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{{Infobox ET}}
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{{EDO intro}}
{{ED intro}}


== Theory ==
== Theory ==
[[File:7edo scale.mp3|thumb|A chromatic 7edo scale on C.]]
[[File:7edo scale.mp3|thumb|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 {{monzo| -2 -14 0 0 7 }}. As a temperament, [[William Lynch]] gives it the name "[[Neutron|Neutron[7]]]" just as the whole tone scale of [[12edo]] is known as "[[Hexe|Hexe[6]]]".


=== Prime harmonics ===
7edo is the basic example of an [[equiheptatonic]] scale, and in terms of tunings with perfect fifths, is essentially the next size up from [[5edo]]. The 7-form is notable as a common structure for many [[5-limit]] systems, including all seven modes of the [[5L 2s|diatonic]] scale—Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian; in 7edo itself, the two sizes of interval in any heptatonic MOS scale are equated, resulting in a [[interval quality|neutral]] feel. All triads are neutral (except if you use suspended triads, which are particularly harsh in 7edo due to the narrowed major second), so functional harmony is almost entirely based on the positions of the chords in the 7edo scale.
{{Harmonics in equal|7}}


=== As an equalized 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 as a leading tone - possibly lending itself to a "sevenplus" scale similar to [[elevenplus]].
7edo unifies the seven [[5L 2s|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 [[interval quality|neutral]] feel somewhere between a [[6edo|whole tone scale]] and major/minor diatonic scale. The second (171.429¢) works well as a basic step for melodic progression.
In terms of just intonation, the 3/2 is flat but usable, but we don't find particularly accurate intervals in pure harmonics outside the 3-limit, which suggests a more melodic approach to just intonation; intervals approximated by each of 7edo's steps include 10/9 for 1 step, 11/9 for 2 steps, 4/3 for 3 steps, and their octave complements. Interestingly, this renders an 8:9:10:11:12 pentad equidistant, from which it can be derived that 7edo supports [[meantone]] (equating the major seconds 10/9 and 9/8) and [[porcupine]] (splitting 4/3 into three equal submajor seconds which simultaneously represent 12/11, 11/10, and 10/9), and is the unique system to do so.  


The step from seventh to octave is too large as a leading tone. Possibly lending itself to a “sevenplus” scale similar to [[elevenplus]].
Due to 7edo's inaccurately tuned [[5/4]] [[major third]] (which is flat by over 40 cents), it supports several exotemperaments in the 5-limit, such as [[dicot]] (which splits the fifth into two equal [[neutral third]]<nowiki/>s, simultaneously representing 5/4 and the [[minor third]] [[6/5]]) and [[mavila]] (which flattens the fifth so that the diatonic "major third" actually approximates 6/5); 6/5 is a slightly more reasonable interpretation of 7edo's third than 5/4, leading to an overall slightly [[minor]] sound.


=== In non-Western traditions ===
In higher limits, this third takes on a new role: as a neutral third, it is a decent approximation of the 13th subharmonic, and as such 7edo can be seen as a 2.3.13 temperament. This third is a near perfect approximation of the interval [[39/32]]; the equation of 16/13 and 39/32 is called [[512/507|harmoneutral]] temperament. In general, the inclusion of 13 allows the pentad discussed earlier to be continued to an 8:9:10:11:12:13 hexad, although the specific interval 13/12 is inaccurate due to the errors adding up in the same direction.  
[[Equiheptatonic]] scales close to 7edo are used in non-western music in some [[African]] cultures<ref>[https://www.britannica.com/art/African-music ''African music'', Encyclopedia Britannica.]</ref> as well as an integral part of early [[Chinese]] music<ref>Robotham, Donald Keith and Gerhard Kubik.</ref>. Also [[Georgian]] music seems to be based on near-equal 7-step scales.  


It has been speculated in ''Indian music: history and structure''<ref>Nambiyathiri, Tarjani. ''[https://archive.org/details/indianmusichistoryandstructureemmietenijenhuisbrill Indian Music History And Structure Emmie Te Nijenhuis Brill]''</ref> 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.
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.  


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).  
7edo represents a 7-step closed [[circle of fifths]], tempering out the Pythagorean chromatic semitone. However, it can also be seen as a circle of neutral thirds, which can be interpreted as 11/9; this is called [[neutron]] temperament.


