Xenharmonic Wiki

8edo: Difference between revisions

← Older edit
VisualWikitext
ArrowHead294 (talk | contribs)
15,213 edits
mNo edit summary
Overthink (talk | contribs)
autopatrolled
2,706 edits
m →Temperaments: this isn't an edo join
 
(23 intermediate revisions by 10 users not shown)
Line 9: Line 9:
== Theory ==
== Theory ==
[[File:8edo scale.mp3|thumb|A chromatic 8edo scale on C.]]
[[File:8edo scale.mp3|thumb|A chromatic 8edo scale on C.]]
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.


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.
=== 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]] 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.


=== Odd harmonics ===
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.
{{Harmonics in equal|8|intervals=odd}}


=== Octave shrinking ===
=== Relationship with the father comma ===
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]].
When 8edo is treated as a very inaccurate 5-limit system, it ends up tempering out the [[Father]] comma, [[16/15]]. In fact, it is the largest edo that tempers this comma. What this means is that intervals 16/15 apart in 8edo map to the same note, such as [[4/3]] being mapped to the same note as [[5/4]].


Pure-octaves 8edo approximates only 2 of the first 11 prime harmonics within 15 cents. 1ed148.5c approximates 5 of them.
Some other odd equivalencies include:


Pure-octaves 8edo approximates only 6 of the first 27 integer harmonics within 20 cents. 1ed148.5c approximates 10 of them.
'''0-2-5''': which can be seen as a minor triad (10:12:15), a sus2 triad (9:8:12), or a major triad in first inversion (5:6:8).


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.
'''0-3-5''': which can be seen as a major triad (4:5:6), a sus4 triad (6:8:9), or a minor chord in second inversion (15:20:24).


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.
=== Odd harmonics ===
{{Harmonics in equal|8|intervals=odd}}


=== Subsets and supersets ===
=== 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 ==
== Intervals ==
Line 36: Line 36:
! rowspan="2" | Cents
! rowspan="2" | Cents
! colspan="3" | JI approximation
! colspan="3" | JI approximation
!Other
|-
|-
! 2.11/3.13/5.19*
! 2.11/3.13/5.19*
! 2.5/3.11/3.13/5*
! 2.5/3.11/3.13/5*
! 10:11:12:13:14*
! 10:11:12:13:14*
!Patent val ⟨8 13 19]
|-
|-
| 0
| 0
Line 46: Line 48:
| 1/1
| 1/1
| 1/1
| 1/1
|1/1, 16/15
|-
|-
| 1
| 1
Line 52: Line 55:
| 12/11
| 12/11
| 12/11, 11/10
| 12/11, 11/10
|10/9, 25/24
|-
|-
| 2
| 2
Line 58: Line 62:
| 6/5
| 6/5
| 6/5
| 6/5
|6/5, 9/8
|-
|-
| 3
| 3
Line 64: Line 69:
| 13/10
| 13/10
| 13/10
| 13/10
|4/3, 5/4
|-
|-
| 4
| 4
Line 70: Line 76:
|  
|  
| 7/5, 10/7
| 7/5, 10/7
|27/20, 25/18, 36/25
|-
|-
| 5
| 5
Line 76: Line 83:
| 20/13
| 20/13
| 20/13
| 20/13
|3/2, 8/5
|-
|-
| 6
| 6
Line 82: Line 90:
| 5/3
| 5/3
| 5/3
| 5/3
|5/3, 16/9
|-
|-
| 7
| 7
Line 88: Line 97:
| 11/6
| 11/6
| 20/11, 11/6
| 20/11, 11/6
|9/5, 48/50
|-
|-
| 8
| 8
Line 94: Line 104:
| 2/1
| 2/1
| 2/1
| 2/1
|2/1, 15/8
|}
|}
<nowiki />* Allows [[inversion]] by 2/1; other interpretations also possible
<nowiki />* Allows [[inversion]] by 2/1; other interpretations also possible
Line 99: Line 110:
== Notation ==
== Notation ==
=== Ups and downs notation ===
=== Ups and downs notation ===
8edo can be notated as a subset of 24edo, using [[Ups and Downs Notation|ups and downs]]. It can also be notated as a subset of 16edo, but this is a less intuitive notation.
8edo can be notated as a subset of 24edo, using [[Ups and downs notation|ups and downs]]. It can also be notated as a subset of 16edo, but this is a less intuitive notation.


