345edo - Revision history

FloraC: Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

2026-03-13T13:31:43Z

Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"

← Older revision		Revision as of 13:31, 13 March 2026
Line 93:		Line 93:
	\| [[Quintosec]] (5-limit)		\| [[Quintosec]] (5-limit)
	\|}		\|}
	<nowiki />* [[Normal ~~lists~~\|Octave-reduced form]], reduced to the first half-octave, and [[~~Normal lists~~\|minimal form]] in parentheses if distinct		<nowiki />* [[Normal forms #Equave-reduced-generator form\|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form\|minimal form]] in parentheses if distinct

	[[Category:Kwai]]		[[Category:Kwai]]

Francium: changed EDO intro to ED intro

2025-02-21T06:29:48Z

changed EDO intro to ED intro

← Older revision		Revision as of 06:29, 21 February 2025
Line 1:		Line 1:
	{{Infobox ET}}		{{Infobox ET}}
	{{~~EDO~~ intro~~\|345~~}}		{{ED intro}}

	== Theory ==		== Theory ==

ArrowHead294 at 18:15, 15 January 2025

2025-01-15T18:15:58Z

← Older revision		Revision as of 18:15, 15 January 2025
Line 9:		Line 9:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	Since 345 factors into ~~3 × 5 × 23~~, 345edo has subset edos {{EDOs\| 3, 5, 15, 23, 69, and 115 }}.		Since 345 factors into {{factorisation\|345}}, 345edo has subset edos {{EDOs\| 3, 5, 15, 23, 69, and 115 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
Line 26:		Line 26:
	\| {{monzo\| 547 -345 }}		\| {{monzo\| 547 -345 }}
	\| {{mapping\| 345 547 }}		\| {{mapping\| 345 547 }}
	\| ~~−0~~.2062		\| −0.2062
	\| 0.2062		\| 0.2062
	\| 5.93		\| 5.93
Line 33:		Line 33:
	\| {{monzo\| 3 -18 11 }}, {{monzo\| 47 -15 -10 }}		\| {{monzo\| 3 -18 11 }}, {{monzo\| 47 -15 -10 }}
	\| {{mapping\| 345 547 801 }}		\| {{mapping\| 345 547 801 }}
	\| ~~−0~~.1050		\| −0.1050
	\| 0.2210		\| 0.2210
	\| 6.35		\| 6.35
Line 40:		Line 40:
	\| 5120/5103, 16875/16807, 68359375/68024448		\| 5120/5103, 16875/16807, 68359375/68024448
	\| {{mapping\| 345 547 801 969 }}		\| {{mapping\| 345 547 801 969 }}
	\| ~~−0~~.2220		\| −0.2220
	\| 0.2788		\| 0.2788
	\| 8.02		\| 8.02
Line 47:		Line 47:
	\| 540/539, 1375/1372, 5120/5103, 1953125/1940598		\| 540/539, 1375/1372, 5120/5103, 1953125/1940598
	\| {{mapping\| 345 547 801 969 1194 }}		\| {{mapping\| 345 547 801 969 1194 }}
	\| ~~−0~~.2773		\| −0.2773
	\| 0.2728		\| 0.2728
	\| 7.84		\| 7.84
Line 54:		Line 54:
	\| 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103		\| 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103
	\| {{mapping\| 345 547 801 969 1194 1277 }}		\| {{mapping\| 345 547 801 969 1194 1277 }}
	\| ~~−0~~.2857		\| −0.2857
	\| 0.2497		\| 0.2497
	\| 7.18		\| 7.18
Line 93:		Line 93:
	\| [[Quintosec]] (5-limit)		\| [[Quintosec]] (5-limit)
	\|}		\|}
	<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

	[[Category:Kwai]]		[[Category:Kwai]]

ArrowHead294: Partial undo

2024-11-16T12:54:30Z

Partial undo

← Older revision		Revision as of 12:54, 16 November 2024
Line 12:		Line 12:

