23rd-octave temperaments - Revision history

FloraC: Text replacement - "{{Technical data page}}

" to "{{Technical data page}}"

2025-04-13T13:46:23Z

Text replacement - "{{Technical data page}}<br><br>" to "{{Technical data page}}"

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	{{Technical data page}}~~<br><br>~~		{{Technical data page}}
	{{Infobox fractional-octave\|23}}		{{Infobox fractional-octave\|23}}
	Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23 are.		Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23 are.

BudjarnLambeth: {{Technical data page}}

2025-04-13T05:29:41Z

{{Technical data page}}<br><br>

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			{{Technical data page}}<br><br>
	{{Infobox fractional-octave\|23}}		{{Infobox fractional-octave\|23}}
	Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23 are.		Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23 are.

ArrowHead294 at 04:00, 14 February 2025

2025-02-14T04:00:38Z

← Older revision		Revision as of 04:00, 14 February 2025
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	{{~~Fractional~~-octave ~~navigation~~\|23}}		{{Infobox fractional-octave\|23}}
	Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23 are.		Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23 are.

BudjarnLambeth: Todo

2024-12-15T02:28:08Z

Todo

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	{{Navbox fractional-octave}}		{{Navbox fractional-octave}}
	{{~~stub~~}}

			{{todo\|inline=1\|expand\|add examples}}

BudjarnLambeth: navbox

2024-12-05T07:21:21Z

navbox

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	~~{{Stub}}~~
	{{Fractional-octave navigation\|23}}		{{Fractional-octave navigation\|23}}
	Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23 are.		Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23 are.
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	However, if we want to guarantee we can find ''all'' primes relative to the same root note, we require a 5*23=115-note [[MOS]], corresponding to [[46L 69s]], that is, to [[2L 3s]] (pentic) with a [[23edo\|1\23 period]], a fascinating "microtonal minification" of the familiar pentatonic scale, using the small step for very subtle commatic variations on notes of 46edo, though few dare to use scales this elaborate, and the [[23L 46s]] scale is likely sufficient for exploring most of the harmony offered (and likely more interesting to play with), so that it might also serve in some strange way as a [[well temperament]] of [[69edo]] (not just 46edo), which is a flat tuning of [[meantone]] supporting many but not all of the equivalences of icositritonic.		However, if we want to guarantee we can find ''all'' primes relative to the same root note, we require a 5*23=115-note [[MOS]], corresponding to [[46L 69s]], that is, to [[2L 3s]] (pentic) with a [[23edo\|1\23 period]], a fascinating "microtonal minification" of the familiar pentatonic scale, using the small step for very subtle commatic variations on notes of 46edo, though few dare to use scales this elaborate, and the [[23L 46s]] scale is likely sufficient for exploring most of the harmony offered (and likely more interesting to play with), so that it might also serve in some strange way as a [[well temperament]] of [[69edo]] (not just 46edo), which is a flat tuning of [[meantone]] supporting many but not all of the equivalences of icositritonic.

			{{Navbox fractional-octave}}
			{{stub}}

BudjarnLambeth at 00:15, 7 October 2024

2024-10-07T00:15:15Z

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	Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23 are.		Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23 are.

	Here are some EDOs with low relative error on ~~low~~ harmonics, which are multiples of 23 - perhaps fertile ground for 23rd-octave temperaments: {{EDOs\|46, 161, 207, 230, 414, 690, 805, 851, 920, 966, and 1012.}}		Here are some EDOs with low relative error on simple harmonics, which are multiples of 23 - perhaps fertile ground for 23rd-octave temperaments: {{EDOs\|46, 161, 207, 230, 414, 690, 805, 851, 920, 966, and 1012.}}

	46 and 1012 are also members of [[zeta]] edo lists.		46 and 1012 are also members of [[zeta]] edo lists.

BudjarnLambeth at 00:13, 7 October 2024

2024-10-07T00:13:44Z

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	{{Stub}}		{{Stub}}
	{{Fractional-octave navigation\|23}}		{{Fractional-octave navigation\|23}}
	Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23, ~~such as~~ {{EDOs\|46, 161, 207 and 1012}} ~~are.~~ 46 and ~~1848~~ are also members of [[zeta]] edo lists.		Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23 are.

			Here are some EDOs with low relative error on low harmonics, which are multiples of 23 - perhaps fertile ground for 23rd-octave temperaments: {{EDOs\|46, 161, 207, 230, 414, 690, 805, 851, 920, 966, and 1012.}}

			46 and 1012 are also members of [[zeta]] edo lists.

	== Icositritonic ==		== Icositritonic ==

BudjarnLambeth at 00:00, 7 October 2024

2024-10-07T00:00:51Z

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	{{Stub}}		{{Stub}}
	{{Fractional-octave navigation\|23}}		{{Fractional-octave navigation\|23}}
	Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 21, such as {{EDOs\|46, 161, 207 and 1012}} are. 46 and 1848 are also members of [[zeta]] edo lists.		Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 23, such as {{EDOs\|46, 161, 207 and 1012}} are. 46 and 1848 are also members of [[zeta]] edo lists.

