8edo: Difference between revisions

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=== Approximation to JI ===

8edo forms an odd and even pitch set of two diminished seventh chords, which when used in combination yield dissonance. The system has been described as a "barbaric" harmonic system, containing no good approximation of harmonics 3, 5, 7, 11, 13, and 17; even so, it does a good job representing the [[just intonation subgroup]]s 2.11/3.13/5, with good intervals of [[13/10]] and an excellent version of [[11/6]]. Stacking the 450-cent interval can result in some semi-consonant chords such as 0-3-6 degrees, although these still are quite dissonant compared to standard root-3rd-P5 triads, which are unavailable in 8edo.

8edo forms an odd and even pitch set of two diminished seventh chords, which when used in combination yield dissonance. The system has been described as a "barbaric" harmonic system, containing no good approximation of harmonics 3, 5, 7, 11, 13, and 17; even so, it does a good job representing the [[just intonation subgroup]] 2.11/3.13/5, with good intervals of [[13/10]] and an excellent version of [[11/6]]. Stacking the 450-cent interval can result in some semi-consonant chords such as 0-3-6 degrees, although these still are quite dissonant compared to standard root-3rd-P5 triads, which are unavailable in 8edo.

Another way of looking at 8edo is to treat a chord of 0-1-2-3-4 degrees (0-150-300-450-600 cents) as approximating harmonics 10:11:12:13:14 (~0-165-316-454-583 cents), which is not too implausible if you can buy that 12edo is a 5-limit temperament. This interpretation would imply that 121/120, 144/143, 169/168, and hence also 36/35 and 66/65, are tempered out.

Another way of looking at 8edo is to treat a chord of 0-1-2-3-4 degrees (0-150-300-450-600 cents) as approximating harmonics 10:11:12:13:14 (~0-165-316-454-583 cents), which is not too implausible if you can buy that 12edo is a 5-limit temperament. This interpretation would imply that 121/120, 144/143, 169/168, and hence also 36/35 and 66/65, are tempered out. The corresponding subgroup is 2.5/3.7/3.11/3.13/3. However, some intervals in this chord, such as [[14/11]] and [[7/6]], are tuned quite inaccurately (over 30 cents off). Nonetheless, the 8-form serves as an underlying structure in many [[non-over-1 temperament]]s.

=== Relationship with the father comma ===

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=== Odd harmonics ===

~~=== Octave shrinking ===~~

~~8edo's approximation of [[JI]] can be improved via [[octave shrinking]]. Compressing 8edo's octave from 1200 [[cent]]s down to 1188 cents gives the tuning called [[1ed148.5c]].~~

~~Pure-octaves 8edo approximates only 2 of the first 11 prime harmonics within 15 cents. 1ed148.5c approximates 5 of them.~~

~~Pure-octaves 8edo approximates only 6 of the first 27 integer harmonics within 20 cents. 1ed148.5c approximates 10 of them.~~

~~In this way, 1ed148.5c (compressed-octaves 8edo) is able to provide a wider and stronger palette of [[consonance]]s compared to pure-octaves 8edo.~~

There are [[zeta peak index]] tunings nearby 8edo as well, however they damage the octave by over 20 cents, rendering them unrecognisable as stretched or compressed 8edo and more like entirely new scales in their own right.

=== Subsets and supersets ===

8edo contains [[2edo]] and [[4edo]] as subsets. Among its supersets are [[16edo]], [[24edo]], [[32edo]], ….

8edo contains [[2edo]] and [[4edo]] as subsets. Among its supersets are [[16edo]], [[24edo]], [[32edo]], … notably including [[72edo]], which expands its 2.11/3.13/5.17/3.19 subgroup into a full 19-limit temperament.

