Harry Partch's 43-tone scale
Encyclopedia
The 43-tone scale is a just intonation
scale with 43 pitches in each octave
, invented and used by Harry Partch
.
The first of Partch's "four concepts" is "The scale of musical intervals
begins with absolute consonance
(1 to 1
) and gradually progresses into an infinitude of dissonance
, the consonance of the intervals decreasing as the odd numbers of their ratios
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.
(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
is the first that is utterly foreign to Western ears. The seventh harmonic is poorly approximated by 12-tone equal temperament
, but it appears in ancient Greek scales, is well-approximated by meantone temperament
, and it is familiar from the barbershop quartet
; the ninth harmonic is comparatively well approximated by equal temperament and it exists in Pythagorean tuning
(because 3 × 3 = 9); but the 11th harmonic falls right in the middle between two pitches of 12-tone equal temperament. Although theorists like Hindemith
and Schoenberg
have suggested that the 11th harmonic is implied by, e.g. F# in the key of C, the fact is that it is simply too far out of tune, and "if the ear does not realize an implication, it does
not exist."
When asked why he did not include 13, Partch replied, "When a hungry man has a large table of aromatic and unusual viands spread before him, he is unlikely to go tramping along the seashore and in the woods for still other exotic fare." In other words, two completely foreign harmonic elements would simply be too much, and Partch was satisfied with 11.
with odd factors up to and including 11, known as the 11-limit tonality diamond
. Note that the inversion
of every interval is also present, so the set is symmetric about the octave.
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
who worked with Partch has pointed out that these added tones form a constant structure of 41 tones with two variables. A constant structure giving one the property of anytime a ratio appears it will be subtended by the same about of steps. In this way Partch resolved his harmonic and melodic symmetry in one of the best ways possible.
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
scale with 43 pitches in each octave
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
, invented and used by Harry Partch
Harry Partch
Harry Partch was an American composer and instrument creator. He was one of the first twentieth-century composers to work extensively and systematically with microtonal scales, writing much of his music for custom-made instruments that he built himself, tuned in 11-limit just intonation.-Early...
.
The first of Partch's "four concepts" is "The scale of musical intervals
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
begins with absolute consonance
Consonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...
(1 to 1
Unison
In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance...
) and gradually progresses into an infinitude of dissonance
Consonance and dissonance
In music, a consonance is a harmony, chord, or interval considered stable, as opposed to a dissonance , which is considered to be unstable...
, the consonance of the intervals decreasing as the odd numbers of their ratios
Just intonation
In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series...
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 limitLimit (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
(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
Harmonic series (music)
Pitched musical instruments are often based on an approximate harmonic oscillator such as a string or a column of air, which oscillates at numerous frequencies simultaneously. At these resonant frequencies, waves travel in both directions along the string or air column, reinforcing and canceling...
is the first that is utterly foreign to Western ears. The seventh harmonic is poorly approximated by 12-tone equal temperament
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...
, but it appears in ancient Greek scales, is well-approximated by meantone temperament
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
, and it is familiar from the barbershop quartet
Barbershop music
Barbershop vocal harmony, as codified during the barbershop revival era , is a style of a cappella, or unaccompanied vocal music characterized by consonant four-part chords for every melody note in a predominantly homophonic texture...
; the ninth harmonic is comparatively well approximated by equal temperament and it exists in Pythagorean tuning
Pythagorean tuning
Pythagorean tuning is a system of musical tuning in which the frequency relationships of all intervals are based on the ratio 3:2. This interval is chosen because it is one of the most consonant...
(because 3 × 3 = 9); but the 11th harmonic falls right in the middle between two pitches of 12-tone equal temperament. Although theorists like Hindemith
Paul Hindemith
Paul Hindemith was a German composer, violist, violinist, teacher, music theorist and conductor.- Biography :Born in Hanau, near Frankfurt, Hindemith was taught the violin as a child...
and Schoenberg
Arnold Schoenberg
Arnold Schoenberg was an Austrian composer, associated with the expressionist movement in German poetry and art, and leader of the Second Viennese School...
have suggested that the 11th harmonic is implied by, e.g. F# in the key of C, the fact is that it is simply too far out of tune, and "if the ear does not realize an implication, it does
not exist."
