Temperament addition - Revision history

ArrowHead294: Update link

2024-11-02T18:46:22Z

Update link

← Older revision		Revision as of 18:46, 2 November 2024
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	Take the case of meantone + porcupine = tetracot from the previous section. What this relationship means is that tetracot is the temperament which doesn't make the meantone comma itself [[vanish]], nor the porcupine comma itself, but instead make whatever comma relates pitches that are exactly one meantone comma plus one porcupine comma apart vanish. And that's the tetracot comma! And on the other hand, for the temperament difference, dicot, this is the temperament that makes neither meantone nor porcupine vanish, but instead the comma that's the size of the difference between them. And that's the dicot comma. So tetracot makes 80/81 × 250/243 vanish, and dicot makes 80/81 × 243/250 vanish.		Take the case of meantone + porcupine = tetracot from the previous section. What this relationship means is that tetracot is the temperament which doesn't make the meantone comma itself [[vanish]], nor the porcupine comma itself, but instead make whatever comma relates pitches that are exactly one meantone comma plus one porcupine comma apart vanish. And that's the tetracot comma! And on the other hand, for the temperament difference, dicot, this is the temperament that makes neither meantone nor porcupine vanish, but instead the comma that's the size of the difference between them. And that's the dicot comma. So tetracot makes 80/81 × 250/243 vanish, and dicot makes 80/81 × 243/250 vanish.

	Similar reasoning is possible for the mapping-rows of mappings — the analogs of the commas of comma bases — but are less intuitive to describe. What's reasonably easy to understand, though, is how temperament addition on maps is essentially navigation of the scale tree for the rank-2 temperament they share; ~~for more information on this, see~~ [[Dave Keenan & Douglas Blumeyer's guide to RTT~~: exploring~~ temperaments#Scale trees]]. So if you understand the effects on individual maps, then you can apply those to changes of maps within a more complex temperament.		Similar reasoning is possible for the mapping-rows of mappings — the analogs of the commas of comma bases — but are less intuitive to describe. What's reasonably easy to understand, though, is how temperament addition on maps is essentially navigation of the scale tree for the rank-2 temperament they share; [[Dave Keenan & Douglas Blumeyer's guide to RTT/Exploring temperaments#Scale trees\|see here]] for more information on this. So if you understand the effects on individual maps, then you can apply those to changes of maps within a more complex temperament.

	Ultimately, these two effects are the primary applications of temperament addition.<ref>It has also been asserted that there exists a connection between temperament addition and "Fokker groups" as discussed on this page: [[Fokker block]], but the connection remains unclear to this author.</ref>		Ultimately, these two effects are the primary applications of temperament addition.<ref>It has also been asserted that there exists a connection between temperament addition and "Fokker groups" as discussed on this page: [[Fokker block]], but the connection remains unclear to this author.</ref>

Cmloegcmluin: /* Fourth example: other side of duality */

2024-06-24T19:48:04Z

Fourth example: other side of duality

← Older revision		Revision as of 19:48, 24 June 2024
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	In order to compare this result directly with our other three results, let's take the dual of this {{rket\|{{map\|6 8 -4 1}} {{map\|0 1 5 10}}}}, which is [{{vector\|22 -15 3 0}} {{vector\|41 -30 2 2}}] (in canonical form), so we can see that's yet a fourth possible result<ref>		In order to compare this result directly with our other three results, let's take the dual of this {{rket\|{{map\|6 8 -4 1}} {{map\|0 1 5 10}}}}, which is [{{vector\|22 -15 3 0}} {{vector\|41 -30 2 2}}] (in canonical form), so we can see that's yet a fourth possible result.<ref>
	It is possible to find a pair of mapping forms for septimal meantone and septimal blackwood that sum to a mapping which is the dual of the comma basis found by summing their canonical comma bases. One example is {{rket\|{{map\|97 152 220 259}} {{map\|-30 -47 -68 -80}}}} + {{rket\|{{map\|-95 -152 -212 -266}} {{map\|30 48 67 84}}}}.</ref>		It is possible to find a pair of mapping forms for septimal meantone and septimal blackwood that sum to a mapping which is the dual of the comma basis found by summing their canonical comma bases. One example is {{rket\|{{map\|97 152 220 259}} {{map\|-30 -47 -68 -80}}}} + {{rket\|{{map\|-95 -152 -212 -266}} {{map\|30 48 67 84}}}}.</ref>

Cmloegcmluin: /* Third example: reordering of canonical form */

2024-06-24T19:47:29Z

Third example: reordering of canonical form

← Older revision		Revision as of 19:47, 24 June 2024
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	And the canonical form of [{{vector\|-2 -2 1 1}} {{vector\|5 -5 0 1}}] is [{{vector\|-7 3 1 0}} {{vector\|5 -5 0 1}}], so that's yet another possible temperament resulting from ~~adding~~ these non-addable temperaments.		And the canonical form of [{{vector\|-2 -2 1 1}} {{vector\|5 -5 0 1}}] is [{{vector\|-7 3 1 0}} {{vector\|5 -5 0 1}}], so that's yet another possible temperament resulting from attempting to add these non-addable temperaments.

