavoured to know in

detail of the tuning of the sh6, and, if pas-

sible, to give a reasonable explanation to the aita-

ke. The result is satisfactory, and a very sim-

ple method is proposed to express numerical- ly the degree of concordance of not only the-

se 11 aitakes but also of almost any accord.

1. Preliminary Consideration. •˜

The Japanese sh6 V has 17 independent

pipes, 2 of which are silent, ", and can give 15 notes as shown in Table 1, pipe and note cor- responding one to one . It is often said" that

hese 15 notes are riot tuned in the 12 equallyt

tempered scale, but are in the Pythagorean

series.

If we put aside the difference of , the

* The eutline of the gagaku is given in European notations: "Cellection of Japanese Musics, 日 本 音 楽 集"edited by Mr . Hisao Tanabe田 辺

尚 雄as the Vol.18 of the "Gesammelte Werke der Welt Musik". Shllnju-sha春 秋 社Publishing Co., 1930. Score of Gagaku, Vol." 1"transcribed by Mr. Sukehiro shiba芝 祐 泰, published from Ryugin-sha龍 吟 社; the succeeding volumes have been published or in

preparation. 1) L. Traynor and S. Kishibe; "The Four Unknown Pipes of the She", Jour. Soc. Res. Asiatic Music, No. 9, 1951, pp. 26-53; this treats in detail the silent pipes of ancient and modern she. Description on the modern she is given in Mr. Hisao Tanabe's work 5) , which

•\1•\ above 15 notes become (in German notation) c g d a eh fis cis gis, the 9 successive notes in the Pythagorean series. As is well known in the theory of pure-intonation, the note of the major above gis, which is obtained by repeating the perfect eight times above c, is his, and is very near to c, exceeding c only by an interval of a skhisma. Thus, if the sho is tuned in the Pythagorean series, we see that 〔1〕The interval between美 (gis2) and比 (c3) is almost the , exceeding the latter only by an interval of one skhisma (1/11 comma) . And, after a little calculation, we see that

2) The intervais between十 (g2) and美 (gis2) , or between比 (c3) and

言 (cis3) are the same, being approximately 5 commas (which is larger than the of the pianoforte) ; and the intervals between工 (cis2)

and〓 (d2) , between下 (fis2) and十 (g3) , between美 (gis2) and行 (a2) ,

betw een七 (h2) and比 (c3) , between言 (cis3) and上 (d3) are the same, being approximately 4 commas (which is smaller than the semitone of the pianoforte) .

•˜ 2. On the Tuning of the Sho.

The results (1) and (2) may serve to verify the assertion that the sho

is tuned in the Pythagorean series. So the writer, in the autumn of 1946,

requested Mr. Yoshihire Oku奥 好 寛, former court-musician3) to examine by his "ears" whether (1) and (2) really hold true. He, an excellent play- er of not only the gagaku-instruments but also of the European flute, who has very good ears, did not give any direct answer, (possibly becau- se he thought his ears were to be tested, or he may have considered that perfectly tuned instruments hardly exist) but gave in detail to the writer the practical method of tuning the sho, as is shown in Table 2, and left the writer to judge from this table.

shall be quoted later.

2) Hisao Tanabe; "Theory of Music, 音 楽 理 論"Revised Version, Kyoritsu Publ. Co. 共 立 出 版 社t 1956, pp. 69-70, is perhaps the latest work.

•\2•\ Table 2. *

In Table 2, we see that the Pythagorean series is adopted except in 3. and 15. In the preliminary consideration the result (1] was supposed to serve as a check, when we perform the tuning of the sh6 by the Pytha- gorean series starting from any one of the 15 notes: but 15. of Table 2 reveals that the major third is used for the tuning itself (not merely as a check) , which the writer least expected. If we interpret the words a little in 3. as a skhisma, from Table 2 we at once get Table 3.4)

3) With whom the writer has been in aquaintance in the Pure-intonation Rese- arch Club directed by the late Dr Shohei Tanaka田 中 正 平.

4) The last colurnn of Table 3 shows the frequency of each nete, when we take e2=652.4sec-1 according to A. Ellis.It happens that乞 (a1) becomes very near to one of the European standards of pitch a1=435.0sec-1

•\3•\ Table 3.

The result of the above tuning differs from the Pythagorean series on- ly in the three notes c3, g2, d3, and the differences are the same for the three, being equal to just one skhisma. Now, judging from the analogy of the cases of other ordinary musical instruments, it would be rare that the tuning should be carried on in the accuracy of a skhisma. If so or

•\ 4 not, we may say that here we have at least an example of tuning which aims at the Pythagorean series within the error of a skhisma.