	== Regular temperament properties ==		== 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		\| 2.3
	\| {{monzo\| 547 -345 }}		\| {{monzo\| 547 -345 }}
	\| {{mapping\| 345 547 }}		\| {{mapping\| 345 547 }}
	\| &~~ndash~~;0.2062		\| −0.2062
	\| 0.2062		\| 0.2062
	\| 5.93		\| 5.93
Line 24:		Line 33:
	\| {{monzo\| 3 -18 11 }}, {{monzo\| 47 -15 -10 }}		\| {{monzo\| 3 -18 11 }}, {{monzo\| 47 -15 -10 }}
	\| {{mapping\| 345 547 801 }}		\| {{mapping\| 345 547 801 }}
	\| &~~ndash~~;0.1050		\| −0.1050
	\| 0.2210		\| 0.2210
	\| 6.35		\| 6.35
Line 31:		Line 40:
	\| 5120/5103, 16875/16807, 68359375/68024448		\| 5120/5103, 16875/16807, 68359375/68024448
	\| {{mapping\| 345 547 801 969 }}		\| {{mapping\| 345 547 801 969 }}
	\| &~~ndash~~;0.2220		\| −0.2220
	\| 0.2788		\| 0.2788
	\| 8.02		\| 8.02
Line 38:		Line 47:
	\| 540/539, 1375/1372, 5120/5103, 1953125/1940598		\| 540/539, 1375/1372, 5120/5103, 1953125/1940598
	\| {{mapping\| 345 547 801 969 1194 }}		\| {{mapping\| 345 547 801 969 1194 }}
	\| &~~ndash~~;0.2773		\| −0.2773
	\| 0.2728		\| 0.2728
	\| 7.84		\| 7.84
Line 45:		Line 54:
	\| 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103		\| 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103
	\| {{mapping\| 345 547 801 969 1194 1277 }}		\| {{mapping\| 345 547 801 969 1194 1277 }}
	\| &~~ndash~~;0.2857		\| −0.2857
	\| 0.2497		\| 0.2497
	\| 7.18		\| 7.18
	~~{{comma basis end}~~}		\|}

	=== Rank-2 temperaments ===		=== 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		\| 1
Line 76:		Line 92:
	\| 1024/891<br />(16/15)		\| 1024/891<br />(16/15)
	\| [[Quintosec]] (5-limit)		\| [[Quintosec]] (5-limit)
	~~{{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

	[[Category:Kwai]]		[[Category:Kwai]]

ArrowHead294 at 03:18, 16 November 2024

2024-11-16T03:18:48Z

← Older revision		Revision as of 03:18, 16 November 2024
Line 12:		Line 12:

	== Regular temperament properties ==		== Regular temperament properties ==
	{~~\| class="wikitable center-4 center-5 center-6"~~		{{comma basis begin}}
	~~! rowspan="2" \| [[Subgroup]]~~
	~~! rowspan="2" \| [[Comma list\|Comma List]]~~
	~~! rowspan="2" \| [[Mapping]]~~
	~~! rowspan="2" \| Optimal<br>8ve Stretch (¢)~~
	~~! colspan="2" \| Tuning Error~~
	\|-
	~~! [[TE error\|Absolute]] (¢)~~
	~~! [[TE simple badness\|Relative]] (%)~~
	\|-		\|-
	\| 2.3		\| 2.3
	\| {{monzo\| 547 -345 }}		\| {{monzo\| 547 -345 }}
	\| {{mapping\| 345 547 }}		\| {{mapping\| 345 547 }}
	\| -0.2062		\| –0.2062
	\| 0.2062		\| 0.2062
	\| 5.93		\| 5.93
Line 32:		Line 24:
	\| {{monzo\| 3 -18 11 }}, {{monzo\| 47 -15 -10 }}		\| {{monzo\| 3 -18 11 }}, {{monzo\| 47 -15 -10 }}
	\| {{mapping\| 345 547 801 }}		\| {{mapping\| 345 547 801 }}
	\| -0.1050		\| –0.1050
	\| 0.2210		\| 0.2210
	\| 6.35		\| 6.35
Line 39:		Line 31:
	\| 5120/5103, 16875/16807, 68359375/68024448		\| 5120/5103, 16875/16807, 68359375/68024448
	\| {{mapping\| 345 547 801 969 }}		\| {{mapping\| 345 547 801 969 }}
	\| -0.2220		\| –0.2220
	\| 0.2788		\| 0.2788
	\| 8.02		\| 8.02
Line 46:		Line 38:
	\| 540/539, 1375/1372, 5120/5103, 1953125/1940598		\| 540/539, 1375/1372, 5120/5103, 1953125/1940598
	\| {{mapping\| 345 547 801 969 1194 }}		\| {{mapping\| 345 547 801 969 1194 }}
	\| -0.2773		\| –0.2773
	\| 0.2728		\| 0.2728
	\| 7.84		\| 7.84
Line 53:		Line 45:
	\| 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103		\| 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103
	\| {{mapping\| 345 547 801 969 1194 1277 }}		\| {{mapping\| 345 547 801 969 1194 1277 }}
	\| -0.2857		\| –0.2857
	\| 0.2497		\| 0.2497
	\| 7.18		\| 7.18
	\|}		{{comma basis end}}

