Semaphore–chromatic equivalence continuum - Revision history

FloraC: Mark as mathematical interest. Cleanup.

2026-01-28T07:51:08Z

Mark as mathematical interest. Cleanup.

← Older revision		Revision as of 07:51, 28 January 2026
Line 1:		Line 1:
			{{Mathematical interest}}

	The '''semaphore–chromatic equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[49/48\|semaphore commas (49/48)]] with a [[25/24\|classic chromatic semitone (25/24)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[decimal]] temperament.		The '''semaphore–chromatic equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[49/48\|semaphore commas (49/48)]] with a [[25/24\|classic chromatic semitone (25/24)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[decimal]] temperament.

	All temperaments in the continuum satisfy {{nowrap\|(49/48)<sup>''n''</sup> ~ 25/24}}. Varying ''n'' results in different temperament families listed in the table below. It converges to [[semaphore]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[7-limit]] temperament families supported by decimal (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.9797965913603088..., and temperaments having ''n'' near this value will be more accurate. As this value is so close to 2, temperaments tempering out the [[2401/2400\|breedsma (2401/2400)]] are unusually accurate. It is even closer to 196/99, but the equivalent comma, while tiny even for an [[unnoticeable comma]] at 0.004907{{cent}}, is unreasonably complex, with a monzo of {{monzo\|-487 -97 -198 392}}.		All temperaments in the continuum satisfy {{nowrap\|(49/48)<sup>''n''</sup> ~ 25/24}}. Varying ''n'' results in different temperament families listed in the table below. It converges to [[semaphore]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[7-limit]] temperament families supported by decimal (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.9797965913603088..., and temperaments having ''n'' near this value will be more accurate. As this value is so close to 2, temperaments tempering out the [[2401/2400\|breedsma (2401/2400)]] are unusually accurate. It is even closer to 196/99, but the equivalent comma, while tiny even for an [[unnoticeable comma]] at 0.004907{{cent}}, is unreasonably complex, with a monzo of {{monzo\|-487 -97 -198 392}}.

	{\| class="wikitable center-1 ~~center-2~~"		{\| class="wikitable center-1"
	\|+ style="font-size: 105%;" \| Temperaments in the continuum		\|+ style="font-size: 105%;" \| Temperaments in the continuum
	\|-		\|-
Line 14:		Line 16:
	\|-		\|-
	\| −1		\| −1
	\| 10 &~~amp; 4 &~~ 12d		\| 4 & 10 & 12d
	\| 1225/1152		\| 1225/1152
	\| {{~~monzo~~\|-7 -2 2 2}}		\| {{Monzo\| -7 -2 2 2 }}
	\|-		\|-
	\| 0		\| 0
	\| [[Dicot]]		\| [[Dicot]] expansion
	\| [[25/24]]		\| [[25/24]]
	\| {{~~monzo~~\|-3 -1 2 0}}		\| {{Monzo\| -3 -1 2 0 }}
	\|-		\|-
	\| 1		\| 1
	\| [[~~Jubilismic temperament\|~~Jubilismic]]		\| [[Jubilismic]]
	\| [[50/49]]		\| [[50/49]]
	\| {{~~monzo~~\|1 0 2 -2}}		\| {{Monzo\| 1 0 2 -2 }}
	\|-		\|-
	\| 2		\| 2
	\| [[Breed (temperament)\|Breed]]		\| [[Breed (temperament)\|Breed]]
	\| [[2401/2400]]		\| [[2401/2400]]
	\| {{~~monzo~~\|-5 -1 -2 4}}		\| {{Monzo\| -5 -1 -2 4 }}
	\|-		\|-
	\| 3		\| 3
	\| 46 &~~amp;~~ 60 ~~& 50~~		\| 46 & 50 & 60
	\| 117649/115200		\| 117649/115200
	\| {{monzo\|-9 -2 -2 6}}		\| {{monzo\|-9 -2 -2 6}}
Line 44:		Line 46:
	\|-		\|-
	\| ∞		\| ∞
	\| [[~~Semaphore~~]]		\| [[Semaphoresmic]]
	\| [[49/48]]		\| [[49/48]]
	\| {{monzo\| -4 -1 0 2}}		\| {{monzo\| -4 -1 0 2 }}
	\|}		\|}

