Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Puhm Juhan Copyright © Scale Tone 43 Partch N Our Door with Our Common Scales

! ! ! !

01#&-!%1#2 01#&-!%1#2 01#&-!%1#2 01#&-!%1#2

"#$!%&'()#! *+!",-$!.)&/$! approximate any 7 or 11 7 or any approximate are 11 limit ratios, leaving 11 are leaving ratios, limit dissonanta almost being and consonant in the context and consonant the context in east two moreeast two incomplete realizing it. AJust ofpinnacle realizing re given exact ratios all and so re given 7/5 cents andratios 583 617 of to supporting 7 ratios, orsupporting to 11 limit the andis made up ofmajor triad an be said to push the bounds of of bounds the push to said be an s with thes made.with choices Partch Partch expanded common Partch our 2, 3 and 35Tworatios. 2,complete and limit 64/45 of 59064/45of andare 610 cents intervals thatup tonalitiesmake intervals the pered minor seventh (1000 cents) is minor seventh peredcents) (1000 onships consonant and harmonic. andconsonant harmonic. onships is considered the of consist to xt the 11 limit ratio of the 4:5:6:7:9:11 ence. of our listening ourexperience,ofthe listening into incorrect to think of the ofminor think seventh to incorrect onalities have theofonalities have the the chord root in 4:5:6:7:9:11 chord. As there before, said 4:5:6:7:9:11 chord. hird above the dominant (3/2 * 6/5 = 9/5 9/5 = 6/5 * (3/2 dominant the above hird of harmony and higher limit harmony are of ratios any fth. All things considered, the 2 and 3 limitfth. 23All things considered, andthe s? I would say it does evenclips Ithough it s? would say

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!! !! tones of Partch’s scale, twelve arePartch’sof 7twelve tonesratios, twelve scale, limit te scale and instruments donotoreven and supportscale remotely te instruments d”. There is one complete “Tonality Diamond”Thered”. is one complete and l at “Tonality o tempering of any intervals whatsoever. intervals All o pitchesof a tempering any rmulating his 43 and his scale rmulating then tone spent the lifetime d with 9/4 dthe of(9/8), 3ratio is part with which limit 11 limit ratio of 11/4 (11/8) forming finally the finally forming (11/8) 11/4 of ratio limit 11 Can deem system. the 11 ratio usical we be harmonic to Certainly on musical init’s own Certainly our it is common system complete tonalities havethe the tonalities the ofchord root in complete m our up to 5 limit music with ourmthe4:5:6:7 5 music with up to limit chord which ed ratio of 600 cents. The 5 limit ratios cents.of5The 600 of 45/32 and limited ratio ratios andbelow equal temperament. not all tonalities are complete. A complete tonality not are all Atonalities complete. complete tonality le would like to pretend. somePossibly 7ratioswould like to limit le c f the thirds and sixths. Thesixths. ffindsall theseandearthe thirds relati Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Puhm Juhan Copyright © Scale Tone 43 Partch n our door with our common scales. As well the well our10/7n As door with commonand our scales. rvals doItapproximate be limitwould rvals not 7 ratios. lities are related to one another by exactly the are oneexactly to sameanother by related lities again, none of our current instruments ourcome of none again, instruments current even close music we listen to is constructed of up toto 5musicratios. is constructed up of listen limit we e. Experience is required to determine in requiredwhat conte e. determineto Experience is with the 3 limit ratios of the perfect fourth and fifourth the andthe of perfect 3 ratios with limit st one-third of a semitone lower (969semitone atem of cents). Our one-third lower st g theg intricacies coming the43toneof and term scale to oot of the chord in the 7Two oot the of chord completein limit t ratios. synthesize or blendthe2, with andlimit 3 5 ratio

"#$!%&'()#!*+!",-$!.)&/$"#$!%&'()#!*+!",-$!.)&/$ 2, and arethat argued 3 5 ratios the limit language "#$!%&'()#!*+!",-$!.)&/$!"#$!%&'()#!*+!",-$!.)&/$ utside ourexperi listeningthe realm andof composing 3 = 16/9, (996 cents)) or further16/9,a3 (996 cents)) removed,minor t =

themselves, hence what Partch calls the “Tonality Diamonthemselves, Partch hencecalls the “Tonality what tonalitiesfour and incomplete have the tonalities r What also see11 will is thatlimit we the tona ratios. complete4:5:6:7:9:11chord.Ten and ten complete in speaksTonalities”though Partch ofas well the “28 nineteen 35ratios.2, limit and quarter-tone between a perfect fourth and the afourthtritonquarter-tone andperfect between the chordmay harmonicbedeemed andthe 43 consonant. Of of the chord? Does 11 synthesize of 11synthesize 4, the 79.chord? with 6, 5, Does and 4:5:6:7:9 chord. One major step further 4:5:6:7:9the major step with chord. comes One is no7 m orin common the 11 ratios experience of our the harmonic seventh ratio of 7/4. This is then expandetheThisharmonicthen 7/4.of ratio seventh is very strongly. Herefirst very thenfro is Partch’s expansion ratios synthesize miraculously with the 5 limit ratios ratioswith theo miraculously 5synthesize limit Does the limit harmonically of 7/4 the ratio7 ratio The of 2ratios (blend) synthesize the limit octave most ofa or are aabove third half of which semitone almost half as close still toalmost once So the half as pure tritone.still close (1018 cents)). There is no 7 limit 7/4 ratio knocking o (1018 7Thereno7/4 limit is knocking cents)). ratio above theare and well below tempersquare 2 of root firmly in the realm of two perfectfirmlyfourths in* 4/the two realm of (4/3 Thetritone and equal minor seventh inte tempered approximating7 7/4,is almo limit of the which ratio

limit ratios, 7 and 11 limit ratios are completely o limit 711 ratios,are ratios and completely limit notharmonicthe least bit,of in regardless what peop what can be considered currentharmonic. Since our 12 no tonality to include 7 and 11 limit ratios. It can betonality tocan 711 limit It ratios. includeand the is aFor starters scale Intonation scale, n pure Just For theareintervals all well. theexact as most part

Intonation, a fairIntonation, aunravelin time ofcan be spent bit Whatfeaturesof are this the of scale? some One can only imagine the years Harry Partch imagine fo spentHarry One the can only years Compendium Musica

“Tonality Diamonds”. We can see the emerging pattern and intricacy of Partch’s choice of the 43 tones of the scale complete with lots of loose ends as somewhere Partch had to draw the line at 43 tones, while it very easily could have been more!

But we haven’t told the whole story and we have to cut the above tonalities by half. The 28 tonalities are made up of 14 tonalities and their 14 inversions. 14 tonalities ascend interval wise from the root note of the 4:5:6:7:9:11 chord but the other 14 descend. Music and intervals are wonderful as they can be transformed upon a mathematical basis. Upwards a perfect fifth (3/2) from 1/1 we get 3/2. Downwards a perfect fifth from 1/1 we get 2/3. Hence an inherent symmetry that Partch calls Otonality –the over tonality or major tonality, and Utonality –the under tonality or minor tonality. Otonality and Utonality are not invented concepts. They are already inherent in the mathematical construct of music, only the labels are new. Hence Partch names these and strictly adheres to this symmetry so in the end the entire 43 note scale is made up of two perfect symmetrical halves related by reflection or inversion. The simplest example of this is that the major triad with the ascending intervals of 5/4 and 6/5 is the Otonality, to the Utonality of the minor triad with descending intervals of 5/4 and 6/5. The major and minor triads are then symmetric in opposite directions.

Without getting overly involved, (that’s why one can read Partch’s book) the Otonality is made up of the chord 4:5:6:7:9:11 with ascending intervals of 5/4, 6/5, 7/6, 9/7 and 11/9. The Utonality is made up of the descending chord 1/4:1/5:1/6:1/7:1/9:1/11 with descending intervals of 5/4, 6/5, 7/6, 9/7 and 11/9. We are not used to thinking about chords descending from a root so we can write the Utonality in ascending form. The Utonality chord ascending would be 1/11:1/9:1/7:1/6:1/5:1/4 with ascending intervals of 11/9, 9/7, 7/6, 6/5 and 5/4. In the simplest sense the pitches are inverted and thus the intervals are in retrograde. As an example, the Otonality (low to high) G B! D F"! A C½# becomes the Utonality (low to high) D½! F A"# C Eb# G . The G is our common inverting note and we can see how the G major triad in inversion becomes the C minor triad. Harmonically transforming a major triad to a minor triad is fine. But, assuming that the Otonality 4:5:6:7:9:11 is even entirely harmonic, is the Utonality (besides the minor triad) of (low to high) 1/11:1/9:1/7:1/6:1/5:1/4 harmonic? I would say not at all. There is a very definite vertical harmonic order in music which is precisely why a 9th is not called a 2nd and an 11th is not called a 4th. You can’t turn an 11th chord upside down and expect it to have the same consonance or harmonicity as it does the right way up. But, with that being said, playing melodically, yes Utonality is as acceptable as Otonality, as well as introducing an entirely new set of satisfying melodic relationships. Partch interestingly refered to his scale as the “Monophonic Fabric”.

What this breaks down to is: 14 major triads, harmonically (and melodically) extended by 7th, 9th and 11th ratios, and 14 minor triads melodically (and dubiously harmonically) extended by under ratios of 1/7, 1/9 and 1/11. Not all triadic extensions are complete. In a nutshell then we have: pure Just Intonation ratios and intervals, 14 major triads, 14 minor triads, expanded tonal resources to 7 and 11 limit ratios, all using 43 notes to the octave. This compares to 12 major and 12 minor tempered (out of tune) triads and no expanded tonal resources using 12 notes to the octave. As well with the Partch scale we have new instruments and as such a new aesthetic.

Of importance is the understanding of Partch’s Primary Ratios and Tonalities. The 6x6 Tonality Diamond determines the 29 Primary Ratios and 12 Primary Tonalities of his system. While many different scales, chords and modes can be constructed out of the 29 ratios of the Tonality Diamond, not a single complete 8 note Just Intonation scale can be constructed. In Just Intonation 8 pitches instead of 7 are required to have available all 6 major/minor triads of a key. We need all 43 ratios to construct 6 complete 8 note Just Intonation scales. This means we have to take into careful consideration the range and capabilities of each of Partch’s instruments. Many cannot play the full 43 tone range of pitches. The 12 Primary tonalities of the Tonality Diamond contain eleven 2, 3 and 5 limit ratios, eight 7 limit ratios and ten 11 limit ratios. The

3 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica remaining 16 tonalities and 14 ratios are called the Secondary Ratios and Tonalities. Only 2 Secondary Tonalities are complete. There is even a small set of 6 Tertiary Tonalities that Partch overlooks that are incomplete missing the 5 limit major third. For colour there are as well many possibilities of irregular “False Ionian” scales among other possible creations.

