8edo: Difference between revisions
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'''8 equal divisions of the octave''' (''' | '''8 equal divisions of the octave''' ('''8edo''') is the [[tuning system]] derived by dividing the [[octave]] into 8 equal steps of 150 [[cent]]s each, or the eighth root of 2. | ||
= Theory = | == Theory == | ||
{| | {{primes in equal|8}} | ||
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; 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]]. | |||
Another way of looking at | 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. | ||
= Notation = | == 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" | ||
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! [[Cent]]s | ! [[Cent]]s | ||
! [[7mu]]s ([[Wikipedia: hexadecimal |hex]]) | ! [[7mu]]s ([[Wikipedia: hexadecimal |hex]]) | ||
! colspan="2" | | ! colspan="2" | 24edo subset notation | ||
! colspan="2" | | ! colspan="2" | 16edo subset notation | ||
|- | |- | ||
| 0 | | 0 | ||
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genchain of 2nds: ...A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8... | genchain of 2nds: ...A1 - A2 - M3 - M4 - M5 - M6 - P7 - P1 - P2 - m3 - m4 - m5 - m6 - d7 - d8... | ||
== Chord Names == | === Chord Names === | ||
[[Ups and Downs Notation #Chords and Chord Progressions|Ups and downs]] can name any | [[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 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: ~2, ^M3, v4, ^5, M6, ~7, ~9, v11 and M13. Thus D7 becomes ^M3, ^5 and ~7. | |||
{| class="wikitable" | {| class="wikitable" | ||
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|} | |} | ||
= JI Intervals = | == JI Intervals == | ||
[[File:8ed2-001.svg|alt=alt : Your browser has no SVG support.]] | [[File:8ed2-001.svg|alt=alt : Your browser has no SVG support.]] | ||
[[:File:8ed2-001.svg|8ed2-001.svg]] | [[:File:8ed2-001.svg|8ed2-001.svg]] | ||
= Commas = | == Commas == | ||
8edo [[tempers out]] the following [[comma]]s. This assumes [[val]] {{val| 8 13 19 22 28 30 }}. | |||
{| class="commatable wikitable center-all left-3 right-5 left-6" | {| class="commatable wikitable center-all left-3 right-5 left-6" | ||
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<references/> | <references/> | ||
= Scala scale file = | == Scala scale file == | ||
For those who use the new tuner Lingot, which accepts scala files, or for anyone else, here is a .scl file of | For those who use the new tuner Lingot, which accepts scala files, or for anyone else, here is a .scl file of 8edo [[:File:08-edo.scl|08-edo.scl]] | ||
0. 1/1 C Unison | 0. 1/1 C Unison | ||
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8. 2/1 C Octave | 8. 2/1 C Octave | ||
== Pathological Modes == | === Pathological Modes === | ||
2 1 1 1 1 1 1 [[1L 6s]] MOS | 2 1 1 1 1 1 1 [[1L 6s]] MOS | ||
= | == Music == | ||
[https://soundcloud.com/overtoneshock/tenacious-chorale-9-edo-and-8-edo-live?in=overtoneshock/sets/xenharmonic-microtonal Tenacious Chorale (only movement II is in 8EDO)] by Stephen Weigel | [https://soundcloud.com/overtoneshock/tenacious-chorale-9-edo-and-8-edo-live?in=overtoneshock/sets/xenharmonic-microtonal Tenacious Chorale (only movement II is in 8EDO)] by Stephen Weigel | ||
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[[:File:octo-icy-pensive(sketch).ogg|octo-icy-pensive(sketch).ogg]] [[:File:octo-icy-pensive-echo(sketch).ogg|octo-icy-pensive-echo(sketch).ogg]]2 versions of the same song, one with echo, one without. (cenobyte) | [[:File:octo-icy-pensive(sketch).ogg|octo-icy-pensive(sketch).ogg]] [[:File:octo-icy-pensive-echo(sketch).ogg|octo-icy-pensive-echo(sketch).ogg]]2 versions of the same song, one with echo, one without. (cenobyte) | ||
= Ear Training = | == 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 = | == See also == | ||
* [[Octatonic scale]] - a scale based on alternating whole and half steps | * [[Octatonic scale]] - a scale based on alternating whole and half steps | ||