643edo: Difference between revisions

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== Theory ==

643edo is ~~uniquely~~ [[consistent]] to the 21-odd-limit, with a generally flat tendency, but the 5th harmonic is only 0.000439 cents sharp as the denominator of a convergent to log25, after [[146edo|146]] and before [[4004edo|4004]]. It tempers out [[32805/32768]] in the 5-limit and [[2401/2400]] in the 7-limit, so that it [[support]]s the [[sesquiquartififths]] temperament. In the 11-limit it tempers out [[3025/3024]] and 151263/151250; in the 13-limit [[1001/1000]], [[1716/1715]] and [[4225/4224]]; in the 17-limit [[1089/1088]], [[1701/1700]], 2431/2430 and [[2601/2600]]; and in the 19-limit 1331/1330, [[1521/1520]], [[1729/1728]], 2376/2375 and 2926/2925. It provides the [[optimal patent val]] for the rank-3 13-limit [[vili]] temperament.

643edo is [[consistency|distinctly consistent]] to the [[21-odd-limit]], with a generally flat tendency, but the [[5/1|5th harmonic]] is only 0.000439 cents sharp as the denominator of a convergent to log25, after [[146edo|146]] and before [[4004edo|4004]]. It [[tempering out|tempers out]] [[32805/32768]] in the 5-limit and [[2401/2400]] in the 7-limit, so that it [[support]]s the [[sesquiquartififths]] temperament. In the 11-limit it tempers out [[3025/3024]] and 151263/151250; in the 13-limit [[1001/1000]], [[1716/1715]] and [[4225/4224]]; in the 17-limit [[1089/1088]], [[1701/1700]], [[2431/2430]] and [[2601/2600]]; and in the 19-limit 1331/1330, [[1521/1520]], [[1729/1728]], 2376/2375 and 2926/2925. It provides the [[optimal patent val]] for the rank-3 13-limit [[vili]] temperament.

=== Prime harmonics ===

=== ~~Miscellaneous properties~~ ===

=== Subsets and supersets ===

643edo is the 117th [[prime edo]].

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| 2.3

| {{monzo| -1019 643 }}

| [{{~~val~~| 643 1019 }}]

| {{mapping| 643 1019 }}

| +0.0771

| 0.0771

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| 2.3.5

| 32805/32768, {{monzo| 1 99 -68 }}

| [{{~~val~~| 643 1019 1493 }}]

| {{mapping| 643 1019 1493 }}

| +0.0513

| 0.7270

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| 2.3.5.7

| 2401/2400, 32805/32768, {{monzo| 9 21 -17 -1 }}

| [{{~~val~~| 643 1019 1493 1805 }}]

| {{mapping| 643 1019 1493 1805 }}

| +0.0600

| 0.0647

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| 2.3.5.7.11

| 2401/2400, 3025/3024, 32805/32768, 391314/390625

| [{{~~val~~| 643 1019 1493 1805 2224 }}]

| {{mapping| 643 1019 1493 1805 2224 }}

| +0.0927

| 0.0874

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| 2.3.5.7.11.13

| 1001/1000, 1716/1715, 3025/3024, 4225/4224, 32805/32768

| [{{~~val~~| 643 1019 1493 1805 2224 2379 }}]

| {{mapping| 643 1019 1493 1805 2224 2379 }}

| +0.1094

| 0.0881

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| 2.3.5.7.11.13.17

| 1001/1000, 1089/1088, 1701/1700, 1716/1715, 2601/2600, 4225/4224

| [{{~~val~~| 643 1019 1493 1805 2224 2379 2628 }}]

|{{mapping| 643 1019 1493 1805 2224 2379 2628 }}

| +0.1094

| 0.0816

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| 2.3.5.7.11.13.17.19

| 1001/1000, 1089/1088, 1521/1520, 1701/1700, 1716/1715, 1729/1728, 2601/2600

| [{{~~val~~| 643 1019 1493 1805 2224 2379 2628 2731 }}]

| {{mapping| 643 1019 1493 1805 2224 2379 2628 2731 }}

| +0.1186

| 0.0801

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

|+Table of rank-2 temperaments by generator

! Periods per ~~Octave~~

! Periods per 8ve

! Generator~~ (Reduced)~~

! Generator*

! Cents~~ (Reduced)~~

! Cents*

! Associated Ratio

! Associated Ratio*

! Temperaments

|-

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| [[Helmholtz]]

|}

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

~~[[Category:Equal divisions of the octave|###]] ~~

[[Category:Sesquiquartififths]]

[[Category:Vili]]

