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8edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{interwiki
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| de = 8edo
: This revision was by author [[User:hstraub|hstraub]] and made on <tt>2008-04-08 05:38:42 UTC</tt>.<br>
| en = 8edo
: The original revision id was <tt>21204965</tt>.<br>
| es =
: The revision comment was: <tt></tt><br>
| ja = 8平均律
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
}}
<h4>Original Wikitext content:</h4>
{{Infobox ET}}
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">=Compositions=  
{{ED intro}}
== Theory ==
[[File:8edo scale.mp3|thumb|A chromatic 8edo scale on C.]]


[[http://www.h-pi.com/mp3/Fantasia8ET.mp3|Fantasia in 8ET]] by Aaron Andrew Hunt
=== Approximation to JI ===
[[http://www.h-pi.com/mp3/Fugue8ET.mp3|Fugue in 8ET]] by Aaron Andrew Hunt</pre></div>
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.
<h4>Original HTML content:</h4>
 
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;8edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Compositions"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Compositions&lt;/h1&gt;
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.
&lt;br /&gt;
 
&lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Fantasia8ET.mp3" rel="nofollow"&gt;Fantasia in 8ET&lt;/a&gt; by Aaron Andrew Hunt&lt;br /&gt;
=== Relationship with the father comma ===
&lt;a class="wiki_link_ext" href="http://www.h-pi.com/mp3/Fugue8ET.mp3" rel="nofollow"&gt;Fugue in 8ET&lt;/a&gt; by Aaron Andrew Hunt&lt;/body&gt;&lt;/html&gt;</pre></div>
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]].
 
Some other odd equivalencies include:
 
'''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).
 
'''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).
 
=== Odd harmonics ===
{{Harmonics in equal|8|intervals=odd}}
 
=== Octave shrinking ===
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 [[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.
 
=== Subsets and supersets ===
8edo contains [[2edo]] and [[4edo]] as subsets. Among its supersets are [[16edo]], [[24edo]], [[32edo]], ….
 
== Intervals ==
{| class="wikitable"
|-
! rowspan="2" | Steps
! rowspan="2" | Cents
! colspan="3" | JI approximation
!Other
|-
! 2.11/3.13/5.19*
! 2.5/3.11/3.13/5*
! 10:11:12:13:14*
!Patent val ⟨8 13 19]
|-
| 0
| 0
| 1/1
| 1/1
| 1/1
|1/1, 16/15
|-
| 1
| 150
| 12/11
| 12/11
| 12/11, 11/10
|10/9, 25/24
|-
| 2
| 300
| 19/16
| 6/5
| 6/5
|6/5, 9/8
|-
| 3
| 450
| 13/10
| 13/10
| 13/10
|4/3, 5/4
|-
| 4
| 600
|
|
| 7/5, 10/7
|27/20, 25/18, 36/25
|-
| 5
| 750
| 20/13
| 20/13
| 20/13
|3/2, 8/5
|-
| 6
| 900
| 32/19
| 5/3
| 5/3
|5/3, 16/9
|-
| 7
| 1050
| 11/6
| 11/6
| 20/11, 11/6
|9/5, 48/50
|-
| 8
| 1200
| 2/1
| 2/1
| 2/1
|2/1, 15/8
|}
<nowiki />* Allows [[inversion]] by 2/1; other interpretations also possible
 
== 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.
 
