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1205edo - Revision history
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FloraC: Birth
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<p>Birth</p>
<p><b>New page</b></p><div>{{Infobox ET}}<br />
{{ED intro}}<br />
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== Theory ==<br />
1205edo shares the same [[3/2|perfect fifth]] with [[241edo]], but the approximation to the higher [[harmonic]]s are very much improved. It is a strong no-17 higher-limit system, fully [[consistent]] to the no-17 [[31-odd-limit]]. <br />
<br />
As an equal temperament, it tempers out {{monzo| 38 -2 -15 }} ([[luna comma]]) in the [[5-limit]], 4375/4374 ([[ragisma]]) in the [[7-limit]], [[support]]ing and providing a great tuning for [[lunatic]]. It further tempers out [[117649/117612]], [[131072/130977]], [[151263/151250]], [[234375/234256]] in the [[11-limit]]; [[2080/2079]], [[4096/4095]], [[4225/4224]] in the [[13-limit]]; and [[3136/3135]], [[3250/3249]], [[5776/5775]], [[6175/6174]] in the [[2.3.5.7.11.13.19 subgroup]]. <br />
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If intervals of 17 are desired, the sharp-tending 1205g [[val]] blends best with the rest, and in fact lends itself as a tuning with a record-low [[31-limit]] [[TE error]]. This maps [[17/16]], [[17/10]], [[17/13]], [[23/17]] and their [[octave complement]]s inconsistently, and tempers out [[2058/2057]] and [[2500/2499]]. The less accurate [[patent val]] tempers out [[1156/1155]], [[2431/2430]] and [[2601/2600]] instead. <br />
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=== Prime harmonics ===<br />
{{Harmonics in equal|1205|columns=11}}<br />
{{Harmonics in equal|1205|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 1205edo (continued)}}<br />
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=== Subsets and supersets ===<br />
Since 1205 factors into primes as {{nowrap| 5 × 241 }}, 1205edo contains [[5edo]] and [[241edo]] as subset edos. <br />
<br />
== Regular temperament properties ==<br />
{| class="wikitable center-4 center-5 center-6"<br />
|-<br />
! rowspan="2" | [[Subgroup]]<br />
! rowspan="2" | [[Comma list]]<br />
! rowspan="2" | [[Mapping]]<br />
! rowspan="2" | Optimal<br>8ve stretch (¢)<br />
! colspan="2" | Tuning error<br />
|-<br />
! [[TE error|Absolute]] (¢)<br />
! [[TE simple badness|Relative]] (%)<br />
|-<br />
| 2.3.5<br />
| {{Monzo| 38 -2 -15 }}, {{monzo| 26 -75 40 }}<br />
| {{Mapping| 1205 1910 2798 }}<br />
| −0.0361<br />
| 0.0309<br />
| 3.10<br />
|-<br />
| 2.3.5.7<br />
| 4375/4374, {{monzo| -1 4 11 -11 }}, {{monzo| 36 -5 0 -10 }}<br />
| {{Mapping| 1205 1910 2798 3383 }}<br />
| −0.0393<br />
| 0.0273<br />
| 2.74<br />
|-<br />
| 2.3.5.7.11<br />
| 4375/4374, 117649/117612, 131072/130977, 234375/234256<br />
| {{Mapping| 1205 1910 2798 3383 4169 }}<br />
| −0.0536<br />
| 0.0376<br />
| 3.78<br />
|-<br />
| 2.3.5.7.11.13<br />
| 2080/2079, 4096/4095, 4375/4374, 117649/117612, 225000/224939<br />
| {{Mapping| 1205 1910 2798 3383 4169 4459 }}<br />
| −0.0433<br />
| 0.0413<br />
| 4.15<br />
|-<br />
| 2.3.5.7.11.13.19<br />
| 2080/2079, 3136/3135, 3250/3249, 4096/4095, 4375/4374, 225000/224939<br />
| {{Mapping| 1205 1910 2798 3383 4169 4459 5119 }}<br />
| −0.0454<br />
| 0.0386<br />
| 3.88<br />
|}<br />
<br />
=== Rank-2 temperaments ===<br />
{| class="wikitable center-all left-5"<br />
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator<br />
|-<br />
! Periods<br>per 8ve<br />
! Generator*<br />
! Cents*<br />
! Associated<br>ratio*<br />
! Temperaments<br />
|-<br />
| 1<br />
| 194\1205<br />
| 193.195<br />
| {{Monzo| -38 5 13 }}<br />
| [[Lunatic]]<br />
|}<br />
<nowiki/>* In [[Normal forms #Minimal-generator form|minimal-generator form]]</div>
FloraC