	=== Rank-2 temperaments ===		=== Rank-2 temperaments ===
	{~~\| class="wikitable center-all left-5"~~		{{rank-2 begin}}
	~~\|+Table of~~ rank-2 ~~temperaments by generator~~
	~~! Periods<br>per 8ve~~
	! Generator*
	! Cents*
	! Associated<br>Ratio*
	~~! Temperaments~~
	\|-		\|-
	\| 1		\| 1
Line 86:		Line 72:
	\|-		\|-
	\| 5		\| 5
	\| 106\345<br>(32\345)		\| 106\345<br />(32\345)
	\| 368.70<br>(111.30)		\| 368.70<br />(111.30)
	\| 1024/891<br>(16/15)		\| 1024/891<br />(16/15)
	\| [[Quintosec]] (5-limit)		\| [[Quintosec]] (5-limit)
	\|}		{{rank-2 end}}
	~~<nowiki>*</nowiki> [[Normal lists\|octave-reduced form]], reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~		{{orf}}

	[[Category:Kwai]]		[[Category:Kwai]]

FloraC: Review

2023-12-15T09:00:05Z

Review

← Older revision		Revision as of 09:00, 15 December 2023
Line 44:		Line 44:
	\|-		\|-
	\| 2.3.5.7.11		\| 2.3.5.7.11
	\| 540/539, ~~3025~~/~~3024~~, 5120/5103, ~~26796875~~/~~26763264~~		\| 540/539, 1375/1372, 5120/5103, 1953125/1940598
	\| {{mapping\| 345 547 801 969 1194 }}		\| {{mapping\| 345 547 801 969 1194 }}
	\| -0.2773		\| -0.2773
Line 51:		Line 51:
	\|-		\|-
	\| 2.3.5.7.11.13		\| 2.3.5.7.11.13
	\| 540/539, 625/624, ~~3025~~/~~3024~~, 5120/5103~~, 4225/4224~~		\| 540/539, 625/624, 1375/1372, 4225/4224, 5120/5103
	\| {{mapping\| 345 547 801 969 1194 1277 }}		\| {{mapping\| 345 547 801 969 1194 1277 }}
	\| -0.2857		\| -0.2857
Line 66:		Line 66:
	! Associated<br>Ratio*		! Associated<br>Ratio*
	! Temperaments		! Temperaments
			\|-
			\| 1
			\| 13\345
			\| 45.22
			\| 250/243
			\| [[Quartonic]] (5-limit)
	\|-		\|-
	\| 1		\| 1

FloraC: Review

2023-12-05T16:07:27Z

Review

← Older revision		Revision as of 16:07, 5 December 2023
Line 3:		Line 3:

	== Theory ==		== Theory ==
	345et is only consistent to the 5-limit~~. Using~~ the patent val~~, it~~ tempers out ~~2460375~~/~~2458624, 134217728/133984375~~, [[~~5120~~/~~5103~~]], ~~95703125~~/~~95551488~~ and ~~359661568~~/~~358722675~~ in the 7-limit; ~~56723625/56689952, 20155392/20131375, 14348907/14348180, 1835008/1830125, 46656/46585, 2734375/2725888, 2657205/2656192,~~ [[~~16384~~/~~16335~~]], ~~514714375~~/~~514434888~~, [[~~540~~/~~539~~]]~~, 9765625/9732096, 369140625/369098752~~, [[~~200704~~/~~200475~~]]~~, 1375/1372, 537109375/536870912~~, [[19712/19683]], ~~968000~~/~~964467~~, [[~~3025~~/~~3024~~]], [[532400/531441]]~~, 202397184~~/~~201768035~~ and ~~20614528~~/~~20588575~~ in the 11-limit. It provides the optimal patent val ~~for [[quartonic~~]] ~~and~~ [[kwai]].		345et is only [[consistent]] to the [[5-odd-limit]], though it has a reasonable [[13-limit]] interpretation using the [[patent val]]. It [[tempering out\|tempers out]] {{monzo\| 3 -18 11 }} (quartonic comma) and {{monzo\| 47 -15 -10 }} (quintosec comma) in the 5-limit; [[5120/5103]], [[16875/16807]], 2460375/2458624, and 68359375/68024448 in the 7-limit; [[540/539]], 1375/1372, [[3025/3024]], [[16384/16335]], [[19712/19683]], 46656/46585, [[200704/200475]], and [[532400/531441]] in the 11-limit; and [[625/624]] and [[4225/4224]] in the 13-limit. It provides the [[optimal patent val]] for 7-limit [[kwai]].

