7edo: Difference between revisions

Line 11:

[[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. 7edo 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.

7edo represents a 7-step closed [[circle of fifths]].

=== Prime harmonics ===

Line 31:

Line 41:

=== 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]]]".

Line 51:

Line 59:

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

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

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

! colspan="5" | Approximated [[JI]] intervals ([[error]] in [[`¢]])

! rowspan="2" | Audio

|-

! [[3-limit]]

!2.3.13

! [[5-limit]]

! [[7-limit]]

Line 63:

Line 72:

| Unison (prime)

| [[1/1]] (just)

|

Line 72:

Line 82:

| Submajor second

|

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

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

Line 81:

Line 92:

| Neutral third

|

|[[39/32]] (+0.374)

[[16/13]] (-16.6)

|

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

| ~~[[39/32]] (+0.374)~~<br />[[11/9]] (-4.551)

| <br />[[11/9]] (-4.551)

| [[File:piano_2_7edo.mp3]]

|-

Line 90:

Line 103:

| Fourth

| [[4/3]] (+16.241)

|

| [[27/20]] (-5.265)

|

Line 99:

Line 113:

| Fifth

| [[3/2]] (-16.241)

|

| [[40/27]] (+5.265)

|

Line 108:

Line 123:

| Neutral sixth

|

|[[13/8]]

(+16.6)

[[64/39]] (-0.374)

|

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

| [[18/11]] (+4.551)<br />~~[[64/39]] (-0.374)~~

| [[18/11]] (+4.551)<br />

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

|-

Line 117:

Line 135:

| Supraminor seventh

|

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

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

Line 126:

Line 145:

| Octave

| [[2/1]] (just)

|

@@ Line 11: / 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. 7edo 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.
+edo represents a 7-step closed [[circle of fifths]].
 === Prime harmonics ===
@@ Line 31: / Line 41: @@
 === 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]]]".
@@ Line 51: / Line 59: @@
 ! rowspan="2" | [[Cent]]s
 ! rowspan="2" | [[Interval region]]
-! colspan="4" | Approximated [[JI]] intervals ([[error]] in [[`¢]])
+! colspan="5" | Approximated [[JI]] intervals ([[error]] in [[`¢]])
 ! rowspan="2" | Audio
 |-
 ! [[3-limit]]
+!2.3.13
 ! [[5-limit]]
 ! [[7-limit]]
@@ Line 63: / Line 72: @@
 | Unison (prime)
 | [[1/1]] (just)
 |
+|
 |
 |
@@ Line 72: / Line 82: @@
 | Submajor second
 |
+|
 | [[10/9]] (-10.975)
 | [[54/49]] (+3.215)
@@ Line 81: / Line 92: @@
 | Neutral third
 |
-|
+|[[39/32]] (+0.374)
+[[16/13]] (-16.6)
+|
 | [[128/105]] (+0.048)
-| [[39/32]] (+0.374)<br />[[11/9]] (-4.551)
+| <br />[[11/9]] (-4.551)
 | [[File:piano_2_7edo.mp3]]
 |-
@@ Line 90: / Line 103: @@
 | Fourth
 | [[4/3]] (+16.241)
+|
 | [[27/20]] (-5.265)
 |
@@ Line 99: / Line 113: @@
 | Fifth
 | [[3/2]] (-16.241)
+|
 | [[40/27]] (+5.265)
 |
@@ Line 108: / Line 123: @@
 | Neutral sixth
 |
-|
+|[[13/8]]
+(+16.6)
+[[64/39]] (-0.374)
+|
 | [[105/64]] (-0.048)
-| [[18/11]] (+4.551)<br />[[64/39]] (-0.374)
+| [[18/11]] (+4.551)<br />
 | [[File:0-857,14 sixth (7-EDO).mp3|frameless]]
 |-
@@ Line 117: / Line 135: @@
 | Supraminor seventh
 |
+|
 | [[9/5]] (+10.975)
 | [[49/27]] (-3.215)
@@ Line 126: / Line 145: @@
 | Octave
 | [[2/1]] (just)
 |
+|
 |
 |