7edo: Difference between revisions
Jump to navigation
Jump to search
Extend commas table to the 13-limit as per xenwolf's note in talk page, talk about temperaments. |
mNo edit summary |
||
| Line 10: | Line 10: | ||
| P5 = 4\7 (686¢) | | P5 = 4\7 (686¢) | ||
| M2 = 1\7 (171¢) | | M2 = 1\7 (171¢) | ||
| | | Semitones = 0\7 : 1\7 | ||
}} | }} | ||
'''7 equal divisions of the octave''' (''' | '''7 equal divisions of the octave''' ('''7EDO''') is the [[tuning system]] derived by dividing the [[octave]] into 7 equal steps of 171.4 [[cent]]s each, or the seventh root of 2. It is the fourth [[prime EDO]], after [[2edo|2EDO]], [[3edo|3EDO]] and [[5edo|5EDO]]. It is the third [[The Riemann Zeta Function and Tuning #Zeta EDO lists|zeta integral EDO]]. | ||
== Theory == | == Theory == | ||
{| class="wikitable center-all" | {| class="wikitable center-all" | ||
! colspan="2" | <!-- empty cell --> | ! colspan="2" | <!-- empty cell --> | ||
| Line 63: | Line 61: | ||
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. 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). | 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. 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 {{monzo| -2 -14 0 0 7 }}. As a temperament, William Lynch gives it the name "Neutron[7]" just as the whole tone scale of [[12edo|12EDO]] is known as "Hexe[6]". | |||
Typically, | 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|14EDO]] or [[21edo|21EDO]]. | ||
The seventh of | 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|harmonic entropy]] than [[Harmonic seventh|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. | 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. | ||
| Line 73: | Line 71: | ||
([http://www.britannica.com/EBchecked/topic/719112/African-music "African music." Encyclopædia Britannica. 2009. Encyclopædia Britannica Online. 05 Jul. 2009.]) | ([http://www.britannica.com/EBchecked/topic/719112/African-music "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 | 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. | ||
=== Differences between distributionally-even scales and smaller | === Differences between distributionally-even scales and smaller EDOs === | ||
{| class="wikitable" | {| class="wikitable" | ||
|+ | |+ | ||
| Line 82: | Line 80: | ||
!s-Nedo | !s-Nedo | ||
|- | |- | ||
|2 | | 2 | ||
|85.714¢ | | 85.714¢ | ||
| -85.714¢ | | -85.714¢ | ||
|- | |- | ||
|3 | | 3 | ||
|114.286¢ | | 114.286¢ | ||
| -57.143¢ | | -57.143¢ | ||
|- | |- | ||
|4 | | 4 | ||
|42.857¢ | | 42.857¢ | ||
| -128.571¢ | | -128.571¢ | ||
|- | |- | ||
|5 | |5 | ||
|102.857¢ | | 102.857¢ | ||
| -68.571¢ | | -68.571¢ | ||
|- | |- | ||
|6 | |6 | ||
|142.857¢ | | 142.857¢ | ||
| -28.571¢ | | -28.571¢ | ||
|} | |} | ||
==Intervals== | == 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. | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
| Line 120: | Line 118: | ||
|171.429 | |171.429 | ||
|second | |second | ||
|6.424¢ from Ptolemy (neutral) second [[11/10]] <br | |6.424¢ from Ptolemy (neutral) second [[11/10]] <br> 3.215¢ from second [[54/49]] <br> -1.006¢ from the 29th subharmonic [[32/29]] <br> -10.975¢ from major second (small whole tone) [[10/9]] | ||
|- | |- | ||
|2 | |2 | ||
|342.857 | |342.857 | ||
|third | |third | ||
|0.374¢ from neutral third [[39/32]] | |0.374¢ from neutral third [[39/32]]<br>-4.55¢ from neutral third [[11/9]] | ||
-4.55¢ from neutral third [[11/9]] | |||
|- | |- | ||
|3 | |3 | ||
|514.286 | |514.286 | ||
|fourth | |fourth | ||
|16.241¢ from just fourth [[4/3]] (498.045¢) <br | |16.241¢ from just fourth [[4/3]] (498.045¢) <br> -5.265¢ from wide fourth [[27/20]] | ||
|- | |- | ||
| 4 | | 4 | ||
|685.714 | |685.714 | ||
|fifth | |fifth | ||
|5.265 ¢ from narrow fifth [[40/27]] <br | |5.265 ¢ from narrow fifth [[40/27]] <br> -16.241¢ from just fifth [[3/2]] (701.955¢) | ||
|- | |- | ||
|5 | |5 | ||
|857.143 | |857.143 | ||
|sixth | |sixth | ||
|4.551¢ from neutral sixth [[18/11]] | |4.551¢ from neutral sixth [[18/11]]<br>-0.374¢ from neutral sixth [[64/39]]<br>-0.048¢ from (neutral sixth) 105/64 | ||
-0.374¢ from neutral sixth [[64/39]] | |||
