167edo: Difference between revisions
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{{Infobox ET}} | {{Infobox ET}} | ||
'''167edo''' is the [[EDO|equal division of the octave]] into 167 parts of 7.18562874251 [[cent]]s each. It [[tempering_out|tempers out]] the [[Würschmidt family|würschmidt comma]], 393216/390625 and 10737418240/10460353203 in the [[5-limit]]; 2401/2400, 3136/3125, and 179200/177147 in the [[7-limit]]; 896/891, 2200/2187, and 3388/3375 in the [[11-limit]]; 325/324, 352/351, 364/363, 1001/1000, and 1716/1715 in the [[13-limit]], providing the [[optimal patent val]] for 11- and 13-limit [[Porwell temperaments|polypyth temperament]]; 256/255, 442/441, 595/594, 715/714, and 936/935 in the [[17-limit]]. It also [[support]]s 11-limit [[Breedsmic temperaments|unthirds temperament]]. | '''167edo''' is the [[EDO|equal division of the octave]] into 167 parts of 7.18562874251 [[cent]]s each. It [[tempering_out|tempers out]] the [[Würschmidt family|würschmidt comma]], 393216/390625 and 10737418240/10460353203 in the [[5-limit]]; [[2401/2400]], [[3136/3125]], and 179200/177147 in the [[7-limit]]; [[896/891]], 2200/2187, and 3388/3375 in the [[11-limit]]; [[325/324]], [[352/351]], [[364/363]], [[1001/1000]], and 1716/1715 in the [[13-limit]], providing the [[optimal patent val]] for 11- and 13-limit [[Porwell temperaments|polypyth temperament]]; [[256/255]], 442/441, [[595/594]], [[715/714]], and [[936/935]] in the [[17-limit]]. It also [[support]]s 11-limit [[Breedsmic temperaments|unthirds temperament]]. | ||
167edo also has a very close approximation to the [[golden magic]] scale. | 167edo also has a very close approximation to the [[golden magic]] scale. | ||
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{{Harmonics in equal|167|intervals=prime|columns=13}} | {{Harmonics in equal|167|intervals=prime|columns=13}} | ||
{{Harmonics in equal|167|intervals=prime|start=14|columns=12}} | {{Harmonics in equal|167|intervals=prime|start=14|columns=12}} | ||
==Regular temperament properties== | |||
{| class="wikitable center-4 center-5 center-6" | |||
! rowspan="2" |[[Subgroup]] | |||
! rowspan="2" |[[Comma list|Comma List]] | |||
! rowspan="2" |[[Mapping]] | |||
! rowspan="2" |Optimal<br>8ve Stretch (¢) | |||
! colspan="2" |Tuning Error | |||
|- | |||
![[TE error|Absolute]] (¢) | |||
![[TE simple badness|Relative]] (%) | |||
|- | |||
|2.3 | |||
|{{monzo|265 -167}} | |||
|{{val|167 265}} | |||
| -0.7056 | |||
| 0.7052 | |||
| 9.81 | |||
|- | |||
|2.3.5 | |||
|{{monzo|17 1 -8}}, {{14 -22 9}} | |||
|{{val|167 265 388}} | |||
| -0.7158 | |||
| 0.5759 | |||
| 8.01 | |||
|- | |||
|2.3.5.7 | |||
|6144/6125, 3136/3125, 179200/177147 | |||
|{{val|167 265 388 469}} | |||
| -0.6467 | |||
| 0.5129 | |||
| 7.14 | |||
|- | |||
|2.3.5.7.11 | |||
|896/891, 2200/2187, 6144/6125, 6250/6237 | |||
|{{val|167 265 388 469 578}} | |||
| -0.6315 | |||
| 0.4598 | |||
| 6.40 | |||
|- | |||
|2.3.5.7.11.13 | |||
|325/324, 352/351, 896/891, 1001/1000, 6656/6615 | |||
|{{val|167 265 388 469 578 618}} | |||
| -0.5349 | |||
| 0.4721 | |||
| 6.57 | |||
|- | |||
|2.3.5.7.11.13.17 | |||
|325/324, 352/351, 896/891, 256/255, 1001/1000, 1225/1224 | |||
|{{val|167 265 388 469 578 618 683}} | |||
| -0.5573 | |||
| 0.4405 | |||
| 6.13 | |||
|} | |||
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | [[Category:Equal divisions of the octave|###]] <!-- 3-digit number --> | ||
[[Category:Prime EDO]] | [[Category:Prime EDO]] | ||