So what then is the legacy of Partch’s scale? It is both historic and contemporary as Partch’s instruments and his music are still being played. Not only that, but new reproductions of his instruments are being made to mount new productions of his stage works and music. Along these lines then, new music for his instruments can be written or by using electronic means, new compositions using his scale can be realized.

One thing still remains lacking in order to fully utilize the 43 tone scale. Partch never created a universal notation for his scale that one can compose abstractly in and understand the relationships of the scale. As Partch said himself, “The status of my notation after four decades of composing for my particular instruments remains unresolved -it is not integrated notation” (Genesis of a Music, page 197). And there we have it, the necessity to have a comprehensive and clear notation. There is nothing more unrealistic than notations people create that may as well be written in Martian. The musical notation system we use not only shows us symbols for pitches and rhythm etc but it is also the gateway to its own theoretical understanding. To learn a new notation, a new notation system that is purely surface symbols, is purely ridiculous. Even Partch when constructing his notations for his various instruments kept in mind that his notation was a reflection or extension of the physical characteristics of the instrument he was notating for.

Any notation that expands our current notation must build upon its foundation. Partch’s 43 tone scale is an expansion of our scale of up to 5 limit ratios and tonalities. There are four distinct groups of pitches in Patch’s 43 tone scale: 1) the 2 and 3 limit ratios, 2) the 5 limit ratios, 3) the 7 limit ratios, and 4) the 11 limit ratios. How to expand our notational resources but keep completely comprehensive the relationship of the pitches in each of the limits? Well to end off, I have created the necessary notation to logically, comprehensively and abstractly compose in 43 tones to the octave. Further to adding a simple set of symbols I have as well developed a polychromatic notation for the Partch 43 tone scale. A completely logical system of coloured notation, in part related to my other polychromatic notations, goes a very, very long way in making the jumble of limits, ratios, tonalities etc and their relationships understandable. All the staves remain the same and we still have 7 letter names and a sharp and a flat. All 2 and 3 limit ratios are written normally. For the 5 limit ratios, a number of pitches have a simple ! or " to show the difference of the 81/80 comma between say, G and G! or G". 7 limit ratios for the most part are around a third of a semitone sharp or flat. 1 1 Hence the 7 limit ratios are all grouped together by #!, #" and also by /6!, /6" symbols in front of the written pitch. The 11 limit ratios for the 2 2 5 5 most part are around a half of a semitone sharp or flat. Hence the 11 limit ratios are grouped together by ½!, ½" and also by /3!, /3" /6!, /6" symbols in front of the written pitch. And so we can comprehensively and logically with full understanding of the 43 tone scale structure compose using our regular notation with just the addition of a few fractional symbols. The fractional symbols are of course approximate and for the most part, also consistent with one another. If we decided to write the pitches using the polychromatic notation we wouldn’t need the fractional symbols at all, creating a very colourful, comprehensive and clear score. The reader is referred to the “Polychromatic Notations and Extended Tonality” chapter for more detail and depth.

So finally in a sense we have completed what Partch didn’t and have created a universal notation to compose abstractly in, whether we apply it to the existing instruments (each with their own notation) or create new acoustic or virtual instruments. Partch designated his 1/1 ratio or pitch as G pitched at 392 hertz or whatever octave he chose G to be in (196hz, 98hz etc). G pitched at 392 is almost perfectly 200 cents below A-440. I have as well completely duplicated all the charts giving the 1/1 ratio or pitch as C if one would like to compose with C as the

4 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

center. There is also a chart comparing Partch’s 43 tone scale to 43 Et, as well as showing the deviations of Partch’s scale from 12Et which is very useful.

As a foolhardy extension of Partch’s scale if we fill in the pitches of the two incomplete Tonality Diamonds we get a 55 or 57 note scale to deal with. The three complete tonality diamonds together do not include the pitches a syntonic comma sharp (81/80) and flat (160/81) of the 1/1 tonic. Partch was wise to stop at 43! As well by extension two different configurations of a 13 limit tonality diamond are given.

The Partch 43 tone scale can be said to be a pinnacle of Just Intonation. With the understanding of the theoretical construction of this scale we can as well further create our own Just Intonation scales necessitated by our own compositional needs. Partch though can be said to be the first to take Just Intonation to such a comprehensive depth creating not only his own tonal language, but the instruments and compositions to realize it. A lifetime of work!

5 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

Partch 43 Tone Scale

1/1 =G =392 * 2^n

A½! D½! E½! G½! ½! approximately 1/2 semitone flat 2 2 1/11 Limit 12/11 16/11 18/11 F# /3! 64/33 /3! approximately 2/3 semitone flat 5 5 150.64 C /6! 648.68 852.59 20/11 1146.73 /6! approximately 5/6 semitone flat 14/11 1035.00 417.51

A"# B"# D"# E"# "# approximately 1/3 semitone sharp 1 1 1 1/7 Limit 8/7 9/7 C# /6# 32/21 12/7 F# /6# /6# approximately 1/6 semitone sharp 231.17 435.08 10/7 729.22 933.13 40/21 Utonality 617.49 1115.53

# 81/80 G# Ab# Bb# C# Eb# F# # syntonic comma (81/80) sharp 1/5 Limit 81/80 16/15 6/5 27/20 8/5 9/5 21.51 111.73 315.64 519.55 813.69 1017.60 (81/80) (81/80) (81/80) (81/80) Bb C F 1/3 Limit G 32/27 4/3 16/9 G 1/1 A294.13 498.04 D E996.09 2/1 3 Limit 0 9/8 3/2 27/16 1200 203.91 701.96 905.87 (81/80) (81/80) (81/80) (81/80) A! B! D! E! F#! G!!syntonic comma (81/80) flat 5 Limit 10/9 5/4 40/27 5/3 15/8 160/81 ! 81/80 182.40 386.31 680.45 884.36 1088.27 1178.49

1 1 1 Otonality Ab /6! Db /6! /6! approximately 1/6 semitone flat 21/20 Bb"! C"! 7/5 Eb"! F"! "! approximately 1/3 semitone flat 7 Limit 84.47 7/6 21/16 582.51 14/9 7/4 266.87 470.78 764.92 968.83

5 5 D /6# /6# approximately 5/6 semitone sharp 2 2 Ab /3# 11/7 /3# approximately 2/3 semitone sharp G½# 11/10 Bb½# C½# 782.49 F½# ½# approximately 1/2 semitone sharp 11 Limit 33/32 165.00 11/9 11/8 11/6 53.27 347.41 551.32 1049.36

6 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

Partch 43 Tone Scale

Harmonic Structure (from which can be derived scales, modes, chords and tonalities) 1/1 =G =392 * 2^n

Primary Ratios -(also the Roots of 3 incomplete Tertiary Tonalities Otonality! C D without the 5 Limit major third) (Bold = Roots) Secondary 4/3 3/2 Ratios 498.04 701.96 (7/6) (9/7) (7/6) D"# (O) A"# (O) E"# (O) B"# 32/21 8/7 12/7 (3/2) 9/7 729.22 231.17 933.13 435.08 1 1 F# /6# (U) C# /6# (7/6) (9/7) (11/9) 5 40/21 10/7 G D /6# $Utonality 2 1115.53 617.49 1/1 11/7 Ab /3# Secondary Otonality! 0782.4911/10 Ratio (11/9) (9/7) (7/6) 165.00 Bb½# (U) F½# (U) C½# (U) G½# 11/9 (3/2) 11/6 11/8 (3/2) 33/32 347.41 1049.36 551.32 53.27 Partch Tuning Notes = 8/7, 11/8 (11/6) (11/9) C D 4/3 3/2 $Utonality 498.04 701.96 (Bold = Roots) Primary Ratios

Secondary Primary Ratios Otonality! Ratios Ab! (O) Eb! (O) Bb! (O) F! (O) C! (O) G# (Bold = Roots) 16/15 8/5 6/5 9/5 27/20 81/80 111.73 813.69 315.64 1017.60 519.55 21.51 Bb (O) F (O) (U) C (O) (U) G (O) (U) D (O) (U) A (U) E 32/27 16/9 4/3 1/1 3/2 9/8 27/16 294.13 996.09 498.04 0 701.96 203.91 905.87 G% (U) D" (U) A" (U) E" (U) B" (U) F#" 160/81 40/27 10/9 5/3 5/4 15/8 $Utonality 1178.49 680.45 182.40 884.36 386.31 1088.27 Secondary (Bold = Roots) Primary Ratios Ratios

Primary Ratios -(also the Roots of 3 incomplete Tertiary Tonalities Otonality! C D without the 5 Limit major third) (Bold = Roots) 4/3 3/2 498.04 701.96 (11/9) (11/6) G½% (O) D½% (O) A½% (O) E½% 64/33 (3/2) 16/11 12/11 (3/2) 18/11 1146.73 648.68 150.64 852.59 2 F# /3% (7/6) (9/7) (11/9) 5 Secondary 20/11 C /6% G $Utonality 1 1 Ratio 1035.00 14/11 1/1 Db /6% (O) Ab /6% Otonality! 417.51 0 7/5 21/20 (11/9) (9/7) (7/6) 582.51 84.47 Eb"% (U) Bb"% (U) F"% (U) C"% 14/9 (3/2) 7/6 7/4 21/16 764.92 266.87 968.83 470.78 Partch Tuning Notes = 7/4, 16/11 (7/6) (9/7) (7/6) C D Secondary 4/3 3/2 Ratios $Utonality 498.04 701.96 (Bold = Roots) Primary Ratios -The 8/7, 11/8, 7/4 and 16/11 intervals are prescribed by Partch to tune the 7 and 11 limit ratios. Without a doubt a difficult task to tune these intervals accurately by ear. After which the rest of the 7 and 11 limit ratios present no problems. The 5 limit ratios present no tuning problems.