When asked why he did not include 13, Partch replied, "When a hungry man has a large table of aromatic and unusual viands spread before him, he is unlikely to go tramping along the seashore and in the woods for still other exotic fare." In other words, two completely foreign harmonic elements would simply be too much, and Partch was satisfied with 11.
Ratios of the 11 Limit
Here are all the ratios within the octaveOctave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
with odd factors up to and including 11, known as the 11-limit tonality diamond
Tonality diamond
In music theory and tuning, a tonality diamond is a two-dimensional diagram of ratios in which one dimension is the Otonality and one the Utonality...
. Note that the inversion
Inversion (music)
In music theory, the word inversion has several meanings. There are inverted chords, inverted melodies, inverted intervals, and inverted voices...
of every interval is also present, so the set is symmetric about the octave.
| Cents Cent (music) The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each... |
0 | 150.6 | 165.0 | 182.4 | 203.9 | 231.2 | 266.9 | 315.6 | 347.4 | 386.3 | 417.5 | 435.1 | 498.0 | 551.3 | 582.5 |
| Ratio Just intonation In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series... |
1/1 Unison In music, the word unison can be applied in more than one way. In general terms, it may refer to two notes sounding the same pitch, often but not always at the same time; or to the same musical voice being sounded by several voices or instruments together, either at the same pitch or at a distance... |
12/11 Neutral second A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:... |
11/10 Neutral second A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:... |
10/9 Major second In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions... |
9/8 Major second In Western music theory, a major second is a musical interval spanning two semitones, and encompassing two adjacent staff positions . For example, the interval from C to D is a major second, as the note D lies two semitones above C, and the two notes are notated on adjacent staff postions... |
8/7 Diminished third In classical music from Western culture, a diminished third is the musical interval produced by narrowing a minor third by a chromatic semitone. For instance, the interval from A to C is a minor third, three semitones wide, and both the intervals from A to C, and from A to C are diminished thirds,... |
7/6 Augmented second In classical music from Western culture, an augmented second is an interval produced by widening a major second by a chromatic semitone. For instance, the interval from C to D is a major second, two semitones wide, and both the intervals from C to D, and from C to D are augmented seconds, spanning... |
6/5 Minor third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major... |
11/9 | 5/4 Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three... |
14/11 | 9/7 Septimal major third In music, the septimal major third , also called the supermajor third and sometimes Bohlen–Pierce third is the musical interval exactly or approximately equal to a just 9:7 ratio of frequencies, or alternately 14:11. It is equal to 435 cents, sharper than a just major third by the septimal... |
4/3 Perfect fourth In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there... |
11/8 Tritone In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones... |
7/5 Septimal tritone The lesser septimal tritone is the interval with ratio 7:5 . The inverse of that interval, the greater septimal tritone, is an interval with ratio 10:7... |
| Cents Cent (music) The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each... |
617.5 | 648.7 | 702.0 | 764.9 | 782.5 | 813.7 | 852.6 | 884.4 | 933.1 | 968.8 | 996.1 | 1017.6 | 1035.0 | 1049.4 | 1200 |
| Ratio Just intonation In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series... |
10/7 Septimal tritone The lesser septimal tritone is the interval with ratio 7:5 . The inverse of that interval, the greater septimal tritone, is an interval with ratio 10:7... |
16/11 | 3/2 Perfect fifth In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones... |
14/9 | 11/7 Minor sixth -Subminor sixth:In music, a subminor sixth or septimal sixth is an interval that is noticeably narrower than a minor sixth but noticeably wider than a diminished sixth.The sub-minor sixth is an interval of a 14:9 ratio or alternately 11:7.... |
8/5 Minor sixth -Subminor sixth:In music, a subminor sixth or septimal sixth is an interval that is noticeably narrower than a minor sixth but noticeably wider than a diminished sixth.The sub-minor sixth is an interval of a 14:9 ratio or alternately 11:7.... |
18/11 Neutral sixth A neutral sixth is a musical interval wider than a minor sixth but narrower than a major sixth . Three distinct intervals may be termed neutral sixths:... |
5/3 Major sixth In classical music from Western culture, a sixth is a musical interval encompassing six staff positions , and the major sixth is one of two commonly occurring sixths. It is qualified as major because it is the largest of the two... |
