7edo

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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

Theory

		prime 2	prime 3	prime 5	prime 7	prime 11	prime 13
error	absolute (¢)	0	-16.24	-43.5	59.7	-37.0	16.6
	relative (%)	0	-9	-25	35	-22	10
nearest edomapping		7	4	2	6	3	5
fifthspan		0	+1	-3	-2	-1	+3

7-edo or "Neutral diatonic" divides the 1200-cent octave into 7 equal parts, making its smallest interval 171.428¢, or the seventh root of 2. It is the fourth prime edo, after 2edo, 3edo and 5edo. It is the third zeta integral edo.

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. 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).

7-tet can be thought of as result of stacking seven 11/9s on top of each other, and then tempering to remove the comma (2^-2 3^-14 11^7). As a temperament, William Lynch gives it the name "Neutron[7]" just as the whole tone scale of 12 ET is known as "Hexe[6]".

Typically, 7-edo 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 14 EDO or 21 EDO.

The seventh of 7-edo 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.

("African music." Encyclopædia Britannica. 2009. Encyclopædia Britannica Online. 05 Jul. 2009)

A Thai xylophone measured by Morton (1974) "varied only plus or minus 5 cents," from 7-TET. 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
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
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

7ed2-001.svg

Observations

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

There is a neutral feel between whole tone scale and major/minor diatonic scale. The second (171.429 c) 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 7 ET with sharp signs being possible to expand to 14 EDO or even 21 EDO.

Commas

7 EDO tempers out the following commas. (Note: This assumes val < 7 11 16 20 24 26 |.)

Ratio	Monzo	Cents	Color Name	Name 1	Name 2	Name 3
2187/2048	[-11 7⟩	113.69	Lawa	Apotome
135/128	[-7 3 1⟩	92.18	Layobi	Major Chroma	Major Limma	Pelagic Comma
25/24	[-3 -1 2⟩	70.67	Yoyo	Chromatic semitone	Chroma
250/243	[1 -5 3⟩	49.17	Triyo	Maximal Diesis	Porcupine Comma
20000/19683	[5 -9 4⟩	27.66	Saquadyo	Minimal Disease	Tricot Comma
81/80	[-4 4 -1⟩	21.51	Gu	Syntonic Comma	Didymos Comma	Meantone Comma
1600000/1594323	[9 -13 5⟩	6.15	Saquinyo	Amity Comma
36/35	[2 2 -1 -1⟩	48.77	Rugu	Septimal Quarter Tone
525/512	[-9 1 2 1⟩	43.41	Lazoyoyo	Avicenna	Avicenna's Enharmonic Diesis
64/63	[6 -2 0 -1⟩	27.26	Ru	Septimal Comma	Archytas' Comma	Leipziger Komma
875/864	[-5 -3 3 1⟩	21.90	Zotriyo	Keema
5120/5103	[10 -6 1 -1⟩	5.76	Saruyo	Hemi Family
6144/6125	[11 1 -3 -2⟩	5.36	Sarurutriyo	Cornwell
4375/4374	[-1 -7 4 1⟩	0.40	Zoquadyo	Ragisma
394839/394762	[47 -7 -7 -7⟩	0.34	Trisa-rugu	Akjaysma	5\7 Octave Comma
100/99	[2 -2 2 0 -1⟩	17.40	Luyoyo	Ptolemies
121/120	[-3 -1 -1 0 2⟩	14.37	Lologu	Batista
176/175	[4 0 -2 -1 1⟩	9.86	Lurugugu	Valinorsma
65536/65219	[16 0 0 -2 -3⟩	8.39	Satrilu-aruru	Organisma
243/242	[-1 5 0 0 -2⟩	7.14	Lulu	Rasta
385/384	[-7 -1 1 1 1⟩	4.50	Lozoyo	Keenanisma
4000/3993	[5 -1 3 0 -3⟩	3.03	Triluyo	Wizard Harry

Music

• Death Giving Monolith by Stephen Weigel (dulcimer and voice)
• Jingle Bells cover! (recorded by Stephen Weigel)
• Pagan's Revenge by Bill Sethares (synthetic gamelan)
• I dream of Tibet ^{[dead link]} by Aaron K. Johnson (electronic swirlies) ^{[dead link]}
• Seven Equal Trio by Robert Walker ((synth) violin, viola, glockenspiel)
• Two-part Invention in 7TET by Aaron Hunt
• Pavouci, Kelt by Milan Guštar
• 7edo Dance by Carlo Serafini
• Nightfire (video) by Carlo Serafini (blog entry)
• Comets Over Flatland 6 by Randy Winchester
• Sävelmä ^{[dead link]} by Sävelmä (long version) by Juhani Nuorvala
• Rock in 7edo by Santiago Cosentino ^{[dead link]}
• Zhaleyka by Dmitry Bazhenov

Ear Training

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