8edo: Difference between revisions
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=== Approximation to JI === | === 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]] | 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 === | === Relationship with the father comma === | ||
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=== Odd harmonics === | === Odd harmonics === | ||
{{Harmonics in equal|8|intervals=odd}} | {{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 == | ||
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== Notation == | == Notation == | ||
=== Ups and downs notation === | === Ups and downs notation === | ||
8edo can be notated as a subset of 24edo, using [[Ups and | 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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== Chord names == | == Chord names == | ||
[[Ups and | [[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. | ||
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== Approximation to JI == | == Approximation to JI == | ||
[[File: | [[File:8ed2-001.svg]] | ||
== Regular temperament properties == | == Regular temperament properties == | ||
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|} | |} | ||
<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 == | ||
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=== 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 [[ | 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 == | ||
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; [[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]] | ||