-0.048¢ from (neutral sixth) 105/64 | |||
|- | |- | ||
|6 | |6 | ||
|1028.571 | |1028.571 | ||
|seventh | |seventh | ||
|10.975¢ from (Didymus) minor seventh [[9/5]] <br | |10.975¢ from (Didymus) minor seventh [[9/5]] <br> -6.424¢ from neutral seventh [[20/11]] <br> -1.006¢ from the 29th harmonic [[29/16]] <br> -3.215¢ from seventh [[49/27]] | ||
|- | |- | ||
|7 | |7 | ||
| Line 161: | Line 155: | ||
[[:File:7ed2-001.svg|7ed2-001.svg]] | [[:File:7ed2-001.svg|7ed2-001.svg]] | ||
==Observations== | == Observations == | ||
Related in a lateral way to traditional Thai music. Subset of [[14edo]] and [[21edo]]. | Related in a lateral way to traditional Thai music. Subset of [[14edo|14EDO]] and [[21edo|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. | 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. | ||
| Line 169: | Line 163: | ||
== Notation== | == Notation== | ||
[[William Lynch]] proposes using numbers 1 through 7 as the nominals of 7edo with sharp signs being possible to expand to | [[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== | == Commas == | ||
7EDO [[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" | ||
| Line 369: | Line 363: | ||
<references /> | <references /> | ||
==Temperaments== | == Temperaments == | ||
7EDO is the first EDO in which regular temperament theory starts to make sense as a way of subdividing the steps into mode of symmetry 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 MOS's of 322 and 2221. 3/7 is on the intersection of [[meantone]] and [[mavila]], and has MOS's of 331 and 21211. | |||
==Music== | == Music == | ||
*[https://soundcloud.com/overtoneshock/death-teasing-monolith-7-edo-premiere Death Giving Monolith] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel] (dulcimer and voice) | *[https://soundcloud.com/overtoneshock/death-teasing-monolith-7-edo-premiere Death Giving Monolith] by [https://www.stephenweigelcomposerperformer.com/ Stephen Weigel] (dulcimer and voice) | ||
| Line 392: | Line 385: | ||
*[https://youtu.be/Z0seAc_Snk4?t=216 "Starfish"] (from 3:36 to 4:28 only) by Sevish (from his 2021 album ''[https://sevish.bandcamp.com/album/bubble Bubble]'') | *[https://youtu.be/Z0seAc_Snk4?t=216 "Starfish"] (from 3:36 to 4:28 only) by Sevish (from his 2021 album ''[https://sevish.bandcamp.com/album/bubble Bubble]'') | ||
==Ear Training== | == Ear Training == | ||
7EDO ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web#list here]. | |||
[[Category:7edo| ]] <!-- main article --> | [[Category:7edo| ]] <!-- main article --> | ||
| Line 403: | Line 396: | ||
[[Category:Prime EDO]] | [[Category:Prime EDO]] | ||
[[Category:Zeta]] | [[Category:Zeta]] | ||
Revision as of 13:47, 13 October 2021
| ← 6edo | 7edo | 8edo → |
(semiconvergent)
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
| 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. 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 11/9s 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.
("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 7EDO. A Ugandan Chopi xylophone measured by Haddon (1952) was also tuned to this system.
Differences between distributionally-even scales and smaller EDOs
| N | L-Nedo | s-Nedo |
|---|---|---|
| 2 | 85.714¢ | -85.714¢ |
| 3 | 114.286¢ | -57.143¢ |
| 4 | 42.857¢ | -128.571¢ |
| 5 | 102.857¢ | -68.571¢ |
| 6 | 142.857¢ | -28.571¢ |
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 -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 |
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 | [-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 | [-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 |
- ↑ 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 mode of symmetry 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 MOS's of 322 and 2221. 3/7 is on the intersection of meantone and mavila, and has MOS's of 331 and 21211.
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
- Clockworkian by Userminusone
- "Fuschiamarine" by Sevish (from his 2021 album Bubble)
- "Starfish" (from 3:36 to 4:28 only) by Sevish (from his 2021 album Bubble)
Ear Training
7EDO ear-training exercises by Alex Ness available here.