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.
=== Prime harmonics ===
{{Harmonics in equal|7}}


A [[Ugandan]], [[Chopi]] xylophone measured by Haddon (1952) was also tuned something close to this.
=== In non-Western traditions ===
[[Equiheptatonic]] scales close to 7edo are used in non-western music in some [[African]] cultures<ref>[https://www.britannica.com/art/African-music ''African music'', Encyclopedia Britannica.]</ref> as well as an integral part of early [[Chinese]] music<ref>Robotham, Donald Keith and Gerhard Kubik.</ref>. Also [[Georgian]] music seems to be based on near-equal 7-step scales.  


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.<ref>Garzoli, John. [http://iftawm.org/journal/oldsite/articles/2015b/Garzoli_AAWM_Vol_4_2.pdf ''The Myth of Equidistance in Thai Tuning.'']</ref>
It has been speculated in ''Indian music: history and structure''<ref>Nambiyathiri, Tarjani. ''[https://archive.org/details/indianmusichistoryandstructureemmietenijenhuisbrill Indian Music History And Structure Emmie Te Nijenhuis Brill]''</ref> that the [[Indian]] three-sruti interval of 165 cents is very similar to one 171-cent step of 7edo.


=== Observations ===
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).
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]].
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{{c}}), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system.


The [[Octave stretch|stretched-octaves]] tuning [[Ed258/128#7ed257/128|7ed257/128]] greatly improves 7edo’s approximation of 3/1, 5/1 and 11/1, at the cost of slightly worsening 2/1 and 7/1, and greatly worsening 13/1. If one is hoping to use 7edo for [[11-limit]] harmonies, then 7ed257/128 is a good choice to make that easier.
A [[Ugandan]], [[Chopi]] xylophone measured by Haddon (1952) was also tuned something close to this.


The stretched 7edo tuning [[zpi|15zpi]] can also be used to improve 7edo’s approximation of JI in the same way.
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.<ref>Garzoli, John. [http://iftawm.org/journal/oldsite/articles/2015b/Garzoli_AAWM_Vol_4_2.pdf ''The Myth of Equidistance in Thai Tuning.'']</ref>


=== Subsets and supersets ===
=== 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.
7edo is the 4th [[prime edo]], after [[5edo]] and before [[11edo]]. It does not contain any nontrivial subset edos, though it contains [[7ed4]]. 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 ==
Line 66: Line 63:
| Unison (prime)
| Unison (prime)
| [[1/1]] (just)
| [[1/1]] (just)
|  
|
|  
|  
|  
|  
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| 171.429
| 171.429
| Submajor second
| Submajor second
|  
|
| [[10/9]] (-10.975)
| [[10/9]] (-10.975)
| [[54/49]] (+3.215)
| [[54/49]] (+3.215)
| [[11/10]] (+6.424)<br />[[32/29]] (-1.006)
| [[11/10]] (+6.424)<br>[[32/29]] (-1.006)
| [[File:0-171,43 second (7-EDO).mp3|frameless]]
| [[File:0-171,43 second (7-EDO).mp3|frameless]]
|-
|-
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| Neutral third
| Neutral third
|  
|  
|  
|
| [[128/105]] (+0.048)
| [[128/105]] (+0.048)
| [[39/32]] (+0.374)<br />[[11/9]] (-4.551)
| [[39/32]] (+0.374)<br>[[16/13]] (-16.6)<br>[[11/9]] (-4.551)
| [[File:piano_2_7edo.mp3]]
| [[File:piano_2_7edo.mp3]]
|-
|-
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|  
|  
| [[105/64]] (-0.048)
| [[105/64]] (-0.048)
| [[18/11]] (+4.551)<br />[[64/39]] (-0.374)
| [[18/11]] (+4.551)<br>[[13/8]] (+16.6)<br>[[64/39]] (-0.374)
| [[File:0-857,14 sixth (7-EDO).mp3|frameless]]
| [[File:0-857,14 sixth (7-EDO).mp3|frameless]]
|-
|-
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| [[9/5]] (+10.975)
| [[9/5]] (+10.975)
| [[49/27]] (-3.215)
| [[49/27]] (-3.215)
| [[29/16]] (-1.006)<br />[[20/11]] (-6.424)
| [[29/16]] (-1.006)<br>[[20/11]] (-6.424)
|[[File:0-1028,57 seventh (7-EDO).mp3|frameless]]
| [[File:0-1028,57 seventh (7-EDO).mp3|frameless]]
|-
|-
| 7
| 7
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{| class="wikitable center-all"
{| class="wikitable center-all"
|+ <span style="font-size: 105%">Notation of 7edo</span>
|+ style="font-size: 105%;" | Notation of 7edo
|-
|-
! rowspan="2" | [[Degree]]
! rowspan="2" | [[Degree]]
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* [[Ups and downs notation]] is identical to circle-of-fifths notation;
* [[Ups and downs notation]] is identical to circle-of-fifths notation;
* Mixed and pure [[sagittal notation]] are identical to circle-of-fifths notation.
* Mixed and pure [[sagittal notation]] are identical to circle-of-fifths notation.
===Sagittal notation===
This notation is a subset of the notations for EDOs [[14edo#Sagittal notation|14]], [[21edo#Sagittal notation|21]], [[28edo#Sagittal notation|28]], [[35edo#Sagittal notation|35]], and [[42edo#Second-best fifth notation|42b]].
<imagemap>
File:7-EDO_Sagittal.svg
desc none
rect 80 0 246 50 [[Sagittal_notation]]
rect 246 0 406 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 246 106 [[Fractional_3-limit_notation#Bad-fifths_limma-fraction_notation | limma-fraction notation]]
default [[File:7-EDO_Sagittal.svg]]
</imagemap>
Because it includes no Sagittal symbols, this Sagittal notation is also a conventional notation.