	=== Odd harmonics ===		=== Odd harmonics ===
Line 9:		Line 9:

	=== Subsets and supersets ===		=== Subsets and supersets ===
	345 factors into 3 × 5 × 23, ~~with~~ subset edos {{EDOs\|3, 5, 15, 23, 69, and 115}}.		Since 345 factors into 3 × 5 × 23, 345edo has subset edos {{EDOs\| 3, 5, 15, 23, 69, and 115 }}.

	== Regular temperament properties ==		== Regular temperament properties ==
	{\| class="wikitable center-4 center-5 center-6"		{\| class="wikitable center-4 center-5 center-6"
	! rowspan="2" \|[[Subgroup]]		! rowspan="2" \| [[Subgroup]]
	! rowspan="2" \|[[Comma list\|Comma List]]		! rowspan="2" \| [[Comma list\|Comma List]]
	! rowspan="2" \|[[Mapping]]		! 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 error\|Absolute]] (¢)
	![[TE simple badness\|Relative]] (%)		! [[TE simple badness\|Relative]] (%)
	\|-		\|-
	\|2.3		\| 2.3
	\|{{monzo\|547 -345}}		\| {{monzo\| 547 -345 }}
	\|{{mapping\|345 547}}		\| {{mapping\| 345 547 }}
	\| -0.2062		\| -0.2062
	\| 0.2062		\| 0.2062
	\| 5.93		\| 5.93
	\|-		\|-
	\|2.3.5		\| 2.3.5
	\|{{monzo\|3 -18 11}}, {{monzo\|47 -15 -10}}		\| {{monzo\| 3 -18 11 }}, {{monzo\| 47 -15 -10 }}
	\|{{mapping\|345 547 801}}		\| {{mapping\| 345 547 801 }}
	\| -0.1050		\| -0.1050
	\| 0.2210		\| 0.2210
	\| 6.35		\| 6.35
	\|-		\|-
	\|2.3.5.7		\| 2.3.5.7
	\|5120/5103, 16875/16807, ~~95703125~~/~~95551488~~		\| 5120/5103, 16875/16807, 68359375/68024448
	\|{{mapping\|345 547 801 969}}		\| {{mapping\| 345 547 801 969 }}
	\| -0.2220		\| -0.2220
	\| 0.2788		\| 0.2788
	\| 8.02		\| 8.02
	\|-		\|-
	\|2.3.5.7.11		\| 2.3.5.7.11
	\|540/539, 3025/3024, 5120/5103, 26796875/26763264		\| 540/539, 3025/3024, 5120/5103, 26796875/26763264
	\|{{mapping\|345 547 801 969 1194}}		\| {{mapping\| 345 547 801 969 1194 }}
	\| -0.2773		\| -0.2773
	\| 0.2728		\| 0.2728
	\| 7.84		\| 7.84
	\|-		\|-
	\|2.3.5.7.11.13		\| 2.3.5.7.11.13
	\|540/539, 625/624, 3025/3024, 5120/5103, 4225/4224		\| 540/539, 625/624, 3025/3024, 5120/5103, 4225/4224
	\|{{mapping\|345 547 801 969 1194 1277}}		\| {{mapping\| 345 547 801 969 1194 1277 }}
	\| -0.2857		\| -0.2857
	\| 0.2497		\| 0.2497
Line 62:		Line 62:
	\|+Table of rank-2 temperaments by generator		\|+Table of rank-2 temperaments by generator
	! Periods<br>per 8ve		! Periods<br>per 8ve
	! Generator~~<br>(reduced)~~*		! Generator*
	! Cents~~<br>(reduced)~~*		! Cents*
	! Associated<br>Ratio*		! Associated<br>Ratio*
	! Temperaments		! Temperaments
	\|-		\|-
	\|1		\| 1
	\|38\345		\| 38\345
	\|132.17		\| 132.17
	\|{{monzo\|-38 5 13}}		\| {{monzo\| -38 5 13 }}
	\|[[Astro]]		\| [[Astro]]
	\|-		\|-
	\|1		\| 1
	\|143\345		\| 143\345
	\|497.39		\| 497.39
	\|4/3		\| 4/3
	\|[[Kwai]]		\| [[Kwai]]
	\|-		\|-
	\|5		\| 5
	\|106\345<br>(32\345)		\| 106\345<br>(32\345)
	\|368.70<br>(111.30)		\| 368.70<br>(111.30)
	\|1024/891<br>(16/15)		\| 1024/891<br>(16/15)
	\|[[~~Qintosec~~]]		\| [[Quintosec]] (5-limit)
	\|}		\|}
			<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>*</nowiki>~~ [[~~Normal lists\|octave-reduced form~~]]~~, reduced to the first half-octave, and [[Normal lists\|minimal form]] in parentheses if it is distinct~~		[[Category:Kwai]]

