7edo

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Prime factorization 7 (prime)
Step size 171.429¢ 
Fifth 4\7 (685.714¢)
(semiconvergent)
Semitones (A1:m2) 0:1 (0¢ : 171.4¢)
Consistency limit 5
Distinct consistency limit 3
Special properties

7 equal divisions of the octave (abbreviated 7edo or 7ed2), also called 7-tone equal temperament (7tet) or 7 equal temperament (7et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 7 equal parts of about 171 ¢ each. Each step represents a frequency ratio of 21/7, or the 7th root of 2.

Theory

A chromatic 7edo scale on C.

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]".

Prime harmonics

Approximation of prime harmonics in 7edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -16.2 -43.5 +59.7 -37.0 +16.6 +66.5 +45.3 +57.4 -1.0 +55.0
Relative (%) +0.0 -9.5 -25.3 +34.9 -21.6 +9.7 +38.8 +26.5 +33.5 -0.6 +32.1
Steps
(reduced)
7
(0)
11
(4)
16
(2)
20
(6)
24
(3)
26
(5)
29
(1)
30
(2)
32
(4)
34
(6)
35
(0)

As an equalized diatonic scale

7edo unifies the seven diatonic scales - Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian - into a single one: 7edo can be used as an interesting diatonic scale choice as well in tunings such as 14edo or 21edo.

There is a neutral feel somewhere between a 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 as a leading tone. Possibly lending itself to a “sevenplus” scale similar to elevenplus.

In non-Western traditions

Equiheptatonic scales close to 7edo are used in non-western music in some African cultures[1] as well as an integral part of early Chinese music[2]. Also Georgian music seems to be based on near-equal 7-step scales.

It has been speculated in Indian music: history and structure[3] that the Indian three-sruti interval of 165 cents is close enough to be mistaken for 171 cents. (or 1.71 semitones), one step of 7edo.

In equiheptatonic systems the desire for harmonic sound may dictate constant adjustments of intonation away from the theoretical interval of 171 cents. (Similar to adaptive just intonation but with equal tuning instead).

One region of 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 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 (5/4, 386 cents), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system.

A Ugandan, Chopi xylophone measured by Haddon (1952) was also tuned something close to this.

It has often been stated that 7edo approximates tunings used in Thai classical music, though this is a myth unsupported by empirical studies of the instruments.[4]

Observations

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.

7edo is the unique intersection of the temperaments of meantone (specifically 3/4-comma meantone) and porcupine.

Subsets and supersets

7edo is the 4th prime edo, after 5edo and before 11edo. Multiples such as 14edo, 21edo, … up to 35edo, share the same tuning of the perfect fifth as 7edo, while improving on other intervals.

Intervals

Intervals of 7edo
Degree Cents Interval region Approximated JI intervals (error in ¢) Audio
3-limit 5-limit 7-limit Other
0 0 Unison (prime) 1/1 (just)
1 171.429 Submajor second 10/9 (-10.975) 54/49 (+3.215) 11/10 (+6.424)
32/29 (-1.006)
2 342.857 Neutral third 128/105 (+0.048) 39/32 (+0.374)
11/9 (-4.551)
3 514.286 Fourth 4/3 (+16.241) 27/20 (-5.265) 35/26 (-0.326)
4 685.714 Fifth 3/2 (-16.241) 40/27 (+5.265) 52/35 (+0.326)
5 857.143 Neutral sixth 105/64 (-0.048) 18/11 (+4.551)
64/39 (-0.374)
6 1028.571 Supraminor seventh 9/5 (+10.975) 49/27 (-3.215) 29/16 (-1.006)
20/11 (-6.424)
7 1200 Octave 2/1 (just)

Notation

The usual notation system for 7edo is the chain-of-fifths notation, which is directly derived from the standard notation used in 12edo.