	== Icositritonic ==		== Icositritonic ==

BudjarnLambeth at 00:00, 7 October 2024

2024-10-07T00:00:37Z

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	{{Stub}}		{{Stub}}
	{{Fractional-octave navigation\|23}}		{{Fractional-octave navigation\|23}}
	Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 21, such as {{EDOs\|46, 161 and 1012}} are. 46 and 1848 are also members of [[zeta]] edo lists.		Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 21, such as {{EDOs\|46, 161, 207 and 1012}} are. 46 and 1848 are also members of [[zeta]] edo lists.

	== Icositritonic ==		== Icositritonic ==

BudjarnLambeth: Added intro

2024-10-06T23:59:32Z

Added intro

← Older revision		Revision as of 23:59, 6 October 2024
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	{{Stub}}		{{Stub}}
	{{Fractional-octave navigation\|23}}		{{Fractional-octave navigation\|23}}
			Although 23edo itself is not particularly accurate for low-complexity harmonics, some temperaments which are multiples of 21, such as {{EDOs\|46, 161 and 1012}} are. 46 and 1848 are also members of [[zeta]] edo lists.

			== Icositritonic ==
	{{See also \| Icositritonic}}		{{See also \| Icositritonic}}

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	* [[23/17]] to 10\23 (-1.6{{cent}})		* [[23/17]] to 10\23 (-1.6{{cent}})
	* [[32/23]] (2.2{{cent}}), [[46/33]] (-1.1{{cent}}) and [[39/28]] (+0.3{{cent}}) to 11\23		* [[32/23]] (2.2{{cent}}), [[46/33]] (-1.1{{cent}}) and [[39/28]] (+0.3{{cent}}) to 11\23
	This serves to define icositritonic in the 23-limit. It is quite accurate so that considering other 23rd-octave temperaments might in time prove unnecessary. Arguably the 69-note [[MOS]] is the place to start, corresponding to [[46L 23s]], essentially a version of 46edo with half of its notes duplicated to allow for better tunings, as (if we pay attention to [[Icositritonic\|its mapping]]) that allows all primes except 13 and 19 to be found relative to the same root note, with ample opportunity for more complex harmony by combining 3 and 5 (found at 1 gen) with other primes which are often found at -1 gen, so that most of the complex harmony is actually accessible directly in [[23edo]], an impressive feat. However, if we want to guarantee we can find ''all'' primes relative to the same root note, we require a 5*23=115-note [[MOS]], corresponding to [[46L 69s]], that is, to [[2L 3s]] (pentic) with a [[23edo\|1\23 period]], a fascinating "microtonal minification" of the familiar pentatonic scale, using the small step for very subtle commatic variations on notes of 46edo, though few dare to use scales this elaborate, and the 23L 46s scale is likely sufficient for exploring most of the harmony offered (and likely more interesting to play with), so that it might also serve in some strange way as a well-temperament of [[69edo]] (not just 46edo), which is a flat tuning of [[meantone]] supporting many but not all of the equivalences of icositritonic.		This serves to define icositritonic in the 23-limit. It is quite accurate so that considering other 23rd-octave temperaments might in time prove unnecessary.

			Arguably the 69-note [[MOS]] is the place to start, corresponding to [[46L 23s]], essentially a version of 46edo with half of its notes duplicated to allow for better tunings, as (if we pay attention to [[Icositritonic\|its mapping]]) that allows all primes except 13 and 19 to be found relative to the same root note, with ample opportunity for more complex harmony by combining 3 and 5 (found at 1 gen) with other primes which are often found at -1 gen, so that most of the complex harmony is actually accessible directly in [[23edo]], an impressive feat.

			However, if we want to guarantee we can find ''all'' primes relative to the same root note, we require a 5*23=115-note [[MOS]], corresponding to [[46L 69s]], that is, to [[2L 3s]] (pentic) with a [[23edo\|1\23 period]], a fascinating "microtonal minification" of the familiar pentatonic scale, using the small step for very subtle commatic variations on notes of 46edo, though few dare to use scales this elaborate, and the [[23L 46s]] scale is likely sufficient for exploring most of the harmony offered (and likely more interesting to play with), so that it might also serve in some strange way as a [[well temperament]] of [[69edo]] (not just 46edo), which is a flat tuning of [[meantone]] supporting many but not all of the equivalences of icositritonic.

23rd-octave temperaments - Revision history

FloraC: Text replacement - "{{Technical data page}}" to "{{Technical data page}}"

BudjarnLambeth: {{Technical data page}}

ArrowHead294 at 04:00, 14 February 2025

BudjarnLambeth: Todo

BudjarnLambeth: navbox

BudjarnLambeth at 00:15, 7 October 2024

BudjarnLambeth at 00:13, 7 October 2024

BudjarnLambeth at 00:00, 7 October 2024

BudjarnLambeth at 00:00, 7 October 2024

BudjarnLambeth: Added intro

FloraC: Text replacement - "{{Technical data page}}

" to "{{Technical data page}}"