== Intervals ==

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! rowspan="2" | Steps

! rowspan="2" | Cents

! colspan="3" | JI approximation

! colspan="4" | JI approximation

!Other

|-

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! 2.5/3.11/3.13/5*

! 10:11:12:13:14*

!val 8d**

!Patent val ⟨8 13 19]

|-

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| 1/1

|1/1

|1/1, 16/15

|-

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| 12/11

| 12/11, 11/10

|14/13, 13/12, 12/11, 11/10

|10/9, 25/24

|-

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| 6/5

|13/11, 6/5

|6/5, 9/8

|-

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| 13/10

|9/7, 13/10

|4/3, 5/4

|-

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|

| 7/5, 10/7

|7/5, 10/7

|27/20, 25/18, 36/25

|-

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| 20/13

|14/9

|3/2, 8/5

|-

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| 5/3

|5/3

|5/3, 16/9

|-

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| 11/6

| 20/11, 11/6

|11/6, 13/7

|9/5, 48/50

|-

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| 2/1

|2/1

|2/1, 15/8

|}

<nowiki />* Allows [[inversion]] by 2/1; other interpretations also possible

<nowiki />** 16 integer limit, relative error < 15 %

== Notation ==

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|}

== Octave stretch and compression ==

8edo's approximation of [[JI]] can be improved via [[octave shrinking]]. Compressing 8edo's octave from 1200 [[cent]]s down to 1187 cents gives the tuning called [[ed12|29ed12]].

Of all prime [[harmonic]]s up to 31, pure-octave 8edo only manages to approximate 2/1 and 19/1 within 15 [[cents]], completely missing all the others.

By contrast, 29ed12 approximates 2/1, 11/1, 13/1, 17/1 and 31/1 all within 15 cents.

Of all integer harmonics up to 30, pure-octave 8edo approximates the following within 20 cents:

* 2, 4, 8, 16, 19, 27.

Of all integer harmonics up to 30, 29ed12 approximates the following within 20 cents:

* 2, 6, 11, 12, 13, 17, 20, 22, 25, 26.

This provides 29ed12 with a comparatively larger, more diverse palette of [[consonance]]s than pure-octaves 8edo.

The nearest [[zeta peak index]] tunings to 8edo don't have an interval within 20 cents of [[2/1]], making them unrecognisable as stretched or compressed 8edo but instead more like entirely new scales in their own right.

; 8edo

* Step size: 150.000{{c}}, octave size: 1200.000{{c}}

{{Harmonics in equal|8|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 8edo}}

{{Harmonics in equal|8|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 8edo (continued)}}

; [[ed12|29ed12]]

* Step size: 148.343{{c}}, octave size: 1186.746{{c}}

{{Harmonics in equal|29|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 29ed12}}

{{Harmonics in equal|29|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 29ed12 (continued)}}

== Scales ==

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=== Temperaments ===

8edo is fairly composite, so the only step that generates a [[mos]] scale that covers every interval other than the 1 is the 3, producing scales of 332 and [[~~3L_2s~~|21212]]. In terms of temperaments, in the 5-limit this is best interpreted as [[~~Father_family|Father~~]], as 8edo is the highest edo that tempers out the diatonic semitone in it's [[patent val]], merging 5/4 and 4/3 into a single interval, which is also the generator. This means major and minor chords are rotations of each other, making them inaccurate but very simple, with even the 5 note mos having 3 of both and providing a functional skeleton of 5-limit harmony, albeit with some very strange enharmonic equivalences. In terms of 7-limit extensions things get even more inaccurate, as the patent val supports [[~~Father_family#Mother|Mother~~]], but the ideal tuning for that is much closer to [[5edo]]. The d val supports septimal father and [[~~Father_family#Pater|Pater~~]], and is much closer to the ideal tuning for both, as the extremely sharp 7 works better with the 3&5. In terms of multi-period temperaments, it makes for a near perfect [[~~Jubilismic_clan#Walid|Walid~~]] or a much less accurate [[~~Dimipent_family#Diminished~~|~~Diminished~~]] scale.

