S-expression - Revision history

Sintel bot: Fix $environments for MathJax v4 (align->aligned, etc.)$

2026-10-27T16:55:04Z

Fix <math> environments for MathJax v4 (align->aligned, etc.)

← Older revision		Revision as of 16:55, 27 October 2026
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	Because of tempering out {{nowrap\|S''k''⋅S(''k'' + 1)⋅…⋅S(''k'' + ''n'' − 1)}}, we require that {{nowrap\|{{sfrac\|''k'' + ''n''\|''k'' + ''n'' − 1}} {{=}} {{sfrac\|''k''\|''k'' − 1}}}} consistently. Therefore, if any superparticular {{sfrac\|''x''\|''x'' − 1}} imbetween (meaning {{nowrap\|''k'' + ''n'' > ''x'' > ''k''}}) is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true:		Because of tempering out {{nowrap\|S''k''⋅S(''k'' + 1)⋅…⋅S(''k'' + ''n'' − 1)}}, we require that {{nowrap\|{{sfrac\|''k'' + ''n''\|''k'' + ''n'' − 1}} {{=}} {{sfrac\|''k''\|''k'' − 1}}}} consistently. Therefore, if any superparticular {{sfrac\|''x''\|''x'' − 1}} imbetween (meaning {{nowrap\|''k'' + ''n'' > ''x'' > ''k''}}) is not tempered to the same tempered interval, it must be mapped to a different tempered interval. But this means that one of the following must be true:

	<math>\displaystyle\begin{~~align~~}		<math>\displaystyle\begin{aligned}
	\operatorname{mapping}\left(\frac{k + n}{k + n - 1}\right) &> \operatorname{mapping}\left(\frac{x}{x - 1}\right) \\		\operatorname{mapping}\left(\frac{k + n}{k + n - 1}\right) &> \operatorname{mapping}\left(\frac{x}{x - 1}\right) \\
	\operatorname{mapping}\left(\frac{k}{k - 1}\right) &< \operatorname{mapping}\left(\frac{x}{x - 1}\right)		\operatorname{mapping}\left(\frac{k}{k - 1}\right) &< \operatorname{mapping}\left(\frac{x}{x - 1}\right)
	\end{~~align~~}</math>		\end{aligned}</math>

	Therefore any superparticular interval {{sfrac\|''x''\|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap\|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap\|S''k''S(''k'' + 1)…*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap\|''k'' − 1}})-odd-limit.		Therefore any superparticular interval {{sfrac\|''x''\|''x'' − 1}} between the extrema must be mapped to the same interval as those extrema in order for a consistent tuning in the ({{nowrap\|''k'' + ''n''}})-odd-limit to even potentially be possible. Another way of phrasing this conclusion is that tempering {{nowrap\|S''k''S(''k'' + 1)…*S(''k'' + ''n'' − 1)}} but not all of the constituent square-particulars limits the possible odd-limit consistency of a temperament to the ({{nowrap\|''k'' − 1}})-odd-limit.
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	<math>\displaystyle		<math>\displaystyle
	\begin{~~align~~}		\begin{aligned}
	S(k-1) \cdot S(k) \cdot S(k+1) &= \left(\frac{\frac{k-1}{k-2}}{\frac{k}{k-1}}\right)\left(\frac{\frac{k}{k-1}}{\frac{k+1}{k}}\right)\left(\frac{\frac{k+1}{k}}{\frac{k+2}{k+1}}\right) \\		S(k-1) \cdot S(k) \cdot S(k+1) &= \left(\frac{\frac{k-1}{k-2}}{\frac{k}{k-1}}\right)\left(\frac{\frac{k}{k-1}}{\frac{k+1}{k}}\right)\left(\frac{\frac{k+1}{k}}{\frac{k+2}{k+1}}\right) \\
	&= \frac{\frac{k-1}{k-2}}{\frac{k+2}{k+1}} \\		&= \frac{\frac{k-1}{k-2}}{\frac{k+2}{k+1}} \\
	&= \frac{(k-1)(k+1)}{(k-2)(k+2)} \\		&= \frac{(k-1)(k+1)}{(k-2)(k+2)} \\
	&= \frac{k^2 - 1}{k^2 - 4}		&= \frac{k^2 - 1}{k^2 - 4}
	\end{~~align~~}		\end{aligned}
	</math>		</math>

