7edo: Difference between revisions

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| ja = 7平均律

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~~{{High priority}}~~

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[[File:7edo scale.mp3|thumb|A chromatic 7edo scale on C.]]

7edo ~~unifies~~ the seven modes of the [[5L 2s|diatonic]] scale—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{{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 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.

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]].

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 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 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.

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 ===

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[[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.

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.

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

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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>

~~=== Observations ===~~

The seventh of 7edo is almost exactly the 29th harmonic ([[29/16]]), which can have a very agreeable sound with harmonic [[timbre]]s. 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]].~~

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]]]".

=== Octave stretch ===

~~The~~ [[~~stretched~~ and compressed tuning|~~stretched~~-octaves]] ~~tuning~~ [[Ed257/128 #7ed257/128|7ed257/128]] greatly improves 7edo's approximation of harmonics 3, 5 and 11, at the cost of slightly worsening 2 and 7, and greatly worsening 13. If one is hoping to use 7edo for [[11-limit]] harmonies, then ~~7ed257/128 is a~~ good ~~choice~~ to make that easier.

[[Stretched and compressed tuning|Stretched-octaves]] tunings such as [[11edt]], [[18ed6]] or [[Ed257/128 #7ed257/128|7ed257/128]] greatly improves 7edo's approximation of harmonics 3, 5 and 11, at the cost of slightly worsening 2 and 7, and greatly worsening 13. If one is hoping to use 7edo for [[11-limit]] harmonies, then these are good choices to make that easier.

The stretched 7edo tuning [[zpi|15zpi]] can also be used to improve ~~7edo’s~~ approximation of JI in a similar way.

The stretched 7edo tuning [[zpi|15zpi]] can also be used to improve 7edo's approximation of JI in a similar way.

=== Subsets and supersets ===

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! rowspan="2" | [[Cent]]s

! rowspan="2" | [[Interval region]]

! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[`¢]])

! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[¢]])

! rowspan="2" | Audio

|-

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| Unison (prime)

| [[1/1]] (just)

|

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| 171.429

| Submajor second

|

| [[10/9]] (-10.975)

| [[54/49]] (+3.215)

| [[11/10]] (+6.424) [[32/29]] (-1.006)

| [[11/10]] (+6.424) [[32/29]] (-1.006)

| [[File:0-171,43 second (7-EDO).mp3|frameless]]

|-

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| Neutral third

|

| [[128/105]] (+0.048)

| [[39/32]] (+0.374) [[11/9]] (-4.551)

| [[39/32]] (+0.374) [[16/13]] (-16.6) [[11/9]] (-4.551)

| [[File:piano_2_7edo.mp3]]

|-

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|

| [[105/64]] (-0.048)

| [[18/11]] (+4.551) [[64/39]] (-0.374)

| [[18/11]] (+4.551) [[13/8]] (+16.6) [[64/39]] (-0.374)

| [[File:0-857,14 sixth (7-EDO).mp3|frameless]]

|-

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| [[9/5]] (+10.975)

| [[49/27]] (-3.215)

| [[29/16]] (-1.006) [[20/11]] (-6.424)

| [[29/16]] (-1.006) [[20/11]] (-6.424)

|[[File:0-1028,57 seventh (7-EDO).mp3|frameless]]

| [[File:0-1028,57 seventh (7-EDO).mp3|frameless]]

|-

| 7

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== 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]]~~

~~=== Zeta peak index ===~~

~~{{ZPI~~

~~| zpi = 15~~

~~| steps = 6.95668765658792~~

~~| step size = 172.495885863671~~

~~| tempered height = 4.166936~~

~~| pure height = 3.940993~~

~~| integral = 1.162332~~

~~| gap = 14.234171~~

~~| octave = 1207.47120104570~~

~~| consistent = 6~~

~~| distinct = 5~~

}}

== Regular temperament properties ==

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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.

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.