	{\| class="wikitable center-1 ~~center-2~~"		{\| class="wikitable center-1"
	\|+ style="font-size: 105%;" \| Temperaments with non-integer n		\|+ style="font-size: 105%;" \| Temperaments with non-integer ''n''
	\|-		\|-
	! rowspan="2" \| ''n''		! rowspan="2" \| ''n''
Line 60:		Line 62:
	\|-		\|-
	\| 196/99		\| 196/99
	\| 10 &~~amp; 4 &~~ 3299cd		\| 4 & 10 & 3299cd
	\| 664 digits		\| 664 digits
	\| {{~~monzo~~\|-487 -97 -198 392}}		\| {{Monzo\| -487 -97 -198 392 }}
	\|}		\|}

	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

ArrowHead294 at 20:49, 12 March 2025

2025-03-12T20:49:34Z

← Older revision		Revision as of 20:49, 12 March 2025
Line 13:		Line 13:
	! Monzo		! Monzo
	\|-		\|-
	\| -1		\| −1
	\| 10 & 4 & 12d		\| 10 & 4 & 12d
	\| 1225/1152		\| 1225/1152

ArrowHead294 at 15:01, 12 March 2025

2025-03-12T15:01:01Z

← Older revision		Revision as of 15:01, 12 March 2025
Line 1:		Line 1:
	The '''~~semaphore-chromatic~~ equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[49/48\|semaphore commas (49/48)]] with a [[25/24\|classic chromatic semitone (25/24)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[decimal]] temperament.		The '''semaphore–chromatic equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[49/48\|semaphore commas (49/48)]] with a [[25/24\|classic chromatic semitone (25/24)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[decimal]] temperament.

			All temperaments in the continuum satisfy {{nowrap\|(49/48)<sup>''n''</sup> ~ 25/24}}. Varying ''n'' results in different temperament families listed in the table below. It converges to [[semaphore]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[7-limit]] temperament families supported by decimal (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.9797965913603088..., and temperaments having ''n'' near this value will be more accurate. As this value is so close to 2, temperaments tempering out the [[2401/2400\|breedsma (2401/2400)]] are unusually accurate. It is even closer to 196/99, but the equivalent comma, while tiny even for an [[unnoticeable comma]] at 0.004907{{cent}}, is unreasonably complex, with a monzo of {{monzo\|-487 -97 -198 392}}.

