301edo: Difference between revisions
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{{Infobox ET | {{Infobox ET}} | ||
{{ED intro}} | |||
}} | |||
== Theory == | == Theory == | ||
301edo is a strong 7-limit system, and distinctly consistent through the [[17-odd-limit]]. | 301edo is a strong 7-limit system, and distinctly [[consistent]] through the [[17-odd-limit]]. The equal temperament [[tempering out|tempers out]] [[32805/32768]] in the 5-limit, [[2401/2400]] in the 7-limit, [[3025/3024]], [[5632/5625]], [[8019/8000]] in the 11-limit, [[729/728]], [[847/845]], [[1001/1000]], [[1716/1715]], [[2200/2197]] in the 13-limit, and 561/560, [[833/832]], [[1089/1088]], [[1156/1155]], 1275/1274 and [[1701/1700]] in the 17-limit. Since it tempers out both 32805/32768 and 2401/2400, it [[support]]s the [[sesquiquartififths]] temperament. | ||
301 | === Prime harmonics === | ||
{{Harmonics in equal|301}} | |||
=== Subsets and supersets === | |||
Since 301 factors into {{factorisation|301}}, 301edo has [[7edo]] and [[43edo]] as its subsets. This is related to the proposal of the deaf French mathematician and acoustician {{w|Joseph Sauveur}} to divide the octave in 43 parts called ''merides'', and those into seven more parts called ''heptamerides''. Back in the days of slide rules and log tables, this made sense since by multiplying the log base ten of the interval in question by 1000, one came close to how many heptamerides it constituted. | |||
301edo also tempers out {{monzo| 168 -43 -43 }} and 5250987/5242880, so it supports the [[Mitonismic temperaments #Meridic|meridic temperament]]. | 301edo also tempers out {{monzo| 168 -43 -43 }} and 5250987/5242880, so it supports the [[Mitonismic temperaments #Meridic|meridic temperament]]. | ||
== 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 31: | Line 27: | ||
| 2.3 | | 2.3 | ||
| {{monzo| -477 301 }} | | {{monzo| -477 301 }} | ||
| | | {{mapping| 301 477 }} | ||
| +0.0927 | | +0.0927 | ||
| 0.0927 | | 0.0927 | ||
| Line 38: | Line 34: | ||
| 2.3.5 | | 2.3.5 | ||
| 32805/32768, {{monzo| 3 45 -32 }} | | 32805/32768, {{monzo| 3 45 -32 }} | ||
| | | {{mapping| 301 477 699 }} | ||
| +0.0048 | | +0.0048 | ||
| 0.1456 | | 0.1456 | ||
| Line 45: | Line 41: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 32805/32768, 1959552/1953125 | | 2401/2400, 32805/32768, 1959552/1953125 | ||
| | | {{mapping| 301 477 699 845 }} | ||
| +0.0085 | | +0.0085 | ||
| 0.1262 | | 0.1262 | ||
| Line 52: | Line 48: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 2401/2400, 3025/3024, 5632/5625, 8019/8000 | | 2401/2400, 3025/3024, 5632/5625, 8019/8000 | ||
| | | {{mapping| 301 477 699 845 1041 }} | ||
| +0.0734 | | +0.0734 | ||
| 0.1720 | | 0.1720 | ||
| Line 59: | Line 55: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 729/728, 847/845, 1001/1000, 1716/1715, 3025/3024 | | 729/728, 847/845, 1001/1000, 1716/1715, 3025/3024 | ||
| | | {{mapping| 301 477 699 845 1041 1114 }} | ||
| +0.0310 | | +0.0310 | ||
| 0.1834 | | 0.1834 | ||
| Line 66: | Line 62: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 561/560, 729/728, 833/832, 847/845, 1001/1000, 1089/1088 | | 561/560, 729/728, 833/832, 847/845, 1001/1000, 1089/1088 | ||
| | | {{mapping| 301 477 699 845 1041 1114 1230 }} | ||
| +0.0721 | | +0.0721 | ||
| 0.1973 | | 0.1973 | ||
| Line 74: | Line 70: | ||
=== 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 84: | Line 81: | ||
| 25\301 | | 25\301 | ||
| 99.67 | | 99.67 | ||
| | | 18/17 | ||
| [[Quintaschis]] | | [[Quintaschis]] | ||
|- | |- | ||
| Line 109: | Line 106: | ||
| 498.34 | | 498.34 | ||
| 4/3 | | 4/3 | ||
| [[Helmholtz]] | | [[Helmholtz (temperament)|Helmholtz]] | ||
|- | |- | ||
| 7 | | 7 | ||
| 125\301<br>(4\301) | | 125\301<br />(4\301) | ||
| 498.34<br>(15.95) | | 498.34<br />(15.95) | ||
| 4/3<br>(245/243) | | 4/3<br />(245/243) | ||
| [[Septant]] | | [[Septant]] | ||
|- | |- | ||
| 43 | | 43 | ||
| 125\301<br>(1\301) | | 125\301<br />(1\301) | ||
| 498.34<br>(3.99) | | 498.34<br />(3.99) | ||
| 4/3<br>(540/539) | | 4/3<br />(540/539) | ||
| [[Meridic]] | | [[Meridic]] | ||
|} | |} | ||
<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:Meridic]] | [[Category:Meridic]] | ||