	==== Fourth example: other side of duality ====		==== Fourth example: other side of duality ====

Cmloegcmluin: /* Initial example: canonical form */

2024-06-24T19:46:56Z

Initial example: canonical form

← Older revision		Revision as of 19:46, 24 June 2024
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	=== Addition on non-addable temperaments ===		=== Addition on non-addable temperaments ===
	==== Initial example: canonical form ====		==== Initial example: canonical form ====
	~~Clearly~~, ~~two non-addable temperaments may still be~~ entry-wise ~~added~~. For example, the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> for the canonical comma bases for septimal meantone [{{vector\|4 -4 1 0}} {{vector\|13 -10 0 1}}] and septimal blackwood [{{vector\|-8 5 0 0}} {{vector\|-6 2 0 1}}] is empty, meaning their <span style="color: #B6321C;"><math>l_{\text{ind}}</math></span> <math>=2</math>, and therefore they aren't addable. Yet we can still do entry-wise addition as if ~~they~~ were:		Even when a pair of temperaments isn’t addable, if they have the same dimensions, that means the matrices representing them have the same shape, and so then there’s nothing stopping us from entry-wise adding them. For example, the <span style="color: #3C8031;"><math>L_{\text{dep}}</math></span> for the canonical comma bases for septimal meantone [{{vector\|4 -4 1 0}} {{vector\|13 -10 0 1}}] and septimal blackwood [{{vector\|-8 5 0 0}} {{vector\|-6 2 0 1}}] is empty, meaning their <span style="color: #B6321C;"><math>l_{\text{ind}}</math></span> <math>=2</math>, and therefore they aren't addable. Yet we can still do entry-wise addition on the matrices that are acting as these temperaments’ comma bases as if the temperaments were addable:


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	\end{array} \right]		\end{array} \right]
	+		+
	\left[ \begin{array} {rr\|r}		\left[ \begin{array} {r\|r}

	-8 & -6 \\		-8 & -6 \\

Cmloegcmluin: /* Initial example: canonical form */

2024-06-24T19:46:04Z

Initial example: canonical form

← Older revision		Revision as of 19:46, 24 June 2024
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	<math>\left[ \begin{array} {~~rrr~~}		<math>\left[ \begin{array} {r\|r}

	4 & 13 \\		4 & 13 \\
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	\end{array} \right]		\end{array} \right]
	+		+
	\left[ \begin{array} {~~rrr~~}		\left[ \begin{array} {rr\|r}

	-8 & -6 \\		-8 & -6 \\
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	\end{array} \right]		\end{array} \right]
	=		=
	\left[ \begin{array} {~~rrr~~}		\left[ \begin{array} {r\|r}

	(4+-8) & (13+-6) \\		(4+-8) & (13+-6) \\
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	\end{array} \right]		\end{array} \right]
	=		=
	\left[ \begin{array} {~~rrr~~}		\left[ \begin{array} {r\|r}

	-4 & 7 \\		-4 & 7 \\
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	# neither Harrison's comma {{vector\|13 -10 0 1}} nor Archytas' comma {{vector\|-6 2 0 1}} vanish, but does make the laruru negative second {{vector\|7 -8 0 2}} vanish.		# neither Harrison's comma {{vector\|13 -10 0 1}} nor Archytas' comma {{vector\|-6 2 0 1}} vanish, but does make the laruru negative second {{vector\|7 -8 0 2}} vanish.

	But while these two monovector additions have worked out individually, the full result cannot truly be said to be the "temperament sum" of septimal meantone and blackwood. And here follows a demonstration why it cannot.		But while these two monovector additions have worked out individually, the full result cannot truly be said to be the "temperament sum" of septimal meantone and blackwood. And here follows a demonstration why it cannot.