=== Alternative notations ===
=== Alternative notations ===
Line 249: Line 260:


== Approximation to JI ==
== Approximation to JI ==
[[File:7ed2-001.svg|alt=alt : Your browser has no SVG support.]]
[[File:7ed2-001.svg]]
 
[[:File:7ed2-001.svg|7ed2-001.svg]]


== Regular temperament properties ==
== Regular temperament properties ==
=== Uniform maps ===
=== Uniform maps ===
{{Uniform map|13|6.5|7.5}}
{{Uniform map|edo=7}}


=== Commas ===
=== Commas ===
7edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 7 11 16 20 24 26 }}.  
7et [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 7 11 16 20 24 26 }}.  


{| class="commatable wikitable center-1 center-2 right-4 center-5"
{| class="commatable wikitable center-1 center-2 right-4 center-5"
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| Triluyo
| Triluyo
| Wizardharry comma
| Wizardharry comma
|-
| 13
| [[14641/13312]]
| {{monzo| -10 0 0 0 4 -1 }}
| 164.74
|
|
|-
| 13
| [[52/49]]
| {{monzo| 2 0 0 -2 0 1 }}
| 102.87
| thoruru unison
| Hammerisma
|-
|-
| 13
| 13
Line 450: Line 473:
|  
|  
| Wilsorma
| Wilsorma
|-
| 13
| [[52/49]]
| {{monzo| 2 0 0 -2 0 1 }}
| 102.87
| thoruru unison
| Hammerisma
|-
| 13
| [[14641/13312]]
| {{monzo| -10 0 0 0 4 -1 }}
| 164.74
|
|
|}
|}


== Temperaments ==
== Temperaments ==
7edo is the first edo in which regular temperament theory starts to make sense as a way of subdividing the steps into [[mos scale]]s, 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-1L''n''s pentatonic mos. This is in part because 7edo is a [[The Riemann zeta function and tuning #Zeta edo lists|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.
7edo is the first edo in which [[regular temperament theory]] starts to make sense as a way of subdividing the steps into [[mos scale]]s, 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-1L''n''s [[pentatonic]] mos. This is in part because 7edo is 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.
 
== Octave stretch ==
What follows is a comparison of stretched-octave 7edo tunings.
 
; 7edo
* Step size: 171.429{{c}}, octave size: 1200.0{{c}}
Pure-octaves 7edo approximates the 2nd, 3rd, 11th and 13th harmonics well for its size, but it's arguable whether it approximates 5 - if it does it does so poorly. It doesn't approximate 7.
{{Harmonics in equal|7|2|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo}}
{{Harmonics in equal|7|2|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7edo (continued)}}
 
; [[WE|7et, 2.3.11.13 WE]]
* Step size: 171.993{{c}}, octave size: 1204.0{{c}}
Stretching the octave of 7edo by around 4{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. The 2.3.11.13 WE tuning and 2.3.11.13 [[TE]] tuning both do this.
{{Harmonics in cet|171.993|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE}}
{{Harmonics in cet|171.993|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.11.13 WE (continued)}}
 
