Starling–sensamagic equivalence continuum - Revision history

FloraC: - data (addressed in the misc. page)

2026-01-17T16:11:34Z

- data (addressed in the misc. page)

@@ Line 78: / Line 78: @@
 * 19 & 46 & 255 ({{nowrap|''n'' {{=}} {{frac|4|3}} {{=}} 1.333…}})
 * 19 & 46 & 183 ({{nowrap|''n'' {{=}} {{frac|3|2}} {{=}} 1.5}})
 [[Category:Sensi]]
 [[Category:Equivalence continua]]

Xenllium at 13:54, 17 January 2026

2026-01-17T13:54:25Z

← Older revision		Revision as of 13:54, 17 January 2026
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	Examples of temperaments with fractional values of ''n'':		Examples of temperaments with fractional values of ''n'':
	* ~~19 & 46 & 94~~ ({{nowrap\|''n'' {{=}} {{frac\|1\|2}} {{=}} 0.5}})		* [[#Sensibeta\|Sensibeta]] ({{nowrap\|''n'' {{=}} {{frac\|1\|2}} {{=}} 0.5}})
	* 19 & 46 & 193 ({{nowrap\|''n'' {{=}} {{frac\|2\|3}} {{=}} 0.666…}})		* 19 & 46 & 193 ({{nowrap\|''n'' {{=}} {{frac\|2\|3}} {{=}} 0.666…}})
	* 19 & 46 & 265 ({{nowrap\|''n'' {{=}} {{frac\|3\|4}} {{=}} 0.75}})		* 19 & 46 & 265 ({{nowrap\|''n'' {{=}} {{frac\|3\|4}} {{=}} 0.75}})
Line 78:		Line 78:
	* 19 & 46 & 255 ({{nowrap\|''n'' {{=}} {{frac\|4\|3}} {{=}} 1.333…}})		* 19 & 46 & 255 ({{nowrap\|''n'' {{=}} {{frac\|4\|3}} {{=}} 1.333…}})
	* 19 & 46 & 183 ({{nowrap\|''n'' {{=}} {{frac\|3\|2}} {{=}} 1.5}})		* 19 & 46 & 183 ({{nowrap\|''n'' {{=}} {{frac\|3\|2}} {{=}} 1.5}})

			== Sensibeta ==
			{{See also\| Sensibeta comma }}

			[[Comma list]]: 1071875/1062882

			{{Mapping\|legend=1\| 1 0 2 -3 \| 0 1 0 4 \| 0 0 3 -5 }}
			: Mapping generators: ~2, ~3, ~175/162

			[[Optimal tuning]]s:
			* [[WE]]: ~2 = 1200.025¢, ~3 = 1902.728¢, ~175/162 = 128.524¢
			* [[CWE]]: ~2 = 1200.000¢, ~3 = 1902.709¢, ~175/162 = 128.530¢

			{{Optimal ET sequence\|legend=1\| 19, 27, 46, 94, 113, 121, 140 }}

			[[Badness]] (Sintel): 7.973

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

Xenllium: Created page with "{{Mathematical interest}} The '''starling–sensamagic equivalence continuum''' is a continuum of 7-limit temperament families which equate a number of starling commas (126/125) with a sensamagic comma (245/243). This continuum is theoretically interesting in that these are all 7-limit temperament families supported by septimal sensi temperament. All temperaments in the continuum sa..."

2026-01-17T05:48:26Z

Created page with "{{Mathematical interest}} The '''starling–sensamagic equivalence continuum''' is a continuum of 7-limit temperament families which equate a number of starling commas (126/125) with a sensamagic comma (245/243). This continuum is theoretically interesting in that these are all 7-limit temperament families supported by septimal sensi temperament. All temperaments in the continuum sa..."

New page

{{Mathematical interest}}

The '''starling–sensamagic equivalence continuum''' is a [[equivalence continuum|continuum]] of 7-limit temperament families which equate a number of [[126/125|starling commas (126/125)]] with a [[245/243|sensamagic comma (245/243)]]. This continuum is theoretically interesting in that these are all 7-limit temperament families supported by [[Sensipent family #Septimal sensi|septimal sensi]] temperament.

All temperaments in the continuum satisfy {{nowrap|(126/125)<sup>''n''</sup> ~ 245/243}}. Varying ''n'' results in different temperament families listed in the table below. It converges to [[Starling temperaments|starling]] 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 septimal sensi (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 1.02868884, and temperaments having ''n'' near this value tend to be the most accurate ones.

{| class="wikitable center-1 center-2"
|+ style="font-size: 105%;" | Temperament families in the continuum
|-
! rowspan="2" | ''n''
! rowspan="2" | Temperament family
! colspan="2" | Comma
|-
! Ratio
! Monzo
|-
| −4
| 19 & 46 & 110c
| [[50824368/48828125]]
| {{Monzo| 4 3 −11 6 }}
|-
| −3
| 19 & 46 & 33
| [[403368/390625]]
| {{Monzo| 3 1 −8 5 }}
|-
| −2
| 19 & 46 & 21
| [[9604/9375]]
| {{Monzo| 2 −1 −5 4 }}
|-
| −1
| [[Sengic family|Sengic]]
| [[686/675]]
| {{Monzo| 1 −3 −2 3 }}
|-
| 0
| [[Sensamagic clan|Sensamagic]]
| [[245/243]]
| {{Monzo| 0 −5 1 2 }}
|-
| 1
| [[Ragismic microtemperaments|Ragismic]]
| [[4375/4374]]
| {{Monzo| −1 −7 4 1 }}
|-
| 2
| [[Sensipent family|Sensipent]]
| [[78732/78125]]
| {{Monzo| 2 9 −7 }}
|-
| 3
| 19 & 46 & 161
| [[9920232/9765625]]
| {{Monzo| 3 11 −10 1 }}
|-
| 4
| 19 & 46 & 173
| [[1249949232/1220703125]]
| {{Monzo| 4 13 −13 2 }}
|-
| …
| …
| …
| …
|-
| ∞
| [[Starling temperaments|Starling]]
| [[126/125]]
| {{Monzo| 1 2 −3 1 }}
|}

Examples of temperaments with fractional values of ''n'':
* 19 & 46 & 94 ({{nowrap|''n'' {{=}} {{frac|1|2}} {{=}} 0.5}})
* 19 & 46 & 193 ({{nowrap|''n'' {{=}} {{frac|2|3}} {{=}} 0.666…}})
* 19 & 46 & 265 ({{nowrap|''n'' {{=}} {{frac|3|4}} {{=}} 0.75}})
* 19 & 46 & 354 ({{nowrap|''n'' {{=}} {{frac|5|4}} {{=}} 1.25}})
* 19 & 46 & 255 ({{nowrap|''n'' {{=}} {{frac|4|3}} {{=}} 1.333…}})
* 19 & 46 & 183 ({{nowrap|''n'' {{=}} {{frac|3|2}} {{=}} 1.5}})

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