342edo: Difference between revisions
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[[684edo]], which doubles 342edo, provides an approximation of harmonic 13 that works well with the flat tendency of its 11-limit mapping. | [[684edo]], which doubles 342edo, provides an approximation of harmonic 13 that works well with the flat tendency of its 11-limit mapping. | ||
== 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 | |||
|- | |||
| [[2568zpi]] | |||
| 341.974850913987 | |||
| 3.50902996753355 | |||
| 13.478611 | |||
| 1.890555 | |||
| 20.767404 | |||
| 342edo | |||
| 1200.08824889647 | |||
| 12 | |||
| 12 | |||
|} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 19: | Line 51: | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br | ! rowspan="2" | Optimal<br>8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 52: | Line 84: | ||
|+ 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* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 77: | Line 109: | ||
|- | |- | ||
| 2 | | 2 | ||
| 124\342<br | | 124\342<br>(47\342) | ||
| 435.09<br | | 435.09<br>(164.91) | ||
| 9/7<br | | 9/7<br>(11/10) | ||
| [[Semisupermajor]] | | [[Semisupermajor]] | ||
|- | |- | ||
| 2 | | 2 | ||
| 142\342<br | | 142\342<br>(29\342) | ||
| 498.25<br | | 498.25<br>(101.75) | ||
| 4/3<br | | 4/3<br>(35/33) | ||
| [[Bipont]] | | [[Bipont]] | ||
|- | |- | ||
| 3 | | 3 | ||
| 71\342<br | | 71\342<br>(43\342) | ||
| 249.12<br | | 249.12<br>(150.88) | ||
| 15/13<br | | 15/13<br>(12/11) | ||
| [[Hemiterm]] | | [[Hemiterm]] | ||
|- | |- | ||
| 6 | | 6 | ||
| 97\342<br | | 97\342<br>(17\342) | ||
| 340.35<br | | 340.35<br>(59.65) | ||
| 162/133<br | | 162/133<br>(88/85) | ||
| [[Semiseptichrome]] | | [[Semiseptichrome]] | ||
|- | |- | ||
| 6 | | 6 | ||
| 142\342<br | | 142\342<br>(28\342) | ||
| 498.25<br | | 498.25<br>(98.25) | ||
| 4/3<br | | 4/3<br>(18/17) | ||
| [[Semiterm]] | | [[Semiterm]] | ||
|- | |- | ||
| 9 | | 9 | ||
| 63\342<br | | 63\342<br>(13\342) | ||
| 221.05<br | | 221.05<br>(45.61) | ||
| 25/22<br | | 25/22<br>(77/75) | ||
| [[Quadraennealimmal]] | | [[Quadraennealimmal]] | ||
|- | |- | ||
| 18 | | 18 | ||
| 71\342<br | | 71\342<br>(5\342) | ||
| 249.12<br | | 249.12<br>(17.54) | ||
| 15/13<br | | 15/13<br>(99/98) | ||
| [[Hemiennealimmal]] | | [[Hemiennealimmal]] | ||
|- | |- | ||
| 38 | | 38 | ||
| 142\342<br | | 142\342<br>(2\342) | ||
| 498.25<br | | 498.25<br>(7.02) | ||
| 4/3<br | | 4/3<br>(225/224) | ||
| [[Hemienneadecal]] | | [[Hemienneadecal]] | ||
|} | |} | ||
<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 | ||