Temperament addition: Difference between revisions

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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.

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>

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=== Addition on non-addable temperaments ===

==== Initial example: canonical form ====

~~Clearly~~, ~~two non-addable temperaments may still be~~ entry-wise ~~added~~. For example, the <math>L_{\text{dep}}</math> 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 <math>l_{\text{ind}}</math> <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 <math>L_{\text{dep}}</math> 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 <math>l_{\text{ind}}</math> <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:

<math>\left[ \begin{array} {~~rrr~~}

<math>\left[ \begin{array} {r|r}

4 & 13 \\

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\end{array} \right]

+

\left[ \begin{array} {~~rrr~~}

\left[ \begin{array} {r|r}

-8 & -6 \\

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\end{array} \right]

=

\left[ \begin{array} {~~rrr~~}

\left[ \begin{array} {r|r}

(4+-8) & (13+-6) \\

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\end{array} \right]

=

\left[ \begin{array} {~~rrr~~}

\left[ \begin{array} {r|r}

-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.

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 ====

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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 ====

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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>

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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.
-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>
@@ Line 475: / Line 475: @@
 === Addition on non-addable temperaments ===
 ==== 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:
-<math>\left[ \begin{array} {rrr}
+<math>\left[ \begin{array} {r|r}
 & 13 \\
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 \end{array} \right]
 +
-\left[ \begin{array} {rrr}
+\left[ \begin{array} {r|r}
 -8 & -6 \\
@@ Line 496: / Line 496: @@
 \end{array} \right]
 =
-\left[ \begin{array} {rrr}
+\left[ \begin{array} {r|r}
 (4+-8) & (13+-6) \\
@@ Line 505: / Line 505: @@
 \end{array} \right]
 =
-\left[ \begin{array} {rrr}
+\left[ \begin{array} {r|r}
 -4 & 7 \\
@@ Line 519: / Line 519: @@
 # 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.
 ==== Second example: alternate form ====
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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 ====
@@ Line 642: / Line 642: @@
-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>