509edo: Difference between revisions

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== Theory ==

509edo has a sharp tendency in lower [[harmonic]]s. The 13-limit [[TE tuning|optimal tuning]] of this temperament is consistent to the 15-integer-limit, so one might want to keep the [[octave compression]] tight.

509edo has a sharp tendency in lower [[harmonic]]s. The [[13-limit]] [[TE tuning|optimal tuning]] of this temperament is consistent to the 15-integer-limit, so one might want to keep the [[octave compression]] tight.

~~The~~ equal temperament [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) in the 5-limit; [[2401/2400]] and 29360128/29296875 in the 7-limit; and [[3025/3024]], [[5632/5625]], [[41503/41472]], 42592/42525, 151263/151250, 172032/171875, 180224/180075, 322102/321489, 422576/421875, 456533/455625, and [[1953125/1948617]] in the 11-limit. It provides the [[optimal patent val]] for [[petrtri]], the 2.11/5.13/5 subgroup temperament tempering out [[2200/2197]].

As an equal temperament, it [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) in the 5-limit; [[2401/2400]] and 29360128/29296875 in the 7-limit; and [[3025/3024]], [[5632/5625]], [[41503/41472]], 42592/42525, 151263/151250, 172032/171875, 180224/180075, 322102/321489, 422576/421875, 456533/455625, and [[1953125/1948617]] in the 11-limit. It provides the [[optimal patent val]] for [[petrtri]], the 2.11/5.13/5 subgroup temperament tempering out [[2200/2197]].

=== Odd harmonics ===

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== 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]] (¢)

! [[TE simple badness|Relative]] (%)

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| {{monzo| 807 -509 }}

| {{mapping| 509 807 }}

| -0.1890

| −0.1890

| 0.1889

| 8.01

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| {{monzo| 9 -13 5 }}, {{monzo| 93 -3 -38 }}

| {{mapping| 509 807 1182 }}

| -0.1729

| −0.1729

| 0.1559

| 6.61

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| 2401/2400, 1600000/1594323, 29360128/29296875

| {{mapping| 509 807 1182 1429 }}

| -0.1415

| −0.1415

| 0.1456

| 6.18

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| 2401/2400, 3025/3024, 5632/5625, 1600000/1594323

| {{mapping| 509 807 1182 1429 1761 }}

| -0.1335

| −0.1335

| 0.1312

| 5.57

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| 2080/2079, 2200/2197, 2401/2400, 3025/3024, 5632/5625

| {{mapping| 509 807 1182 1429 1761 1884 }}

| -0.1618

| −0.1618

| 0.1354

| 5.74

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| 1225/1224, 2080/2079, 2200/2197, 2401/2400, 2431/2430, 4914/4913

| {{mapping| 509 807 1182 1429 1761 1884 2081 }}

| -0.1784

| −0.1784

| 0.1318

| 5.59

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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 8ve

|-

! Periods per 8ve

! Generator*

! Cents*

! Associated ~~Ratio~~*

! Associated ratio*

! Temperaments

|-

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| [[Amity]]

|}

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

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

== Music ==

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|509}}
+{{ED intro}}
 == Theory ==
-edo has a sharp tendency in lower [[harmonic]]s. The 13-limit [[TE tuning|optimal tuning]] of this temperament is consistent to the 15-integer-limit, so one might want to keep the [[octave compression]] tight.
+edo has a sharp tendency in lower [[harmonic]]s. The [[13-limit]] [[TE tuning|optimal tuning]] of this temperament is consistent to the 15-integer-limit, so one might want to keep the [[octave compression]] tight.
-The equal temperament [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) in the 5-limit; [[2401/2400]] and 29360128/29296875 in the 7-limit; and [[3025/3024]], [[5632/5625]], [[41503/41472]], 42592/42525, 151263/151250, 172032/171875, 180224/180075, 322102/321489, 422576/421875, 456533/455625, and [[1953125/1948617]] in the 11-limit. It provides the [[optimal patent val]] for [[petrtri]], the 2.11/5.13/5 subgroup temperament tempering out [[2200/2197]].
+As an equal temperament, it [[tempering out|tempers out]] 1600000/1594323 ([[amity comma]]) in the 5-limit; [[2401/2400]] and 29360128/29296875 in the 7-limit; and [[3025/3024]], [[5632/5625]], [[41503/41472]], 42592/42525, 151263/151250, 172032/171875, 180224/180075, 322102/321489, 422576/421875, 456533/455625, and [[1953125/1948617]] in the 11-limit. It provides the [[optimal patent val]] for [[petrtri]], the 2.11/5.13/5 subgroup temperament tempering out [[2200/2197]].
 === Odd harmonics ===
@@ Line 15: / Line 15: @@
 == 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<br>8ve Stretch (¢)
+! rowspan="2" | Optimal<br />8ve stretch (¢)
-! colspan="2" | Tuning Error
+! colspan="2" | Tuning error
 |-
 ! [[TE error|Absolute]] (¢)
 ! [[TE simple badness|Relative]] (%)
@@ Line 27: / Line 28: @@
 | {{monzo| 807 -509 }}
 | {{mapping| 509 807 }}
-| -0.1890
+| −0.1890
 | 0.1889
 | 8.01
@@ Line 34: / Line 35: @@
 | {{monzo| 9 -13 5 }}, {{monzo| 93 -3 -38 }}
 | {{mapping| 509 807 1182 }}
-| -0.1729
+| −0.1729
 | 0.1559
 | 6.61
@@ Line 41: / Line 42: @@
 | 2401/2400, 1600000/1594323, 29360128/29296875
 | {{mapping| 509 807 1182 1429 }}
-| -0.1415
+| −0.1415
 | 0.1456
 | 6.18
@@ Line 48: / Line 49: @@
 | 2401/2400, 3025/3024, 5632/5625, 1600000/1594323
 | {{mapping| 509 807 1182 1429 1761 }}
-| -0.1335
+| −0.1335
 | 0.1312
 | 5.57
@@ Line 55: / Line 56: @@
 | 2080/2079, 2200/2197, 2401/2400, 3025/3024, 5632/5625
 | {{mapping| 509 807 1182 1429 1761 1884 }}
-| -0.1618
+| −0.1618
 | 0.1354
 | 5.74
@@ Line 62: / Line 63: @@
 | 1225/1224, 2080/2079, 2200/2197, 2401/2400, 2431/2430, 4914/4913
 | {{mapping| 509 807 1182 1429 1761 1884 2081 }}
-| -0.1784
+| −0.1784
 | 0.1318
 | 5.59
@@ Line 69: / Line 70: @@
 === 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 8ve
+|-
+! Periods<br />per 8ve
 ! Generator*
 ! Cents*
-! Associated<br>Ratio*
+! Associated<br />ratio*
 ! Temperaments
 |-
@@ Line 100: / Line 102: @@
 | [[Amity]]
 |}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+<nowiki />* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
 == Music ==