68edo: Difference between revisions
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{{Infobox ET | {{Infobox ET | ||
| Prime factorization = 2<sup>2</sup> × 17 | | Prime factorization = 2<sup>2</sup> × 17 | ||
| Step size = 17. | | Step size = 17.6471¢ | ||
| Fifth = 40\68 (705. | | Fifth = 40\68 (705.9¢) (→ [[17edo|10\17]]) | ||
| Semitones = 8:4 (141. | | Semitones = 8:4 (141.2¢ : 70.6¢) | ||
| Consistency = 9 | | Consistency = 9 | ||
}} | }} | ||
The '''68 equal divisions of the octave''' ('''68edo'''), or the '''68(-tone) equal temperament''' ('''68tet''', '''68et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 68 [[equal]]ly-sized steps. Each step represents a frequency ratio of 17.65 [[cent]]s. | The '''68 equal divisions of the octave''' ('''68edo'''), or the '''68(-tone) equal temperament''' ('''68tet''', '''68et''') when viewed from a [[regular temperament]] perspective, is the tuning system derived by dividing the [[octave]] into 68 [[equal]]ly-sized steps. Each step represents a frequency ratio of 17.65 [[cent]]s. | ||
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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 for in [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly. | 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 for in [[11-limit]]; though it's certainly usable for that purpose, it does not represent the 11-limit diamond [[consistent]]ly. | ||
As a 7-limit system it tempers out [[ | As a 7-limit system it 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]]. It is a sharp-tending system, with the third, fifth and seventh harmonics all sharp. | ||
The 3rd degree of 68edo can be used as a generator for [[23edo and octave stretching|stretched 23edo]]. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65 cents). | The 3rd degree of 68edo can be used as a generator for [[23edo and octave stretching|stretched 23edo]]. It results in a 23edo scale with octaves stretched by 1 step of 68edo (octaves of 1217.65 cents). | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{ | {{Harmonics in equal|68}} | ||
== Intervals == | == Intervals == | ||
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| 1200.00 | | 1200.00 | ||
| 2/1 | | 2/1 | ||
|} | |||
== Regular temperament properties == | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" | Subgroup | |||
! rowspan="2" | [[Comma list]] | |||
! rowspan="2" | [[Mapping]] | |||
! rowspan="2" | Optimal<br>8ve stretch (¢) | |||
! colspan="2" | Tuning error | |||
|- | |||
! [[TE error|Absolute]] (¢) | |||
! [[TE simple badness|Relative]] (%) | |||
|- | |||
| 2.3.5.7 | |||
| 245/243, 2048/2025, 2401/2400 | |||
| [{{val| 68 108 158 191 }}] | |||
| -0.983 | |||
| 0.915 | |||
| 5.19 | |||
|} | |} | ||