{| class="wikitable center-all"
|-
! Edostep
! [[Cent]]s
! colspan="2" | 24edo subset notation<br />([[Enharmonic unisons in ups and downs notation|EUs:]] vvA1 and d2)
! colspan="2" | 16edo subset notation<br />(major narrower than minor)
! colspan="2" | 16edo subset notation<br />(major wider than minor)
! [[3L&nbsp;2s]] notation<br />({{nowrap|J {{=}} 1/1}})
! Audio
|-
| 0
| 0¢
| P1
| D
| P1
| D
| P1
| D
| J
| [[File:0-0 unison.mp3|frameless]]
|-
| 1
| 150
| ~2
| vE
| M2
| E
| m2
| E
| K
| [[File:0-150 (8-EDO).mp3|frameless]]
|-
| 2
| 300
| m3
| F
| M3
| F#
| m3
| Fb
| K#, Lb
| [[File:0-300 (12-EDO).mp3|frameless]]
|-
| 3
| 450
| ^M3 / v4
| ^F# / vG
| d3 / A4
| Fb / G#
| A3, D4
| F#, Gb
| L
| [[File:0-450 (8-EDO).mp3|frameless]]
|-
| 4
| 600
| A4, d5
| G#, Ab
| d4, A5
| Gb, A#
| A4, D5
| G#, Ab
| M
| [[File:0-600 (12-EDO).mp3|frameless]]
|-
| 5
| 750
| ^5, vm6
| ^A, vBb
| d5, A6
| Ab, B#
| A5, d6
| A#, Bb
| M#, Nb
| [[File:0-750 (8-EDO).mp3|frameless]]
|-
| 6
| 900
| M6
| B
| m6
| Bb
| M6
| B#
| N
| [[File:0-900 (12-EDO).mp3|frameless]]
|-
| 7
| 1050
| ~7
| ^C
| m7
| C
| M7
| C
| N#, Jb
| [[File:0-1050 (8-EDO).mp3|frameless]]
|-
| 8
| 1200
| P8
| D
| P8
| D
| P8
| D
| J
| [[File:0-1200 octave.mp3|frameless]]
|}
 
This is a heptatonic notation generated by 5ths (5th meaning 3/2). Alternative notations include pentatonic 5th-generated, octatonic, and heptatonic 2nd-generated.
 
'''<u>Pentatonic 5th-generated</u>: D * E G * A C * D'''  (generator = 5\8 = perfect 5thoid)
 
D - D#/Eb - E - G - G#/Ab - A - C - C#/Db - D
 
P1 - A1/ms3 - Ms3 - P4d - A4d/d5d - P5d - ms7 - Ms7/d8d - P8d  (s = sub-, d = -oid, see [[5edo#Alternative notations|5edo]] notation)
 
pentatonic genchain of 5ths: ...Cb - Gb - Db - Ab - Eb - C - G - D - A - E - C# - G# - D# - A# - E#...
 
pentatonic genchain of 5ths: ...d8d - d5d - ms3 - ms7 - P4d - P1 - P5d - Ms3 - Ms7 - A4d - A1... (s = sub-, d = -oid)
 
[[Enharmonic unison]]: ds3
 
'''<u>Octatonic:</u>''' '''A B C D E F G H A'''  (every interval is a generator)
 
P1 - P2 - P3 - P4 - P5 - P6 - P7 - P8 - P9
 
Enharmonic unison: none
 
'''<u>Heptatonic 2nd-generated</u>: D E F G * A B C D'''  (generator = 1\8 = perfect 2nd = 150¢)
 
D - E - F - G - G#/Ab - A -B - C - D
 
P1 - P2 - m3 - M3/m4 - M4/m5 - M5/m6 - M6 - P7 - P8
 
genchain of 2nds: ...D# - E# - F# - G# - A - B - C - D - E - F - G - Ab - Bb - Cb - Db...
 
genchain of 2nds: ...A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8...
 
Enharmonic unison: d2
 
===Sagittal notation===
This notation is a subset of the notations for EDOs [[24edo#Sagittal notation|24]], [[48edo#Sagittal notation|48]], and [[72edo#Sagittal notation|72]].
====Evo flavor====
 
<imagemap>
File:8-EDO_Evo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 392 0 552 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 392 106 [[24-EDO#Sagittal_notation | 24-EDO notation]]
default [[File:8-EDO_Evo_Sagittal.svg]]
</imagemap>
 
====Revo flavor====
 
<imagemap>
File:8-EDO_Revo_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 400 0 560 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 400 106 [[24-EDO#Sagittal_notation | 24-EDO notation]]
default [[File:8-EDO_Revo_Sagittal.svg]]
</imagemap>
 
====Evo-SZ flavor====
 
<imagemap>
File:8-EDO_Evo-SZ_Sagittal.svg
desc none
rect 80 0 300 50 [[Sagittal_notation]]
rect 376 0 536 80 [https://sagittal.org#periodic-table Periodic table of EDOs with sagittal notation]
rect 20 80 376 106 [[24-EDO#Sagittal_notation | 24-EDO notation]]
default [[File:8-EDO_Evo-SZ_Sagittal.svg]]
</imagemap>
 
Because it contains no Sagittal symbols, this Evo-SZ Sagittal notation is also a Stein-Zimmerman notation.
 
