171edo: Difference between revisions
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=== Subsets and supersets === | === Subsets and supersets === | ||
171 factors into primes as {{nowrap|3<sup>2</sup> | 171 factors into primes as {{nowrap|3<sup>2</sup> × 19}}, and 171edo shares the nearly pure [[7/6]] of [[9edo]] and the nearly pure [[6/5]] of [[19edo]], with every 7-limit interval expressible in terms of 2, 6/5, 7/6, and any one of primes 3, 5, or 7. [[342edo]], which doubles 171, provides an excellent correction for the 11th harmonic. | ||
[[342edo]], which doubles 171, provides an excellent correction for the 11th harmonic. | |||
== Intervals == | == Intervals == | ||
{{Main| 171edo/Intervals }} | {{Main| 171edo/Intervals }} | ||
== Approximation to JI == | |||
=== Zeta peak index === | |||
{| class="wikitable center-all" | |||
|- | |||
! colspan="3" | Tuning | |||
! colspan="3" | Strength | |||
! colspan="2" | Closest edo | |||
! colspan="2" | Integer limit | |||
|- | |||
! ZPI | |||
! Steps per octave | |||
! Step size (cents) | |||
! Height | |||
! Integral | |||
! Gap | |||
! Edo | |||
! Octave (cents) | |||
! Consistent | |||
! Distinct | |||
|- | |||
| [[1114zpi]] | |||
| 170.995891689006 | |||
| 7.01771246166817 | |||
| 11.076998 | |||
| 1.652856 | |||
| 19.091741 | |||
| 171edo | |||
| 1200.02883094526 | |||
| 14 | |||
| 14 | |||
|} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
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|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
|- | |- | ||
! Periods<br | ! Periods<br>per 8ve | ||
! Generator* | ! Generator* | ||
! Cents* | ! Cents* | ||
! Associated<br | ! Associated<br>ratio* | ||
! Temperament | ! Temperament | ||
|- | |- | ||
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| [[Enneadecal]] | | [[Enneadecal]] | ||
|} | |} | ||
<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | <nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct | ||
== Scales == | == Scales == | ||