; [[18ed6]]
* Step size: 172.331{{c}}, octave size: 1206.3{{c}}
Stretching the octave of 7edo by around 6{{c}} results in much improved primes 3, 5 and 7, but much worse primes 11 and 13. The tuning 18ed6 does this.
{{Harmonics in equal|18|6|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6}}
{{Harmonics in equal|18|6|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 18ed6 (continued)}}
 
; [[WE|7et, 2.3.5.11.13 WE]]
* Step size: 172.390{{c}}, octave size: 1206.7{{c}}
Stretching the octave of 7edo by around 7{{c}} results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 [[TE]] tuning both do this.
{{Harmonics in cet|172.390|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE}}
{{Harmonics in cet|172.390|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)}}
 
; [[zpi|15zpi]]
* Step size: 172.495{{c}}, octave size: 1207.5{{c}}
Stretching the octave of 7edo by around 7.5{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 15zpi does this.
{{Harmonics in cet|172.495|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi}}
{{Harmonics in cet|172.495|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 15zpi (continued)}}
 
; [[11edt]]
* Step size: 172.905{{c}}, octave size: 1210.3{{c}}
Stretching the octave of 7edo by around NNN{{c}} results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 11edt does this.
{{Harmonics in equal|11|3|1|columns=11|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt}}
{{Harmonics in equal|11|3|1|columns=12|start=12|collapsed=true|intervals=integer|title=Approximation of harmonics in 11edt (continued)}}
 
== Instruments ==
* [[Lumatone mapping for 7edo]]


== Music ==
== Music ==
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<references />
<references />


[[Category:3-limit record edos|#]] <!-- 1-digit number -->
[[Category:7-tone scales]]
[[Category:7-tone scales]]
[[Category:Macrotonal]]

Latest revision as of 23:45, 21 August 2025

← 6edo 7edo 8edo →
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 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 is the basic example of an equiheptatonic scale, and in terms of tunings with perfect fifths, is essentially the next size up from 5edo. The 7-form is notable as a common structure for many 5-limit systems, including all seven modes of the diatonic scale—Ionian, Dorian, Phrygian, Lydian, Mixolydian, Aeolian, and Locrian; in 7edo itself, the two sizes of interval in any heptatonic MOS scale are equated, resulting in a neutral feel. All triads are neutral (except if you use suspended triads, which are particularly harsh in 7edo due to the narrowed major second), so functional harmony is almost entirely based on the positions of the chords in the 7edo 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 terms of just intonation, the 3/2 is flat but usable, but we don't find particularly accurate intervals in pure harmonics outside the 3-limit, which suggests a more melodic approach to just intonation; intervals approximated by each of 7edo's steps include 10/9 for 1 step, 11/9 for 2 steps, 4/3 for 3 steps, and their octave complements. Interestingly, this renders an 8:9:10:11:12 pentad equidistant, from which it can be derived that 7edo supports meantone (equating the major seconds 10/9 and 9/8) and porcupine (splitting 4/3 into three equal submajor seconds which simultaneously represent 12/11, 11/10, and 10/9), and is the unique system to do so.

Due to 7edo's inaccurately tuned 5/4 major third (which is flat by over 40 cents), it supports several exotemperaments in the 5-limit, such as dicot (which splits the fifth into two equal neutral thirds, simultaneously representing 5/4 and the minor third 6/5) and mavila (which flattens the fifth so that the diatonic "major third" actually approximates 6/5); 6/5 is a slightly more reasonable interpretation of 7edo's third than 5/4, leading to an overall slightly minor sound.

In higher limits, this third takes on a new role: as a neutral third, it is a decent approximation of the 13th subharmonic, and as such 7edo can be seen as a 2.3.13 temperament. This third is a near perfect approximation of the interval 39/32; the equation of 16/13 and 39/32 is called harmoneutral temperament. In general, the inclusion of 13 allows the pentad discussed earlier to be continued to an 8:9:10:11:12:13 hexad, although the specific interval 13/12 is inaccurate due to the errors adding up in the same direction.

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 represents a 7-step closed circle of fifths, tempering out the Pythagorean chromatic semitone. However, it can also be seen as a circle of neutral thirds, which can be interpreted as 11/9; this is called neutron temperament.

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)

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 very similar to one 171-cent 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 ¢), 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]

Subsets and supersets

7edo is the 4th prime edo, after 5edo and before 11edo. It does not contain any nontrivial subset edos, though it contains 7ed4. 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)
16/13 (-16.6)
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)
13/8 (+16.6)
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:

Sagittal notation

This notation is a subset of the notations for EDOs 14, 21, 28, 35, and 42b.