== 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).
 
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.
 
{| class="wikitable"
|-
! Chord edosteps
! Chord notes
! Full name
! Abbreviated name
! Homonyms
|-
| 0 – 3 – 5
| D ^F♯ ^A
| D^(^5)
| D
| ^F♯m or vGm
|-
| 0 – 2 – 5
| D F ^A
| Dm(^5)
| Dm
| ^A or vB♭
|-
| 0 – 3 – 5 – 7
| D ^F♯ ^A ^C
| D^7(^5)
| D7
| ^F♯m♯11 or vGm♯11
|-
| 0 – 3 – 5 – 6
| D ^F♯ ^A B
| D6(^3,^5)
| D6
| Bm7 and vG,♯9
|-
| 0 – 2 – 5 – 7
| D F ^A ^C
| Dm,~7(^5)
| Dm7
| F6 and vB♭,♯9
|-
| 0 – 2 – 5 – 6
| D F ^A B
| Dm6(^5)
| Dm6
| Bm7(♭5)
|-
| 0 – 2 – 4 – 7
| D F A♭ ^C
| Ddim,~7
| Dm7(♭5)
| Fm6
|}
 
== Approximation to JI ==
[[File:8ed2-001.svg]]
 
== Regular temperament properties ==
=== Uniform maps ===
{{Uniform map|edo=8}}
 
=== Commas ===
8et [[tempering out|tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 8 13 19 22 28 30 }}.
 
{| class="commatable wikitable center-all left-3 right-4 left-6"
! [[Harmonic limit|Prime<br />limit]]
! [[Ratio]]<ref>Ratios longer than 10 digits are presented by placeholders with informative hints</ref>
! [[Monzo]]
! [[Cent]]s
! [[Color name]]
! Name(s)
|-
| 3
| [[8192/6561]]
| {{monzo| 13 -8 }}
| 384.35
| sawa 4th
| Pythagorean diminished fourth
|-
| 5
| [[16/15]]
| {{monzo| 4 -1 -1 }}
| 111.73
| gu 2nd
| Father comma, classic limma
|-
| 5
| [[648/625]]
| {{monzo| 3 4 -4 }}
| 62.57
| Quadgu
| Diminished comma, major diesis
|-
| 5
| [[250/243]]
| {{monzo| 1 -5 3 }}
| 49.17
| Triyo
| Porcupine comma, maximal diesis
|-
| 5
| [[78732/78125]]
| {{monzo| 2 9 -7 }}
| 13.40
| Sepgu
| Sensipent comma, medium semicomma
|-
| 7
| [[64/63]]
| {{monzo| 6 -2 0 -1 }}
| 27.26
| Ru
| Septimal comma, Archytas' comma, Leipziger Komma
|-
| 7
| [[875/864]]
| {{monzo| -5 -3 3 1 }}
| 21.90
| Zotriyo
| Keema
|-
| 7
| <abbr title="321489/320000">(12 digits)</abbr>
| {{monzo| -9 8 -4 2 }}
| 8.04
| Labizogugu
| [[Varunisma]]
|-
| 7
| [[6144/6125]]
| {{monzo| 11 1 -3 -2 }}
| 5.36
| Sarurutrigu
| Porwell comma
|-
| 11
| [[100/99]]
| {{monzo| 2 -2 2 0 -1 }}
| 17.40
| Luyoyo
| Ptolemisma
|-
| 11
| [[121/120]]
| {{monzo| -3 -1 -1 0 2 }}
| 14.37
| Lologu
| Biyatisma
|-
| 11
| [[176/175]]
| {{monzo| 4 0 -2 -1 1 }}
| 9.86
| Lorugugu
| Valinorsma
|-
| 11
| [[65536/65219]]
| {{monzo| 16 0 0 -2 -3 }}
| 8.39
| Satrilu-aruru
| Orgonisma
|-
| 11
| [[385/384]]
| {{monzo| -7 -1 1 1 1 }}
| 4.50
| Lozoyo
| Keenanisma
|-
| 11
| [[4000/3993]]
| {{monzo| 5 -1 3 0 -3 }}
| 3.03
| Triluyo
| Wizardharry comma
|-
| 13
| [[40/39]]
| {{monzo| 3 -1 1 0 0 -1 }}
| 43.83
| Thuyo unison
| Tridecimal minor diesis
|}
<references/>
 
== 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 {{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.
 