Eufalesio: More S-expressions

2026-10-14T23:04:43Z

More S-expressions

← Older revision		Revision as of 23:04, 14 October 2026
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	\|S12/(S9/S11)		\|S12/(S9/S11)
	\|S11⋅S12/S9		\|S11⋅S12/S9
	\|		\|S40⋅S41⋅S42⋅S43⋅S44
	\|(S8/S9)/(S64⋅S65)		\|(S8/S9)/(S64⋅S65)
	\|-		\|-
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	\|[[Septischisma\|<small>33554432/33480783</small>]]*		\|[[Septischisma\|<small>33554432/33480783</small>]]*
	\|(S8/S9)²/S15		\|(S8/S9)²/S15
	\|		\|(S8/S9)/(S27/S64)
	\|(S16/S18)³/(S19/S20)		\|(S16/S18)³/(S19/S20)
	\|S49⋅S55⋅(S64²⋅S65)²		\|S49⋅S55⋅(S64²⋅S65)²
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	\|[[32805/32768]]*		\|[[32805/32768]]*
	\|		\|
	\|		\|S27/S64
	\|S15/(S8/S9)		\|S15/(S8/S9)
	\|S19/(S16/S18)²		\|S19/(S16/S18)²
	\|-		\|-
	\|[[1001/1000]]		\|[[1001/1000]]
			\|Cp10
	\|		\|
	~~\|Cp10~~
	\|√{{Overline\|S26⋅S49⋅S99}}		\|√{{Overline\|S26⋅S49⋅S99}}
	\|S49⋅(S64⋅S65)⋅S99		\|S49⋅(S64⋅S65)⋅S99
Line 3,559:		Line 3,559:
	\|		\|
	\|S99/S351		\|S99/S351
	\|~~S55⋅S65~~		\|S55/S65
	\|-		\|-
	\| [[25921/25920]]		\| [[25921/25920]]

FloraC: /* Sk/S(k + 1) (ultraparticulars) */ no need for this section title

2026-09-17T12:18:23Z

Sk/S(k + 1) (ultraparticulars): no need for this section title

← Older revision		Revision as of 12:18, 17 September 2026
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	== S''k''/S(''k'' + 1) (ultraparticulars) ==		== S''k''/S(''k'' + 1) (ultraparticulars) ==
	~~=== Motivational example ===~~
	Often it is desirable to make consecutive superparticular intervals equidistant. This has a number of nice consequences, many of which not explained here—see the motivation section for each infinite family of commas defined on this page.		Often it is desirable to make consecutive superparticular intervals equidistant. This has a number of nice consequences, many of which not explained here—see the motivation section for each infinite family of commas defined on this page.

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	From this, it can be deduced that {{nowrap\| (6/5)<sup>3</sup> ~ [[7/4]] }}, because one of the 6/5's can be lowered by S6 to 7/6 and another of the 6/5's can be raised by S5 to 5/4. Then because we have tempered S5 and S6 together, we have lowered and raised by the same amount, so the result of {{nowrap\| (7/6)⋅(6/5)⋅(5/4) {{=}} 7/4 }} must be the same as the result of {{nowrap\| (6/5)⋅(6/5)⋅(6/5)}} in this temperament.		From this, it can be deduced that {{nowrap\| (6/5)<sup>3</sup> ~ [[7/4]] }}, because one of the 6/5's can be lowered by S6 to 7/6 and another of the 6/5's can be raised by S5 to 5/4. Then because we have tempered S5 and S6 together, we have lowered and raised by the same amount, so the result of {{nowrap\| (7/6)⋅(6/5)⋅(5/4) {{=}} 7/4 }} must be the same as the result of {{nowrap\| (6/5)⋅(6/5)⋅(6/5)}} in this temperament.

	The reader is encouraged to familiarize themself with the structure of this argument, as it generalizes to arbitrary S''k'' (→ [[S-expression/~~Advanced_results~~#~~Mathematical_derivations~~]]); the algebraic proof is tedious, but the intuition is the same:		The reader is encouraged to familiarize themself with the structure of this argument, as it generalizes to arbitrary S''k'' (→ [[S-expression/Advanced results #Mathematical derivations]]); the algebraic proof is tedious, but the intuition is the same:

	$$ \frac{k+2}{k+1} \leftarrow S(k+1)~Sk \rightarrow \frac{k+1}{k} \leftarrow S(k+1)~Sk \rightarrow \frac{k}{k-1} $$		$$ \frac{k+2}{k+1} \leftarrow S(k+1)~Sk \rightarrow \frac{k+1}{k} \leftarrow S(k+1)~Sk \rightarrow \frac{k}{k-1} $$

FloraC: Make cubeparticulars a subsection of ultraparticulars, by Godtone's request