	All temperaments in the continuum satisfy (49/48)<sup>''n''</sup> ~ 25/24. Varying ''n'' results in different temperament families listed in the table below. It converges to [[semaphore]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[7-limit]] temperament families supported by decimal (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.9797965913603088..., and temperaments having ''n'' near this value will be more accurate. As this value is so close to 2, temperaments tempering out the [[2401/2400\|breedsma (2401/2400)]] are unusually accurate. It is even closer to 196/99, but the equivalent comma, while tiny even for an [[unnoticeable comma]] at 0.004907 cents, is unreasonably complex, with a monzo of {{monzo\|-487 -97 -198 392}}.
	{\| class="wikitable center-1 center-2"		{\| class="wikitable center-1 center-2"
	\|+Temperaments in the continuum		\|+ style="font-size: 105%;" \| Temperaments in the continuum
	\|-		\|-
	! rowspan="2" \|''n''		! rowspan="2" \| ''n''
	! rowspan="2" \|Temperament		! rowspan="2" \| Temperament
	! colspan="2" \|Comma		! colspan="2" \| Comma
	\|-		\|-
	!Ratio		! Ratio
	!Monzo		! Monzo
	\|-		\|-
	\| -1		\| -1
	\|10 & 4 & 12d		\| 10 & 4 & 12d
	\|1225/1152		\| 1225/1152
	\|{{monzo\|-7 -2 2 2}}		\| {{monzo\|-7 -2 2 2}}
	\|-		\|-
	\|0		\| 0
	\|[[Dicot]]		\| [[Dicot]]
	\|[[25/24]]		\| [[25/24]]
	\|{{monzo\|-3 -1 2 0}}		\| {{monzo\|-3 -1 2 0}}
	\|-		\|-
	\|1		\| 1
	\|[[Jubilismic temperament\|Jubilismic]]		\| [[Jubilismic temperament\|Jubilismic]]
	\|[[50/49]]		\| [[50/49]]
	\|{{monzo\|1 0 2 -2}}		\| {{monzo\|1 0 2 -2}}
	\|-		\|-
	\|2		\| 2
	\|[[Breed (temperament)\|Breed]]		\| [[Breed (temperament)\|Breed]]
	\|[[2401/2400]]		\| [[2401/2400]]
	\|{{monzo\|-5 -1 -2 4}}		\| {{monzo\|-5 -1 -2 4}}
	\|-		\|-
	\|3		\| 3
	\|46 & 60 & 50		\| 46 & 60 & 50
	\|117649/115200		\| 117649/115200
	\|{{monzo\|-9 -2 -2 6}}		\| {{monzo\|-9 -2 -2 6}}
	\|-		\|-
	\|…		\| …
	\|…		\| …
	\|…		\| …
	\|…		\| …
	\|-		\|-
	\|∞		\| ∞
	\|[[Semaphore]]		\| [[Semaphore]]
	\|[[49/48]]		\| [[49/48]]
	\|{{monzo\| -4 -1 0 2}}		\| {{monzo\| -4 -1 0 2}}
	\|}		\|}

	{\| class="wikitable center-1 center-2"		{\| class="wikitable center-1 center-2"
	\|+Temperaments with non-integer n		\|+ style="font-size: 105%;" \| Temperaments with non-integer n
	\|-		\|-
	! rowspan="2" \|''n''		! rowspan="2" \| ''n''
	! rowspan="2" \|Temperament		! rowspan="2" \| Temperament
	! colspan="2" \|Comma		! colspan="2" \| Comma
	\|-		\|-
	!Ratio		! Ratio
	!Monzo		! Monzo
	\|-		\|-
	\|196/99		\| 196/99
	\|10 & 4 & 3299cd		\| 10 & 4 & 3299cd
	\|664 digits		\| 664 digits
	\|{{monzo\|-487 -97 -198 392}}		\| {{monzo\|-487 -97 -198 392}}
	\|}		\|}

	[[Category:Equivalence continua]]		[[Category:Equivalence continua]]

ArrowHead294: ArrowHead294 moved page Semaphore-chromatic equivalence continuum to Semaphore–chromatic equivalence continuum

2025-03-12T14:59:17Z

ArrowHead294 moved page Semaphore-chromatic equivalence continuum to Semaphore–chromatic equivalence continuum

← Older revision	Revision as of 14:59, 12 March 2025
(No difference)

Jerdle: Added the temperament associated with this tiny, complex comma.

2024-10-28T23:33:10Z

Added the temperament associated with this tiny, complex comma.

@@ Line 46: / Line 46: @@
 |[[49/48]]
 |{{monzo| -4 -1 0 2}}
 |}
 [[Category:Equivalence continua]]

Jerdle: Added content on -487 -97 -198 392.

2024-10-28T23:26:15Z

Added content on -487 -97 -198 392.