8edo is fairly composite, so the only step that generates a [[mos]] scale that covers every interval other than the 1 is the 3, producing scales of 332 and [[3L 2s|21212]]. In terms of temperaments, in the 5-limit this is best interpreted as [[father]], as 8edo is the highest edo that tempers out the diatonic semitone in it's [[patent val]], merging 5/4 and 4/3 into a single interval, which is also the generator. This means major and minor chords are rotations of each other, making them inaccurate but very simple, with even the 5 note mos having 3 of both and providing a functional skeleton of 5-limit harmony, albeit with some very strange enharmonic equivalences. In terms of 7-limit extensions things get even more inaccurate, as the patent val supports [[mother]], but the ideal tuning for that is much closer to [[5edo]]. The 8d val supports septimal father and [[pater]], and is much closer to the ideal tuning for both, as the extremely sharp 7 works better with the 3 and 5. In terms of multi-period temperaments, it makes for a near perfect [[walid]] or a much less accurate [[diminished (temperament)|diminished]] scale.

== Instruments ==

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; [[City of the Asleep]]

* [http://ia600607.us.archive.org/3/items/Transcendissonance/10Malebolge-CityOfTheAsleep.mp3 "Malebolge"], from [https://cityoftheasleep.bandcamp.com/album/transcendissonance ''Transcendissonance''] (2011)

; [[Bryan Deister]]

* [https://www.youtube.com/shorts/qk_kMCCXpss ''microtonal improvisation in 8edo''] (2024)

; [[Milan Guštar]]

