253edo: Difference between revisions

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== Theory ==

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. It 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.

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 harmonic]]s from 5 to 17 are all slightly flat. As an equal temperament, it [[tempering out|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.

=== Prime harmonics ===

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=== Subsets and supersets ===

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

Since 253 factors into 11 × 23, and has subset edos [[11edo]] and [[23edo]]. [[1012edo]] divides 253edo's step size into 4 equal parts and provides a good approximation of the 13-limit.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list~~|Comma List~~]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal ~~8ve~~ ~~Stretch~~ (¢)

! rowspan="2" | Optimal 8ve stretch (¢)

! colspan="2" | Tuning ~~Error~~

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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| 2.3

| {{monzo| 401 -253 }}

| [{{~~val~~| 253 401 }}]

| {{mapping| 253 401 }}

| -0.007

| −0.007

| 0.007

| 0.14

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| 2.3.5

| 32805/32768, {{monzo| -4 -37 27 }}

| [{{~~val~~| 253 401 587 }}]

| {{mapping| 253 401 587 }}

| +0.300

| 0.435

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| 2.3.5.7

| 2401/2400, 32805/32768, 390625/387072

| [{{~~val~~| 253 401 587 710 }}]

| {{mapping| 253 401 587 710 }}

| +0.335

| 0.381

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| 2.3.5.7.11

| 385/384, 1375/1372, 4000/3993, 19712/19683

| [{{~~val~~| 253 401 587 710 875 }}]

| {{mapping| 253 401 587 710 875 }}

| +0.333

| 0.341

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| 2.3.5.7.11.13

| 325/324, 385/384, 1375/1372, 1575/1573, 2200/2197

| [{{~~val~~| 253 401 587 710 875 936 }}]

| {{mapping| 253 401 587 710 875 936 }}

| +0.323

| 0.312

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| 2.3.5.7.11.13.17

| 325/324, 375/374, 385/384, 595/594, 1275/1274, 2200/2197

| [{{~~val~~| 253 401 587 710 875 936 1034 }}]

| {{mapping| 253 401 587 710 875 936 1034 }}

| +0.298

| 0.295

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=== Rank-2 temperaments ===

{| class="wikitable center-all left-5"

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per ~~Octave~~

|-

! Generator~~ (Reduced)~~

! Periods per 8ve

! Cents~~ (Reduced)~~

! Generator*

! Associated ~~Ratio~~

! Cents*

! Associated ratio*

! Temperaments

|-

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| 498.02

| 4/3

| [[Helmholtz]]

| [[Helmholtz (temperament)|Helmholtz]]

|-

| 1

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| [[Cotritone]]

|}

<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct

== Scales ==

* 63 32 63 63 32: [[3L 2s|~~Pentic~~]]

* 63 32 63 63 32: One of many [[3L 2s|pentic]] scales available

* 43 43 19 43 43 43 19: [[Helmholtz]][7]

* 43 43 19 43 43 43 19: [[Helmholtz (temperament)|Helmholtz]][7]

* 41 41 24 41 41 41 24: [[Meantone]][7]

* 35 35 35 35 35 35 35 8: [[Porcupine]][8]

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* 20 20 20 11 20 20 20 20 11 20 20 20 20 11: [[11L 3s|Ketradektriatoh scale]]

[[Category:3-limit record edos|###]]

[[Category:Tertiaschis]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|253}}
+{{ED intro}}
 == Theory ==
-edo 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. It 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.
+edo 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 harmonic]]s from 5 to 17 are all slightly flat. As an equal temperament, it [[tempering out|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.
 === Prime harmonics ===
@@ Line 9: / Line 9: @@
 === Subsets and supersets ===
-= 11 × 23, and has subset edos [[11edo]] and [[23edo]].
+Since 253 factors into 11 × 23, and has subset edos [[11edo]] and [[23edo]]. [[1012edo]] divides 253edo's step size into 4 equal parts and provides a good approximation of the 13-limit.
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list|Comma List]]
+! rowspan="2" | [[Comma list]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal 8ve <br>Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 24: / Line 25: @@
 | 2.3
 | {{monzo| 401 -253 }}
-| [{{val| 253 401 }}]
+| {{mapping| 253 401 }}
-| -0.007
+| −0.007
 | 0.007
 | 0.14
@@ Line 31: / Line 32: @@
 | 2.3.5
 | 32805/32768, {{monzo| -4 -37 27 }}
-| [{{val| 253 401 587 }}]
+| {{mapping| 253 401 587 }}
 | +0.300
 | 0.435
@@ Line 38: / Line 39: @@
 | 2.3.5.7
 | 2401/2400, 32805/32768, 390625/387072
-| [{{val| 253 401 587 710 }}]
+| {{mapping| 253 401 587 710 }}
 | +0.335
 | 0.381
@@ Line 45: / Line 46: @@
 | 2.3.5.7.11
 | 385/384, 1375/1372, 4000/3993, 19712/19683
-| [{{val| 253 401 587 710 875 }}]
+| {{mapping| 253 401 587 710 875 }}
 | +0.333
 | 0.341
@@ Line 52: / Line 53: @@
 | 2.3.5.7.11.13
 | 325/324, 385/384, 1375/1372, 1575/1573, 2200/2197
-| [{{val| 253 401 587 710 875 936 }}]
+| {{mapping| 253 401 587 710 875 936 }}
 | +0.323
 | 0.312
@@ Line 59: / Line 60: @@
 | 2.3.5.7.11.13.17
 | 325/324, 375/374, 385/384, 595/594, 1275/1274, 2200/2197
-| [{{val| 253 401 587 710 875 936 1034 }}]
+| {{mapping| 253 401 587 710 875 936 1034 }}
 | +0.298
 | 0.295
@@ Line 67: / Line 68: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per Octave
+|-
-! Generator<br>(Reduced)
+! Periods<br />per 8ve
-! Cents<br>(Reduced)
+! Generator*
-! Associated<br>Ratio
+! Cents*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 90: / Line 92: @@
 | 498.02
 | 4/3
-| [[Helmholtz]]
+| [[Helmholtz (temperament)|Helmholtz]]
 |-
 | 1
@@ Line 98: / Line 100: @@
 | [[Cotritone]]
 |}
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Scales ==
-* 63 32 63 63 32: [[3L 2s|Pentic]]
+* 63 32 63 63 32: One of many [[3L 2s|pentic]] scales available
-* 43 43 19 43 43 43 19: [[Helmholtz]][7]
+* 43 43 19 43 43 43 19: [[Helmholtz (temperament)|Helmholtz]][7]
 * 41 41 24 41 41 41 24: [[Meantone]][7]
 * 35 35 35 35 35 35 35 8: [[Porcupine]][8]
@@ Line 109: / Line 112: @@
 * 20 20 20 11 20 20 20 20 11 20 20 20 20 11: [[11L 3s|Ketradektriatoh scale]]
+[[Category:3-limit record edos|###]] <!-- 3-digit number -->
 [[Category:Tertiaschis]]