← Older revision		Revision as of 23:26, 28 October 2024
Line 1:		Line 1:
	The '''semaphore-chromatic equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[49/48\|semaphore commas (49/48)]] with a [[25/24\|classic chromatic semitone (25/24)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[decimal]] temperament.		The '''semaphore-chromatic equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[49/48\|semaphore commas (49/48)]] with a [[25/24\|classic chromatic semitone (25/24)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[decimal]] temperament.

	All temperaments in the continuum satisfy (49/48)<sup>''n''</sup> ~ 25/24. Varying ''n'' results in different temperament families listed in the table below. It converges to [[semaphore]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[7-limit]] temperament families supported by decimal (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.9797965913603088..., and temperaments having ''n'' near this value will be more accurate. As this value is so close to 2, temperaments tempering out the [[2401/2400\|breedsma (2401/2400)]] are unusually accurate.		All temperaments in the continuum satisfy (49/48)<sup>''n''</sup> ~ 25/24. Varying ''n'' results in different temperament families listed in the table below. It converges to [[semaphore]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[7-limit]] temperament families supported by decimal (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.9797965913603088..., and temperaments having ''n'' near this value will be more accurate. As this value is so close to 2, temperaments tempering out the [[2401/2400\|breedsma (2401/2400)]] are unusually accurate. It is even closer to 196/99, but the equivalent comma, while tiny even for an [[unnoticeable comma]] at 0.004907 cents, is unreasonably complex, with a monzo of {{monzo\|-487 -97 -198 392}}.
	{\| class="wikitable center-1 center-2"		{\| class="wikitable center-1 center-2"
	\|+Temperaments in the continuum		\|+Temperaments in the continuum

Jerdle: Created page. While dicot-semaphore-blackwood already exists, it is a user page and explicitly works with 10edo, while this works with decimal and is a mainspace page.

2024-10-28T23:18:02Z

Created page. While dicot-semaphore-blackwood already exists, it is a user page and explicitly works with 10edo, while this works with decimal and is a mainspace page.

New page

The '''semaphore-chromatic equivalence continuum''' is a continuum of 7-limit rank-3 temperament families which equate a number of [[49/48|semaphore commas (49/48)]] with a [[25/24|classic chromatic semitone (25/24)]]. This continuum is theoretically interesting in that these are all 7-limit rank-3 temperament families supported by [[decimal]] temperament.

All temperaments in the continuum satisfy (49/48)<sup>''n''</sup> ~ 25/24. Varying ''n'' results in different temperament families listed in the table below. It converges to [[semaphore]] as ''n'' approaches infinity. If we allow non-integer and infinite ''n'', the continuum describes the set of all [[7-limit]] temperament families supported by decimal (due to it being the unique rank-2 temperament that tempers both commas and thus tempers all combinations of them). The just value of ''n'' is approximately 1.9797965913603088..., and temperaments having ''n'' near this value will be more accurate. As this value is so close to 2, temperaments tempering out the [[2401/2400|breedsma (2401/2400)]] are unusually accurate.
{| class="wikitable center-1 center-2"
|+Temperaments in the continuum
|-
! rowspan="2" |''n''
! rowspan="2" |Temperament
! colspan="2" |Comma
|-
!Ratio
!Monzo
|-
| -1
|10 & 4 & 12d
|1225/1152
|{{monzo|-7 -2 2 2}}
|-
|0
|[[Dicot]]
|[[25/24]]
|{{monzo|-3 -1 2 0}}
|-
|1
|[[Jubilismic temperament|Jubilismic]]
|[[50/49]]
|{{monzo|1 0 2 -2}}
|-
|2
|[[Breed (temperament)|Breed]]
|[[2401/2400]]
|{{monzo|-5 -1 -2 4}}
|-
|3
|46 & 60 & 50
|117649/115200
|{{monzo|-9 -2 -2 6}}
|-
|…
|…
|…
|…
|-
|∞
|[[Semaphore]]
|[[49/48]]
|{{monzo| -4 -1 0 2}}
|}
[[Category:Equivalence continua]]