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]]

Revision as of 07:12, 21 April 2023

← 166edo 167edo 168edo →
Prime factorization 167 (prime)
Step size 7.18563 ¢ 
Fifth 98\167 (704.192 ¢)
Semitones (A1:m2) 18:11 (129.3 ¢ : 79.04 ¢)
Consistency limit 7
Distinct consistency limit 7

167edo is the equal division of the octave into 167 parts of 7.18562874251 cents each. It tempers out the 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 polypyth temperament; 256/255, 442/441, 595/594, 715/714, and 936/935 in the 17-limit. It also supports 11-limit unthirds temperament.

167edo also has a very close approximation to the golden magic scale.

167edo is the 39th prime EDO.


Approximation of prime harmonics in 167edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37 41
Error Absolute (¢) +0.00 +2.24 +1.71 +1.23 +1.98 +0.19 +2.83 -2.90 -3.12 -2.03 -2.52 +0.15 +2.08
Relative (%) +0.0 +31.1 +23.8 +17.2 +27.5 +2.7 +39.4 -40.4 -43.5 -28.3 -35.1 +2.1 +28.9
Steps
(reduced) 		167
(0) 		265
(98) 		388
(54) 		469
(135) 		578
(77) 		618
(117) 		683
(15) 		709
(41) 		755
(87) 		811
(143) 		827
(159) 		870
(35) 		895
(60)
Approximation of prime harmonics in 167edo
Harmonic 43 47 53 59 61 67 71 73 79 83 89 97
Error Absolute (¢) -1.34 +2.76 +3.14 -2.88 -3.11 -0.27 -0.06 +2.15 +1.93 +2.65 -3.22 -1.33
Relative (%) -18.6 +38.4 +43.7 -40.1 -43.3 -3.7 -0.8 +29.9 +26.9 +36.8 -44.7 -18.5
Steps
(reduced) 		906
(71) 		928
(93) 		957
(122) 		982
(147) 		990
(155) 		1013
(11) 		1027
(25) 		1034
(32) 		1053
(51) 		1065
(63) 		1081
(79) 		1102
(100)

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢) 		Tuning Error
Absolute (¢) Relative (%)
2.3 [265 -167⟩ ⟨167 265] -0.7056 0.7052 9.81
2.3.5 [17 1 -8⟩, Template:14 -22 9 ⟨167 265 388] -0.7158 0.5759 8.01
2.3.5.7 6144/6125, 3136/3125, 179200/177147 ⟨167 265 388 469] -0.6467 0.5129 7.14
2.3.5.7.11 896/891, 2200/2187, 6144/6125, 6250/6237 ⟨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 ⟨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 ⟨167 265 388 469 578 618 683] -0.5573 0.4405 6.13
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