Francium: Created page with "{{Infobox ET}} {{EDO intro|345}} == Theory == 345et is only consistent to the 5-limit. Using the patent val, it tempers out 2460375/2458624, 134217728/133984375, [[5120/5103]..."

2023-12-02T22:09:16Z

Created page with "{{Infobox ET}} {{EDO intro|345}} == Theory == 345et is only consistent to the 5-limit. Using the patent val, it tempers out 2460375/2458624, 134217728/133984375, [[5120/5103]..."

New page

{{Infobox ET}}
{{EDO intro|345}}

== Theory ==
345et is only consistent to the 5-limit. Using the patent val, it tempers out 2460375/2458624, 134217728/133984375, [[5120/5103]], 95703125/95551488 and 359661568/358722675 in the 7-limit; 56723625/56689952, 20155392/20131375, 14348907/14348180, 1835008/1830125, 46656/46585, 2734375/2725888, 2657205/2656192, [[16384/16335]], 514714375/514434888, [[540/539]], 9765625/9732096, 369140625/369098752, [[200704/200475]], 1375/1372, 537109375/536870912, [[19712/19683]], 968000/964467, [[3025/3024]], [[532400/531441]], 202397184/201768035 and 20614528/20588575 in the 11-limit. It provides the optimal patent val for [[quartonic]] and [[kwai]].

=== Odd harmonics ===
{{Harmonics in equal|345}}

=== Subsets and supersets ===
345 factors into 3 × 5 × 23, with subset edos {{EDOs|3, 5, 15, 23, 69, and 115}}.

== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
! rowspan="2" |[[Subgroup]]
! rowspan="2" |[[Comma list|Comma List]]
! rowspan="2" |[[Mapping]]
! rowspan="2" |Optimal 8ve Stretch (¢)
! colspan="2" |Tuning Error
|-
![[TE error|Absolute]] (¢)
![[TE simple badness|Relative]] (%)
|-
|2.3
|{{monzo|547 -345}}
|{{mapping|345 547}}
| -0.2062
| 0.2062
| 5.93
|-
|2.3.5
|{{monzo|3 -18 11}}, {{monzo|47 -15 -10}}
|{{mapping|345 547 801}}
| -0.1050
| 0.2210
| 6.35
|-
|2.3.5.7
|5120/5103, 16875/16807, 95703125/95551488
|{{mapping|345 547 801 969}}
| -0.2220
| 0.2788
| 8.02
|-
|2.3.5.7.11
|540/539, 3025/3024, 5120/5103, 26796875/26763264
|{{mapping|345 547 801 969 1194}}
| -0.2773
| 0.2728
| 7.84
|-
|2.3.5.7.11.13
|540/539, 625/624, 3025/3024, 5120/5103, 4225/4224
|{{mapping|345 547 801 969 1194 1277}}
| -0.2857
| 0.2497
| 7.18
|}

=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+Table of rank-2 temperaments by generator
! Periods per 8ve
! Generator (reduced)*
! Cents (reduced)*
! Associated Ratio*
! Temperaments
|-
|1
|38\345
|132.17
|{{monzo|-38 5 13}}
|[[Astro]]
|-
|1
|143\345
|497.39
|4/3
|[[Kwai]]
|-
|5
|106\345 (32\345)
|368.70 (111.30)
|1024/891 (16/15)
|[[Qintosec]]
|}

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