== Instruments ==
A [[Lumatone mapping for 8edo]] is available.
 
== Music ==
; [[Abnormality]]
* [https://www.youtube.com/watch?v=tqgRYQxsA9U ''mushroom''] (2024)
 
; [[Cenobyte]]
* [[:File:8edo-pre-improv.ogg|8edo-pre-improv]] (2011)
* [[:File:darkreflections(sketch).ogg|darkreflections(sketch)]] (2011)
* [[:File:skiphop(sketch).ogg|skiphop(sketch)]] (2011)
* [[:File:octo-icy-pensive(sketch).ogg|octo-icy-pensive(sketch)]], [[:File:octo-icy-pensive-echo(sketch).ogg|octo-icy-pensive-echo(sketch)]] (2011)
 
; [[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)
 
; [[Milan Guštar]]
* [http://www.uvnitr.cz/flaoyg/forgotten_works/spendliky.html ''Špendlíky''] (1988-2005)
 
; [[Hideya]]
* [https://www.youtube.com/watch?v=dxjBt4Umbw4 ''Like yamashiro''] (2021)
* [https://www.youtube.com/watch?v=h75K1KOb5is ''Like Ensor's paintings''] (2023)
 
; [[Nathan Ho]]
* [https://www.youtube.com/watch?v=DR1SSW-kQqs ''8edo minimal techno in SuperCollider''] (2023)
 
; [[Aaron Andrew Hunt]]
* [https://aaronandrewhunt.bandcamp.com/track/fantasia-fugue-a4-in-8et "Fantasia & Fugue a4 in 8ET"], from [https://aaronandrewhunt.bandcamp.com/album/the-equal-tempered-keyboard ''The Equal-Tempered Keyboard''] (1999-2022)
 
; [[NullPointerException Music]]
* [https://www.youtube.com/watch?v=CryT-HTzrZE "Disorient"], from [https://www.youtube.com/playlist?list=PLg1YtcJbLxnwTJkG4m0BWZWxIHj7ScdNn ''Edolian''] (2020)
* [https://www.youtube.com/watch?v=AErBW21bljk ''Towards Dissolution''] (2020)
 
; [[User:Phanomium|Phanomium]]
* [https://www.youtube.com/watch?v=22OKO6e6nGk ''Octahedron''] (2024)
 
; [[Carlo Serafini]]
* ''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]]
* [https://www.youtube.com/watch?v=0mpDeL7iybI ''News To Me''] (2011)
* [https://www.youtube.com/watch?v=HvDEoZ4LIdc ''Tomorrow Night (pop music from 2045)''] (2011)
 
; [[Ron Sword]]
* [http://www.ronsword.com/sounds/ronsword_8edo_improv.mp3 ''Acoustic Improvisation in 8EDO'']{{dead link}}
 
; [[Stephen Weigel]]
* [https://soundcloud.com/overtoneshock/tenacious-chorale-9-edo-and-8-edo-live ''Tenacious Chorale'': Movement 2] (2016)
 
; [[Randy Winchester]]
* [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 ==
8edo ear-training exercises by Alex Ness available [https://drive.google.com/a/playgroundsessions.com/folderview?id=0BwsXD8q2VCYUamtVWEgyRFA5alU&usp=sharing#list here].
 
== See also ==
* [[Octatonic scale]] - a scale based on alternating whole and half steps
* [[Fendo family]] - temperaments closely related to 8edo
 
[[Category:8-tone scales]]
[[Category:Listen]]
Retrieved from "https://en.xen.wiki/w/8edo"