3. Preliminary Considerations on the Aitake.5' •˜

The ordinary way of playing the shO is to sound 6 or 5 pipes together, 6) thus producing peculiar sorts of accord called aitake. There are 11 sorts of the aitake, which have been used in the court-music since the ancient times. Except in the two cases, the name of each aitake is the same as that of the lowest pipe: examine the left margin of each table (4-1) , (4-2) ,

4-11) , whose principal part shall be explained in the ---, (next section,

§ 4.Among these ll sorts of the aitake, the 6 sorts乞, 一, 行, 〓, 乙, 十

双) are accepted as concords i. e. consonant accords, and the 5 sorts比, (

下, 十, 工, 美as discords i. e. dissonant accords.7) Of course, even the so-called dissonant aitakes are not disagreeable to the ear, although they have some more peculiarity than the so-called consonant ones, as they have been used in actual music from ancient times. The above classification into consonant and dissonant aitakes has been made merely in a meaning somewhat resembling to that of the Europe- an harmony. Therefore, it can be easily foreseen that even when we suc- ceed to give the order of the degree of concordance to the 11 aitakes, probably we can not draw the line of demarcation between consonant and dissonant aitakes, so long as we have no other reason. If we examine the constructions of the consonant aitakes, putting aside the difference of octave, we see that the乞is made up from the 4 succ- essive notes a e h fis of the Pythagorean series; and the-.t行, 九, 乙 are from the 5 successive notes d a g h fis; finally the十 (双) is from

5) For the aitake合 竹, detailed descriptions are given by Mr. Hisao Tanabe: "Lectures on the Japanese Musics, 日 本 音 楽 講 話", published from Iwanami-sho-

ten岩 波 書 店1919, pp. 445-450.

6) In the te-utsuri手 移 り (fingering to change the accords) , several pipes sound successively, i. e. not simultaneously.And, in the fuki-tome吹 止 め (a sort of

Coda er Cadenza}, etc, , only 2 or 3 pipes are sounded tegether, or only l pipe alone sounds.

7) In the work 5) , the aitake行is classified as dissonant, but in thelater work the same author classifies the ll aitakes int。 three kinds, (1) perfectty con- cordant: 乞, 九, 乙; (2) ) fairly concordant: 一, 行, 十 (双) ; (3) dissenant: 十, 下, 工,

美, 比.Mr. Yoshihiro Oku also informed the writer that the行is accepted as

•\5•\ the 5 successive notes g d a e h. Therefore, there occurs no semitone in any of the 5 consonant aitakes. On the other hand, one semitone-interval not putting aside the difference of octave, i. e. semitone-interval ( itself) is

always contained in every dissonant aitake; in the工and the美, there are two. From this observation, we are led to the view that the distinc- tion between dissonant—and consonant-aitake may have relation with the existence and nonexistence of semitone-interval. In the European classical music, are sounded sometimes two notes of a semitone-interval simultaneously; I) its result is never disagreeable, but it produces a peculiar "timbre-effect". In modern musics, there are many works which utilize this timbre-effect. Perhaps a cause of this effect is the beat between the two notes, whose frequencies are comparatively near to each other; if two notes in the wholetone-interval are sounded together, this timbre-effect becomes smaller in general, whose amount depending upon the absolute pitches and the kinds of instruments. In the case of the sho, we may say that the limit for this effect lies between semitone and who- letone. The property of the so-called dissonant aitake of the shO may owe greatly to this "timbre-effect" due to the existence of semitone-interval. As we have already seen, the constructions of even the so-called con- sonant aitakes are afar from those of European major—and minor-chords. However, even in the European classical music, there are cases where 3 or 4 successive notes of the Pythagorean series are sounded simultaneous- ly or successively, in some ways which may be considered as concords; " and it is well known that the similar methods are widely utilized by some composers of modern musics. Therefore, to try to give a reasonable expla- nation to the aitake takes into concern also the European music.

concord. 8) The only examples the writer has encountered are the two places in Bach's "Preludio con Fughetta, e -moll" (in the 79-th and 83-th tacts of the Fughet- ta) ; probably many other examples may be found. Cf. the explanation-exam- ples Nos. 249-252 of Stephan Krehl' s "Harmonielehre" (1922) . 9) The examples the writer has encountered are: one accord in Bach' s "Chro- matische Phantasie und Fuge" (the top of the 50-th tact of the Phantasie) ; several accords in Haydn's "Jahreszeiten" (choral parts) ; a phrase in Beetho- ven' s "Symphonic No. 6" (beginning from the 5-th tact of the Finale) ; and a phrase in Chopin' s "EU-1de Ges-dur" (the 33-36-th, 37-th tacts) .