== Instruments ==

* [[Lumatone mapping for 7edo]]

== Music ==

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[[Category:3-limit record edos|#]]

[[Category:7-tone scales]]

@@ Line 5: / Line 5: @@
 | ja = 7平均律
 }}
-{{High priority}}
 {{Infobox ET}}
 {{ED intro}}
@@ Line 12: / Line 11: @@
 [[File:7edo scale.mp3|thumb|A chromatic 7edo scale on C.]]
-edo unifies the seven modes of the [[5L 2s|diatonic]] scale—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{{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]].
+edo 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.
+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]].
+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 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 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.
+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.
+edo 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 ===
@@ Line 20: / Line 31: @@
 [[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.
+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.
 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{{c}}), especially at points of rest. In this manner, the principles of equidistance and harmonic euphony are accommodated within one tonal-harmonic system.
@@ Line 29: / Line 40: @@
 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>
-=== Observations ===
-The seventh of 7edo is almost exactly the 29th harmonic ([[29/16]]), which can have a very agreeable sound with harmonic [[timbre]]s. 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.
-edo is the unique intersection of the temperaments of [[meantone]] (specifically [[3/4-comma meantone]]) and [[porcupine]].
-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|Neutron[7]]]" just as the whole tone scale of [[12edo]] is known as "[[hexe|Hexe[6]]]".
 === Octave stretch ===
-The [[stretched and compressed tuning|stretched-octaves]] tuning [[Ed257/128 #7ed257/128|7ed257/128]] greatly improves 7edo's approximation of harmonics 3, 5 and 11, at the cost of slightly worsening 2 and 7, and greatly worsening 13. If one is hoping to use 7edo for [[11-limit]] harmonies, then 7ed257/128 is a good choice to make that easier.
+[[Stretched and compressed tuning|Stretched-octaves]] tunings such as [[11edt]], [[18ed6]] or [[Ed257/128 #7ed257/128|7ed257/128]] greatly improves 7edo's approximation of harmonics 3, 5 and 11, at the cost of slightly worsening 2 and 7, and greatly worsening 13. If one is hoping to use 7edo for [[11-limit]] harmonies, then these are good choices to make that easier.
-The stretched 7edo tuning [[zpi|15zpi]] can also be used to improve 7edo’s approximation of JI in a similar way.
+The stretched 7edo tuning [[zpi|15zpi]] can also be used to improve 7edo's approximation of JI in a similar way.
 === Subsets and supersets ===
@@ Line 52: / Line 56: @@
 ! rowspan="2" | [[Cent]]s
 ! rowspan="2" | [[Interval region]]
-! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[`¢]])
+! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[¢]])
 ! rowspan="2" | Audio
 |-
@@ Line 64: / Line 68: @@
 | Unison (prime)
 | [[1/1]] (just)
 |
 |
 |
@@ Line 72: / Line 76: @@
 | 171.429
 | Submajor second
 |
 | [[10/9]] (-10.975)
 | [[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]]
 |-
@@ Line 82: / Line 86: @@
 | Neutral third
 |
 |
 | [[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]]
 |-
@@ Line 111: / Line 115: @@
 |
 | [[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]]
 |-
@@ Line 120: / Line 124: @@
 | [[9/5]] (+10.975)
 | [[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
@@ Line 261: / Line 265: @@
 == 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]]
-=== Zeta peak index ===
-{{ZPI
-| zpi = 15
-| steps = 6.95668765658792
-| step size = 172.495885863671
-| tempered height = 4.166936
-| pure height = 3.940993
-| integral = 1.162332
-| gap = 14.234171
-| octave = 1207.47120104570
-| consistent = 6
-| distinct = 5
-}}
 == Regular temperament properties ==
@@ Line 497: / Line 485: @@
 /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.
-/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.
+/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.
+== Instruments ==
+* [[Lumatone mapping for 7edo]]
 == Music ==
@@ Line 512: / Line 503: @@
 <references />
+[[Category:3-limit record edos|#]] <!-- 1-digit number -->
 [[Category:7-tone scales]]