7 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

Partch 43 Tone Scale

1/1 =G =392 * 2^n

01 - 422 - 413 - 404 - 395 - 386 - 377 - 368 - 359 - 34 10 - 33 81/80 55/54 56/55 64/63 45/44 121/120 100/99 81/80 64/63 49/48 21.51 31.77 31.19 27.26 38.91 14.37 17.40 21.51 27.26 35.70

2 Primary G A½! Ab /3" A! AA#" Bb#! Ratios 1/1 12/11 11/10 10/9 9/8 8/7 7/6 0 150.64 165.00 182.40 203.91 231.17 266.87

1 Secondary G" G½" Ab /6! Ab" Ratios 81/80 33/32 21/20 16/15 21.51 53.27 84.47 111.73

1 Secondary G! G½! F# /6" F#! Ratios 160/81 64/33 40/21 15/8 1178.49 1146.73 1115.53 1088.27

2 Primary G F½" F# /3! F" F F#! E#" Ratios 2/1 11/6 20/11 9/5 16/9 7/4 12/7 1200 1049.36 1035.00 1017.60 996.09 968.83 933.13

False Ionian Scales 1/1 =G =392 * 2^n

1 1 A#" B#" C# /6" D#" E#" F# /6" A½! A#" 8/7 9/8 9/7 10/9 10/7 16/15 32/21 9/8 12/7 10/9 40/21 63/55 12/11 22/21 8/7 231.17 435.08 617.49 729.22 933.13 1115.53 150.64 231.17

2 D½! E½! F# /3! G½! A½! Bb½" C½" D½! 16/11 9/8 18/11 10/9 20/11 16/15 64/33 9/8 12/11 121/ 11/9 9/8 11/8 128/ 16/11 648.68 852.59 1035.00 1146.73 150.64 108 347.41 551.32 121 648.68

1 DEF#! G A B! C# /6" D 3/2 9/8 27/16 10/9 15/8 16/15 1/1 9/8 9/8 10/9 5/4 8/7 10/7 21/20 3/2 701.96 905.87 1088.27 0 203.91 386.31 617.49 701.96

1 Ab" Bb" CDb/6! Eb" F G Ab" 16/15 9/8 6/5 10/9 4/3 21/20 7/5 8/7 8/5 10/9 16/9 9/8 1/1 16/15 16/15 111.73 315.64 498.04 582.51 813.69 996.09 0 111.73

8 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

11 - 32 12 - 31 13 - 30 14 - 29 15 - 28 16 - 27 17 - 26 18 - 25 19 - 24 20 - 23 21 - 22 64/63 81/80 55/54 45/44 56/55 99/98 49/48 64/63 81/80 55/54 56/55 27.26 21.51 31.77 38.91 31.19 17.58 35.70 27.26 21.51 31.77 31.19

5 1 Bb! Bb½! B" C /6" B#! C C½! Db /6" 6/5 11/9 5/4 14/11 9/7 4/3 11/8 7/5 315.64 347.41 386.31 417.51 435.08 498.04 551.32 582.51

Bb C#" C! 32/27 21/16 27/20 294.13 470.78 519.55

50/49 34.98

E D#! D" 27/16 32/21 40/27 905.87 729.22 680.45

5 1 E" E½" Eb! D /6! Eb#" DD½" C# /6! 5/3 18/11 8/5 11/7 14/9 3/2 16/11 10/7 884.36 852.59 813.69 782.49 764.92 701.96 648.68 617.49

False Ionian Scales 1/1 =G =392 * 2^n

1 5 1 D#! E#! F# /6! G! A#! C /6" C# /6! D#! 32/21 9/8 12/7 10/9 40/21 1701/ 81/80 640/ 8/7 49/44 14/11 55/49 10/7 16/15 32/21 729.22 933.13 1115.53 1600 21.51 567 231.17 417.51 617.49 729.22

5 Bb C D" D /6! FG" A" Bb 32/27 9/8 4/3 10/9 40/27 297/ 11/7 112/ 16/9 10/9 160/81 9/8 10/9 16/15 32/27 294.13 498.04 680.45 280 782.49 99 996.09 1178.49 182.40 294.13

1 1 1 Db /6" Eb#" F#" F#" Ab /6" Bb#" C#" Db /6" 7/5 10/9 14/9 9/8 7/4 15/14 15/8 28/25 21/20 10/9 7/6 9/8 21/16 16/15 7/5 582.51 764.92 968.83 1088.27 84.47 266.87 470.78 582.51

C! DEF! G! A B#! C! 27/20 10/9 3/2 9/8 27/16 16/15 9/5 9/8 81/80 10/9 9/8 8/7 9/7 21/20 27/20 519.55 701.96 905.87 1017.60 21.51 203.91 435.08 519.55

9 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

Partch 28 Tonalities - Tonality Structures and Roots

The Two Tonality Structures Inversely Related 1/1 =G =392 * 2^n (being also the Roots of the 12 Primary Tonalities) Utonality Otonality

2 Limit 5 Limit 3 Limit 7 Limit 3^2 Limit 11 Limit

D½! (11/9) F (9/7) A"# (7/6) C (6/5) Eb# (5/4) G (5/4) B! (6/5) D (7/6) F"! (9/7) A (11/9) C½# 16/11 347.41 16/9 435.08 8/7 266.87 4/3 315.64 8/5 386.31 1/1 386.31 5/4 315.64 3/2 266.87 7/4 435.08 9/8 347.41 11/8 648.68 996.09 231.17 498.04 813.69 0 386.31 701.96 968.83 203.91 551.32

1 / 11 Limit 1 / 3^2 Limit 1 / 7 Limit 1 / 3 Limit 1 / 5 Limit 1 / 2 Limit

The Roots of the 16 Secondary Tonalities (The Roots of the 6 Tertiary Tonalities)

Otonality Complete Utonality Otonality No 5 Limit major third Utonality

D C G½! G½" 3/2 4/3 64/33 33/32 701.96 498.04 1146.73 53.27

Otonality Incomplete Utonality Otonality Incomplete Utonality E#" Bb#! 12/7 7/6 Bb# E! F# A! 933.13 266.87 6/5 5/3 9/5 10/9 315.64 884.36 1017.60 182.40 A½! F½" 12/11 11/6 D#" C#! C# D! 150.64 1049.36 32/21 21/16 27/20 40/27 729.22 470.78 519.55 680.45

1 1 Ab# F#! Db /6! C# /6" 16/15 15/8 7/5 10/7 111.73 1088.27 582.51 617.49

Bb E 32/27 27/16 294.13 905.87

10 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

Partch 28 Tonalities - The Primary Tonalities and Ratios

The 12 Primary Tonalities and 29 Primary Ratios - (6 Otonalities + 6 Utonalities)

Tonality Diamond 1/1 =G =392 * 2^n Tonality Diamond "Reversum" 1/1 =G =392 * 2^n

(11 * 2m) / x x / (11 * 2n) C½! D½" 11/8 16/11 m n (9 * 2 ) / x 551.32 x / (9 * 2 ) 648.68 2 2 A Ab /3! Utonalities F F# /3" Otonalities 9/8 11/10 16/9 20/11 m n (7 * 2 ) / x 203.91 165.00 x / (7 * 2 ) 996.09 1035.00 F#" F! F½! A#! A" A½" 7/4 9/5 11/6 8/7 10/9 12/11 m n (3 * 2 ) / x 968.83 1017.60 1049.36 x / (3 * 2 ) 231.17 182.40 150.64 1 5 1 5 D Db /6" DD/6! C C# /6! CC/6" 3/2 7/5 3/2 (9/6) 11/7 4/3 10/7 4/3 (12/9) 14/11 m n (5 * 2 ) / x 701.96 582.51 701.96 782.49 x / (5 * 2 ) 498.04 617.49 498.04 417.51 B! Bb! Bb#" B#! Bb½! Eb" E" E#! Eb#" E½" 5/4 6/5 7/6 9/7 11/9 8/5 5/3 12/7 14/9 18/11 m n (2 ) / x 386.31 315.64 266.87 435.08 347.41 x / (2 ) 813.69 884.36 933.13 764.92 852.59 G GGGGG G GGGGG 1/1 5/5 3/3 7/7 9/9 11/11 1/1 5/5 3/3 7/7 9/9 11/11 000000 000000 n m x / (2 ) Eb" E" E#! Eb#" E½" (2 ) / x B! Bb! Bb#" B#! Bb½! 8/5 5/3 12/7 14/9 18/11 5/4 6/5 7/6 9/7 11/9 813.69 884.36 933.13 764.92 852.59 386.31 315.64 266.87 435.08 347.41 n 1 5 m 1 5 x / (5 * 2 ) C C# /6! CC/6" (5 * 2 ) / x D Db /6" DD/6! 4/3 10/7 4/3 (12/9) 14/11 3/2 7/5 3/2 (9/6) 11/7 498.04 617.49 498.04 417.51 701.96 582.51 701.96 782.49 n m x / (3 * 2 ) A#! A" A½" (3 * 2 ) / x F#" F! F½! 8/7 10/9 12/11 7/4 9/5 11/6 231.17 182.40 150.64 968.83 1017.60 1049.36 n 2 m 2 x / (7 * 2 ) F F# /3" (7 * 2 ) / x A Ab /3! Utonalities 16/9 20/11 Otonalities 9/8 11/10 996.09 1035.00 203.91 165.00 x / (9 * 2n) D½" (9 * 2m) / x C½! 16/11 11/8 648.68 551.32 x / (11 * 2n) (11 * 2m) / x

11 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

Partch 28 Tonalities - Secondary Tonalities and Ratios

The 16 Secondary Tonalities and 14 Secondary Ratios Incomplete Tonality Diamonds on C and D 1/1 =G =392 * 2^n

-1 Complete and 4 Incomplete Secondary Tonalities - (3 Otonalities + 2 Utonalities) -plus 3 incomplete Tertiary Tonalities without the 5 Limit major third - (1 Otonalities + 2 Utonalities) 2 Complete Secondary Tonalities m (12 + 2 complete Tonalities in total - 7 Otonalities + 7 Utonalities) (11 * 2 ) / x F½! D Otonality 11/6 m D F#" A C#" E G½! (9 * 2 ) / x 1049.36 3/2 15/8 9/8 21/16 27/16 33/32 (D) 701.96 1088.27 203.91 470.78 905.87 53.27 Otonalities 3/2 m (7 * 2 ) / x 701.96 Bb#" Bb! Bb½! C Utonality 7/6 6/5 11/9 m G½" Bb D#! F Ab! C (3 * 2 ) / x 266.87 315.64 347.41 64/33 32/27 32/21 16/9 16/15 4/3 (G) G 1146.73 294.13 729.22 996.09 111.73 498.04 1/1 1/1 m (5 * 2 ) / x 0 0 E! Eb! Eb#" E#! 5/3 8/5 14/9 12/7 m (2 ) / x 884.36 813.69 764.92 933.13 Ratios that don't belong to any of the 14 complete Tonalities C (U) CCCCC 4/3 4/3 4/3 4/3 4/3 4/3 1 F# /6! 498.04 498.04 498.04 498.04 498.04 498.04 n 40/21 x / (2 ) Ab" A" A#! A½" 1115.53 16/15 10/9 8/7 12/11 G! C! 111.73 182.40 231.17 150.64 n 1 81/80 27/20 x / (5 * 2 ) (F) F# /6! F 21.51 519.55 16/9 40/21 16/9 996.09 1115.53 996.09 n x / (3 * 2 ) D#! D" D½" 32/21 40/27 16/11 D" G" 729.22 680.45 648.68 n 40/27 160/81 x / (7 * 2 ) Bb 680.45 1178.49 Utonalities 32/27 1 Ab /6" 294.13 n 21/20 x / (9 * 2 ) G½" 84.47 64/33 1146.73 x / (11 * 2n)

