89edo: Difference between revisions
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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]]. | 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 [[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 [[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. | ||
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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. | 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 the 11th in the {{w|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. (φ | 89edo is the 11th in the {{w|Fibonacci sequence}}, which means its 55th step approximates logarithmic φ (i.e. {{nowrap|1200(φ − 1){{c}}}} within a fraction of a cent. | ||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 44: | Line 44: | ||
| 32805/32768, 10077696/9765625 | | 32805/32768, 10077696/9765625 | ||
| {{mapping| 89 141 207 }} | | {{mapping| 89 141 207 }} | ||
| | | −0.500 | ||
| 1.098 | | 1.098 | ||
| 8.15 | | 8.15 | ||
| Line 51: | Line 51: | ||
| 126/125, 1728/1715, 32805/32768 | | 126/125, 1728/1715, 32805/32768 | ||
| {{mapping| 89 141 207 250 }} | | {{mapping| 89 141 207 250 }} | ||
| | | −0.550 | ||
| 0.955 | | 0.955 | ||
| 7.08 | | 7.08 | ||
| Line 58: | Line 58: | ||
| 126/125, 176/175, 243/242, 16384/16335 | | 126/125, 176/175, 243/242, 16384/16335 | ||
| {{mapping| 89 141 207 250 308 }} | | {{mapping| 89 141 207 250 308 }} | ||
| | | −0.526 | ||
| 0.855 | | 0.855 | ||
| 6.35 | | 6.35 | ||
| Line 103: | Line 103: | ||
| [[Grackle]] | | [[Grackle]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if | <nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||
| Line 112: | Line 112: | ||
== Music == | == Music == | ||
; [[Francium]] | ; [[Francium]] | ||
* [https://www.youtube.com/watch?v=5Du9RfDUqCs ''Singing Golden Myna''] (2022) | * [https://www.youtube.com/watch?v=5Du9RfDUqCs ''Singing Golden Myna''] (2022) – myna[11] in 89edo | ||
[[Category:Listen]] | [[Category:Listen]] | ||
[[Category:Myna]] | [[Category:Myna]] | ||
[[Category:Thrush]] | [[Category:Thrush]] | ||