89edo: Difference between revisions
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{{Infobox ET | {{Infobox ET}} | ||
{{ED intro}} | |||
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== Theory == | == Theory == | ||
89edo has a [[3/1|harmonic 3]] less than a cent flat and a [[5/1|harmonic 5]] less than five cents sharp, with a [[7/1|7]] two cents sharp and an [[11/1|11]] 1.5 cents sharp. It thus delivers reasonably good 11-limit harmony and very good no-fives harmony along with the very useful approximations represented by its commas. On a related note, a notable characteristic of this edo is that it is the lowest in a series of four consecutive edos to temper out [[quartisma]]. | |||
It [[tempering out|tempers out]] 32805/32768 ([[schisma]]) in the 5-limit; [[126/125]], [[1728/1715]], and [[2401/2400]] in the 7-limit; and [[176/175]], [[243/242]], [[441/440]] and [[540/539]] in the 11-limit. It is an especially good tuning for the [[myna]] temperament, both in the [[7-limit]], tempering out 126/125 and 1728/1715, and in the [[11-limit]], where 176/175 is tempered out also. It is likewise a good tuning for the rank-3 temperament [[thrush]], tempering out 126/125 and 176/175. | |||
The [[13-limit]] is a little tricky as [[13/1|13]] is tuned distinctly flat, tempering out [[169/168]], [[364/363]], [[729/728]], [[832/825]], and [[1287/1280]]. [[13/10]] and [[15/13]] are particularly out of tune in this system, each being about 9 cents off. The alternative 89f val fixes that but tunes [[13/8]] much sharper, conflating it with [[18/11]]. It tempers out [[144/143]], [[196/195]], [[351/350]], and [[352/351]] instead, and [[support]]s 13-limit myna and thrush. However, [[58edo]] is a better tuning for those purposes. | |||
The [[17/1|17]] and [[19/1|19]] are tuned fairly well, making it [[consistent]] to the no-13 [[21-odd-limit]]. The equal temperament tempers out [[256/255]] and [[561/560]] in the 17-limit; and [[171/170]], [[361/360]], [[513/512]], and [[1216/1215]] in the 19-limit. | |||
89edo is | 89edo is the 11th in the {{w|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. 1200{{nowrap|(φ − 1)}}{{c}} within a fraction of a cent. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|89}} | {{Harmonics in equal|89}} | ||
=== Subsets and supersets === | |||
89edo is the 24th [[prime edo]], following [[83edo]] and before [[97edo]]. | |||
== Intervals == | |||
{{Interval table}} | |||
== Notation == | |||
=== Ups and downs notation === | |||
89edo can be notated using [[ups and downs notation]] using [[Helmholtz–Ellis]] accidentals: | |||
{{Sharpness-sharp8}} | |||
== Approximation to JI == | |||
=== Zeta peak index === | |||
{{ZPI | |||
| zpi = 497 | |||
| steps = 89.0229355804124 | |||
| step size = 13.4796723133902 | |||
| tempered height = 7.567368 | |||
| pure height = 7.158697 | |||
| integral = 1.124501 | |||
| gap = 16.042570 | |||
| octave = 1199.69083589172 | |||
| consistent = 12 | |||
| distinct = 12 | |||
}} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | |- | ||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
| Line 31: | Line 57: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -141 89 }} | | {{monzo| -141 89 }} | ||
| | | {{mapping| 89 141 }} | ||
| +0.262 | | +0.262 | ||
| 0.262 | | 0.262 | ||
| Line 38: | Line 64: | ||
| 2.3.5 | | 2.3.5 | ||
| 32805/32768, 10077696/9765625 | | 32805/32768, 10077696/9765625 | ||
| | | {{mapping| 89 141 207 }} | ||
| | | −0.500 | ||
| 1.098 | | 1.098 | ||
| 8.15 | | 8.15 | ||
| Line 45: | Line 71: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 126/125, 1728/1715, 32805/32768 | | 126/125, 1728/1715, 32805/32768 | ||
| | | {{mapping| 89 141 207 250 }} | ||
| | | −0.550 | ||
| 0.955 | | 0.955 | ||
| 7.08 | | 7.08 | ||
| Line 52: | Line 78: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 126/125, 176/175, 243/242, 16384/16335 | | 126/125, 176/175, 243/242, 16384/16335 | ||
| | | {{mapping| 89 141 207 250 308 }} | ||
| | | −0.526 | ||
| 0.855 | | 0.855 | ||
| 6.35 | | 6.35 | ||
|} | |} | ||
=== Rank-2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br />per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br />ratio* | |||
! Temperaments | |||
|- | |||
| 1 | |||
| 13\89 | |||
| 175.28 | |||
| 72/65 | |||
| [[Sesquiquartififths]] / [[sesquart]] | |||
|- | |||
| 1 | |||
| 21\89 | |||
| 283.15 | |||
| 13/11 | |||
| [[Neominor]] | |||
|- | |||
| 1 | |||
| 23\89 | |||
| 310.11 | |||
| 6/5 | |||
| [[Myna]] | |||
|- | |||
| 1 | |||
| 29\89 | |||
| 391.01 | |||
| 5/4 | |||
| [[Amigo]] | |||
|- | |||
| 1 | |||
| 37\89 | |||
| 498.87 | |||
| 4/3 | |||
| [[Grackle]] | |||
|} | |||
<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | |||
== Scales == | == Scales == | ||
| Line 63: | Line 131: | ||
* [[Myna15]] | * [[Myna15]] | ||
[[ | == Instruments == | ||
[ | ; Lumatone | ||
[[Category: | ''See [[Lumatone mapping for 89edo]].'' | ||
== Music == | |||
; [[Bryan Deister]] | |||
* [https://www.youtube.com/watch?v=2JNIeqvXKlM ''microtonal improvisation in 89edo''] (2025) | |||
; [[Francium]] | |||
* [https://www.youtube.com/watch?v=5Du9RfDUqCs ''Singing Golden Myna''] (2022) – myna[11] in 89edo | |||
[[Category:Listen]] | |||
[[Category:Myna]] | [[Category:Myna]] | ||
[[Category:Thrush]] | [[Category:Thrush]] | ||