12 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

-1 Complete and 4 Incomplete Secondary Tonalities - (2 Otonalities + 3 Utonalities) Remaining 6 Incomplete Secondary Tonalities 1/1 =G =392 * 2^n -plus 3 incomplete Tertiary Tonalities without the 5 Limit major third - (2 Otonalities + 1 Utonalities)

(11 * 2m) / x G½! C! EG! 33/32 Otonality 27/20 27/16 81/80 m (9 * 2 ) / x 53.27 519.55 905.87 21.51 E Otonalities 27/16 G" Bb D" m (7 * 2 ) / x 905.87 Utonality 160/81 32/27 40/27 C#" C! C½! 1178.49 294.13 680.45 21/16 27/20 11/8 m (3 * 2 ) / x 470.78 519.55 551.32 1 (A) Ab /6" A 9/8 21/20 9/8 F! AC! G! m (5 * 2 ) / x 203.91 84.47 203.91 Otonality 9/5 9/8 27/20 81/80 F#" F! F#" F½! 1017.60 203.91 519.55 21.51 15/8 9/5 7/4 11/6 m (2 ) / x 1088.27 1017.60 968.83 1049.36 G" D" F A" D (O) DDDDD Utonality 160/81 40/27 16/9 10/9 3/2 3/2 3/2 3/2 3/2 3/2 1178.49 680.45 996.09 182.40 701.96 701.96 701.96 701.96 701.96 701.96 n x / (2 ) Bb! B" B#! Bb#" 6/5 5/4 9/7 7/6 1 1 315.64 386.31 435.08 266.87 Db /6" F#" Ab /6" n x / (5 * 2 ) (G) G Otonality 7/5 7/4 21/20 1/1 1/1 582.51 968.83 84.47 0 0 n 1 1 x / (3 * 2 ) E#! E" E½" F# /6! A#! C# /6! 12/7 5/3 18/11 Utonality 40/21 8/7 10/7 933.13 884.36 852.59 1115.53 231.17 617.49 n x / (7 * 2 ) (C) Utonalities 4/3 498.04 x / (9 * 2n) A½" 12/11 150.64 x / (11 * 2n)

13 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

Partch 43 Tone Scale Complete

Extended Tonality Diamond 1/1 =G =392 * 2^n

12 Primary, 16 Secondary (and 6 Tertiary) Tonalities 14 Complete Tonalities - (7 Otonalities + 7 Utonalities)

G! G½! 81/80 33/32 21.51 53.27 E Otonalities 27/16 905.87 C"# C! 21/16 27/20 m 470.78 519.55 (11 * 2 ) / x 1 (A) Ab /6# C½! 9/8 21/20 11/8 m 203.91 84.47 (9 * 2 ) / x 551.32 2 F#! F! A Ab /3! 15/8 9/5 9/8 11/10 m 1088.27 1017.60 (7 * 2 ) / x 203.91 165.00 D (O) D F"# F! F½! 3/2 3/2 7/4 9/5 11/6 m 701.96 701.96 (3 * 2 ) / x 968.83 1017.60 1049.36 1 5 Bb" D Db /6# DD/6! 6/5 3/2 7/5 3/2 (9/6) 11/7 m 315.64 (5 * 2 ) / x 701.96 582.51 701.96 782.49 B! Bb! Bb"# B"! Bb½! 5/4 6/5 7/6 9/7 11/9 m (2 ) / x 386.31 315.64 266.87 435.08 347.41 G GGGGG 1/1 5/5 3/3 7/7 9/9 11/11 000000 n x / (2 ) Eb" E# E"! Eb"# E½# 8/5 5/3 12/7 14/9 18/11 813.69 884.36 933.13 764.92 852.59 n 1 5 E! x / (5 * 2 ) C C# /6! CC/6# 5/3 4/3 10/7 4/3 (12/9) 14/11 884.36 498.04 617.49 498.04 417.51 n C (U) C x / (3 * 2 ) A"! A# A½# 4/3 4/3 8/7 10/9 12/11 498.04 498.04 231.17 182.40 150.64 n 2 Ab" A# x / (7 * 2 ) F F# /3# 16/15 10/9 16/9 20/11 111.73 182.40 996.09 1035.00 1 n (F) F# /6! x / (9 * 2 ) D½# 16/9 40/21 16/11 996.09 1115.53 648.68 n D"! D# x / (11 * 2 ) 32/21 40/27 729.22 680.45 Bb Utonalities 32/27 294.13 G# G½# 160/81 64/33 1178.49 1146.73

14 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

13 Limit Tonality Diamond

13 Limit Tonality Diamond by seconds 13 Limit Tonality Diamond by thirds - (7 Otonalities + 7 Utonalities) - (7 Otonalities + 7 Utonalities)

1/1 =G =392 * 2^n 1/1 =G =392 * 2^n

(7 * 2m) / x (13 * 2m) / x 2 F!" Eb /5# 7/4 13/8 m m (13 * 2 ) / x 968.83 (11 * 2 ) / x 840.53 2 3 Eb /5# Eb!" C½# Cb /5# 13/8 14/9 11/8 13/10 m m (3 * 2 ) / x 840.53 764.92 (9 * 2 ) / x 551.32 454.21 2 1 2 2 D Db /5# Db /6" A Ab /3# Ab /5# Otonalities 3/2 13/9 7/5 Otonalities 9/8 11/10 13/12 m m (11 * 2 ) / x 701.96 636.62 582.51 (7 * 2 ) / x 203.91 165.00 138.57 3 5 7 C½# CCb/5# C /6" F!" F# F½# F /10# 11/8 4/3 (12/9) 13/10 14/11 7/4 9/5 11/6 13/7 m m (5 * 2 ) / x 551.32 498.04 454.21 417.51 (3 * 2 ) / x 968.83 1017.60 1049.36 1071.70 9 1 5 2 B! Bb½# Bb# Bb /10# Bb!" D Db /6" DD/6# Db /5# 5/4 11/9 6/5 13/11 7/6 3/2 7/5 3/2 (9/6) 11/7 13/9 m m (9 * 2 ) / x 386.31 347.41 315.64 289.21 266.87 (5 * 2 ) / x 701.96 582.51 701.96 782.49 636.62 2 2 7 9 A A" Ab /3# A½" Ab /5# A /10" B! Bb# Bb!" B!# Bb½# Bb /10# 9/8 10/9 11/10 12/11 13/12 14/13 5/4 6/5 7/6 9/7 11/9 13/11 m m (2 ) / x 203.91 182.40 165.00 150.64 138.57 128.30 (2 ) / x 386.31 315.64 266.87 435.08 347.41 289.21 G GG GGGG G GGGGGG 1/1 9/9 5/5 11/11 3/3 13/13 7/7 1/1 5/5 3/3 7/7 9/9 11/11 13/13 000 0000 0000000 n 2 2 7 n 9 x / (2 ) F F# F# /3" F½# F# /5" F /10# x / (2 ) Eb" E" E!# Eb!" E½" E /10" 16/9 9/5 20/11 11/6 24/13 13/7 8/5 5/3 12/7 14/9 18/11 22/13 996.09 1017.60 1035.00 1049.36 1061.43 1071.70 813.69 884.36 933.13 764.92 852.59 910.79 n 9 n 1 5 2 x / (9 * 2 ) Eb" E½" E" E /10" E!# x / (5 * 2 ) C C# /6# CC/6" C# /5" 8/5 18/11 5/3 22/13 12/7 4/3 10/7 4/3 (12/9) 14/11 18/13 813.69 852.59 884.36 910.79 933.13 498.04 617.49 498.04 417.51 563.38 n 3 5 n 7 x / (5 * 2 ) D½" DD# /5" D /6# x / (3 * 2 ) A!# A" A½" A /10" 16/11 3/2 (9/6) 20/13 11/7 8/7 10/9 12/11 14/13 648.68 701.96 745.79 782.49 231.17 182.40 150.64 128.30 n 2 1 n 2 2 x / (11 * 2 ) C C# /5" C# /6# x / (7 * 2 ) F F# /3" F# /5" Utonalities 4/3 18/13 10/7 Utonalities 16/9 20/11 24/13 498.04 563.38 617.49 996.09 1035.00 1061.43 n 2 n 3 x / (3 * 2 ) B /5" B!# x / (9 * 2 ) D½" D# /5" 16/13 9/7 16/11 20/13 359.47 435.08 648.68 745.79 n n 2 x / (13 * 2 ) A!# x / (11 * 2 ) B /5" 8/7 16/13 231.17 359.47 x / (7 * 2n) x / (13 * 2n)