― 6 From the above considerations, we see that the attempt to explain the

aitake is sufficiently interesting but is fairly difficult. If we remain in the

ordinary explanation based on the major- and minor-chords, probably we

can do nothing more, but it is also clear that we can not put aside these

common chords.

4. The •˜Structure of each Aitake as an Accord.

The writer thinks it adequate to rely upon the following view:

A) All the intervals contained in an aitake of the shO must ( be those

which can be accepted as accordant.

This postulate is quite right, because it states only the necessary conditi-

ons that an aitake should be an accord. the fact which has been suffici-

ently proved by the actual music. But, if we insist upon this postulaton,

we soon recognize that we must take as accordant far wider sorts of in-

tervals than hitherto ordinarily accepted. First of all, let us examine whe-

ther this extension of the scope of accordant intervals is within a convin-

cible limit.

Putting aside the difference of octave, the notes of the shO may be re-

garded as the 9 successive ones of the Pythagorean series

c g d a eh fis cis gis,

and the last one gis can be considered either a note in the series or a

note standing in the relation of major third with c, so long as we neglect

the interval of a skhisma. In any concordant aitake, there appear no more

notes than 5 successive ones of the above series. Among the so-called

dissonant aitakes, each of the following four

工: d a e h x cis gis 下: d a x hfisXgis 十: g d a e hfis 比: c x d a e h fis

has not more notes than 7 successive ones; and the same holds true for 美: c x d a x h fis x gis if we consider gis stands outside of the Pythagorean series. Therefore,

we know that all the Pythagorean intervals contained in the 11 aitakes can be constructed from the following interval-ratios i. e. frequency-ratios

•\7•\ which are the ones of 7 successive Pythagorean notes e. g. c g d a e h fis These last three or four ratios may never be said to be simple, but the complicated Pythagorean interval-ratios concerning the above four have fairly accurate approximate values as indicated by () :

Pythagorean semitone:

Pythagorean :

Pythagorean major third:

Pythagorean augmented fourth:

Thus, when we put aside the difference of octave, any interval contained in each aitake has at most the complexity of 27/19 (the "sum" s of the numerator and denominator being equal to 46) 1E, so long as we neglect the small intervals of the order of 1/4 comma. Table 4 shows11) ,everywhere within the error of the order of 1/4 com- ma (which is smaller than the kleisma) , all the intervals contained in each aitake, not putting aside the difference of octave. For example, in the aitake t, the interval-ratios of et, a3, h', e3, and fist against the lowest

Table 4.

乞 ( 4-1)

一 ( 4-2)

•\8•\ note a1 are 3/2, 2/1, 9/4, 3/1, and 27/8 respectively; those of a1, a2, h2, e3, fis3 against e2 are 2/3, 4/3, 3/2, 2/1. 9/4 respectively; ---. In this way, against any one of the 6 notes contained in theŒî, the interval-ratios of the remaining 5 notes are shown in the table (4-1) . In the case

10) This method of expressing the complexity of an interval, is stated in detail in the writer's previous paper: "Twice Intermediate Tuning", Jour. Soc. Res. Asiatic Music, No 10-11, 1952. In that paper, for the sake of brevity, only the intervals not exceeding the octave were tabulated. When we abandon this restriction, we get instead of the Table D of that paper the following one, which also does not contradict to the experience

the gothics being the very well aquainted intervals, the italics the ones util- ized by the late Dr. ShOhei Tanaka or Paul Hindemith, and the first column the overtone series. This remark is necessary for the discussion in the succeeding paragraphs. 11) All the l i. e. 1/1 in the diagonal line were not abolished only for the sake of reference and external appearance; these places occupied by the letter 1 must be left empty. Similarly, the above half (upwards from the diagonal) is also superfluous, each element being the reciprocal of the corresponding one in the lower half, thus counting each interval once more; nevertheless, it does no harm to the present consideration. but it serves to check the calcu- lation.

•\9•\ 〓 ( 4-3)

乙 ( 4-4)

十 ( 4-5) 双)(

行 ( 4-6)

工 ( 4-7)

下 ( 4-8)

•\ 10•\ 十 ( 4-9)

美 ( 4-10)

比 ( 4-11)

of the乞, the above given ratios are all strictly Pythagorean. But, more complicated ratios appearing in the cases of the other aitakes have gene- rally simpler approximate values12) within the above mentioned accuracy, and were substituted by the latter ones, as indicated by () .

