253edo: Difference between revisions

253edo is consistent to the 17-odd-limit, approximating the fifth by 148\253 (0.021284 cents sharper than the just 3/2), and the prime harmonics from 5 to 17 are all slightly flat. The equal temperament tempers out 32805/32768 in the 5-limit; 2401/2400 in the 7-limit; 385/384, 1375/1372 and 4000/3993 in the 11-limit; 325/324, 1575/1573 and 2200/2197 in the 13-limit; 375/374 and 595/594 in the 17-limit. It provides the optimal patent val for the tertiaschis temperament, and a good tuning for the sesquiquartififths temperament in the higher limits.

Approximation of prime harmonics in 253edo
Harmonic		2	3	5	7	11	13	17	19	23	29	31
Error	Absolute (¢)	+0.00	+0.02	-2.12	-1.24	-1.12	-1.00	-0.61	+1.30	-2.19	-0.33	-1.95
Error	Relative (%)	+0.0	+0.4	-44.8	-26.1	-23.6	-21.1	-12.8	+27.4	-46.1	-6.9	-41.2
Steps (reduced)		253 (0)	401 (148)	587 (81)	710 (204)	875 (116)	936 (177)	1034 (22)	1075 (63)	1144 (132)	1229 (217)	1253 (241)

253 = 11 × 23, and has subset edos 11edo and 23edo.

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[401 -253⟩	[⟨253 401]]	−0.007	0.007	0.14
2.3.5	32805/32768, [-4 -37 27⟩	[⟨253 401 587]]	+0.300	0.435	9.16
2.3.5.7	2401/2400, 32805/32768, 390625/387072	[⟨253 401 587 710]]	+0.335	0.381	8.03
2.3.5.7.11	385/384, 1375/1372, 4000/3993, 19712/19683	[⟨253 401 587 710 875]]	+0.333	0.341	7.19
2.3.5.7.11.13	325/324, 385/384, 1375/1372, 1575/1573, 2200/2197	[⟨253 401 587 710 875 936]]	+0.323	0.312	6.58
2.3.5.7.11.13.17	325/324, 375/374, 385/384, 595/594, 1275/1274, 2200/2197	[⟨253 401 587 710 875 936 1034]]	+0.298	0.295	6.22

Table of rank-2 temperaments by generator
Periods per 8ve	Generator*	Cents*	Associated ratio*	Temperaments
1	35\253	166.01	11/10	Tertiaschis
1	37\253	175.49	448/405	Sesquiquartififths
1	105\253	498.02	4/3	Helmholtz
1	123\253	583.40	7/5	Cotritone

* Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if it is distinct

63 32 63 63 32: One of many pentic scales available
43 43 19 43 43 43 19: Helmholtz[7]
41 41 24 41 41 41 24: Meantone[7]
35 35 35 35 35 35 35 8: Porcupine[8]
33 33 33 11 33 33 33 33 11: "The Hendecapliqued superdiatonic of the Icositriphony"
31 31 31 18 31 31 31 31 18: Mavila[9]
26 26 15 26 26 26 15 26 26 26 15: Sensi[11]
20 20 20 11 20 20 20 20 11 20 20 20 20 11: Ketradektriatoh scale

@@ Line 12: / Line 12: @@
 == Regular temperament properties ==
-{{comma basis begin}}
+{| 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
@@ Line 55: / Line 64: @@
 | 0.295
 | 6.22
-{{comma basis end}}
+|}
 === Rank-2 temperaments ===
-{{rank-2 begin}}
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperaments
 |-
 | 1
@@ Line 83: / Line 99: @@
 | 7/5
 | [[Cotritone]]
-{{rank-2 end}}
+|}
-{{orf}}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 == Scales ==