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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=== 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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Line 49:

| 1/1

|1/1

|1/1, 16/15

|-

Line 55:

Line 57:

| 12/11

| 12/11, 11/10

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

|10/9, 25/24

|-

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Line 65:

| 6/5

|13/11, 6/5

|6/5, 9/8

|-

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Line 73:

| 13/10

|9/7, 13/10

|4/3, 5/4

|-

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Line 81:

|

| 7/5, 10/7

|7/5, 10/7

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

|-

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Line 89:

| 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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~~== Scales ==~~

== Octave stretch and compression ==

~~=== Scala file ===~~

~~Here is a .scl file of 8edo: [[:File:08-EDO.scl|08-edo.scl]]~~

~~<pre>~~

~~! 08-EDO.scl~~

!

~~8 EDO~~

8

!

~~150.00~~

~~300.00~~

~~450.00~~

~~600.00~~

~~750.00~~

~~900.00~~

~~1050.00~~

~~2/1~~

~~</pre>~~

~~=== 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]], 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 {{nowrap| 3 & 5 }}. In terms of multi-period temperaments, it makes for a near perfect [[walid]] or a much less accurate [[diminished (temperament)|diminished]] scale.

=== 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]].

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; 8edo

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

{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 8edo}}

{{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)}}

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{{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 ==

=== Scala file ===

Here is a .scl file of 8edo: [[:File:08-EDO.scl|08-edo.scl]]

<pre>

! 08-EDO.scl

!

8 EDO

8

!

150.00

300.00

450.00

600.00

750.00

900.00

1050.00

2/1

</pre>

=== 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]], 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 544:

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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 28: / Line 28: @@
 === 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 35: / Line 35: @@
 ! rowspan="2" | Steps
 ! rowspan="2" | Cents
-! colspan="3" | JI approximation
+! colspan="4" | JI approximation
 !Other
 |-
@@ Line 41: / Line 41: @@
 ! 2.5/3.11/3.13/5*
 ! 10:11:12:13:14*
+!val 8d**
 !Patent val ⟨8 13 19]
 |-
@@ Line 48: / Line 49: @@
 | 1/1
 | 1/1
+|1/1
 |1/1, 16/15
 |-
@@ Line 55: / Line 57: @@
 | 12/11
 | 12/11, 11/10
+|14/13, 13/12, 12/11, 11/10
 |10/9, 25/24
 |-
@@ Line 62: / Line 65: @@
 | 6/5
 | 6/5
+|13/11, 6/5
 |6/5, 9/8
 |-
@@ Line 69: / Line 73: @@
 | 13/10
 | 13/10
+|9/7, 13/10
 |4/3, 5/4
 |-
@@ Line 76: / Line 81: @@
 |
 | 7/5, 10/7
+|7/5, 10/7
 |27/20, 25/18, 36/25
 |-
@@ Line 83: / Line 89: @@
 | 20/13
 | 20/13
+|14/9
 |3/2, 8/5
 |-
@@ Line 90: / Line 97: @@
 | 5/3
 | 5/3
+|5/3
 |5/3, 16/9
 |-
@@ Line 97: / Line 105: @@
 | 11/6
 | 20/11, 11/6
+|11/6, 13/7
 |9/5, 48/50
 |-
@@ Line 104: / 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 479: / Line 491: @@
 <references/>
-== Scales ==
+== Octave stretch and compression ==
-=== Scala file ===
-Here is a .scl file of 8edo: [[:File:08-EDO.scl|08-edo.scl]]
-<pre>
-! 08-EDO.scl
-!
-EDO
-!
-.00
-.00
-.00
-.00
-.00
-.00
-.00
-/1
-</pre>
-=== 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]], 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 {{nowrap| 3 & 5 }}. In terms of multi-period temperaments, it makes for a near perfect [[walid]] or a much less accurate [[diminished (temperament)|diminished]] scale.
-=== 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]].
@@ Line 521: / Line 510: @@
 ; 8edo
 * Step size: 150.000{{c}}, octave size: 1200.000{{c}}
-{{Harmonics in equal|12|2|1|intervals=integer|columns=11|collapsed=true|title=Approximation of harmonics in 8edo}}
+{{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)}}
@@ Line 528: / Line 517: @@
 {{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 ==
+=== Scala file ===
+Here is a .scl file of 8edo: [[:File:08-EDO.scl|08-edo.scl]]
+<pre>
+! 08-EDO.scl
+!
+EDO
+!
+.00
+.00
+.00
+.00
+.00
+.00
+.00
+/1
+</pre>
+=== 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]], 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 544: / 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]]