7edo

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7edo
Prime factorization 7 (prime)
Step size 171.429¢
Fifth 4\7 (686¢)
Major 2nd 1\7 (171¢)
Semitones (A1:m2) 0:1 (0¢ : 171¢)
Consistency limit 5

7 equal divisions of the octave (7edo) is the tuning system derived by dividing the octave into 7 equal steps of 171.4 cents each, or the seventh root of 2. It is the fourth prime edo, after 2edo, 3edo and 5edo. It is the third zeta integral edo.

Theory

Approximation of prime harmonics in 7edo
Harmonic 2 3 5 7 11 13 17 19 23 29
Error absolute (¢) +0.0 -16.2 -43.5 +59.7 -37.0 +16.6 +66.5 +45.3 +57.4 -1.0
relative (%) +0 -9 -25 +35 -22 +10 +39 +26 +34 -1
Steps
(reduced)
7
(0)
11
(4)
16
(2)
20
(6)
24
(3)
26
(5)
29
(1)
30
(2)
32
(4)
34
(6)
A chromatic 7edo scale on C.

Equal-heptatonic scales are used in non-western music in African cultures as well as an integral part of early Thai and early Chinese music[1]. Also Georgian music seems to be based on near-equal 7-step scales. It has been speculated in "Indian music:history and structure", that the Indian three-sruti interval of 165 cents is close enough to be mistaken for 171 cents. (or 1.71 semitones).

7edo can be thought of as the result of stacking seven 11/9's on top of each other, and then tempering to remove the comma [-2 -14 0 0 7. As a temperament, William Lynch gives it the name "Neutron[7]" just as the whole tone scale of 12edo is known as "Hexe[6]".

Typically, 7edo exists as the tuning for pentatonic scales in traditional Thai music with the other two pitches acting as auxiliary tones. However, it can be used as an interesting diatonic scale choice as well in tunings such as 14edo or 21edo.

The seventh of 7edo is almost exactly the 29th harmonic (29/16), which can have a very agreeable sound with harmonic timbres. However it also finds itself nested between ratios such as 20/11 and 9/5, which gives it considerably higher harmonic entropy than 7/4, a much simpler overtone seventh.

Similarly, in equi-heptatonic systems the desire for harmonic sound may dictate constant adjustments of intonation away from the theoretical interval of 171 cents. One of the most impressive areas in Africa in which a pen-equidistant heptatonic scale is combined with a distinctively harmonic style based on singing in intervals of thirds plus fifths, or thirds plus fourths, is the eastern Angolan culture area. This music is heptatonic and non-modal; i.e., there is no concept of major or minor thirds as distinctive intervals. In principle all the thirds are neutral, but in practice the thirds rendered by the singers often approximate natural major thirds (386 cents), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system. For the notation of such music, a seven-line stave is most appropriate, with each horizontal line representing one pitch level.

A Thai xylophone measured by Morton (1974) "varied only plus or minus 5 cents," from 7edo. A Ugandan Chopi xylophone measured by Haddon (1952) was also tuned to this system.

Intervals

7edo can be notated on a five-line staff without accidentals. There is no distinction between major or minor; each pitch class is unique.

Interval Cents interval name The "neighborhood" of just intervals Audio
0 0.000 unison / prime exactly 1/1
1 171.429 second 6.424¢ from Ptolemy (neutral) second 11/10
3.215¢ from second 54/49
-1.006¢ from the 29th subharmonic 32/29
-10.975¢ from major second (small whole tone) 10/9
2 342.857 third 0.374¢ from neutral third 39/32
-4.55¢ from neutral third 11/9
3 514.286 fourth 16.241¢ from just fourth 4/3 (498.045¢)
-5.265¢ from wide fourth 27/20
4 685.714 fifth 5.265 ¢ from narrow fifth 40/27
-16.241¢ from just fifth 3/2 (701.955¢)
5 857.143 sixth 4.551¢ from neutral sixth 18/11
-0.374¢ from neutral sixth 64/39
-0.048¢ from (neutral sixth) 105/64
6 1028.571 seventh 10.975¢ from (Didymus) minor seventh 9/5
-6.424¢ from neutral seventh 20/11
-1.006¢ from the 29th harmonic 29/16
-3.215¢ from seventh 49/27
7 1200 octave exactly 2/1

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7ed2-001.svg

Observations

Related in a lateral way to traditional Thai music. Subset of 14edo and 21edo.