Through these tables, the most complicated interval-ratio is 38/27 (the "sum" s being 65) , which appears in the T and the ӟ . Although this in- terval, the Pythagorean diminished fifth, is fairly complicated, we must consider that, at least in the sho, accordancy of such interval has long been proved by the practice of the gagaku.

Hitherto, we pursued the structures of the aitakes from the stand-point of the tuning of the shb. Here, the writer desires to propose to reverse the consideration: We might say that the accords at which the aitakes aim are just what the above tables (4-1) , (4-2) , ---, (4-11) show, and the

12) Which, in most cases, can be obtained without trial, if we use the method of continued fraction. But, to avoid any possible ambiguity, the writer prepared a table of all the fractions whose s' s are between 2 and 109, and consulted this table.

•\ 11•\ tuning of the sho, either by the strict Pythagorean series or by Table 3 itself, is merely an approximate way to realize these accords. If so, we must regard Table 4 as exact. In the foregoing considerations, we naturally assumed that: B) Any interval which is sufficiently near to a certain (accordant inter- val can be accepted as the latter. To what extent of difference this postulate should hold, depends not only upon the requirements of the hearers i.e. the ability of distinguishing the intervals and the attitude of hearing, but also upon the kinds of the ins- truments. In the pianoforte it is well known that the admissible difference can reach to more than 2/3 commas.

•˜ 5. An Attempt to Express the Degree of Concordance.

Once knowing the structures of the aitakes as in the tables (4-1) , (4-2) ---,

4-11) , we naturally feel the temptation to compare their degrees of comp- (

lexity. Obviously, the simplest method is to take the arithmetical mean

value of the values of s's of all the interval-ratios contained in each aita- ke. For example, in the case of the Δ, the s' s of the interval-ratios in

the table (4-1) become

which is a symmetrical matrix without the diagonal elements. 13) The sum 2S of these 30 integers s' s is 364; dividing this 2S by 30, we get the mean value s=12.333--- Similar calculation for each aitake gives Table 5, where the aitakes are arranged in the order of s. This result well agrees, on the whole, with the fact. The values of s for the so-called consonant aitakes are compa- ratively small and are situated in the former part of the table, those for

13) For the reason of abandoning these elements, cf. 11)

12 Table 5.

the so-called dissonant aitakes are comparatively great and are situated in the latter half. And, as can be seen in the Fig., the distribution of is uniform on the whole, therefore the distinction between the so-called consonant and dissonant aitakes can not be done from the values of g; this also agrees with the anticipation in •˜ 3.

Table 5

Table 6

Table 7

It is obvious that the above method can be applied not only for the aitake of the sho, but also for any accord consisting of many notes (in- cluding the case of only two notes14) ). We may expect that: The smaller the value of s' is, the greater the concordancy becomes, whether the notes of the accord are sounded simultaneously or successively. It must be in- teresting to calculate g at least for the examples quoted in 9) , and further

14) In this case N becomes always an integer, the "sum" s itself of the numer- ator and denominator of the interval-ratio.

•\ 13•\ for the similar accords used by modern composers e. g. Hoist, Debussy,

Sahon berg, and Dr. Mitsukuri –¥•ì•H‹g, and compare the values with those of the sho; but the notes of some of these opus are at present inaccessible to the writer, so the writer has to leave them for future opportunities. Here are given in Tables 6 and 7 the values of g for some Table 6.

From the Compositions given in the Appendix of the late Dr. Shohei Tanaka' s "Foundation of Japanese Harmony, 日 本 和 声 の 基 礎", Sogen-sha (創 元 社) , 1940:

The two accords 1 and 2 in the above table are tetrads, but if we consider them as triads, we get:

thus we see that the degree of accordance becomes more inferior than in the case of tetrads. This result is worth noticing.

•\ 14•\ Table 7.

From the Examples Nos . 15, 16 of Stephan Krehl' s "Harmonielehre" (1922) :

The accord (d) , for example, of the above table is an Unterkldnge under the Haupttont a; if we construct the corresponding Oberklange upon the same Haupt- ton a1, we get cis1 a1 e2 e3; these two accords have quite the same S and s, because their corresponding interval-ratios are in the reciprocal relation. Thus we see that this paper agrees with the famous theory founded and developed by Oettingen and Riemann. The triads which are obtained from the 4 tetrads of the above table by omit- ting one of the two notes in the octave relation, have sometimes smaller values of N than the original tetrads but in general greater:

very simple examples. From the Fig., where the values of g of Tables 5, 6, 7 are compared, we can see that although the aitakes of the sha have quite different composition from that of ordinary European chords, they are not so particular from the point of view of concordancy.

30-th July, 1957 at TOHOKU UNIVERSITY NORTH COLLEGE, Kita-shichiban-chO, Sendal, Japan.

