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The 43-tone scale is a
just intonation In music, just intonation or pure intonation is a musical tuning, tuning system in which the space between notes' frequency, frequencies (called interval (music), intervals) is a natural number, whole number ratio, ratio. Intervals spaced in thi ...
scale with 43 pitches in each
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
. It is based on an eleven-limit tonality diamond, similar to the seven-limit diamond previously devised by Max Friedrich Meyer and refined by
Harry Partch Harry Partch (June 24, 1901 – September 3, 1974) was an American composer, music theorist, and creator of unique musical instruments. He composed using scales of unequal intervals in just intonation, and was one of the first 20th-century com ...
. The first of Partch's "four concepts" is "The scale of musical intervals begins with absolute
consonance In music, consonance and dissonance are categorizations of simultaneous or successive sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness, unple ...
( 1 to 1) and gradually progresses into an infinitude of dissonance, the consonance of the intervals decreasing as the odd numbers of their
ratios In mathematics, a ratio () shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to th ...
increase." Almost all of Partch's music is written in the 43-tone scale, and although most of his instruments can play only subsets of the full scale, he used it as an all-encompassing framework.


Construction

Partch chose the 11 limit (i.e. all rational numbers with odd factors of numerator and denominator not exceeding 11) as the basis of his music, because the 11th
harmonic In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the ''fundamental frequency'' of a periodic signal. The fundamental frequency is also called the ''1st har ...
is the first that is utterly foreign to Western ears. The seventh harmonic is poorly approximated by 12-tone
equal temperament An equal temperament is a musical temperament or Musical tuning#Tuning systems, tuning system that approximates Just intonation, just intervals by dividing an octave (or other interval) into steps such that the ratio of the frequency, frequencie ...
, but it appears in ancient Greek scales, is well-approximated by
meantone temperament Meantone temperaments are musical temperaments; that is, a variety of Musical tuning#Tuning systems, tuning systems constructed, similarly to Pythagorean tuning, as a sequence of equal fifths, both rising and descending, scaled to remain within th ...
, and it is familiar from the
barbershop quartet A barbershop quartet is a group of four singers who sing music in the barbershop style, characterized by four-part harmony without instrumental accompaniment (a cappella). The four voices are: the lead, the vocal part which typically carries t ...
; the ninth harmonic is comparatively well approximated by equal temperament and it exists in
Pythagorean tuning Pythagorean tuning is a system of musical tuning in which the frequency ratios of all intervals are determined by choosing a sequence of fifthsBruce Benward and Marilyn Nadine Saker (2003). ''Music: In Theory and Practice'', seventh editi ...
(because 3 × 3 = 9); but the 11th harmonic falls right in the middle between two pitches of 12-tone equal temperament (551.3 cents). Although theorists like Hindemith and Schoenberg have suggested that the 11th harmonic is implied by, e.g. F in the key of C, Partch's opinion is that it is simply too far out of tune, and "if the ear does not realize an implication, it does not exist."


Ratios of the 11 limit

Here are all the ratios within the
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
with odd factors up to and including 11, known as the 11-limit
tonality diamond In music theory and musical tuning, tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is called the Otonality and Utonality, otonality and the other is called the utonality.Rasch, Rudolph (2000). "A Word or ...
. Note that the inversion of every interval is also present, so the set is symmetric about the octave.


Filling in the gaps

There are two reasons why the 11-limit ratios by themselves would not make a good scale. First, the scale only contains a complete set of chords ( otonalities and utonalities) based on one tonic pitch. Second, it contains large gaps, between the tonic and the two pitches to either side, as well as several other places. Both problems can be solved by filling in the gaps with "multiple-number ratios", or intervals obtained from the product or quotient of other intervals within the 11 limit. Together with the 29 ratios of the 11 limit, these 14 multiple-number ratios make up the full 43-tone scale.
Erv Wilson Ervin Wilson (June 11, 1928 – December 8, 2016) was a Mexican/ American (dual citizen) music theorist. Early life Ervin Wilson was born iColonia Pacheco a small village in the remote mountains of northwest Chihuahua, Mexico, where he lived u ...
who worked with Partch has pointed out that these added tones form a constant structure of 41 tones with two variables.page 11 A constant structure giving one the property of anytime a ratio appears it will be subtended by the same number of steps. In this way Partch resolved his harmonic and melodic symmetry in one of the best ways possible.


Other Partch scales

The 43-tone scale was published in ''
Genesis of a Music ''Genesis of a Music'' is a book first published in 1949 by microtonal composer Harry Partch (1901–1974). Partch first presents a polemic against both equal temperament and the long history of stagnation in the teaching of music; according ...
'', and is sometimes known as the Genesis scale, or Partch's pure scale. Other scales he used or considered include a 29 tone scale for adapted viola from 1928; 29, 37, and 55 tone scales from an unpublished manuscript titled "Exposition of Monophony" from 1928; 33, a 39 tone scale proposed for a keyboard, and a 41 tone scale and an alternative 43 tone scale from "Exposition of Monophony". Besides the 11 limit diamond, he also published 5 and 13 limit diamonds, and in an unpublished manuscript worked out a 17 limit diamond. Erv Wilson who did the original drawings in Partch's ''Genesis of a Music'' has made a series of diagrams of Partch's diamond as well as others like Diamonds.


References

Sources * * {{Musical tuning Harry Partch 11-limit tuning and intervals