Because the Pythagorean apotome (2187/2048) is tempered out, sharps (♯) and flats (♭) are redundant in 7edo. Therefore, 7edo can be notated on a five-line staff without accidentals. Alternatively, a seven-line stave can be used, with each horizontal line representing one pitch level. There is no distinction between major or minor, so every interval has the interval quality "perfect" instead.

Notation of 7edo
Degree Cents Chain-of-fifths notation
Diatonic interval names Note names (on D)
0 0 Perfect unison (P1) D
1 171.429 Perfect second (P2) E
2 342.857 Perfect third (P3) F
3 514.286 Perfect fourth (P4) G
4 685.714 Perfect fifth (P5) A
5 857.143 Perfect sixth (P6) B
6 1028.571 Perfect seventh (P7) C
7 1200 Perfect octave (P8) D

In 7edo:

Alternative notations

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.

Solfege

Solfege of 7edo
Degree Cents Standard solfege
(movable do)
Uniform solfege
(1 vowel)
0 0 Do Da
1 171.429 Re Ra
2 342.857 Mi Ma
3 514.286 Fa Fa
4 685.714 So Sa
5 857.143 La La
6 1028.571 Ti Ta
7 1200 Do Da

Approximation to JI

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7ed2-001.svg

Regular temperament properties

Uniform maps

13-limit uniform maps between 6.5 and 7.5
Min. size Max. size Wart notation Map
6.5000 6.5040 7bbccdddeeeefff 7 10 15 18 22 24]
6.5040 6.5898 7bbccdddeefff 7 10 15 18 23 24]
6.5898 6.6208 7bbccdeefff 7 10 15 19 23 24]
6.6208 6.6248 7bbccdeef 7 10 15 19 23 25]
6.6248 6.6755 7ccdeef 7 11 15 19 23 25]
6.6755 6.7930 7deef 7 11 16 19 23 25]
6.7930 6.8911 7df 7 11 16 19 24 25]
6.8911 6.9460 7d 7 11 16 19 24 26]
6.9460 7.0821 7 7 11 16 20 24 26]
7.0821 7.1062 7e 7 11 16 20 25 26]
7.1062 7.1613 7ce 7 11 17 20 25 26]
7.1613 7.2557 7ceff 7 11 17 20 25 27]
7.2557 7.3022 7bceff 7 12 17 20 25 27]
7.3022 7.3712 7bcddeff 7 12 17 21 25 27]
7.3712 7.4315 7bcddeeeff 7 12 17 21 26 27]
7.4315 7.5000 7bcddeeeffff 7 12 17 21 26 28]

Commas

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

Prime
limit
Ratio[note 1] Monzo Cents Color name Name(s)
3 2187/2048 [-11 7 113.69 Lawa Whitewood comma, apotome, Pythagorean chroma
5 135/128 [-7 3 1 92.18 Layobi Mavila comma, major chroma
5 25/24 [-3 -1 2 70.67 Yoyo Dicot comma, classic chroma
5 250/243 [1 -5 3 49.17 Triyo Porcupine comma, maximal diesis
5 20000/19683 [5 -9 4 27.66 Saquadyo Tetracot comma, minimal diesis
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 (14 digits) [9 7 -5 -3 75.64 Triru-aquingu Superpine comma
7 36/35 [2 2 -1 -1 48.77 Rugu Mint comma, 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 comma
7 6144/6125 [11 1 -3 -2 5.36 Sarurutriyo Porwell comma
7 4375/4374 [-1 -7 4 1 0.40 Zoquadyo Ragisma
7 (30 digits) [47 -7 -7 -7 0.34 Trisa-rugu Akjaysma
11 33/32 [-5 1 0 0 1 53.27 Ilo Io comma, undecimal quartertone
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 comma
13 27/26 [-1 3 0 0 0 -1 65.33 Thu 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

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

See also: Category:7edo tracks

Ear training

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

Notes

  1. Ratios longer than 10 digits are presented by placeholders with informative hints

References