436edo: Difference between revisions

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== Theory ==

436edo is [[consistent]] to the [[23-odd-limit]]. The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, ~~harmonics~~ from 3 to 37 are all flat. It tempers out [[32805/32768]] and {{monzo| 1 -68 46 }} in the 5-limit; [[390625/388962]], 420175/419904, and [[2100875/2097152]] in the 7-limit; 1375/1372, [[6250/6237]], [[41503/41472]], and 322102/321489 in the 11-limit; [[625/624]], [[1716/1715]], [[2080/2079]], [[10648/10647]], and 15379/15360 in the 13-limit; [[715/714]], [[1089/1088]], [[1225/1224]], 1275/1274, [[2025/2023]], and 11271/11264 in the 17-limit; 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit.

436edo is [[consistent]] to the [[23-odd-limit]]. The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, [[harmonic]]s from 3 to 37 are all flat.

436edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 [[subgroup]].

The equal temperament [[tempering out|tempers out]] [[32805/32768]] and {{monzo| 1 -68 46 }} in the 5-limit; [[390625/388962]], 420175/419904, and [[2100875/2097152]] in the 7-limit; 1375/1372, [[6250/6237]], [[41503/41472]], and 322102/321489 in the 11-limit; [[625/624]], [[1716/1715]], [[2080/2079]], [[10648/10647]], and 15379/15360 in the 13-limit; [[715/714]], [[1089/1088]], [[1225/1224]], [[1275/1274]], [[2025/2023]], and 11271/11264 in the 17-limit; 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit. It [[support]]s and gives a good tuning to [[quadrant]]. It also supports [[tsaharuk]], but [[171edo]] is better suited for that purpose.

436edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 [[subgroup]].

=== Prime harmonics ===

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=== Subsets and supersets ===

Since 436 factors into ~~22 × 109~~, 436edo has subset edos {{EDOs| 2, 4, 109, and 218 }}.

Since 436 factors into {{factorization|436}}, 436edo has subset edos {{EDOs| 2, 4, 109, and 218 }}.

[[1308edo]], which divides ~~the~~ edostep into three, is a [[zeta gap edo]] and is consistent in the 21-odd-limit.

[[1308edo]], which divides its edostep into three, is a [[zeta gap edo]] and is consistent in the 21-odd-limit.

== Regular temperament properties ==

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| 2.3

| {{monzo| -691 436 }}

| [{{~~val~~| 436 691 }}]

| {{mapping| 436 691 }}

| +0.0379

| 0.0379

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| 2.3.5

| 32805/32768, {{monzo| 1 -68 46 }}

| [{{~~val~~| 436 691 1012 }}]

| {{mapping| 436 691 1012 }}

| +0.1678

| 0.1863

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| 2.3.5.7

| 32805/32768, 390625/388962, 420175/419904

| [{{~~val~~| 436 691 1012 1224 }}]

| {{mapping| 436 691 1012 1224 }}

| +0.1275

| 0.1758

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| 2.3.5.7.11

| 1375/1372, 6250/6237, 32805/32768, 41503/41472

| [{{~~val~~| 436 691 1012 1224 1508 }}]

| {{mapping| 436 691 1012 1224 1508 }}

| +0.1517

| 0.1645

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| 2.3.5.7.11.13

| 625/624, 1375/1372, 2080/2079, 10648/10647, 15379/15360

| [{{~~val~~| 436 691 1012 1224 1508 1613 }}]

| {{mapping| 436 691 1012 1224 1508 1613 }}

| +0.1749

| 0.1589

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| 2.3.5.7.11.13.17

| 625/624, 715/714, 1089/1088, 1225/1224, 2431/2430, 10648/10647

| [{{~~val~~| 436 691 1012 1224 1508 1613 1782 }}]

| {{mapping| 436 691 1012 1224 1508 1613 1782 }}

| +0.1628

| 0.1501

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| 2.3.5.7.11.13.17.19

| 625/624, 715/714, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1729/1728

| [{{~~val~~| 436 691 1012 1224 1508 1613 1782 1852 }}]

| {{mapping| 436 691 1012 1224 1508 1613 1782 1852 }}

| +0.1503

| 0.1443

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[Quadrant]]