15 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

Extended 55 or 57 Tone Scale

Completing the Tonality Diamonds on C and D 1/1 =G =392 * 2^n

- 10 Complete Secondary Tonalities plus 6 Complete Tertiary Tonalities - (8 Otonalities + 8 Utonalities) - 28 Complete Tonalities in total (as well further increasing the number of incomplete Tonalities) - 14 Extra Ratios Required (11 * 2m) / x (11 * 2m) / x G½! F½! 33/32 11/6 m m (9 * 2 ) / x 53.27 (9 * 2 ) / x 1049.36 2 2 E Eb /3! (D) Db /3! Otonalities 27/16 33/20 Otonalities 3/2 22/15 m m (7 * 2 ) / x 905.87 866.96 (7 * 2 ) / x 701.96 663.05 C"# C! C½! Bb"# Bb! Bb½! 21/16 27/20 11/8 7/6 6/5 11/9 m m (3 * 2 ) / x 470.78 519.55 551.32 (3 * 2 ) / x 266.87 315.64 347.41 1 5 1 5 (A) Ab /6# AA/6! (G) Gb /6# GG/6! 9/8 21/20 9/8 33/28 1/1 28/15 1/1 22/21 m m (5 * 2 ) / x 203.91 84.47 203.91 284.45 (5 * 2 ) / x 01080.56080.54 F#! F! F"# F#"! F½! E! Eb! Eb"# E"! Eb½! 15/8 9/5 7/4 27/14 11/6 5/3 8/5 14/9 12/7 44/27 m m (2 ) / x 1088.27 1017.60 968.83 1137.04 1049.36 (2 ) / x 884.36 813.69 764.92 933.13 845.45 D (O) D D D D D C (U) C C C C C 3/2 3/2 3/2 3/2 3/2 3/2 4/3 4/3 4/3 4/3 4/3 4/3 701.96 701.96 701.96 701.96 701.96 701.96 498.04 498.04 498.04 498.04 498.04 498.04 n n x / (2 ) Bb" B# B"! Bb"# B½# x / (2 ) Ab" A# A"! Ab"# A½# 6/5 5/4 9/7 7/6 27/22 16/15 10/9 8/7 28/27 12/11 315.64 386.31 435.08 266.87 354.55 111.73 182.40 231.17 62.96 150.64 n 1 5 n 1 5 x / (5 * 2 ) (G) G# /6! G G /6# x / (5 * 2 ) (F) F# /6! F F /6# 1/1 15/14 1/1 21/11 16/9 40/21 16/9 56/33 0 119.44 0 1119.46 996.09 1115.53 996.09 915.55 n n x / (3 * 2 ) E"! E# E½# x / (3 * 2 ) D"! D# D½# 12/7 5/3 18/11 32/21 40/27 16/11 933.13 884.36 852.59 729.22 680.45 648.68 n 2 n 2 x / (7 * 2 ) (C) C# /3# x / (7 * 2 ) Bb B /3# Utonalities 4/3 15/11 Utonalities 32/27 40/33 498.04 536.95 294.13 333.04 x / (9 * 2n) A½# x / (9 * 2n) G½# 12/11 64/33 150.64 1146.73 x / (11 * 2n) x / (11 * 2n)

Harmonic Structure (from which can be derived scales, modes, chords and tonalities) -Adding 14 Extra Ratios from the Completed Tonality Diamonds on C and D 1/1 =G =392 * 2^n -G!, G" not used in any three Tonality Diamonds

Otonality$ Ab" (O) Eb" (O) Bb" (O) F" (O) C" (O) ( G! ) (Bold = Roots) 16/15 8/5 6/5 9/5 27/20 81/80 111.73 813.69 315.64 1017.60 519.55 21.51 Bb (O) F (O) (U) C (O) (U) G (O) (U) D (O) (U) A (U) E 32/27 16/9 4/3 1/1 3/2 9/8 27/16 294.13 996.09 498.04 0 701.96 203.91 905.87 ( G# ) (U) D! (U) A! (U) E! (U) B! (U) F#! 160/81 40/27 10/9 5/3 5/4 15/8 %Utonality 1178.49 680.45 182.40 884.36 386.31 1088.27 (Bold = Roots)

16 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

Otonality! C G D (Bold = Roots) 4/3 1/1 3/2 498.04 0 701.96 (7/6) (9/7) (7/6) (9/7) (7/6) (9/7) D"# (O) A"# (O) E"# (O) B"# F#"# 32/21 8/7 12/7 9/7 (3/2) 27/14 729.22 231.17 933.13 435.08 1137.04 1 1 1 F# /6# (U) C# /6# G# /6# 40/21 10/7 15/14 1115.53 617.49 119.44

D 3/2 701.96 (27/22) (11/9) C G 2 2 2 4/3 1/1 Db /3# Ab /3# Eb /3# 498.04 0 22/15 11/10 33/20 (11/9) (27/22) (11/9) 663.05 165.00 866.96 Eb½# Bb½# (U) F½# (U) C½# (U) G½# 44/27 (3/2) 11/9 11/6 11/8 33/32 845.45 347.41 1049.36 551.32 53.27 (9/7) (7/6) (9/7) (7/6) (9/7) (7/6) 5 5 5 G /6# D /6# A /6# 22/21 (3/2) 11/7 (3/2) 33/28 $Utonality

80.54 782.49 284.45 (Bold = Roots)

5 5 5 Otonality! F /6% C /6% G /6% (Bold = Roots) 56/33 14/11 21/11 915.55 417.51 1119.46 (7/6) (9/7) (7/6) (9/7) (7/6) (9/7) G½% (O) D½% (O) A½% (O) E½% B½% 64/33 16/11 12/11 18/11 (3/2) 27/22 1146.73 648.68 150.64 852.59 354.55 2 2 2 B /3% F# /3% C# /3% (11/9) (27/22) (11/9) 40/33 20/11 15/11 G D 333.04 1035.00 536.95 1/1 3/2 0701.96 (11/9) (27/22) C 4/3 498.04

1 1 1 Gb /6% Db /6% (O) Ab /6% 28/15 7/5 21/20 1080.56 582.51 84.47 Ab"% Eb"% (U) Bb"% (U) F"% (U) C"% 28/27 (3/2) 14/9 7/6 7/4 21/16 62.96 764.92 266.87 968.83 470.78 (9/7) (7/6) (9/7) (7/6) (9/7) (7/6) C G D 4/3 1/1 3/2 $Utonality 498.04 0 701.96 (Bold = Roots)

17 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

Partch 43 Tone Scale

1/1 =C

D½! G½! A½! C½! ½! approximately 1/2 semitone flat 2 2 1/11 Limit 12/11 16/11 18/11 B /3! 64/33 /3! approximately 2/3 semitone flat 5 5 150.64 F /6! 648.68 852.59 20/11 1146.73 /6! approximately 5/6 semitone flat 14/11 1035.00 417.51

D"# E"# G"# A"# "# approximately 1/3 semitone sharp 1 1 1 1/7 Limit 8/7 9/7 F# /6# 32/21 12/7 B /6# /6# approximately 1/6 semitone sharp 231.17 435.08 10/7 729.22 933.13 40/21 Utonality 617.49 1115.53

# 81/80 C# Db# Eb# F# Ab# Bb# # syntonic comma (81/80) sharp 1/5 Limit 81/80 16/15 6/5 27/20 8/5 9/5 21.51 111.73 315.64 519.55 813.69 1017.60 (81/80) (81/80) (81/80) (81/80) Eb F Bb 1/3 Limit C 32/27 4/3 16/9 C 1/1 D294.13 498.04 G A996.09 2/1 3 Limit 0 9/8 3/2 27/16 1200 203.91 701.96 905.87 (81/80) (81/80) (81/80) (81/80) D! E! G! A! B! C!!syntonic comma (81/80) flat 5 Limit 10/9 5/4 40/27 5/3 15/8 160/81 ! 81/80 182.40 386.31 680.45 884.36 1088.27 1178.49

1 1 1 Otonality Db /6! Gb /6! /6! approximately 1/6 semitone flat 21/20 Eb"! F"! 7/5 Ab"! Bb"! "! approximately 1/3 semitone flat 7 Limit 84.47 7/6 21/16 582.51 14/9 7/4 266.87 470.78 764.92 968.83

5 5 G /6# /6# approximately 5/6 semitone sharp 2 2 Db /3# 11/7 /3# approximately 2/3 semitone sharp C½# 11/10 Eb½# F½# 782.49 Bb½# ½# approximately 1/2 semitone sharp 11 Limit 33/32 165.00 11/9 11/8 11/6 53.27 347.41 551.32 1049.36

18 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

Partch 43 Tone Scale

Harmonic Structure (from which can be derived scales, modes, chords and tonalities) 1/1 =C

Primary Ratios -(also the Roots of 3 incomplete Tertiary Tonalities Otonality! F G without the 5 Limit major third) (Bold = Roots) Secondary 4/3 3/2 Ratios 498.04 701.96 (7/6) (9/7) (7/6) G"# (O) D"# (O) A"# (O) E"# 32/21 8/7 12/7 (3/2) 9/7 729.22 231.17 933.13 435.08 1 1 B /6# (U) F# /6! (7/6) (9/7) (11/9) 5 40/21 10/7 C G /6# $Utonality 2 1115.53 617.49 1/1 11/7 Db /3# Secondary Otonality! 0782.4911/10 Ratio (11/9) (9/7) (7/6) 165.00 Eb½# (U) Bb½# (U) F½# (U) C½# 11/9 (3/2) 11/6 11/8 (3/2) 33/32 347.41 1049.36 551.32 53.27 Partch Tuning Notes = 8/7, 11/8 (11/6) (11/9) F G 4/3 3/2 $Utonality 498.04 701.96 (Bold = Roots) Primary Ratios

Secondary Primary Ratios Otonality! Ratios Db! (O) Ab! (O) Eb! (O) Bb! (O) F! (O) C# (Bold = Roots) 16/15 8/5 6/5 9/5 27/20 81/80 111.73 813.69 315.64 1017.60 519.55 21.51 Eb (O) Bb (O) (U) F (O) (U) C (O) (U) G (O) (U) D (U) A 32/27 16/9 4/3 1/1 3/2 9/8 27/16 294.13 996.09 498.04 0 701.96 203.91 905.87 C% (U) G" (U) D" (U) A" (U) E" (U) B" 160/81 40/27 10/9 5/3 5/4 15/8 $Utonality 1178.49 680.45 182.40 884.36 386.31 1088.27 Secondary (Bold = Roots) Primary Ratios Ratios

Primary Ratios -(also the Roots of 3 incomplete Tertiary Tonalities Otonality! F G without the 5 Limit major third) (Bold = Roots) 4/3 3/2 498.04 701.96 (11/9) (11/6) C½% (O) G½% (O) D½% (O) A½% 64/33 (3/2) 16/11 12/11 (3/2) 18/11 1146.73 648.68 150.64 852.59 2 B /3% (7/6) (9/7) (11/9) 5 Secondary 20/11 F /6% C $Utonality 1 1 Ratio 1035.00 14/11 1/1 Gb /6% (O) Db /6% Otonality! 417.51 0 7/5 21/20 (11/9) (9/7) (7/6) 582.51 84.47 Ab"% (U) Eb"% (U) Bb"% (U) F"% 14/9 (3/2) 7/6 7/4 21/16 764.92 266.87 968.83 470.78 Partch Tuning Notes = 7/4, 16/11 (7/6) (9/7) (7/6) F G Secondary 4/3 3/2 Ratios $Utonality 498.04 701.96 (Bold = Roots) Primary Ratios -The 8/7, 11/8, 7/4 and 16/11 intervals are prescribed by Partch to tune the 7 and 11 limit ratios. Without a doubt a difficult task to tune these intervals accurately by ear. After which the rest of the 7 and 11 limit ratios present no problems. The 5 limit ratios present no tuning problems.

