296edo: Difference between revisions
→Theory: +essentially tempered chords and adopt new template |
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" |
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{{Infobox ET | {{Infobox ET}} | ||
{{ED intro}} | |||
}} | |||
== Theory == | == Theory == | ||
In the 5-limit, 296et not only tempers out the [[semicomma]] of 5-limit orwell (orson) temperament, 2109375/2097152, it also provides its [[optimal patent val]], and tempers out the minortone comma, {{monzo| -16 35 -17 }}. It is also an interesting temperament in higher limits, being distinctly consistent through to the 15-odd-limit. In the 7-limit it tempers out 4375/4374 ([[ragisma]]), 16875/16807 (mirkwai), and 118098/117649 (stearnsma), [[support | In the 5-limit, 296et not only [[tempering out|tempers out]] the [[semicomma]] of 5-limit orwell (orson) temperament, 2109375/2097152, it also provides its [[optimal patent val]], and tempers out the minortone comma, {{monzo| -16 35 -17 }}. It is also an interesting temperament in higher limits, being [[consistency|distinctly consistent]] through to the [[15-odd-limit]]. In the 7-limit it tempers out 4375/4374 ([[ragisma]]), 16875/16807 (mirkwai), and 118098/117649 (stearnsma), [[support]]ing 7-limit [[octoid]] and [[sabric]]. In the 11-limit, [[540/539]], 1375/1372, [[3025/3024]], [[4000/3993]], [[6250/6237]] and [[9801/9800]]; in the 13-limit, [[625/624]], [[729/728]], [[1575/1573]], [[1716/1715]], [[2080/2079]], and [[6656/6655]], so that it also supports the 11- and 13-limit versions of octoid. It allows [[swetismic chords]] and [[squbemic chords]] in the 13-odd-limit, in addition to [[nicolic chords]] in the 15-odd-limit. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|296|columns=11}} | {{Harmonics in equal|296|columns=11}} | ||
=== Subsets and supersets === | |||
Since 296 factors into {{factorisation|296}}, 296edo has subset edos {{EDOs| 2, 4, 8, 37, 74 and 148 }}. | |||
== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
! rowspan="2" | Subgroup | |- | ||
! rowspan="2" | [[Subgroup]] | |||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve stretch (¢) | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
| Line 29: | Line 25: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -469 296 }} | | {{monzo| -469 296 }} | ||
| | | {{mapping| 296 469 }} | ||
| +0.1904 | | +0.1904 | ||
| 0.1905 | | 0.1905 | ||
| Line 36: | Line 32: | ||
| 2.3.5 | | 2.3.5 | ||
| 2109375/2097152, {{monzo| -16 35 -17 }} | | 2109375/2097152, {{monzo| -16 35 -17 }} | ||
| | | {{mapping| 296 469 687 }} | ||
| +0.2962 | | +0.2962 | ||
| 0.2158 | | 0.2158 | ||
| Line 43: | Line 39: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 4375/4374, 16875/16807, 2100875/2097152 | | 4375/4374, 16875/16807, 2100875/2097152 | ||
| | | {{mapping| 296 469 687 831 }} | ||
| +0.2138 | | +0.2138 | ||
| 0.2350 | | 0.2350 | ||
| Line 50: | Line 46: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 540/539, 1375/1372, 4000/3993, 2100875/2097152 | | 540/539, 1375/1372, 4000/3993, 2100875/2097152 | ||
| | | {{mapping| 296 469 687 831 1024 }} | ||
| +0.1691 | | +0.1691 | ||
| 0.2284 | | 0.2284 | ||
| Line 57: | Line 53: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 540/539, 625/624, 729/728, 1375/1372, 15379/15360 | | 540/539, 625/624, 729/728, 1375/1372, 15379/15360 | ||
| | | {{mapping| 296 469 687 831 1024 1095 }} | ||
| +0.2012 | | +0.2012 | ||
| 0.2206 | | 0.2206 | ||
| Line 65: | Line 61: | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
{| class="wikitable center-all left-5" | {| class="wikitable center-all left-5" | ||
|+Table of rank-2 temperaments by generator | |+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | ||
! Periods<br>per | |- | ||
! Generator | ! Periods<br />per 8ve | ||
! Cents | ! Generator* | ||
! Associated<br>ratio | ! Cents* | ||
! Associated<br />ratio* | |||
! Temperaments | ! Temperaments | ||
|- | |- | ||
| Line 76: | Line 73: | ||
| 182.43 | | 182.43 | ||
| 10/9 | | 10/9 | ||
| [[ | | [[Mitonic]] | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 82: | Line 79: | ||
| 271.62 | | 271.62 | ||
| 75/64 | | 75/64 | ||
| [[ | | [[Sabric]] | ||
|- | |- | ||
| 1 | | 1 | ||
| Line 97: | Line 94: | ||
|- | |- | ||
| 8 | | 8 | ||
| 144\296<br>(4\296) | | 144\296<br />(4\296) | ||
| 583.78<br>(16.22) | | 583.78<br />(16.22) | ||
| 7/5<br>(126/125) | | 7/5<br />(126/125) | ||
| [[Octoid]] | | [[Octoid]] | ||
|- | |||
| 37 | |||
| 67\296<br />(3\296) | |||
| 271.62<br />(12.16) | |||
| 117/100<br />(?) | |||
| [[Dzelic]] | |||
|} | |} | ||
<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct | |||
[[Category: | [[Category:Sabric]] | ||