@@ Line 11: / Line 11: @@
 === Approximation to JI ===
-edo forms an odd and even pitch set of two diminished seventh chords, which when used in combination yield dissonance. The system has been described as a "barbaric" harmonic system, containing no good approximation of harmonics 3, 5, 7, 11, 13, and 17; even so, it does a good job representing the [[just intonation subgroup]]s 2.11/3.13/5, with good intervals of [[13/10]] and an excellent version of [[11/6]]. Stacking the 450-cent interval can result in some semi-consonant chords such as 0-3-6 degrees, although these still are quite dissonant compared to standard root-3rd-P5 triads, which are unavailable in 8edo.
+edo forms an odd and even pitch set of two diminished seventh chords, which when used in combination yield dissonance. The system has been described as a "barbaric" harmonic system, containing no good approximation of harmonics 3, 5, 7, 11, 13, and 17; even so, it does a good job representing the [[just intonation subgroup]] 2.11/3.13/5, with good intervals of [[13/10]] and an excellent version of [[11/6]]. Stacking the 450-cent interval can result in some semi-consonant chords such as 0-3-6 degrees, although these still are quite dissonant compared to standard root-3rd-P5 triads, which are unavailable in 8edo.
-Another way of looking at 8edo is to treat a chord of 0-1-2-3-4 degrees (0-150-300-450-600 cents) as approximating harmonics 10:11:12:13:14 (~0-165-316-454-583 cents), which is not too implausible if you can buy that 12edo is a 5-limit temperament. This interpretation would imply that 121/120, 144/143, 169/168, and hence also 36/35 and 66/65, are tempered out.
+Another way of looking at 8edo is to treat a chord of 0-1-2-3-4 degrees (0-150-300-450-600 cents) as approximating harmonics 10:11:12:13:14 (~0-165-316-454-583 cents), which is not too implausible if you can buy that 12edo is a 5-limit temperament. This interpretation would imply that 121/120, 144/143, 169/168, and hence also 36/35 and 66/65, are tempered out. The corresponding subgroup is 2.5/3.7/3.11/3.13/3. However, some intervals in this chord, such as [[14/11]] and [[7/6]], are tuned quite inaccurately (over 30 cents off). Nonetheless, the 8-form serves as an underlying structure in many [[non-over-1 temperament]]s.
 === Relationship with the father comma ===
@@ Line 26: / Line 26: @@
 === Odd harmonics ===
 {{Harmonics in equal|8|intervals=odd}}
-=== Octave shrinking ===
-edo's approximation of [[JI]] can be improved via [[octave shrinking]]. Compressing 8edo's octave from 1200 [[cent]]s down to 1188 cents gives the tuning called [[1ed148.5c]].
-Pure-octaves 8edo approximates only 2 of the first 11 prime harmonics within 15 cents. 1ed148.5c approximates 5 of them.
-Pure-octaves 8edo approximates only 6 of the first 27 integer harmonics within 20 cents. 1ed148.5c approximates 10 of them.
-In this way, 1ed148.5c (compressed-octaves 8edo) is able to provide a wider and stronger palette of [[consonance]]s compared to pure-octaves 8edo.
-There are [[zeta peak index]] tunings nearby 8edo as well, however they damage the octave by over 20 cents, rendering them unrecognisable as stretched or compressed 8edo and more like entirely new scales in their own right.
 === Subsets and supersets ===
-edo contains [[2edo]] and [[4edo]] as subsets. Among its supersets are [[16edo]], [[24edo]], [[32edo]], ….
+edo contains [[2edo]] and [[4edo]] as subsets. Among its supersets are [[16edo]], [[24edo]], [[32edo]], … notably including [[72edo]], which expands its 2.11/3.13/5.17/3.19 subgroup into a full 19-limit temperament.
 == Intervals ==
@@ Line 46: / Line 35: @@
 ! rowspan="2" | Steps
 ! rowspan="2" | Cents
-! colspan="3" | JI approximation
+! colspan="4" | JI approximation
 !Other
 |-
@@ Line 52: / Line 41: @@
 ! 2.5/3.11/3.13/5*
 ! 10:11:12:13:14*
+!val 8d**
 !Patent val ⟨8 13 19]
 |-
@@ Line 59: / Line 49: @@
 | 1/1
 | 1/1
+|1/1
 |1/1, 16/15
 |-
@@ Line 66: / Line 57: @@
 | 12/11
 | 12/11, 11/10
+|14/13, 13/12, 12/11, 11/10
 |10/9, 25/24
 |-
@@ Line 73: / Line 65: @@
 | 6/5
 | 6/5
+|13/11, 6/5
 |6/5, 9/8
 |-
@@ Line 80: / Line 73: @@
 | 13/10
 | 13/10
+|9/7, 13/10
 |4/3, 5/4
 |-
@@ Line 87: / Line 81: @@
 |
 | 7/5, 10/7
+|7/5, 10/7
 |27/20, 25/18, 36/25
 |-
@@ Line 94: / Line 89: @@
 | 20/13
 | 20/13
+|14/9
 |3/2, 8/5
 |-
@@ Line 101: / Line 97: @@
 | 5/3
 | 5/3
+|5/3
 |5/3, 16/9
 |-
@@ Line 108: / Line 105: @@
 | 11/6
 | 20/11, 11/6
+|11/6, 13/7
 |9/5, 48/50
 |-
@@ Line 115: / Line 113: @@
 | 2/1
 | 2/1
+|2/1
 |2/1, 15/8
 |}
 <nowiki />* Allows [[inversion]] by 2/1; other interpretations also possible
