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]]. | ||
89et [[tempering out|tempers out]] the commas [[126/125]], [[1728/1715]], | 89et [[tempering out|tempers out]] the commas 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. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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| 2.3 | | 2.3 | ||
| {{monzo| -141 89 }} | | {{monzo| -141 89 }} | ||
| | | {{mapping| 89 141 }} | ||
| +0.262 | | +0.262 | ||
| 0.262 | | 0.262 | ||
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| 2.3.5 | | 2.3.5 | ||
| 32805/32768, 10077696/9765625 | | 32805/32768, 10077696/9765625 | ||
| | | {{mapping| 89 141 207 }} | ||
| -0.500 | | -0.500 | ||
| 1.098 | | 1.098 | ||
| Line 43: | Line 43: | ||
| 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 | | -0.550 | ||
| 0.955 | | 0.955 | ||
| Line 50: | Line 50: | ||
| 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 | | -0.526 | ||
| 0.855 | | 0.855 | ||
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|- | |- | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperament | ! Temperament | ||
|- | |- | ||
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| [[Grackle]] | | [[Grackle]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
== Scales == | == Scales == | ||
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[[Category:Myna]] | [[Category:Myna]] | ||
[[Category:Thrush]] | [[Category:Thrush]] | ||
[[Category:Listen]] | [[Category:Listen]] | ||