19 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

Partch 43 Tone Scale

1/1 =C

01 - 422 - 413 - 404 - 395 - 386 - 377 - 368 - 359 - 34 10 - 33 81/80 55/54 56/55 64/63 45/44 121/120 100/99 81/80 64/63 49/48 21.51 31.77 31.19 27.26 38.91 14.37 17.40 21.51 27.26 35.70

2 Primary C D½! Db /3" D! DD#" Eb#! Ratios 1/1 12/11 11/10 10/9 9/8 8/7 7/6 0 150.64 165.00 182.40 203.91 231.17 266.87

1 Secondary C" C½" Db /6! Db" Ratios 81/80 33/32 21/20 16/15 21.51 53.27 84.47 111.73

1 Secondary C! C½! B /6" B! Ratios 160/81 64/33 40/21 15/8 1178.49 1146.73 1115.53 1088.27

2 Primary C Bb½" B /3! Bb" Bb Bb#! A#" Ratios 2/1 11/6 20/11 9/5 16/9 7/4 12/7 1200 1049.36 1035.00 1017.60 996.09 968.83 933.13

False Ionian Scales 1/1 =C

1 1 D#" E#" F# /6" G#" A#" B /6" D½! D#" 8/7 9/8 9/7 10/9 10/7 16/15 32/21 9/8 12/7 10/9 40/21 63/55 12/11 22/21 8/7 231.17 435.08 617.49 729.22 933.13 1115.53 150.64 231.17

2 G½! A½! B /3! C½! D½! Eb½" F½" G½! 16/11 9/8 18/11 10/9 20/11 16/15 64/33 9/8 12/11 121/ 11/9 9/8 11/8 128/ 16/11 648.68 852.59 1035.00 1146.73 150.64 108 347.41 551.32 121 648.68

1 GAB! C D E! F# /6" G 3/2 9/8 27/16 10/9 15/8 16/15 1/1 9/8 9/8 10/9 5/4 8/7 10/7 21/20 3/2 701.96 905.87 1088.27 0 203.91 386.31 617.49 701.96

1 Db" Eb" FGb/6! Ab" Bb C Db" 16/15 9/8 6/5 10/9 4/3 21/20 7/5 8/7 8/5 10/9 16/9 9/8 1/1 16/15 16/15 111.73 315.64 498.04 582.51 813.69 996.09 0 111.73

20 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 Compendium Musica

5 1 Eb! Eb½! E" F /6" E#! F F½! Gb /6" 6/5 11/9 5/4 14/11 9/7 4/3 11/8 7/5 315.64 347.41 386.31 417.51 435.08 498.04 551.32 582.51

Eb F#" F! 32/27 21/16 27/20 294.13 470.78 519.55

50/49 34.98

A G#! G" 27/16 32/21 40/27 905.87 729.22 680.45

5 1 A" A½" Ab! G /6! Ab#" GG½" F# /6! 5/3 18/11 8/5 11/7 14/9 3/2 16/11 10/7 884.36 852.59 813.69 782.49 764.92 701.96 648.68 617.49

False Ionian Scales 1/1 =C

1 5 1 G#! A#! B /6! C! D#! F /6" F# /6! G#! 32/21 9/8 12/7 10/9 40/21 1701/ 81/80 640/ 8/7 49/44 14/11 55/49 10/7 16/15 32/21 729.22 933.13 1115.53 1600 21.51 567 231.17 417.51 617.49 729.22

5 Eb F G" G /6! Bb C" D" Eb 32/27 9/8 4/3 10/9 40/27 297/ 11/7 112/ 16/9 10/9 160/81 9/8 10/9 16/15 32/27 294.13 498.04 680.45 280 782.49 99 996.09 1178.49 182.40 294.13

1 1 1 Gb /6" Ab#" Bb#" B" Db /6" Eb#" F#" Gb /6" 7/5 10/9 14/9 9/8 7/4 15/14 15/8 28/25 21/20 10/9 7/6 9/8 21/16 16/15 7/5 582.51 764.92 968.83 1088.27 84.47 266.87 470.78 582.51

F! GABb! C! D E#! F! 27/20 10/9 3/2 9/8 27/16 16/15 9/5 9/8 81/80 10/9 9/8 8/7 9/7 21/20 27/20 519.55 701.96 905.87 1017.60 21.51 203.91 435.08 519.55

Partch 28 Tonalities - Tonality Structures and Roots

The Two Tonality Structures Inversely Related 1/1 =C (being also the Roots of the 12 Primary Tonalities) Utonality Otonality

2 Limit 5 Limit 3 Limit 7 Limit 3^2 Limit 11 Limit

G½! (11/9) Bb (9/7) D"# (7/6) F (6/5) Ab# (5/4) C (5/4) E! (6/5) G (7/6) Bb"! (9/7) D (11/9) F½# 16/11 347.41 16/9 435.08 8/7 266.87 4/3 315.64 8/5 386.31 1/1 386.31 5/4 315.64 3/2 266.87 7/4 435.08 9/8 347.41 11/8 648.68 996.09 231.17 498.04 813.69 0 386.31 701.96 968.83 203.91 551.32

1 / 11 Limit 1 / 3^2 Limit 1 / 7 Limit 1 / 3 Limit 1 / 5 Limit 1 / 2 Limit

The Roots of the 16 Secondary Tonalities (The Roots of the 6 Tertiary Tonalities)

Otonality Complete Utonality Otonality No 5 Limit major third Utonality

G F C½! C½" 3/2 4/3 64/33 33/32 701.96 498.04 1146.73 53.27

Otonality Incomplete Utonality Otonality Incomplete Utonality A#" Eb#! 12/7 7/6 Eb# A! Bb# D! 933.13 266.87 6/5 5/3 9/5 10/9 315.64 884.36 1017.60 182.40 D½! Bb½" 12/11 11/6 G#" F#! F# G! 150.64 1049.36 32/21 21/16 27/20 40/27 729.22 470.78 519.55 680.45

1 1 Db# B! Gb /6! F# /6" 16/15 15/8 7/5 10/7 111.73 1088.27 582.51 617.49

Eb A 32/27 27/16 294.13 905.87

Partch 28 Tonalities - The Primary Tonalities and Ratios

The 12 Primary Tonalities and 29 Primary Ratios - (6 Otonalities + 6 Utonalities)

Tonality Diamond 1/1 =C Tonality Diamond "Reversum" 1/1 =C

(11 * 2m) / x x / (11 * 2n) F½! G½" 11/8 16/11 m n (9 * 2 ) / x 551.32 x / (9 * 2 ) 648.68 2 2 D Db /3! Utonalities Bb B /3" Otonalities 9/8 11/10 16/9 20/11 m n (7 * 2 ) / x 203.91 165.00 x / (7 * 2 ) 996.09 1035.00 Bb#" Bb! Bb½! D#! D" D½" 7/4 9/5 11/6 8/7 10/9 12/11 m n (3 * 2 ) / x 968.83 1017.60 1049.36 x / (3 * 2 ) 231.17 182.40 150.64 1 5 1 5 G Gb /6" GG/6! F F# /6! FF/6" 3/2 7/5 3/2 (9/6) 11/7 4/3 10/7 4/3 (12/9) 14/11 m n (5 * 2 ) / x 701.96 582.51 701.96 782.49 x / (5 * 2 ) 498.04 617.49 498.04 417.51 E! Eb! Eb#" E#! Eb½! Ab" A" A#! Ab#" A½" 5/4 6/5 7/6 9/7 11/9 8/5 5/3 12/7 14/9 18/11 m n (2 ) / x 386.31 315.64 266.87 435.08 347.41 x / (2 ) 813.69 884.36 933.13 764.92 852.59 C CCCCC C CCCCC 1/1 5/5 3/3 7/7 9/9 11/11 1/1 5/5 3/3 7/7 9/9 11/11 000000 000000 n m x / (2 ) Ab" A" A#! Ab#" A½" (2 ) / x E! Eb! Eb#" E#! Eb½! 8/5 5/3 12/7 14/9 18/11 5/4 6/5 7/6 9/7 11/9 813.69 884.36 933.13 764.92 852.59 386.31 315.64 266.87 435.08 347.41 n 1 5 m 1 5 x / (5 * 2 ) F F# /6! FF/6" (5 * 2 ) / x G Gb /6" GG/6! 4/3 10/7 4/3 (12/9) 14/11 3/2 7/5 3/2 (9/6) 11/7 498.04 617.49 498.04 417.51 701.96 582.51 701.96 782.49 n m x / (3 * 2 ) D#! D" D½" (3 * 2 ) / x Bb#" Bb! Bb½! 8/7 10/9 12/11 7/4 9/5 11/6 231.17 182.40 150.64 968.83 1017.60 1049.36 n 2 m 2 x / (7 * 2 ) Bb B /3" (7 * 2 ) / x D Db /3! Utonalities 16/9 20/11 Otonalities 9/8 11/10 996.09 1035.00 203.91 165.00 x / (9 * 2n) G½" (9 * 2m) / x F½! 16/11 11/8 648.68 551.32 x / (11 * 2n) (11 * 2m) / x

Partch 28 Tonalities - Secondary Tonalities and Ratios

The 16 Secondary Tonalities and 14 Secondary Ratios Incomplete Tonality Diamonds on F and G 1/1 =C

2 Complete Secondary Tonalities -1 Complete and 4 Incomplete Secondary Tonalities - (3 Otonalities + 2 Utonalities) (12 + 2 complete Tonalities in total - 7 Otonalities + 7 Utonalities) -plus 3 incomplete Tertiary Tonalities without the 5 Limit major third - (1 Otonalities + 2 Utonalities)