Sagittal notationPeriodic table of EDOs with sagittal notationlimma-fraction notation

Because it includes no Sagittal symbols, this Sagittal notation is also a conventional notation.

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

Regular temperament properties

Uniform maps

13-limit uniform maps between 6.8 and 7.2
Min. size Max. size Wart notation Map
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]

Commas

7et 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 14641/13312 [-10 0 0 0 4 -1⟩ 164.74
13 52/49 [2 0 0 -2 0 1⟩ 102.87 thoruru unison Hammerisma
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

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 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.

Octave stretch

What follows is a comparison of stretched-octave 7edo tunings.

7edo
  • Step size: 171.429 ¢, octave size: 1200.0 ¢

Pure-octaves 7edo approximates the 2nd, 3rd, 11th and 13th harmonics well for its size, but it's arguable whether it approximates 5 - if it does it does so poorly. It doesn't approximate 7.

Approximation of harmonics in 7edo
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +0.0 -16.2 +0.0 -43.5 -16.2 +59.7 +0.0 -32.5 -43.5 -37.0 -16.2
Relative (%) +0.0 -9.5 +0.0 -25.3 -9.5 +34.9 +0.0 -18.9 -25.3 -21.6 -9.5
Steps
(reduced) 		7
(0) 		11
(4) 		14
(0) 		16
(2) 		18
(4) 		20
(6) 		21
(0) 		22
(1) 		23
(2) 		24
(3) 		25
(4)
Approximation of harmonics in 7edo (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +16.6 +59.7 -59.7 +0.0 +66.5 -32.5 +45.3 -43.5 +43.5 -37.0 +57.4 -16.2
Relative (%) +9.7 +34.9 -34.8 +0.0 +38.8 -18.9 +26.5 -25.3 +25.4 -21.6 +33.5 -9.5
Steps
(reduced) 		26
(5) 		27
(6) 		27
(6) 		28
(0) 		29
(1) 		29
(1) 		30
(2) 		30
(2) 		31
(3) 		31
(3) 		32
(4) 		32
(4)
7et, 2.3.11.13 WE
  • Step size: 171.993 ¢, octave size: 1204.0 ¢

Stretching the octave of 7edo by around 4 ¢ results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. The 2.3.11.13 WE tuning and 2.3.11.13 TE tuning both do this.

Approximation of harmonics in 7et, 2.3.11.13 WE
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +4.0 -10.0 +7.9 -34.4 -6.1 +71.0 +11.9 -20.1 -30.5 -23.5 -2.1
Relative (%) +2.3 -5.8 +4.6 -20.0 -3.5 +41.3 +6.9 -11.7 -17.7 -13.7 -1.2
Step 7 11 14 16 18 20 21 22 23 24 25
Approximation of harmonics in 7et, 2.3.11.13 WE (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +31.3 +75.0 -44.5 +15.8 +82.8 -16.1 +62.3 -26.5 +61.0 -19.5 +75.5 +1.8
Relative (%) +18.2 +43.6 -25.8 +9.2 +48.2 -9.4 +36.2 -15.4 +35.5 -11.4 +43.9 +1.1
Step 26 27 27 28 29 29 30 30 31 31 32 32
18ed6
  • Step size: 172.331 ¢, octave size: 1206.3 ¢

Stretching the octave of 7edo by around 6 ¢ results in much improved primes 3, 5 and 7, but much worse primes 11 and 13. The tuning 18ed6 does this.

Approximation of harmonics in 18ed6
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.3 -6.3 +12.6 -29.0 +0.0 +77.8 +18.9 -12.6 -22.7 -15.4 +6.3
Relative (%) +3.7 -3.7 +7.3 -16.8 +0.0 +45.1 +11.0 -7.3 -13.2 -8.9 +3.7
Steps
(reduced) 		7
(7) 		11
(11) 		14
(14) 		16
(16) 		18
(0) 		20
(2) 		21
(3) 		22
(4) 		23
(5) 		24
(6) 		25
(7)
Approximation of harmonics in 18ed6 (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +40.1 +84.1 -35.3 +25.3 -79.7 -6.3 +72.4 -16.4 +71.5 -9.1 -86.0 +12.6
Relative (%) +23.3 +48.8 -20.5 +14.7 -46.2 -3.7 +42.0 -9.5 +41.5 -5.3 -49.9 +7.3
Steps
(reduced) 		26
(8) 		27
(9) 		27
(9) 		28
(10) 		28
(10) 		29
(11) 		30
(12) 		30
(12) 		31
(13) 		31
(13) 		31
(13) 		32
(14)
7et, 2.3.5.11.13 WE
  • Step size: 172.390 ¢, octave size: 1206.7 ¢

Stretching the octave of 7edo by around 7 ¢ results in much improved primes 3, 5 and 11, but much worse primes 7 and 13. Its 2.3.5.11.13 WE tuning and 2.3.5.11.13 TE tuning both do this.