There is a neutral feel between whole tone scale and major/minor diatonic scale. The second (171.429¢) works well as a basic step for melodic progression.

The step from seventh to octave is too large for the leading tone.

Notation

William Lynch proposes using numbers 1 through 7 as the nominals of 7edo with sharp signs being possible to expand to 14edo or even 21edo.

Commas

7edo tempers out the following commas. This assumes val 7 11 16 20 24 26].

Prime
Limit
Ratio[1] Monzo Cents Color Name Name(s)
3 2187/2048 [-11 7 113.69 Lawa Apotome, Pythagorean chromatic semitone
5 135/128 [-7 3 1 92.18 Layobi Major chroma, major limma, pelogic comma
5 25/24 [-3 -1 2 70.67 Yoyo Classic chromatic semitone, dicot comma
5 250/243 [1 -5 3 49.17 Triyo Maximal diesis, porcupine comma
5 20000/19683 [5 -9 4 27.66 Saquadyo Minimal diesis, tetracot comma
5 81/80 [-4 4 -1 21.51 Gu Syntonic comma, Didymus comma, meantone comma
5 (14 digits) [9 -13 5 6.15 Saquinyo Amity comma
7 36/35 [2 2 -1 -1 48.77 Rugu Septimal quartertone
7

525/512

[-9 1 2 1 43.41 Lazoyoyo Avicennma, Avicenna's enharmonic diesis
7 64/63 [6 -2 0 -1 27.26 Ru Septimal comma, Archytas' comma, Leipziger Komma
7 875/864 [-5 -3 3 1 21.90 Zotriyo Keema
7 5120/5103 [10 -6 1 -1 5.76 Saruyo Hemifamity
7 6144/6125 [11 1 -3 -2 5.36 Sarurutriyo Porwell
7 4375/4374 [-1 -7 4 1 0.40 Zoquadyo Ragisma
7 (30 digits) [47 -7 -7 -7 0.34 Trisa-rugu Akjaysma, 5\7 octave comma
11 100/99 [2 -2 2 0 -1 17.40 Luyoyo Ptolemisma
11

121/120

[-3 -1 -1 0 2 14.37 Lologu Biyatisma
11 176/175 [4 0 -2 -1 1 9.86 Lurugugu Valinorsma
11 65536/65219 [16 0 0 -2 -3 8.39 Satrilu-aruru Orgonisma
11 243/242 [-1 5 0 0 -2 7.14 Lulu Rastma
11 385/384 [-7 -1 1 1 1 4.50 Lozoyo Keenanisma
11 4000/3993 [5 -1 3 0 -3 3.03 Triluyo Wizardharry
13 27/26 [-1 3 0 0 0 -1 65.33 thu unison small tridecimal third tone
13 65/64 [-6 0 1 0 0 1 26.84 wilsorma
13 52/49 [2 0 0 -2 0 1 102.87 thoruru unison hammerisma
13 14641/13312 [-10 0 0 0 4 -1 164.74
  1. Ratios longer than 10 digits are presented by placeholders with informative hints

Temperaments

7edo is the first edo in which regular temperament theory starts to make sense as a way of subdividing the steps into mos scales, with three different ways of dividing it, although there is still quite a lot of ambiguity as each step can be considered as the sharp extreme of one temperament or the flat end of another. 1/7 can be considered the intersection of sharp porcupine and flat tetracot temperaments, as three steps makes a 4th and four a 5th. 2/7 can be interpreted as critically flat Mohajira or critically sharp amity, and creates mosses of 322 and 2221. 3/7 is on the intersection of meantone and mavila, and has MOS's of 331 and 21211, making 7edo the first edo with a non-equalized, non-1Lns pentatonic mos. This is in part because 7edo is a strict zeta edo (close to low-complexity JI for its size), and is the second edo with a good fifth for its size (after 5edo), the fifth serving as a generator for the edo's meantone and mavila interpertations.

Music

Ear training

7edo ear-training exercises by Alex Ness available here.