G Otonality (11 * 2m) / x G B! D F"! A C½# Bb½# 3/2 15/8 9/8 21/16 27/16 33/32 11/6 m 701.96 1088.27 203.91 470.78 905.87 53.27 (9 * 2 ) / x 1049.36 (G) Otonalities 3/2 m F Utonality (7 * 2 ) / x 701.96 C½! Eb G"# Bb Db# F Eb"! Eb# Eb½# 64/33 32/27 32/21 16/9 16/15 4/3 7/6 6/5 11/9 m 1146.73 294.13 729.22 996.09 111.73 498.04 (3 * 2 ) / x 266.87 315.64 347.41 (C) C 1/1 1/1 m (5 * 2 ) / x 0 0 A! Ab# Ab"! A"# Ratios that don't belong to any of the 14 complete Tonalities 5/3 8/5 14/9 12/7 m (2 ) / x 884.36 813.69 764.92 933.13 1 B /6# F (U) FFFFF 40/21 4/3 4/3 4/3 4/3 4/3 4/3 1115.53 498.04 498.04 498.04 498.04 498.04 498.04 n C# F# x / (2 ) Db" D! D"# D½! 81/80 27/20 16/15 10/9 8/7 12/11 21.51 519.55 111.73 182.40 231.17 150.64 n 1 x / (5 * 2 ) (Bb) B /6# Bb 16/9 40/21 16/9 996.09 1115.53 996.09 n G! C! x / (3 * 2 ) G"# G! G½! 40/27 160/81 32/21 40/27 16/11 680.45 1178.49 729.22 680.45 648.68 1 n Db /6! x / (7 * 2 ) Eb 21/20 Utonalities 32/27 84.47 294.13 x / (9 * 2n) C½! 64/33 1146.73 x / (11 * 2n)

-1 Complete and 4 Incomplete Secondary Tonalities - (3 Otonalities + 2 Utonalities) Remaining 6 Incomplete Secondary Tonalities 1/1 =C -plus 3 incomplete Tertiary Tonalities without the 5 Limit major third - (1 Otonalities + 2 Utonalities)

(11 * 2m) / x C½! F! AC! 33/32 Otonality 27/20 27/16 81/80 m (9 * 2 ) / x 53.27 519.55 905.87 21.51 A Otonalities 27/16 C" Eb G" m (7 * 2 ) / x 905.87 Utonality 160/81 32/27 40/27 F#" F! F½! 1178.49 294.13 680.45 21/16 27/20 11/8 m (3 * 2 ) / x 470.78 519.55 551.32 1 (D) Db /6" D 9/8 21/20 9/8 Bb! DF! C! m (5 * 2 ) / x 203.91 84.47 203.91 Otonality 9/5 9/8 27/20 81/80 B" Bb! Bb#" Bb½! 1017.60 203.91 519.55 21.51 15/8 9/5 7/4 11/6 m (2 ) / x 1088.27 1017.60 968.83 1049.36 C" G" Bb D" G (O) GGGGG Utonality 160/81 40/27 16/9 10/9 3/2 3/2 3/2 3/2 3/2 3/2 1178.49 680.45 996.09 182.40 701.96 701.96 701.96 701.96 701.96 701.96 n x / (2 ) Eb! E" E#! Eb#" 6/5 5/4 9/7 7/6 315.64 386.31 435.08 266.87 n 1 1 x / (5 * 2 ) (C) C Otonality Gb /6" Bb#" Db /6" 1/1 1/1 7/5 7/4 21/20 0 0 582.51 968.83 84.47 x / (3 * 2n) A#! A" A½" 1 1 12/7 5/3 18/11 Utonality B /6! D#! F# /6! 933.13 884.36 852.59 40/21 8/7 10/7 n x / (7 * 2 ) (F) 1115.53 231.17 617.49 Utonalities 4/3 498.04 x / (9 * 2n) D½" 12/11 150.64 x / (11 * 2n)

Partch 43 Tone Scale Complete

Extended Tonality Diamond 1/1 =C

12 Primary, 16 Secondary (and 6 Tertiary) Tonalities 14 Complete Tonalities - (7 Otonalities + 7 Utonalities)

C! C½! 81/80 33/32 21.51 53.27 A Otonalities 27/16 905.87 F"# F! 21/16 27/20 m 470.78 519.55 (11 * 2 ) / x 1 DDb/6# F½! 9/8 21/20 11/8 m 203.91 84.47 (9 * 2 ) / x 551.32 2 B! Bb! D Db /3! 15/8 9/5 9/8 11/10 m 1088.27 1017.60 (7 * 2 ) / x 203.91 165.00 G (O) G Bb"# Bb! Bb½! 3/2 3/2 7/4 9/5 11/6 m 701.96 701.96 (3 * 2 ) / x 968.83 1017.60 1049.36 1 5 Eb" G Gb /6# GG/6! 6/5 3/2 7/5 3/2 (9/6) 11/7 m 315.64 (5 * 2 ) / x 701.96 582.51 701.96 782.49 E! Eb! Eb"# E"! Eb½! 5/4 6/5 7/6 9/7 11/9 m (2 ) / x 386.31 315.64 266.87 435.08 347.41 C CCCCC 1/1 5/5 3/3 7/7 9/9 11/11 000000 n x / (2 ) Ab" A# A"! Ab"# A½# 8/5 5/3 12/7 14/9 18/11 813.69 884.36 933.13 764.92 852.59 n 1 5 A! x / (5 * 2 ) F F# /6! FF/6# 5/3 4/3 10/7 4/3 (12/9) 14/11 884.36 498.04 617.49 498.04 417.51 n F (U) F x / (3 * 2 ) D"! D# D½# 4/3 4/3 8/7 10/9 12/11 498.04 498.04 231.17 182.40 150.64 n 2 Db" D# x / (7 * 2 ) Bb B /3# 16/15 10/9 16/9 20/11 111.73 182.40 996.09 1035.00 1 n Bb B /6! x / (9 * 2 ) G½# 16/9 40/21 16/11 996.09 1115.53 648.68 n G"! G# x / (11 * 2 ) 32/21 40/27 729.22 680.45 Eb Utonalities 32/27 294.13 C# C½# 160/81 64/33 1178.49 1146.73

13 Limit Tonality Diamond

13 Limit Tonality Diamond by seconds 13 Limit Tonality Diamond by thirds - (7 Otonalities + 7 Utonalities) - (7 Otonalities + 7 Utonalities)

1/1 =C 1/1 =C

(7 * 2m) / x (13 * 2m) / x 2 Bb!" Ab /5# 7/4 13/8 m m (13 * 2 ) / x 968.83 (11 * 2 ) / x 840.53 2 3 Ab /5# Ab!" F½# Fb /5# 13/8 14/9 11/8 13/10 m m (3 * 2 ) / x 840.53 764.92 (9 * 2 ) / x 551.32 454.21 2 1 2 2 G Gb /5# Gb /6" D Db /3# Db /5# Otonalities 3/2 13/9 7/5 Otonalities 9/8 11/10 13/12 m m (11 * 2 ) / x 701.96 636.62 582.51 (7 * 2 ) / x 203.91 165.00 138.57 3 5 7 F½# FFb/5# F /6" Bb!" Bb# Bb½# Bb /10# 11/8 4/3 (12/9) 13/10 14/11 7/4 9/5 11/6 13/7 m m (5 * 2 ) / x 551.32 498.04 454.21 417.51 (3 * 2 ) / x 968.83 1017.60 1049.36 1071.70 9 1 5 2 E! Eb½# Eb# Eb /10# Eb!" G Gb /6" GG/6# Gb /5# 5/4 11/9 6/5 13/11 7/6 3/2 7/5 3/2 (9/6) 11/7 13/9 m m (9 * 2 ) / x 386.31 347.41 315.64 289.21 266.87 (5 * 2 ) / x 701.96 582.51 701.96 782.49 636.62 2 2 7 9 D D" Db /3# D½" Db /5# D /10" E! Eb# Eb!" E!# Eb½# Eb /10# 9/8 10/9 11/10 12/11 13/12 14/13 5/4 6/5 7/6 9/7 11/9 13/11 m m (2 ) / x 203.91 182.40 165.00 150.64 138.57 128.30 (2 ) / x 386.31 315.64 266.87 435.08 347.41 289.21 C CC CCCC C CCCCCC 1/1 9/9 5/5 11/11 3/3 13/13 7/7 1/1 5/5 3/3 7/7 9/9 11/11 13/13 000 0000 0000000 n 2 2 7 n 9 x / (2 ) Bb Bb# B /3" Bb½# B /5" Bb /10# x / (2 ) Ab" A" A!# Ab!" A½" A /10" 16/9 9/5 20/11 11/6 24/13 13/7 8/5 5/3 12/7 14/9 18/11 22/13 996.09 1017.60 1035.00 1049.36 1061.43 1071.70 813.69 884.36 933.13 764.92 852.59 910.79 n 9 n 1 5 2 x / (9 * 2 ) Ab" A½" A" A /10" A!# x / (5 * 2 ) F F# /6# FF/6" F# /5" 8/5 18/11 5/3 22/13 12/7 4/3 10/7 4/3 (12/9) 14/11 18/13 813.69 852.59 884.36 910.79 933.13 498.04 617.49 498.04 417.51 563.38 n 3 5 n 7 x / (5 * 2 ) G½" GG# /5" G /6# x / (3 * 2 ) D!# D" D½" D /10" 16/11 3/2 (9/6) 20/13 11/7 8/7 10/9 12/11 14/13 648.68 701.96 745.79 782.49 231.17 182.40 150.64 128.30 n 2 1 n 2 2 x / (11 * 2 ) F F# /5" F# /6# x / (7 * 2 ) Bb B /3" B /5" Utonalities 4/3 18/13 10/7 Utonalities 16/9 20/11 24/13 498.04 563.38 617.49 996.09 1035.00 1061.43 n 2 n 3 x / (3 * 2 ) E /5" E!# x / (9 * 2 ) G½" G# /5" 16/13 9/7 16/11 20/13 359.47 435.08 648.68 745.79 n n 2 x / (13 * 2 ) D!# x / (11 * 2 ) E /5" 8/7 16/13 231.17 359.47 x / (7 * 2n) x / (13 * 2n)

