68edo: Difference between revisions
Remove the zeta peak table that had no explanation or context |
→Regular temperament properties: + associated ratios; this fixes the alignment. Note hemiwuerschmidt |
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68edo's step is half of the step size of [[34edo]], which does well in the 5-limit but not so well in the 7-limit, and one quarter the size of [[17edo]], which does well in the [[3-limit]], but not so well in the [[5-limit]]. The luck continues: 68 is a strong [[7-limit]] system, but does not do as well in the [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly since [[11/9]] is not mapped to its best approximation. However, 68edo maps many higher primes better than it does 11 (specifically 13 and 23 inherited from 17edo, 17 inherited from 34edo, and 19 and 31 new to 68edo), notably being [[consistent]] in the entire no-11s 25-[[odd limit]] add-31. It achieves this by having a consistent sharp tendency among all primes up to 31, save 11 and 29. Therefore, a slight octave compression, such as in [[158ed5]] or [[191ed7]], can improve upon the accuracy of 68edo's harmonic series. | 68edo's step is half of the step size of [[34edo]], which does well in the 5-limit but not so well in the 7-limit, and one quarter the size of [[17edo]], which does well in the [[3-limit]], but not so well in the [[5-limit]]. The luck continues: 68 is a strong [[7-limit]] system, but does not do as well in the [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly since [[11/9]] is not mapped to its best approximation. However, 68edo maps many higher primes better than it does 11 (specifically 13 and 23 inherited from 17edo, 17 inherited from 34edo, and 19 and 31 new to 68edo), notably being [[consistent]] in the entire no-11s 25-[[odd limit]] add-31. It achieves this by having a consistent sharp tendency among all primes up to 31, save 11 and 29. Therefore, a slight octave compression, such as in [[158ed5]] or [[191ed7]], can improve upon the accuracy of 68edo's harmonic series. | ||
As a 7-limit system, 68et [[tempering out|tempers out]] [[2048/2025]], [[245/243]], [[4000/3969]], [[15625/15552]], [[3136/3125]], [[6144/6125]], and [[2401/2400]]. It [[support]]s [[octacot]], [[shrutar]], [[hemiwürschmidt]], [[hemikleismic]], [[clyde]] and [[neptune]] temperaments, and supplies the [[optimal patent val]] for 11-limit [[hemikleismic]]. | As a 7-limit system, 68et [[tempering out|tempers out]] [[2048/2025]], [[245/243]], [[4000/3969]], [[15625/15552]], [[3136/3125]], [[6144/6125]], and [[2401/2400]]. It [[support]]s [[octacot]], [[shrutar]], [[hemiwürschmidt]], [[hemikleismic]], [[clyde]] and [[Neptune (temperament)|neptune]] temperaments, and supplies the [[optimal patent val]] for 11-limit [[hemikleismic]]. | ||
The 3rd degree of 68edo can be used as a generator for [[23edo and octave stretching|stretched 23edo]], which also acts as the [[quartkeenlig]] temperament tempering out the quartisma, 385/384 and 6250/6237. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65{{c}}). It also works as a [[22L 1s]] MOS of the quartkeenlig temperament. | The 3rd degree of 68edo can be used as a generator for [[23edo and octave stretching|stretched 23edo]], which also acts as the [[quartkeenlig]] temperament tempering out the quartisma, 385/384 and 6250/6237. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65{{c}}). It also works as a [[22L 1s]] MOS of the quartkeenlig temperament. | ||
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=== Interval mappings === | === Interval mappings === | ||
{{Q-odd-limit intervals}} | {{Q-odd-limit intervals}} | ||