|}

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

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is [[consistent]] to the [[23-odd-limit]]. The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, harmonics from 3 to 37 are all flat. It tempers out [[32805/32768]] and {{monzo| 1 -68 46 }} in the 5-limit; [[390625/388962]], 420175/419904, and [[2100875/2097152]] in the 7-limit; 1375/1372, [[6250/6237]], [[41503/41472]], and 322102/321489 in the 11-limit; [[625/624]], [[1716/1715]], [[2080/2079]], [[10648/10647]], and 15379/15360 in the 13-limit; [[715/714]], [[1089/1088]], [[1225/1224]], 1275/1274, [[2025/2023]], and 11271/11264 in the 17-limit; 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit.
+edo is [[consistent]] to the [[23-odd-limit]]. The [[patent val]] of 436edo has a distinct flat tendency, in the sense that if the [[octave]] is pure, [[harmonic]]s from 3 to 37 are all flat.
-edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 [[subgroup]].
+The equal temperament [[tempering out|tempers out]] [[32805/32768]] and {{monzo| 1 -68 46 }} in the 5-limit; [[390625/388962]], 420175/419904, and [[2100875/2097152]] in the 7-limit; 1375/1372, [[6250/6237]], [[41503/41472]], and 322102/321489 in the 11-limit; [[625/624]], [[1716/1715]], [[2080/2079]], [[10648/10647]], and 15379/15360 in the 13-limit; [[715/714]], [[1089/1088]], [[1225/1224]], [[1275/1274]], [[2025/2023]], and 11271/11264 in the 17-limit; 1331/1330, [[1445/1444]], [[1521/1520]], 1540/1539, [[1729/1728]], 4394/4389, and 4875/4864 in the 19-limit; 875/874, 897/896, 1105/1104, 1863/1862, 2024/2023, 2185/2184, 2300/2299, and 2530/2527 in the 23-limit. It [[support]]s and gives a good tuning to [[quadrant]]. It also supports [[tsaharuk]], but [[171edo]] is better suited for that purpose.
+edo is accurate for some intervals including [[3/2]], [[7/4]], [[11/10]], [[13/10]], [[18/17]], and [[19/18]], so it is especially suitable for the 2.3.7.11/5.13/5.17.19 [[subgroup]].
 === Prime harmonics ===
@@ Line 11: / Line 13: @@
 === Subsets and supersets ===
-Since 436 factors into 2<sup>2</sup> × 109, 436edo has subset edos {{EDOs| 2, 4, 109, and 218 }}.
+Since 436 factors into {{factorization|436}}, 436edo has subset edos {{EDOs| 2, 4, 109, and 218 }}.
-[[1308edo]], which divides the edostep into three, is a [[zeta gap edo]] and is consistent in the 21-odd-limit.
+[[1308edo]], which divides its edostep into three, is a [[zeta gap edo]] and is consistent in the 21-odd-limit.
 == Regular temperament properties ==
@@ Line 28: / Line 30: @@
 | 2.3
 | {{monzo| -691 436 }}
-| [{{val| 436 691 }}]
+| {{mapping| 436 691 }}
 | +0.0379
 | 0.0379
@@ Line 35: / Line 37: @@
 | 2.3.5
 | 32805/32768, {{monzo| 1 -68 46 }}
-| [{{val| 436 691 1012 }}]
+| {{mapping| 436 691 1012 }}
 | +0.1678
 | 0.1863
@@ Line 42: / Line 44: @@
 | 2.3.5.7
 | 32805/32768, 390625/388962, 420175/419904
-| [{{val| 436 691 1012 1224 }}]
+| {{mapping| 436 691 1012 1224 }}
 | +0.1275
 | 0.1758
@@ Line 49: / Line 51: @@
 | 2.3.5.7.11
 | 1375/1372, 6250/6237, 32805/32768, 41503/41472
-| [{{val| 436 691 1012 1224 1508 }}]
+| {{mapping| 436 691 1012 1224 1508 }}
 | +0.1517
 | 0.1645
@@ Line 56: / Line 58: @@
 | 2.3.5.7.11.13
 | 625/624, 1375/1372, 2080/2079, 10648/10647, 15379/15360
-| [{{val| 436 691 1012 1224 1508 1613 }}]
+| {{mapping| 436 691 1012 1224 1508 1613 }}
 | +0.1749
 | 0.1589
@@ Line 63: / Line 65: @@
 | 2.3.5.7.11.13.17
 | 625/624, 715/714, 1089/1088, 1225/1224, 2431/2430, 10648/10647
-| [{{val| 436 691 1012 1224 1508 1613 1782 }}]
+| {{mapping| 436 691 1012 1224 1508 1613 1782 }}
 | +0.1628
 | 0.1501
@@ Line 70: / Line 72: @@
 | 2.3.5.7.11.13.17.19
 | 625/624, 715/714, 1089/1088, 1225/1224, 1331/1330, 1445/1444, 1729/1728
-| [{{val| 436 691 1012 1224 1508 1613 1782 1852 }}]
+| {{mapping| 436 691 1012 1224 1508 1613 1782 1852 }}
 | +0.1503
 | 0.1443
@@ Line 80: / Line 82: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
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 | [[Quadrant]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct