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

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.

Theory

		prime 2	prime 3	prime 5	prime 7	prime 11	prime 13
error	absolute (¢)	0.0	-16.2	-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

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. This assumes val ⟨7 11 16 20 24 26].

Prime Limit	Ratios	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	1600000/1594323	[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	…	[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

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.