	==== Second example: alternate form ====		==== Second example: alternate form ====

Inthar at 22:32, 9 February 2024

2024-02-09T22:32:26Z

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	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Math]]		[[Category:Math]]
	[[Category:~~Articles~~ with proofs]]		[[Category:Pages with proofs]]

BudjarnLambeth: Category:Articles with proofs

2024-01-29T01:21:44Z

Category:Articles with proofs

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	[[Category:Terms]]		[[Category:Terms]]
	[[Category:Math]]		[[Category:Math]]
			[[Category:Articles with proofs]]

Cmloegcmluin: nondecomposable → indecomposable (seems to be the preferred term in mathematical settings)

2023-12-02T04:39:57Z

nondecomposable → indecomposable (seems to be the preferred term in mathematical settings)

← Older revision		Revision as of 04:39, 2 December 2023
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	==== How it looks with multivectors ====		==== How it looks with multivectors ====
	We've now observed the outcome when adding non-addable temperaments using the matrix approach. It's instructive to observe how it works with multivectors as well. The canonical multicommas for septimal meantone and septimal blackwood are {{multicomma\|12 -13 4 10 -4 1}} and {{multicomma\|14 0 -8 0 5 0}}, respectively. When we add these, we get {{multicomma\|26 -13 -4 10 1 1}}. What temperament is this — does it match with any of the four comma bases we've already found? Let's check by converting it back to matrix form. Oh, wait — we can't. This is what we call a [[decomposability\|~~nondecomposable~~]] multivector. In other words, there is no set of vectors that could be wedged together to produce this multivector. This is the way that multivectors convey to us that there is no true temperament sum of these two temperaments.		We've now observed the outcome when adding non-addable temperaments using the matrix approach. It's instructive to observe how it works with multivectors as well. The canonical multicommas for septimal meantone and septimal blackwood are {{multicomma\|12 -13 4 10 -4 1}} and {{multicomma\|14 0 -8 0 5 0}}, respectively. When we add these, we get {{multicomma\|26 -13 -4 10 1 1}}. What temperament is this — does it match with any of the four comma bases we've already found? Let's check by converting it back to matrix form. Oh, wait — we can't. This is what we call a [[decomposability\|indecomposable]] multivector. In other words, there is no set of vectors that could be wedged together to produce this multivector. This is the way that multivectors convey to us that there is no true temperament sum of these two temperaments.

	=== Further explanations ===		=== Further explanations ===

Fredg999: Heading levels (level 1 reserved for page title), misc. edits

2023-09-04T19:37:55Z

Heading levels (level 1 reserved for page title), misc. edits

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Cmloegcmluin: interval basis → domain basis (thanks Jukka!)

2023-05-04T00:30:30Z

interval basis → domain basis (thanks Jukka!)

← Older revision		Revision as of 00:30, 4 May 2023
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	Temperament addition is only possible for temperaments with the same [[dimensions]], that is, the same [[rank]] and [[dimensionality]] (and therefore, by the [[rank-nullity theorem]], also the same [[nullity]]). The reason for this is visually obvious: without the same <math>d</math>, <math>r</math>, and <math>n</math> (dimensionality, rank, and nullity, respectively), the numeric representations of the temperament — such as matrices and multivectors — will not have the same proportions, and therefore their entries will be unable to be matched up one-to-one. From this condition it also follows that the result of temperament addition will be a new temperament with the same <math>d</math>, <math>r</math>, and <math>n</math> as the input temperaments.		Temperament addition is only possible for temperaments with the same [[dimensions]], that is, the same [[rank]] and [[dimensionality]] (and therefore, by the [[rank-nullity theorem]], also the same [[nullity]]). The reason for this is visually obvious: without the same <math>d</math>, <math>r</math>, and <math>n</math> (dimensionality, rank, and nullity, respectively), the numeric representations of the temperament — such as matrices and multivectors — will not have the same proportions, and therefore their entries will be unable to be matched up one-to-one. From this condition it also follows that the result of temperament addition will be a new temperament with the same <math>d</math>, <math>r</math>, and <math>n</math> as the input temperaments.

	==The temperaments share the same ~~interval subspace~~==		==The temperaments share the same domain basis==

	If you're unfamiliar with [[~~interval~~ basis\|~~interval subspaces~~]], then you can probably safely assume your temperaments are in the same subspace, because they should be in the default, standard, [[prime limit\|prime-limit]] interval subspace. If they're not, change them to be on the same interval subspace if you can, and then come back to temperament addition.		If you're unfamiliar with [[domain basis\|domain bases]], then you can probably safely assume your temperaments are in the same subspace, because they should be in the default, standard, [[prime limit\|prime-limit]] interval subspace. If they're not, change them to be on the same interval subspace if you can, and then come back to temperament addition.

	==The temperaments are addable==		==The temperaments are addable==