{| class="wikitable center-all"
{| class="wikitable center-all"
Line 281: Line 292:


== Chord names ==
== Chord names ==
[[Ups and Downs Notation #Chords and Chord Progressions|Ups and downs]] can name any 8edo chord. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).  
[[Ups and downs notation #Chords and Chord Progressions|Ups and downs]] can name any 8edo chord. Alterations are always enclosed in parentheses, additions never are. An up, down or mid immediately after the chord root affects the 3rd, 6th, 7th, and/or the 11th (every other note of a stacked-3rds chord 6-1-3-5-7-9-11-13).  


8edo chords are very ambiguous, with many chord homonyms. Even the major and minor triads are [[Chord homonym|homonyms]]. Chord components usually default to M2, M3, P4, P5, M6, m7, M9, P11 and M13. Thus D7 has a M3, P5 and m7. 8-edo chord names using 24edo subset names are greatly simplified by using different defaults: instead of the conventional M2, M3, P4, P5, M6, m7, M9, P11 and M13, we have  ~2, ^M3, v4, ^5, M6, ~7, ~9, v11 and M13. Thus D7 becomes ^M3, ^5 and ~7.
8edo chords are very ambiguous, with many chord homonyms. Even the major and minor triads are [[Chord homonym|homonyms]]. Chord components usually default to M2, M3, P4, P5, M6, m7, M9, P11 and M13. Thus D7 has a M3, P5 and m7. 8-edo chord names using 24edo subset names are greatly simplified by using different defaults: instead of the conventional M2, M3, P4, P5, M6, m7, M9, P11 and M13, we have  ~2, ^M3, v4, ^5, M6, ~7, ~9, v11 and M13. Thus D7 becomes ^M3, ^5 and ~7.
Line 337: Line 348:


== Approximation to JI ==
== Approximation to JI ==
[[File:8ed2-001.svg|alt=alt : Your browser has no SVG support.]]
[[File:8ed2-001.svg]]
 
[[:File:8ed2-001.svg|8ed2-001.svg]]


== Regular temperament properties ==
== Regular temperament properties ==
=== Uniform maps ===
=== Uniform maps ===
{{Uniform map|13|7.5|8.5}}
{{Uniform map|edo=8}}


=== Commas ===
=== Commas ===
Line 469: Line 478:
|}
|}
<references/>
<references/>
== 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 ==
== Scales ==
Line 491: Line 527:


=== Temperaments ===
=== 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 ==
A [[Lumatone mapping for 8edo]] is available.


== Music ==
== Music ==
Line 505: Line 544:
; [[City of the Asleep]]
; [[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)
* [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]]
; [[Milan Guštar]]
Line 528: Line 570:
; [[Carlo Serafini]]
; [[Carlo Serafini]]
* ''FunkEight 1'' (2013) – [http://www.seraph.it/blog_files/86557ece0348a8d042346d5e1b13e2d9-166.html blog] | [http://www.seraph.it/dep/det/FunkEight1.mp3 play]
* ''FunkEight 1'' (2013) – [http://www.seraph.it/blog_files/86557ece0348a8d042346d5e1b13e2d9-166.html blog] | [http://www.seraph.it/dep/det/FunkEight1.mp3 play]
; [[Sevish]]
* [https://youtu.be/1fpYEVEdcaE ''Reckoner''] (2025)


; [[Jake Sherman]]
; [[Jake Sherman]]
Line 541: Line 586:
; [[Randy Winchester]]
; [[Randy Winchester]]
* [https://archive.org/details/jamendo-005173/07.mp3 "7. 8 / octave"], from ''[[Comets Over Flatland]]'' (2007)
* [https://archive.org/details/jamendo-005173/07.mp3 "7. 8 / octave"], from ''[[Comets Over Flatland]]'' (2007)
; [[User:Fitzgerald_Lee|Fitzgerald Lee]]
* [https://youtu.be/zwRDfjLzXkU Jonky Jazz] (2025)


== Ear training ==
== Ear training ==
Line 547: Line 595:
== See also ==
== See also ==
* [[Octatonic scale]] - a scale based on alternating whole and half steps
* [[Octatonic scale]] - a scale based on alternating whole and half steps
* [[Fendo family]] - temperaments closely related to 8edo


[[Category:8edo| ]] <!-- main article -->
[[Category:Equal divisions of the octave|#]] <!-- 1-digit number -->
[[Category:8-tone scales]]
[[Category:8-tone scales]]
[[Category:Listen]]
[[Category:Listen]]
[[Category:Macrotonal]]
Retrieved from "https://en.xen.wiki/w/8edo"