7edo: Difference between revisions

Line 10:

| P5 = 4\7 (686¢)

| M2 = 1\7 (171¢)

| m2 = 1\7 ~~(171¢)~~

| Semitones = 0\7 : 1\7

~~| A1 = 0~~\7 ~~(0¢)~~

}}

'''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~~|prime edo~~]], after [[2edo]], [[3edo]] and [[5edo]]. It is the third [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral ~~edo~~]].

'''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 ==

{| class="wikitable center-all"

! colspan="2" |

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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]] is known as "Hexe[6]".

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, ~~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]].

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 ~~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.

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.

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([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 ~~7edo~~. A Ugandan Chopi xylophone measured by Haddon (1952) was also tuned to this system.

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~~ ===

=== Differences between distributionally-even scales and smaller EDOs ===

{| class="wikitable"

|+

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!s-Nedo

|-

|2

| 2

|85.714¢

| 85.714¢

| -85.714¢

|-

|3

| 3

|114.286¢

| 114.286¢

| -57.143¢

|-

|4

| 4

|42.857¢

| 42.857¢

| -128.571¢

|-

|5

|102.857¢

| 102.857¢

| -68.571¢

|-

|6

|142.857¢

| 142.857¢

| -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.

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"

|-

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|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]]

|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]]

|0.374¢ from neutral third [[39/32]] -4.55¢ from neutral third [[11/9]]

-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]]

|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¢)

|5.265 ¢ 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]]

|4.551¢ from neutral sixth [[18/11]] -0.374¢ from neutral sixth [[64/39]] -0.048¢ from (neutral sixth) 105/64

-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]]

|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

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[[: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.

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== 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]]~~.

[[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 }}.

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"

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==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.

~~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 4 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)

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*[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].

7EDO ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web#list here].

[[Category:7edo| ]]

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[[Category:Prime EDO]]

[[Category:Zeta]]

~~[[Category:todo:unify precision]]~~

@@ Line 10: / Line 10: @@
 | P5 = 4\7 (686¢)
 | M2 = 1\7 (171¢)
-| m2 = 1\7 (171¢)
+| Semitones = 0\7 : 1\7
-| A1 = 0\7 (0¢)
 }}
-'''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|prime edo]], after [[2edo]], [[3edo]] and [[5edo]]. It is the third [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]].
+'''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 ==
 {| class="wikitable center-all"
 ! 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).
-edo 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]] is known as "Hexe[6]".
+EDO 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, 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]].
+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 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.
+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.
@@ Line 73: / Line 71: @@
 ([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 7edo. A Ugandan Chopi xylophone measured by Haddon (1952) was also tuned to this system.
+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 ===
+=== Differences between distributionally-even scales and smaller EDOs ===
 {| class="wikitable"
 |+
@@ Line 82: / Line 80: @@
 !s-Nedo
 |-
-|2
+| 2
-|85.714¢
+| 85.714¢
 | -85.714¢
 |-
-|3
+| 3
-|114.286¢
+| 114.286¢
 | -57.143¢
 |-
-|4
+| 4
-|42.857¢
+| 42.857¢
 | -128.571¢
 |-
 |5
-|102.857¢
+| 102.857¢
 | -68.571¢
 |-
 |6
-|142.857¢
+| 142.857¢
 | -28.571¢
 |}
-==Intervals==
+== Intervals ==
-edo can be notated on a five-line staff without accidentals. There is no distinction between major or minor; each pitch class is unique.
+EDO 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"
 |-
@@ Line 120: / Line 118: @@
 |171.429
 |second
-|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]]
+|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
 |342.857
 |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
 |514.286
 |fourth
-|16.241¢ from just fourth [[4/3]] (498.045¢) <br /> -5.265¢ from wide fourth [[27/20]]
+|16.241¢ from just fourth [[4/3]] (498.045¢) <br> -5.265¢ from wide fourth [[27/20]]
 |-
 | 4
 |685.714
 |fifth
-|5.265 ¢ from narrow fifth [[40/27]] <br /> -16.241¢ from just fifth [[3/2]] (701.955¢)
+|5.265 ¢ from narrow fifth [[40/27]] <br> -16.241¢ from just fifth [[3/2]] (701.955¢)
 |-
 |5
 |857.143
 |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
 |1028.571
 |seventh
-|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]]
+|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
@@ Line 161: / Line 155: @@
 [[: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.
@@ Line 169: / Line 163: @@
 == 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]].
+[[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 ==
-edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 7 11 16 20 24 26 }}.
+EDO [[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"
@@ Line 369: / Line 363: @@
 <references />
-==Temperaments==
+== Temperaments ==
+EDO 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.
-edo 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 4 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)
@@ 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]'')
-==Ear Training==
+== Ear Training ==
-edo ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web#list here].
+EDO ear-training exercises by Alex Ness available [https://drive.google.com/folderview?id=0BwsXD8q2VCYUT3VEZUVmeVZUcmc&usp=drive_web#list here].
 [[Category:7edo| ]] <!-- main article -->
@@ Line 403: / Line 396: @@
 [[Category:Prime EDO]]
 [[Category:Zeta]]
-[[Category:todo:unify precision]]