354edo: Difference between revisions
m →Theory: quality of higher harmonics |
Rework; cleanup; clarify the title row of the rank-2 temp table |
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== Theory == | == Theory == | ||
354edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by tempering out the [[schisma]] and the [[parakleisma]], but the approximation to higher | 354edo is [[enfactoring|enfactored]] in the 5-limit, with the same tuning as [[118edo]], defined by [[tempering out]] the [[schisma]] and the [[parakleisma]], but the approximation to higher [[harmonic]]s are much improved. In the 7-limit, the equal temperament tempers out 118098/117649 (stearnsma), 250047/250000 ([[landscape comma]]), and 703125/702464 ([[meter]]); in the 11-limit, [[540/539]], and [[4000/3993]]; in the 13-limit, [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], [[4096/4095]], and [[4225/4224]]. It provides the [[optimal patent val]] for [[stearnscape]], the 72 & 282 temperament, and 13- and 17-limit [[terminator]], the 171 & 183 temperament. | ||
=== Prime harmonics === | === Prime harmonics === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
Since 354 factors into | Since 354 factors into {{factorization|354}}, 354edo has subset edos {{EDOs| 2, 3, 6, 59, 118, and 177 }}. | ||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 24: | Line 24: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 32805/32768, 118098/117649, 250047/250000 | | 32805/32768, 118098/117649, 250047/250000 | ||
| | | {{mapping| 354 561 822 994 }} | ||
| -0.0319 | | -0.0319 | ||
| 0.1432 | | 0.1432 | ||
| Line 31: | Line 31: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 540/539, 4000/3993, 32805/32768, 137781/137500 | | 540/539, 4000/3993, 32805/32768, 137781/137500 | ||
| | | {{mapping| 354 561 822 994 1225 }} | ||
| -0.0963 | | -0.0963 | ||
| 0.1817 | | 0.1817 | ||
| Line 38: | Line 38: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 540/539, 729/728, 1575/1573, 4096/4095, 31250/31213 | | 540/539, 729/728, 1575/1573, 4096/4095, 31250/31213 | ||
| | | {{mapping| 354 561 822 994 1225 1310 }} | ||
| -0.0871 | | -0.0871 | ||
| 0.1671 | | 0.1671 | ||
| Line 45: | Line 45: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 4096/4095 | | 540/539, 729/728, 936/935, 1156/1155, 1575/1573, 4096/4095 | ||
| | | {{mapping| 354 561 822 994 1225 1310 1447 }} | ||
| -0.0791 | | -0.0791 | ||
| 0.1559 | | 0.1559 | ||
| Line 52: | Line 52: | ||
| 2.3.5.7.11.13.17.19 | | 2.3.5.7.11.13.17.19 | ||
| 540/539, 729/728, 936/935, 969/968, 1156/1155, 1445/1444, 1521/1520 | | 540/539, 729/728, 936/935, 969/968, 1156/1155, 1445/1444, 1521/1520 | ||
| | | {{mapping| 354 561 822 994 1225 1310 1447 1504 }} | ||
| -0.0926 | | -0.0926 | ||
| 0.1509 | | 0.1509 | ||
| Line 64: | Line 64: | ||
|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 79: | Line 79: | ||
| 498.31<br>(98.31) | | 498.31<br>(98.31) | ||
| 4/3<br>(18/17) | | 4/3<br>(18/17) | ||
| [[Term]] / terminator | | [[Term (temperament)|Term]] / terminator | ||
|- | |- | ||
| 6 | | 6 | ||
| Line 99: | Line 99: | ||
| [[Oganesson]] | | [[Oganesson]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||
[[Category:Stearnscape]] | |||