12/7 Supermajor sixth In music and tuning, a supermajor sixth is a supermajor interval that is noticeably wider than a major sixth but noticeably narrower than an augmented sixth, and may be a just interval of 12:7.... |
7/4 Augmented sixth In classical music from Western culture, an augmented sixth is an interval produced by widening a major sixth by a chromatic semitone. For instance, the interval from C to A is a major sixth, nine semitones wide, and both the intervals from C to A, and from C to A are augmented sixths, spanning... |
16/9 Minor seventh In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the minor seventh is one of two commonly occurring sevenths. The minor quality specification identifies it as being the smallest of the two: the minor seventh spans ten semitones, the... |
9/5 Minor seventh In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the minor seventh is one of two commonly occurring sevenths. The minor quality specification identifies it as being the smallest of the two: the minor seventh spans ten semitones, the... |
20/11 Neutral seventh A neutral seventh is a musical interval wider than a minor seventh but narrower than a major seventh . Four distinct intervals may be termed neutral sevenths:* A septimal neutral seventh has a ratio of 64:35 or about 1045 cents.... |
11/6 Neutral seventh A neutral seventh is a musical interval wider than a minor seventh but narrower than a major seventh . Four distinct intervals may be termed neutral sevenths:* A septimal neutral seventh has a ratio of 64:35 or about 1045 cents.... |
2/1 Octave In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"... |
Filling in the gaps
There are two reasons why the 11-limit ratios by themselves would not make a good scale. First, the scale contains only complete chords (otonalities and utonalities) based on one tonicTonic (music)
In music, the tonic is the first scale degree of the diatonic scale and the tonal center or final resolution tone. The triad formed on the tonic note, the tonic chord, is thus the most significant chord...
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.
| Cents Cent (music) The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each... |
0 | 21.5 | 53.2 | 84.5 | 111.7 | 150.6 |
| Ratio Just intonation In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series... |
1/1 | 81/80 Syntonic comma In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents... |
33/32 | 21/20 Septimal chromatic semitone In music, a septimal chromatic semitone or minor semitone is the interval 21:20 . It is about 84.47 cents. The septimal chromatic semitone may be derived from the harmonic series as the interval between the twentieth and twenty-first harmonics.... |
16/15 Semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically.... |
12/11 |
| Cents Cent (music) The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each... |
266.9 | 294.1 | 315.6 |
| Ratio Just intonation In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series... |
7/6 | 32/27 | 6/5 |
| Cents Cent (music) The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each... |
435.1 | 470.8 | 498.0 | 519.5 | 551.3 |
| Ratio Just intonation In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series... |
9/7 | 21/16 | 4/3 | 27/20 Wolf interval In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament... |
11/8 |
| Cents Cent (music) The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each... |
648.7 | 680.5 | 702.0 | 729.2 | 764.9 |
| Ratio Just intonation In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series... |
16/11 | 40/27 Wolf interval In music theory, the wolf fifth is a particularly dissonant musical interval spanning seven semitones. Strictly, the term refers to an interval produced by a specific tuning system, widely used in the sixteenth and seventeenth centuries: the quarter-comma meantone temperament... |
3/2 | 32/21 | 14/9 |
| Cents Cent (music) The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each... |
884.4 | 905.9 | 933.1 |
| Ratio Just intonation In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series... |
5/3 | 27/16 | 12/7 |
| Cents Cent (music) The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each... |
1049.4 | 1088.3 | 1115.5 | 1146.8 | 1178.5 | 1200 |
| Ratio Just intonation In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series... |
11/6 | 15/8 Major seventh In classical music from Western culture, a seventh is a musical interval encompassing seven staff positions , and the major seventh is one of two commonly occurring sevenths. It is qualified as major because it is the larger of the two... |
40/21 | 64/33 | 160/81 | 2/1 |
Together with the 29 ratios of the 11 limit, these 14 multiple-number ratios make up the full 43-tone scale.
Erv Wilson
Erv Wilson
Ervin Wilson is a Mexican/American music theorist. Despite his avoidance of academia, Wilson has been influential on those interested in microtonal music and just intonation, especially in the areas of scale, keyboard, and notation design...
who worked with Partch has pointed out that these added tones form a constant structure of 41 tones with two variables. A constant structure giving one the property of anytime a ratio appears it will be subtended by the same about of steps. In this way Partch resolved his harmonic and melodic symmetry in one of the best ways possible.