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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]].
{{EDO intro|167}}
 
== Theory ==
167et [[tempering out|tempers out]] the [[würschmidt comma]], 393216/390625, and the leapday comma, {{monzo| 31 -21 1 }}, 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 the 11-limit [[Breedsmic temperaments #Unthirds|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.


167edo is the 39th [[prime EDO]].
=== Prime harmonics ===
{{Harmonics in equal|167|intervals=prime|columns=12}}
{{Harmonics in equal|167|intervals=prime|columns=12|start=13|title=Approximation of prime harmonics in 167edo (continued)|collapsed=1}}


{{Harmonics in equal|167|intervals=prime|columns=13}}
=== Subsets and supersets ===
{{Harmonics in equal|167|intervals=prime|start=14|columns=12}}
167edo is the 39th [[prime edo]].


==Regular temperament properties==
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" | [[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" | [[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" | [[Mapping]]
! rowspan="2" |Optimal<br>8ve Stretch (¢)
! rowspan="2" | Optimal<br>8ve Stretch (¢)
! colspan="2" |Tuning Error
! colspan="2" | Tuning Error
|-
|-
![[TE error|Absolute]] (¢)
! [[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
! [[TE simple badness|Relative]] (%)
|-
|-
|2.3
| 2.3
|{{monzo|265 -167}}
| {{monzo| 265 -167 }}
|{{val|167 265}}
| {{val| 167 265 }}
| -0.7056
| -0.7056
| 0.7052
| 0.7052
| 9.81
| 9.81
|-
|-
|2.3.5
| 2.3.5
|{{monzo|17 1 -8}}, {{monzo|14 -22 9}}
| 393216/390625, {{monzo| 31 -21 1 }}
|{{val|167 265 388}}
| {{val| 167 265 388 }}
| -0.7158
| -0.7158
| 0.5759
| 0.5759
| 8.01
| 8.01
|-
|-
|2.3.5.7
| 2.3.5.7
|6144/6125, 3136/3125, 179200/177147
| 2401/2400, 3136/3125, 179200/177147
|{{val|167 265 388 469}}
| {{val| 167 265 388 469 }}
| -0.6467
| -0.6467
| 0.5129
| 0.5129
| 7.14
| 7.14
|-
|-
|2.3.5.7.11
| 2.3.5.7.11
|896/891, 2200/2187, 6144/6125, 6250/6237
| 896/891, 2200/2187, 2401/2400, 3136/3125
|{{val|167 265 388 469 578}}
| {{val| 167 265 388 469 578 }}
| -0.6315
| -0.6315
| 0.4598
| 0.4598
| 6.40
| 6.40
|-
|-
|2.3.5.7.11.13
| 2.3.5.7.11.13
|325/324, 352/351, 896/891, 1001/1000, 6656/6615
| 325/324, 352/351, 364/363, 1001/1000, 1716/1715
|{{val|167 265 388 469 578 618}}
| {{val| 167 265 388 469 578 618 }}
| -0.5349
| -0.5349
| 0.4721
| 0.4721
| 6.57
| 6.57
|-
|-
|2.3.5.7.11.13.17
| 2.3.5.7.11.13.17
|325/324, 352/351, 896/891, 256/255, 1001/1000, 1225/1224
| 256/255, 325/324, 352/351, 364/363, 442/441, 1001/1000
|{{val|167 265 388 469 578 618 683}}
| {{val| 167 265 388 469 578 618 683 }}
| -0.5573
| -0.5573
| 0.4405
| 0.4405
| 6.13
| 6.13
|}
|}
[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
[[Category:Prime EDO]]

Revision as of 10:05, 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

Template:EDO intro

Theory

167et tempers out the würschmidt comma, 393216/390625, and the leapday comma, [31 -21 1⟩, 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 the 11-limit unthirds temperament.

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

Prime harmonics

Approximation of prime harmonics in 167edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31 37
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
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
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)
Approximation of prime harmonics in 167edo (continued)
Harmonic 41 43 47 53 59 61 67 71 73 79 83 89
Error Absolute (¢) +2.08 -1.34 +2.76 +3.14 -2.88 -3.11 -0.27 -0.06 +2.15 +1.93 +2.65 -3.22
Relative (%) +28.9 -18.6 +38.4 +43.7 -40.1 -43.3 -3.7 -0.8 +29.9 +26.9 +36.8 -44.7
Steps
(reduced) 		895
(60) 		906
(71) 		928
(93) 		957
(122) 		982
(147) 		990
(155) 		1013
(11) 		1027
(25) 		1034
(32) 		1053
(51) 		1065
(63) 		1081
(79)

Subsets and supersets

167edo is the 39th prime edo.

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 393216/390625, [31 -21 1⟩ ⟨167 265 388] -0.7158 0.5759 8.01
2.3.5.7 2401/2400, 3136/3125, 179200/177147 ⟨167 265 388 469] -0.6467 0.5129 7.14
2.3.5.7.11 896/891, 2200/2187, 2401/2400, 3136/3125 ⟨167 265 388 469 578] -0.6315 0.4598 6.40
2.3.5.7.11.13 325/324, 352/351, 364/363, 1001/1000, 1716/1715 ⟨167 265 388 469 578 618] -0.5349 0.4721 6.57
2.3.5.7.11.13.17 256/255, 325/324, 352/351, 364/363, 442/441, 1001/1000 ⟨167 265 388 469 578 618 683] -0.5573 0.4405 6.13
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