8edo: Difference between revisions
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=== Octave shrinking === | === Octave shrinking === | ||
8edo's approximation of [[JI]] can be improved via [[octave shrinking]]. Compressing 8edo's octave from 1200 [[cent]]s down to | 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 === | === Subsets and supersets === | ||
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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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=== 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 {{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 == | == Music == | ||