~~[[Category:Prime EDO]]~~

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is uniquely [[consistent]] to the 21-odd-limit, with a generally flat tendency, but the 5th harmonic is only 0.000439 cents sharp as the denominator of a convergent to log<sub>2</sub>5, after [[146edo|146]] and before [[4004edo|4004]]. It tempers out [[32805/32768]] in the 5-limit and [[2401/2400]] in the 7-limit, so that it [[support]]s the [[sesquiquartififths]] temperament. In the 11-limit it tempers out [[3025/3024]] and 151263/151250; in the 13-limit [[1001/1000]], [[1716/1715]] and [[4225/4224]]; in the 17-limit [[1089/1088]], [[1701/1700]], 2431/2430 and [[2601/2600]]; and in the 19-limit 1331/1330, [[1521/1520]], [[1729/1728]], 2376/2375 and 2926/2925. It provides the [[optimal patent val]] for the rank-3 13-limit [[vili]] temperament.
+edo is [[consistency|distinctly consistent]] to the [[21-odd-limit]], with a generally flat tendency, but the [[5/1|5th harmonic]] is only 0.000439 cents sharp as the denominator of a convergent to log<sub>2</sub>5, after [[146edo|146]] and before [[4004edo|4004]]. It [[tempering out|tempers out]] [[32805/32768]] in the 5-limit and [[2401/2400]] in the 7-limit, so that it [[support]]s the [[sesquiquartififths]] temperament. In the 11-limit it tempers out [[3025/3024]] and 151263/151250; in the 13-limit [[1001/1000]], [[1716/1715]] and [[4225/4224]]; in the 17-limit [[1089/1088]], [[1701/1700]], [[2431/2430]] and [[2601/2600]]; and in the 19-limit 1331/1330, [[1521/1520]], [[1729/1728]], 2376/2375 and 2926/2925. It provides the [[optimal patent val]] for the rank-3 13-limit [[vili]] temperament.
 === Prime harmonics ===
 {{Harmonics in equal|643|columns=11}}
-=== Miscellaneous properties ===
+=== Subsets and supersets ===
 edo is the 117th [[prime edo]].
@@ Line 24: / Line 24: @@
 | 2.3
 | {{monzo| -1019 643 }}
-| [{{val| 643 1019 }}]
+| {{mapping| 643 1019 }}
 | +0.0771
 | 0.0771
@@ Line 31: / Line 31: @@
 | 2.3.5
 | 32805/32768, {{monzo| 1 99 -68 }}
-| [{{val| 643 1019 1493 }}]
+| {{mapping| 643 1019 1493 }}
 | +0.0513
 | 0.7270
@@ Line 38: / Line 38: @@
 | 2.3.5.7
 | 2401/2400, 32805/32768, {{monzo| 9 21 -17 -1 }}
-| [{{val| 643 1019 1493 1805 }}]
+| {{mapping| 643 1019 1493 1805 }}
 | +0.0600
 | 0.0647
@@ Line 45: / Line 45: @@
 | 2.3.5.7.11
 | 2401/2400, 3025/3024, 32805/32768, 391314/390625
-| [{{val| 643 1019 1493 1805 2224 }}]
+| {{mapping| 643 1019 1493 1805 2224 }}
 | +0.0927
 | 0.0874
@@ Line 52: / Line 52: @@
 | 2.3.5.7.11.13
 | 1001/1000, 1716/1715, 3025/3024, 4225/4224, 32805/32768
-| [{{val| 643 1019 1493 1805 2224 2379 }}]
+| {{mapping| 643 1019 1493 1805 2224 2379 }}
 | +0.1094
 | 0.0881
@@ Line 59: / Line 59: @@
 | 2.3.5.7.11.13.17
 | 1001/1000, 1089/1088, 1701/1700, 1716/1715, 2601/2600, 4225/4224
-| [{{val| 643 1019 1493 1805 2224 2379 2628 }}]
+|{{mapping| 643 1019 1493 1805 2224 2379 2628 }}
 | +0.1094
 | 0.0816
@@ Line 66: / Line 66: @@
 | 2.3.5.7.11.13.17.19
 | 1001/1000, 1089/1088, 1521/1520, 1701/1700, 1716/1715, 1729/1728, 2601/2600
-| [{{val| 643 1019 1493 1805 2224 2379 2628 2731 }}]
+| {{mapping| 643 1019 1493 1805 2224 2379 2628 2731 }}
 | +0.1186
 | 0.0801
@@ Line 75: / Line 75: @@
 {| class="wikitable center-all left-5"
 |+Table of rank-2 temperaments by generator
-! Periods<br>per Octave
+! Periods<br>per 8ve
-! Generator<br>(Reduced)
+! Generator*
-! Cents<br>(Reduced)
+! Cents*
-! Associated<br>Ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
@@ Line 93: / Line 93: @@
 | [[Helmholtz]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Sesquiquartififths]]
 [[Category:Vili]]
-[[Category:Prime EDO]]