{{Q-odd-limit intervals|68.1|apx=val|header=none|tag=none|title=15-odd-limit intervals by 68e val mapping}} | |||
== Regular temperament properties == | == Regular temperament properties == | ||
| Line 369: | Line 370: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 245/243, 2048/2025, 2401/2400 | | 245/243, 2048/2025, 2401/2400 | ||
| {{ | | {{Mapping| 68 108 158 191 }} | ||
| −0.983 | | −0.983 | ||
| 0.915 | | 0.915 | ||
| Line 376: | Line 377: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 121/120, 176/175, 245/243, 1375/1372 | | 121/120, 176/175, 245/243, 1375/1372 | ||
| {{ | | {{Mapping| 68 108 158 191 235 }} | ||
| −0.541 | | −0.541 | ||
| 1.206 | | 1.206 | ||
| Line 383: | Line 384: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 121/120, 176/175, 196/195, 245/243, 275/273 | | 121/120, 176/175, 196/195, 245/243, 275/273 | ||
| {{ | | {{Mapping| 68 108 158 191 235 252 }} | ||
| −0.745 | | −0.745 | ||
| 1.191 | | 1.191 | ||
| Line 390: | Line 391: | ||
| 2.3.5.7.11.13.17 | | 2.3.5.7.11.13.17 | ||
| 121/120, 136/135, 154/153, 176/175, 196/195, 275/273 | | 121/120, 136/135, 154/153, 176/175, 196/195, 275/273 | ||
| {{ | | {{Mapping| 68 108 158 191 235 252 278 }} | ||
| −0.671 | | −0.671 | ||
| 1.118 | | 1.118 | ||
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| 2.3.5.7.11.13.17.19 | | 2.3.5.7.11.13.17.19 | ||
| 121/120, 136/135, 154/153, 190/189, 176/175, 196/195, 275/273 | | 121/120, 136/135, 154/153, 190/189, 176/175, 196/195, 275/273 | ||
| {{ | | {{Mapping| 68 108 158 191 235 252 278 289 }} | ||
| −0.661 | | −0.661 | ||
| 1.046 | | 1.046 | ||
| 5.93 | | 5.93 | ||
|} | |} | ||
=== Rank 2 temperaments === | |||
{| class="wikitable center-all left-5" | |||
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator | |||
|- | |||
! Periods<br>per 8ve | |||
! Generator* | |||
! Cents* | |||
! Associated<br>ratio* | |||
! Temperament | |||
|- | |||
| 1 | |||
| 5\68 | |||
| 88.24 | |||
| 20/19 | |||
| [[Octacot]] | |||
|- | |||
| 1 | |||
| 11\68 | |||
| 194.12 | |||
| 28/25 | |||
| [[Hemiwürschmidt]] (68e) / hemiwur (68) | |||
|- | |||
| 2 | |||
| 3\68 | |||
| 52.94 | |||
| 33/32 | |||
| [[Shrutar]] | |||
|- | |||
| 4 | |||
| 6\68 | |||
| 105.88 | |||
| 17/16 | |||
| [[Bidia]] | |||
|} | |||
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | |||
== Scales == | == Scales == | ||
: ''See also: [[List of MOS scales in 68edo]], [[34edo #Scales]], and [[17edo #Scales]].'' | : ''See also: [[List of MOS scales in 68edo]], [[34edo #Scales]], and [[17edo #Scales]].'' | ||
* | * [[Deeptone]][7]: 10 10 9 10 10 10 9 | ||
* [[Hemiwur]][31]: 2 2 3 2 2 2 2 3 2 2 2 2 3 2 2 2 2 2 3 2 2 2 2 3 2 2 2 2 3 2 2 | |||
* Inverse half octave: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7 | * Inverse half octave{{clarify}}: 4 4 7 4 4 4 4 7 4 4 7 4 4 4 4 7 | ||
* | * Negative semitone: 14 14 -1 14 14 14 -1 (''E is sharper than F, and B is sharper than C'') | ||
* Quartkeenlig[23] ( | * [[Octacot]][27]: 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 2 3 | ||
* [[Quartkeenlig]][23] (stretched [[23edo]]): 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 | |||
* [[Shrutar]][22]: 3 3 3 3 3 4 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3 | |||
* [[Superpyth]] quarter octave{{clarify}}: 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 3 3 1 3 3 3 1 | |||
== Instruments == | == Instruments == | ||
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[[Category:Shrutar]] | [[Category:Shrutar]] | ||
[[Category:Quartismic]] | [[Category:Quartismic]] | ||
[[Category:Todo:add rank 2 temperaments table]] | |||