Approximation of harmonics in 7et, 2.3.5.11.13 WE
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +6.7 -5.7 +13.5 -28.1 +1.1 +79.0 +20.2 -11.3 -21.3 -14.0 +7.8
Relative (%) +3.9 -3.3 +7.8 -16.3 +0.6 +45.8 +11.7 -6.6 -12.4 -8.1 +4.5
Step 7 11 14 16 18 20 21 22 23 24 25
Approximation of harmonics in 7et, 2.3.5.11.13 WE (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +41.6 +85.7 -33.7 +26.9 -78.0 -4.6 +74.2 -14.6 +73.3 -7.2 -84.2 +14.5
Relative (%) +24.1 +49.7 -19.6 +15.6 -45.3 -2.7 +43.0 -8.5 +42.5 -4.2 -48.8 +8.4
Step 26 27 27 28 28 29 30 30 31 31 31 32
15zpi
  • Step size: 172.495 ¢, octave size: 1207.5 ¢

Stretching the octave of 7edo by around 7.5 ¢ results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 15zpi does this.

Approximation of harmonics in 15zpi
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +7.5 -4.5 +14.9 -26.4 +3.0 +81.1 +22.4 -9.0 -18.9 -11.4 +10.4
Relative (%) +4.3 -2.6 +8.7 -15.3 +1.7 +47.0 +13.0 -5.2 -11.0 -6.6 +6.0
Step 7 11 14 16 18 20 21 22 23 24 25
Approximation of harmonics in 15zpi (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +44.3 -84.0 -30.9 +29.9 -75.1 -1.6 +77.3 -11.5 +76.6 -4.0 -80.9 +17.9
Relative (%) +25.7 -48.7 -17.9 +17.3 -43.5 -0.9 +44.8 -6.6 +44.4 -2.3 -46.9 +10.4
Step 26 26 27 28 28 29 30 30 31 31 31 32
11edt
  • Step size: 172.905 ¢, octave size: 1210.3 ¢

Stretching the octave of 7edo by around NNN ¢ results in much improved primes 3, 5 and 11, but much worse primes 2, 7 and 13. The tuning 11edt does this.

Approximation of harmonics in 11edt
Harmonic 2 3 4 5 6 7 8 9 10 11 12
Error Absolute (¢) +10.3 +0.0 +20.7 -19.8 +10.3 -83.6 +31.0 +0.0 -9.5 -1.6 +20.7
Relative (%) +6.0 +0.0 +12.0 -11.5 +6.0 -48.4 +17.9 +0.0 -5.5 -0.9 +12.0
Steps
(reduced) 		7
(7) 		11
(0) 		14
(3) 		16
(5) 		18
(7) 		19
(8) 		21
(10) 		22
(0) 		23
(1) 		24
(2) 		25
(3)
Approximation of harmonics in 11edt (continued)
Harmonic 13 14 15 16 17 18 19 20 21 22 23 24
Error Absolute (¢) +55.0 -73.3 -19.8 +41.3 -63.6 +10.3 -83.3 +0.8 -83.6 +8.7 -68.2 +31.0
Relative (%) +31.8 -42.4 -11.5 +23.9 -36.8 +6.0 -48.2 +0.5 -48.4 +5.1 -39.5 +17.9
Steps
(reduced) 		26
(4) 		26
(4) 		27
(5) 		28
(6) 		28
(6) 		29
(7) 		29
(7) 		30
(8) 		30
(8) 		31
(9) 		31
(9) 		32
(10)

Instruments

Music

Main article: 7edo/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

  1. ↑ African music, Encyclopedia Britannica.
  2. ↑ Robotham, Donald Keith and Gerhard Kubik.
  3. ↑ Nambiyathiri, Tarjani. Indian Music History And Structure Emmie Te Nijenhuis Brill
  4. ↑ Garzoli, John. The Myth of Equidistance in Thai Tuning.
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