+<nowiki />** 16 integer limit, relative error < 15 %
 == Notation ==
@@ Line 489: / Line 490: @@
 |}
 <references/>
+== Octave stretch and compression ==
+edo's approximation of [[JI]] can be improved via [[octave shrinking]]. Compressing 8edo's octave from 1200 [[cent]]s down to 1187 cents gives the tuning called [[ed12|29ed12]].
+Of all prime [[harmonic]]s up to 31, pure-octave 8edo only manages to approximate 2/1 and 19/1 within 15 [[cents]], completely missing all the others.
+By contrast, 29ed12 approximates 2/1, 11/1, 13/1, 17/1 and 31/1 all within 15 cents.
+Of all integer harmonics up to 30, pure-octave 8edo approximates the following within 20 cents:
+* 2, 4, 8, 16, 19, 27.
+Of all integer harmonics up to 30, 29ed12 approximates the following within 20 cents:
+* 2, 6, 11, 12, 13, 17, 20, 22, 25, 26.
+This provides 29ed12 with a comparatively larger, more diverse palette of [[consonance]]s than pure-octaves 8edo.
+The nearest [[zeta peak index]] tunings to 8edo don't have an interval within 20 cents of [[2/1]], making them unrecognisable as stretched or compressed 8edo but instead more like entirely new scales in their own right.
+; 8edo
+* Step size: 150.000{{c}}, octave size: 1200.000{{c}}
+{{Harmonics in equal|8|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 8edo}}
+{{Harmonics in equal|8|2|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 8edo (continued)}}
+; [[ed12|29ed12]]
+* Step size: 148.343{{c}}, octave size: 1186.746{{c}}
+{{Harmonics in equal|29|12|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 29ed12}}
+{{Harmonics in equal|29|12|1|intervals=integer|columns=12|start=12|collapsed=true|title=Approximation of harmonics in 29ed12 (continued)}}
 == Scales ==
@@ Line 511: / Line 539: @@
 === Temperaments ===
-edo is fairly composite, so the only step that generates a [[mos]] scale that covers every interval other than the 1 is the 3, producing scales of 332 and [[3L_2s|21212]]. In terms of temperaments,  in the 5-limit this is best interpreted as [[Father_family|Father]], as 8edo is the highest edo that tempers out the diatonic semitone in it's [[patent val]], merging 5/4 and 4/3 into a single interval, which is also the generator. This means major and minor chords are rotations of each other, making them inaccurate but very simple, with even the 5 note mos having 3 of both and providing a functional skeleton of 5-limit harmony, albeit with some very strange enharmonic equivalences. In terms of 7-limit extensions things get even more inaccurate, as the patent val supports [[Father_family#Mother|Mother]], but the ideal tuning for that is much closer to [[5edo]]. The d val supports septimal father and [[Father_family#Pater|Pater]], and is much closer to the ideal tuning for both, as the extremely sharp 7 works better with the 3&5. In terms of multi-period temperaments, it makes for a near perfect [[Jubilismic_clan#Walid|Walid]] or a much less accurate [[Dimipent_family#Diminished|Diminished]] scale.
+edo is fairly composite, so the only step that generates a [[mos]] scale that covers every interval other than the 1 is the 3, producing scales of 332 and [[3L 2s|21212]]. In terms of temperaments, in the 5-limit this is best interpreted as [[father]], as 8edo is the highest edo that tempers out the diatonic semitone in it's [[patent val]], merging 5/4 and 4/3 into a single interval, which is also the generator. This means major and minor chords are rotations of each other, making them inaccurate but very simple, with even the 5 note mos having 3 of both and providing a functional skeleton of 5-limit harmony, albeit with some very strange enharmonic equivalences. In terms of 7-limit extensions things get even more inaccurate, as the patent val supports [[mother]], but the ideal tuning for that is much closer to [[5edo]]. The 8d val supports septimal father and [[pater]], and is much closer to the ideal tuning for both, as the extremely sharp 7 works better with the 3 and 5. In terms of multi-period temperaments, it makes for a near perfect [[walid]] or a much less accurate [[diminished (temperament)|diminished]] scale.
 == Instruments ==
@@ Line 528: / Line 556: @@
 ; [[City of the Asleep]]
 * [http://ia600607.us.archive.org/3/items/Transcendissonance/10Malebolge-CityOfTheAsleep.mp3 "Malebolge"], from [https://cityoftheasleep.bandcamp.com/album/transcendissonance ''Transcendissonance''] (2011)
+; [[Bryan Deister]]
+* [https://www.youtube.com/shorts/qk_kMCCXpss ''microtonal improvisation in 8edo''] (2024)
 ; [[Milan Guštar]]