2026-09-17T12:17:04Z

Make cubeparticulars a subsection of ultraparticulars, by Godtone's request

Show changes

Eufalesio: Added olympia and refined more examples list

2026-09-04T13:40:36Z

Added olympia and refined more examples list

← Older revision		Revision as of 13:40, 4 September 2026
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	\| [[364/363]]*		\| [[364/363]]*
	\| S14/(S11/S13)		\| S14/(S11/S13)
	\|S13⋅S14/S11		\|(S13⋅S14)/S11
	\|S24⋅S25⋅S26/S22		\|S24⋅S25⋅S26/S22
	\|(S9/S11)/S27		\|(S9/S11)/S27
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	\|		\|
	\|S49⋅(S64⋅S65)		\|S49⋅(S64⋅S65)
	\|S64⋅S65/S26⋅S99		\|(S64⋅S65)/S26⋅S99
	\|-		\|-
	\|[[1216/1215]]		\|[[1216/1215]]
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	\|S49⋅S55		\|S49⋅S55
	\|S65⋅(S76⋅S77²)		\|S65⋅(S76⋅S77²)
			\|-
			\|<small>[[131072/130977]]</small>
			\|S64²⋅S65
			\|
			\|S32/S63
			\|
	\|-		\|-
	\|[[43904/43875]]		\|[[43904/43875]]
Line 3,550:		Line 3,556:
	\|		\|
	\|(S76⋅S77)/S64		\|(S76⋅S77)/S64
	\|(S18⋅S19/S16)/(S12/S14)		\|((S18⋅S19)/S16)/(S12/S14)
	\|-		\|-
	\|[[10648/10647]]*		\|[[10648/10647]]*

Godtone: /* Examples */ schisma

2026-09-04T03:10:19Z

Examples: schisma

← Older revision		Revision as of 03:10, 4 September 2026
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	\|		\|
	\|S49⋅(S64⋅S65)²⋅S99		\|S49⋅(S64⋅S65)²⋅S99
			\|-
			\|[[32805/32768]]*
			\|
			\|
			\|S15/(S8/S9)
			\|S19/(S16/S18)²
	\|-		\|-
	\|[[1001/1000]]		\|[[1001/1000]]

Godtone: /* Examples */

2026-09-04T03:07:27Z

Examples

@@ Line 3,551: / Line 3,551: @@
 |S99/S351
 |S55⋅S65
 |-
 | [[25921/25920]]

Godtone: /* Examples */ order fix

2026-09-04T03:05:15Z

Examples: order fix

← Older revision		Revision as of 03:05, 4 September 2026
Line 3,383:		Line 3,383:
	\|		\|
	\|(S26⋅S27)²/S351		\|(S26⋅S27)²/S351
			\|-
			\|[[15625/15552]]
			\|S25²⋅S26
			\|
			\|S15⋅(S25/S27)
			\|
	\|-		\|-
	\|[[225/224]]		\|[[225/224]]
Line 3,419:		Line 3,425:
	\|		\|
	\|(S8/S9)/(S64²/S65)		\|(S8/S9)/(S64²/S65)
			\|-
			\|[[Septischisma\|<small>33554432/33480783</small>]]*
			\|(S8/S9)²/S15
			\|
			\|(S16/S18)³/(S19/S20)
			\|S49⋅S55⋅(S64²⋅S65)²
	\|-		\|-
	\| [[540/539]]		\| [[540/539]]

Godtone: /* Examples */ order fix

2026-09-04T02:52:58Z

Examples: order fix

← Older revision		Revision as of 02:52, 4 September 2026
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	\|		\|
	\|		\|
			\|-
			\|[[41503/41472]]*
			\|
			\|
			\|S49⋅S55
			\|S65⋅(S76⋅S77²)
	\|-		\|-
	\|[[43904/43875]]		\|[[43904/43875]]
Line 3,473:		Line 3,479:
	\|S49⋅S64		\|S49⋅S64
	\|S56⋅(S76⋅S77)		\|S56⋅(S76⋅S77)
	\|-
	\|[[41503/41472]]*
	\|
	\|
	~~\|S49⋅S55~~
	~~\|S65⋅(S76⋅S77²)~~
	\|-		\|-
	\|[[2080/2079]]		\|[[2080/2079]]

Godtone: /* Examples */ order fix

2026-09-04T02:49:25Z

Examples: order fix

← Older revision		Revision as of 02:49, 4 September 2026
Line 3,473:		Line 3,473:
	\|S49⋅S64		\|S49⋅S64
	\|S56⋅(S76⋅S77)		\|S56⋅(S76⋅S77)
			\|-
			\|[[41503/41472]]*
			\|
			\|
			\|S49⋅S55
			\|S65⋅(S76⋅S77²)
	\|-		\|-
	\|[[2080/2079]]		\|[[2080/2079]]
Line 3,551:		Line 3,557:
	\|S15/(S8/S9)		\|S15/(S8/S9)
	\|S19/(S16/S18)²		\|S19/(S16/S18)²
	\|-
	\|[[41503/41472]]*
	\|
	\|
	~~\|S49⋅S55~~
	~~\|S65⋅(S76⋅S77²)~~
	\|-		\|-
	\| [[25921/25920]]		\| [[25921/25920]]

S-expression - Revision history

Sintel bot: Fix environments for MathJax v4 (align->aligned, etc.)

Eufalesio: More S-expressions

FloraC: /* Sk/S(k + 1) (ultraparticulars) */ no need for this section title

FloraC: Make cubeparticulars a subsection of ultraparticulars, by Godtone's request

Eufalesio: Added olympia and refined more examples list

Godtone: /* Examples */ schisma

Godtone: /* Examples */

Godtone: /* Examples */ order fix

Godtone: /* Examples */ order fix

Godtone: /* Examples */ order fix

Sintel bot: Fix $environments for MathJax v4 (align->aligned, etc.)$