15 Remarks: 1)( The greatest prime number contained in Table 4 is 19, therefore the aitakes can be considered to belong to the pure-intonation of the 7-th order.* The interval-ratios used for Table 6 or 7 are, of course, those of the pure-intonation of the 3-rd or the 2-nd order respectively. 2) Even when we take in Table 4 either the exact interval-ratios ( in the Pythagorean series or those given by Table 3, of course we can also calculate the corresponding values of g, but it may be rather ridiculous if we trust the so great values literally as the measure of the concor- dancy of the aitakes, although the more acute the "ear" is, the greater becomes the sense of discordance, when the intervals are not strict. 3) If we caluculate the values of .t for further examples of the (Stephan Krehl's "Harmonielehre" than those quoted in Table 7, we know that the result, so far as the writer has examined, is also satisfactory in general. Cf. Table 7' and the Fig., a little below the place for Table 7, where the result is shown by the following marks. o: Verminderte and übermil ssige Dreikange from Nos. 137, 139 and 143-146; x: Neben- dissonanzen from Nos. 249, 259, 275, 279-280;

•E: Charakteristische Dissonanzen from Nos. 110, 137, 139, 222; •¬: Doppel-

seitige Akkorde from No. 357.

The four accords 17., 18., 21., 22., which involve the major or

the , are treated as dissonant in the classical harmony, but

their values of s are smaller than that of the concordant accord (d) of

Table 7. It may be expected that we may find many other similar cases,

but it hardly can be regarded as a defect of the method of •˜ 5. Of cour-

se, in the European classical music, it is adequate in general to regard

only the Dur- and Moll-Dreiklang as concordant accords, but at the same

time, numerous special cases are also known where this view does not

ho!d; therefore, if we say strictly, this view is too narrow and defective

* Greatest prime number: 2 3 5 7 11 13 17 19 Order of pure-intonation: 0-th 1-st 2-nd 3-rd 4-th 5-th 6-th 7-th + For examples, Cf. 9) and the übermassige Dreiklange in F. List's "Faust Symphonie" quoted by Stephan Krehl himself.

•\ 16•\ even in the explanation of the classics. In modern musics, the scope of concordant accords is extending itself very rapidly. Some examples of the accords involving the major sixth or the treated as conco- rdant are found in Nos. 585, 697-719, 798 or in Nos. (438, ) 477, (483-488, )

491, 552-553 of Mr. Y. Matsudaira (松 平 頼 則) 's "Modern Harmony, 近 代 和

声 学", published from Ongaku-no-Tomo-sha, 音 楽 之 友 社, 1955.

Probably all the accords o and x can be used as concords, if we pay suitable attentions to the instrumentation.

4) The method of •˜ 5 is a ( very simple solution of the problem refered to in the foot-note (7) of the writer's previous paper 10) . In this "solution", not avoiding to adopt the non-whole number for no "auxiliary princi- ples" are needed.

Table 7'.

Further Examples from the Stephan Krehl's "Harmonielehre":

from No. 110

from Nos. 137, 139

from Nos. 143-146

from No. 222

•\ 17•\ from No. 249

from No. 259

from No. 275

from Nos. 279, 280

from No. 357

In the above calculation, every interval-ratio which has a simpler approximate value within the error smaller than 114 comma (< a kleisma) is substituted by the latter, under the same meaning as explained in the case of Table 4. The numbering with the half bracket ) shows that such substitution has taken place. If we change some of the above accords by employing the natural seventh system, we get:

•\ 18•\ It is interesting to compare each new accord with the original one. The first two ones shall be astonishingly more harmonious than the original.

FOUR WORKS OF THE RECIPIENTS OF ONGAKU-NO-TOMO-SHA PRIZE AT JAPAN CONTEMPORARY MUSIC FESTIVAL

TORU TAKEMITSU : Le son-calligraphi # 1 8 Instruments a cordes \ 80 Recipient of the first prize in the composition concours of the second Contemporary Music Festival at Karuizawa 1958. 8 instruments (4 Vin., 2 Vla., 2 Vcl.) devided in two sections perform a small pieces with only 32 measures.

SHIN'ICHI MATSUSHITA : Composition da Camera per 8

•c•c\ 360

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players.

YOSHIRO IRINO : Composition for Violin and cello,

•c•c\ 150

Won award in the third festival in 1959. Irino is one of

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•c•c\ 100

Recipient of the award in composition field of the third

festival. Two movements "Continu" and "Incidental".

Ongaku-no-Tomo Sha Inc. Kagurazaka Shinjuku, Tokyo

•\19•\