167edo: Difference between revisions

Revision as of 13:22, 21 October 2023

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.

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
Error	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
Error	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)

167edo is the 39th prime edo.

Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	Tuning Error
Subgroup	Comma List	Mapping	Optimal 8ve Stretch (¢)	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

"way too random partying" from Helayo EP (2023) – Spotify | Bandcamp | YouTube – helayo in 167edo tuning

@@ Line 27: / Line 27: @@
 | 2.3
 | {{monzo| 265 -167 }}
-| {{val| 167 265 }}
+| {{mapping| 167 265 }}
 | -0.7056
 | 0.7052
@@ Line 34: / Line 34: @@
 | 2.3.5
 | 393216/390625, {{monzo| 31 -21 1 }}
-| {{val| 167 265 388 }}
+| {{mapping| 167 265 388 }}
 | -0.7158
 | 0.5759
@@ Line 41: / Line 41: @@
 | 2.3.5.7
 | 2401/2400, 3136/3125, 179200/177147
-| {{val| 167 265 388 469 }}
+| {{mapping| 167 265 388 469 }}
 | -0.6467
 | 0.5129
@@ Line 48: / Line 48: @@
 | 2.3.5.7.11
 | 896/891, 2200/2187, 2401/2400, 3136/3125
-| {{val| 167 265 388 469 578 }}
+| {{mapping| 167 265 388 469 578 }}
 | -0.6315
 | 0.4598
@@ Line 55: / Line 55: @@
 | 2.3.5.7.11.13
 | 325/324, 352/351, 364/363, 1001/1000, 1716/1715
-| {{val| 167 265 388 469 578 618 }}
+| {{mapping| 167 265 388 469 578 618 }}
 | -0.5349
 | 0.4721
@@ Line 62: / Line 62: @@
 | 2.3.5.7.11.13.17
 | 256/255, 325/324, 352/351, 364/363, 442/441, 1001/1000
-| {{val| 167 265 388 469 578 618 683 }}
+| {{mapping| 167 265 388 469 578 618 683 }}
 | -0.5573
 | 0.4405
@@ Line 69: / Line 69: @@
 == Music ==
-* [https://www.youtube.com/watch?v=33T11NI7EQQ way too random partying] by Francium
+; [[User:Francium|Francium]]
+* "way too random partying" from ''Helayo EP'' (2023) – [https://open.spotify.com/track/4yf5R4eVOxK2fgZEZRfCqU Spotify] | [https://francium223.bandcamp.com/track/way-too-random-partying Bandcamp] | [https://youtu.be/33T11NI7EQQ?si=mZ57p2EN4uvPCVo7 YouTube] – helayo in 167edo tuning