! "#$!%&'()#! *+!",-$!.)&/$! ! ! ! 01#&-!%1#2 !!! "#$!%&'()#!*+!",-$!.)&/$"#$!%&'()#!*+!",-$!.)&/$!!!! One can only imagine the years Harry Partch spent formulating his 43 tone scale and then the lifetime spent realizing it. A pinnacle of Just Intonation, a fair bit of time can be spent unraveling the intricacies of the 43 tone scale and coming to terms with the choices Partch made. What are some of the features of this scale? For starters the scale is a pure Just Intonation scale, no tempering of any intervals whatsoever. All pitches are given exact ratios and so all intervals are exact as well. For the most part all the music we listen to is constructed of up to 5 limit ratios. Partch expanded our common tonality to include 7 and 11 limit ratios. It can be argued that 2, 3 and 5 limit ratios are the language of harmony and any higher limit ratios are not harmonic in the least bit, regardless of what people would like to pretend. Possibly some 7 limit ratios can be said to push the bounds of what can be considered harmonic. Since our current 12 note scale and instruments do not even remotely support or approximate any 7 or 11 limit ratios, 7 and 11 limit ratios are completely outside the realm of our listening and composing experience. The equal tempered tritone and minor seventh intervals do not approximate 7 limit ratios. It would be incorrect to think of the minor seventh approximating the 7 limit ratio of 7/4, which is almost one-third of a semitone lower (969 cents). Our tempered minor seventh (1000 cents) is firmly in the realm of two perfect fourths (4/3 * 4/3 = 16/9, (996 cents)) or further removed, a minor third above the dominant (3/2 * 6/5 = 9/5 (1018 cents)). There is no 7 limit 7/4 ratio knocking on our door with our common scales. As well the 10/7 and 7/5 ratios of 617 and 583 cents are well above and below the square root of 2 tempered ratio of 600 cents. The 5 limit ratios of 45/32 and 64/45 of 590 and 610 cents are almost half as close still to the pure tritone. So once again, none of our current instruments come even close to supporting 7 or 11 limit ratios, most of which are a third or half of a semitone above and below equal temperament. The 2 limit ratios of the octave synthesize (blend) with the 3 limit ratios of the perfect fourth and fifth. All things considered, the 2 and 3 limit ratios synthesize miraculously with the 5 limit ratios of the thirds and sixths. The ear finds all these relationships consonant and harmonic. Does the 7 limit ratio of the 7/4 ratio harmonically synthesize or blend with the 2, 3 and 5 limit ratios? I would say it does even though it clips very strongly. Here then is Partch’s first expansion from our up to 5 limit music with the 4:5:6:7 chord which is made up of the major triad and the harmonic seventh ratio of 7/4. This is then expanded with the 3 limit ratio of 9/4 (9/8), which is part of our listening experience, into the 4:5:6:7:9 chord. One major step further comes with the 11 limit ratio of 11/4 (11/8) forming finally the 4:5:6:7:9:11 chord. As said before, there is no experience of the 7 or 11 ratios in our common musical system. Can we deem the 11 ratio to be harmonic and consonant in the context of the chord? Does 11 synthesize with 4, 5, 6, 7 and 9. Certainly on it’s own in our common musical system it is dissonant being almost a quarter-tone between a perfect fourth and the tritone. Experience is required to determine in what context the 11 limit ratio of the 4:5:6:7:9:11 chord may be deemed harmonic and consonant. Of the 43 tones of Partch’s scale, twelve are 7 limit ratios, twelve are 11 limit ratios, leaving nineteen 2, 3 and 5 limit ratios. Partch as well speaks of the “28 Tonalities” though not all tonalities are complete. A complete tonality is considered to consist of the complete 4:5:6:7:9:11 chord. Ten complete and ten incomplete tonalities have the root of the chord in the 2, 3 and 5 limit ratios. Two complete tonalities and four incomplete tonalities have the root of the chord in the 7 limit ratios. Two complete tonalities have the root of the chord in the 11 limit ratios. What we will also see is that the tonalities are related to one another by exactly the same intervals that make up the tonalities themselves, hence what Partch calls the “Tonality Diamond”. There is one complete “Tonality Diamond” and at least two more incomplete 2 Partch 43 Tone Scale © Copyright Juhan Puhm 2016 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.