Extended 55 or 57 Tone Scale

Completing the Tonality Diamonds on F and G 1/1 =C

- 10 Complete Secondary Tonalities plus 6 Complete Tertiary Tonalities - (8 Otonalities + 8 Utonalities) - 28 Complete Tonalities in total (as well further increasing the number of incomplete Tonalities) - 14 Extra Ratios Required (11 * 2m) / x (11 * 2m) / x C½! Bb½! 33/32 11/6 m m (9 * 2 ) / x 53.27 (9 * 2 ) / x 1049.36 2 2 A Ab /3! (G) Gb /3! Otonalities 27/16 33/20 Otonalities 3/2 22/15 m m (7 * 2 ) / x 905.87 866.96 (7 * 2 ) / x 701.96 663.05 F"# F! F½! Eb"# Eb! Eb½! 21/16 27/20 11/8 7/6 6/5 11/9 m m (3 * 2 ) / x 470.78 519.55 551.32 (3 * 2 ) / x 266.87 315.64 347.41 1 5 1 5 (D) Db /6# DD/6! (C) Cb /6# CC/6! 9/8 21/20 9/8 33/28 1/1 28/15 1/1 22/21 m m (5 * 2 ) / x 203.91 84.47 203.91 284.45 (5 * 2 ) / x 01080.56080.54 B! Bb! Bb"# B"! Bb½! A! Ab! Ab"# A"! Ab½! 15/8 9/5 7/4 27/14 11/6 5/3 8/5 14/9 12/7 44/27 m m (2 ) / x 1088.27 1017.60 968.83 1137.04 1049.36 (2 ) / x 884.36 813.69 764.92 933.13 845.45 G (O) G G G G G F (U) F F F F F 3/2 3/2 3/2 3/2 3/2 3/2 4/3 4/3 4/3 4/3 4/3 4/3 701.96 701.96 701.96 701.96 701.96 701.96 498.04 498.04 498.04 498.04 498.04 498.04 n n x / (2 ) Eb" E# E"! Eb"# E½# x / (2 ) Db" D# D"! Db"# D½# 6/5 5/4 9/7 7/6 27/22 16/15 10/9 8/7 28/27 12/11 315.64 386.31 435.08 266.87 354.55 111.73 182.40 231.17 62.96 150.64 n 1 5 n 1 5 x / (5 * 2 ) (C) C# /6! C C /6# x / (5 * 2 ) (Bb) B /6! Bb Bb /6# 1/1 15/14 1/1 21/11 16/9 40/21 16/9 56/33 0 119.44 0 1119.46 996.09 1115.53 996.09 915.55 n n x / (3 * 2 ) A"! A# A½# x / (3 * 2 ) G"! G# G½# 12/7 5/3 18/11 32/21 40/27 16/11 933.13 884.36 852.59 729.22 680.45 648.68 n 2 n 2 x / (7 * 2 ) (F) F# /3# x / (7 * 2 ) Eb E /3# Utonalities 4/3 15/11 Utonalities 32/27 40/33 498.04 536.95 294.13 333.04 x / (9 * 2n) D½# x / (9 * 2n) C½# 12/11 64/33 150.64 1146.73 x / (11 * 2n) x / (11 * 2n)

Harmonic Structure (from which can be derived scales, modes, chords and tonalities) - Adding 14 Extra Ratios from the Completed Tonality Diamonds on F and G 1/1 =C -C!, C" not used in any three Tonality Diamonds

Otonality$ Db" (O) Ab" (O) Eb" (O) Bb" (O) F" (O) ( C! ) (Bold = Roots) 16/15 8/5 6/5 9/5 27/20 81/80 111.73 813.69 315.64 1017.60 519.55 21.51 Eb (O) Bb (O) (U) F (O) (U) C (O) (U) G (O) (U) D (U) A 32/27 16/9 4/3 1/1 3/2 9/8 27/16 294.13 996.09 498.04 0 701.96 203.91 905.87 ( C# ) (U) G! (U) D! (U) A! (U) E! (U) B! 160/81 40/27 10/9 5/3 5/4 15/8 %Utonality 1178.49 680.45 182.40 884.36 386.31 1088.27 (Bold = Roots)

Otonality! F C G (Bold = Roots) 4/3 1/1 3/2 498.04 0 701.96 (7/6) (9/7) (7/6) (9/7) (7/6) (9/7) G"# (O) D"# (O) A"# (O) E"# B"# 32/21 8/7 12/7 9/7 (3/2) 27/14 729.22 231.17 933.13 435.08 1137.04 1 1 1 B /6# (U) F# /6# C# /6# 40/21 10/7 15/14 1115.53 617.49 119.44

G 3/2 701.96 (27/22) (11/9) F C 2 2 2 4/3 1/1 Gb /3# Db /3# Ab /3# 498.04 0 22/15 11/10 33/20 (11/9) (27/22) (11/9) 663.05 165.00 866.96 Ab½# Eb½# (U) Bb½# (U) F½# (U) C½# 44/27 (3/2) 11/9 11/6 11/8 33/32 845.45 347.41 1049.36 551.32 53.27 (9/7) (7/6) (9/7) (7/6) (9/7) (7/6) 5 5 5 C /6# G /6# D /6# 22/21 (3/2) 11/7 (3/2) 33/28 $Utonality

80.54 782.49 284.45 (Bold = Roots)

5 5 5 Otonality! Bb /6% F /6% C /6% (Bold = Roots) 56/33 14/11 21/11 915.55 417.51 1119.46 (7/6) (9/7) (7/6) (9/7) (7/6) (9/7) C½% (O) G½% (O) D½% (O) A½% E½% 64/33 16/11 12/11 18/11 (3/2) 27/22 1146.73 648.68 150.64 852.59 354.55 2 2 2 E /3% B /3% F# /3% (11/9) (27/22) (11/9) 40/33 20/11 15/11 C G 333.04 1035.00 536.95 1/1 3/2 0701.96 (11/9) (27/22) F 4/3 498.04

1 1 1 Cb /6% Gb /6% (O) Db /6% 28/15 7/5 21/20 1080.56 582.51 84.47 Db"% Ab"% (U) Eb"% (U) Bb"% (U) F"% 28/27 (3/2) 14/9 7/6 7/4 21/16 62.96 764.92 266.87 968.83 470.78 (9/7) (7/6) (9/7) (7/6) (9/7) (7/6) F C G 4/3 1/1 3/2 $Utonality 498.04 0 701.96 (Bold = Roots)

Partch 43 Tone Scale Compared to 43ET and 12ET 1/1 =G =392 * 2^n or 1/1 =C

43ET +/- 43ET +/- 43ET +/- 43ET +/- Partch Scale 43ET Ratio Cents from 12ET 5 Limit from Just 7 Limit from Just 11 Limit from Just +/- from 12ET Avg.-> 7.65 Avg.-> 6.58 Avg.-> 6.56 2^(43/43) 2 1200 0 2/1 1200 0 01200 2^(42/43) 1.968019 1172.09 -27.91 160/81 1178.49 -6.40 -21.51 1200 2^(41/43) 1.936549 1144.19 44.19 64/33 1146.73 -2.54 46.73 1100 2^(40/43) 1.905583 1116.28 16.28 40/21 1115.53 0.75 15.53 1100 2^(39/43) 1.875112 1088.37 -11.63 15/8 1088.27 0.10 -11.73 1100 2^(38/43) 1.845128 1060.47 -39.53 11/6 1049.36 11.10 49.36 1000 2^(37/43) 1.815624 1032.56 32.56 20/11 1035.00 -2.44 35.00 1000 2^(36/43) 1.786591 1004.65 4.65 ! 9/5 1017.60 -12.95 17.60 1000 2^(35/43) 1.758022 976.74 -23.26 16/9 996.09 8.56 7/4 968.83 7.92 (16/9) -3.91 1000 2^(34/43) 1.729911 948.84 48.84 (7/4) -31.17 1000 2^(33/43) 1.702249 920.93 20.93 12/7 933.13 -12.20 33.13 900 2^(32/43) 1.675029 893.02 -6.98 ! 27/16 905.87 -12.84 5.87 900 2^(31/43) 1.648244 865.12 -34.88 5/3 884.36 8.66 -15.64 900 2^(30/43) 1.621888 837.21 37.21 18/11 852.59 -15.38 -47.41 900 2^(29/43) 1.595953 809.30 9.30 8/5 813.69 -4.38 13.69 800 2^(28/43) 1.570433 781.40 -18.60 11/7 782.49 -1.10 -17.51 800 2^(27/43) 1.545321 753.49 -46.51 14/9 764.92 -11.43 -35.08 800 2^(26/43) 1.520611 725.58 25.58 32/21 729.22 -3.64 29.22 700 2^(25/43) 1.496296 697.67 -2.33 3/2 701.96 -4.28 1.96 700 2^(24/43) 1.472369 669.77 -30.23 40/27 680.45 -10.68 -19.55 700 2^(23/43) 1.448825 641.86 41.86 16/11 648.68 -6.82 48.68 600 2^(22/43) 1.425658 613.95 13.95 10/7 617.49 -3.53 17.49 600 2^(21/43) 1.402861 586.05 -13.95 7/5 582.51 3.53 -17.49 600 2^(20/43) 1.380429 558.14 -41.86 11/8 551.32 6.82 -48.68 600 2^(19/43) 1.358355 530.23 30.23 27/20 519.55 10.68 19.55 500 2^(18/43) 1.336634 502.33 2.33 4/3 498.04 4.28 -1.96 500 2^(17/43) 1.315261 474.42 -25.58 21/16 470.78 3.64 -29.22 500 2^(16/43) 1.294229 446.51 46.51 9/7 435.08 11.43 35.08 400 2^(15/43) 1.273534 418.60 18.60 14/11 417.51 1.10 17.51 400 2^(14/43) 1.253169 390.70 -9.30 5/4 386.31 4.38 -13.69 400 2^(13/43) 1.233131 362.79 -37.21 11/9 347.41 15.38 47.41 300 2^(12/43) 1.213412 334.88 34.88 6/5 315.64 -8.66 15.64 300 2^(11/43) 1.194009 306.98 6.98 ! 32/27 294.13 12.84 -5.87 300 2^(10/43) 1.174916 279.07 -20.93 7/6 266.87 12.20 -33.13 300 2^(9/43) 1.156129 251.16 -48.84 (8/7) 31.17 200 2^(8/43) 1.137642 223.26 23.26 9/8 203.91 -8.56 8/7 231.17 -7.92 (9/8) 3.91 200 2^(7/43) 1.119450 195.35 -4.65 ! 10/9 182.40 12.95 -17.60 200 2^(6/43) 1.101550 167.44 -32.56 11/10 165.00 2.44 -35.00 200 2^(5/43) 1.083936 139.53 39.53 12/11 150.64 -11.10 -49.36 200 2^(4/43) 1.066603 111.63 11.63 16/15 111.73 -0.10 11.73 100 2^(3/43) 1.049547 83.72 -16.28 21/20 84.47 -0.75 -15.53 100 2^(2/43) 1.032765 55.81 -44.19 33/32 53.27 2.54 -46.73 100 2^(1/43) 1.016250 27.91 27.91 81/80 21.51 6.40 21.51 0 2^(0/43) 1 00 1/100 00

Partch 43 Tone Scale Compared to 53ET 1/1 =G =392 * 2^n or 1/1 =C

53ET +/- 53ET +/- 53ET +/- 53ET +/- 53ET Ratio Cents from 12ET 5 Limit from Just 7 Limit from Just 11 Limit from Just Avg.-> 0.92 Avg.-> 5.24 Avg.-> 8.00 2^ 1 00 1/100