Temperament addition: Difference between revisions
Cmloegcmluin (talk | contribs) reorder sections |
Cmloegcmluin (talk | contribs) LaTeX examples in opening |
||
| Line 3: | Line 3: | ||
'''Temperament arithmetic''' is the general name for either the '''temperament sum''' or the '''temperament difference''', which are two closely related operations on [[regular temperaments]]. Basically, to do temperament arithmetic means to match up the entries of temperament vectors and then add or subtract them individually. The result is a new temperament that has similar properties to the original temperaments. | '''Temperament arithmetic''' is the general name for either the '''temperament sum''' or the '''temperament difference''', which are two closely related operations on [[regular temperaments]]. Basically, to do temperament arithmetic means to match up the entries of temperament vectors and then add or subtract them individually. The result is a new temperament that has similar properties to the original temperaments. | ||
For example, the sum of [[12-ET]] and [[7-ET]] is [[19-ET]] because {{map|12 19 28}} + {{map|7 11 16}}={{map|(12+7) (19+11) (28+16)}}={{map|19 30 44}}, and the difference of 12-ET and 7-ET is 5-ET because {{map|12 19 28}} - {{map|7 11 16}}={{map|(12-7) (8-11) (12-16)}}={{map|5 8 12}} | For example, the sum of [[12-ET]] and [[7-ET]] is [[19-ET]] because {{map|12 19 28}} + {{map|7 11 16}}={{map|(12+7) (19+11) (28+16)}}={{map|19 30 44}}, and the difference of 12-ET and 7-ET is 5-ET because {{map|12 19 28}} - {{map|7 11 16}}={{map|(12-7) (8-11) (12-16)}}={{map|5 8 12}}. | ||
Temperament sums and differences can also be found using commas; for example meantone + porcupine=tetracot because {{vector|4 -4 1}} + {{vector|1 -5 3}}={{vector|(4+1) (-4+-5) (1+3)}}={{vector|5 -9 4}} and meantone - porcupine=dicot because {{vector|4 -4 1}} - {{vector|1 -5 3}}={{vector|(4-1) (-4--5) (1-3)}}={{vector|3 1 -2}}. We could write this in ratio form — replacing addition with multiplication and subtraction with division — as 80/81 × 250/243=20000/19683 and 80/81 ÷ 250/243=25/24, respectively. The similarity in these temperaments can be seen in how all of them are supported by 7-ET. | <math>\left[ \begin{array} {rrr} | ||
12 & 19 & 28 \\ | |||
\end{array} \right] | |||
+ | |||
\left[ \begin{array} {rrr} | |||
7 & 11 & 16 \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
(12+7) & (19+11) & (28+16) \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
19 & 30 & 44 \\ | |||
\end{array} \right]</math> | |||
<math>\left[ \begin{array} {rrr} | |||
12 & 19 & 28 \\ | |||
\end{array} \right] | |||
- | |||
\left[ \begin{array} {rrr} | |||
7 & 11 & 16 \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
(12-7) & (19-11) & (28-16) \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
5 & 8 & 12 \\ | |||
\end{array} \right]</math> | |||
We can write these using [[wart notation]] as 12p + 7p=19p and 12p - 7p=5p, respectively. The similarity in these temperaments can be seen in how, like both 12-ET and 7-ET, 19-ET (their sum) and 5-ET (their difference) both also support [[meantone temperament]]. | |||
Temperament sums and differences can also be found using commas; for example meantone + porcupine=tetracot because {{vector|4 -4 1}} + {{vector|1 -5 3}}={{vector|(4+1) (-4+-5) (1+3)}}={{vector|5 -9 4}} and meantone - porcupine=dicot because {{vector|4 -4 1}} - {{vector|1 -5 3}}={{vector|(4-1) (-4--5) (1-3)}}={{vector|3 1 -2}}. | |||
<math>\left[ \begin{array} {rrr} | |||
4 \\ | |||
-4 \\ | |||
1 \\ | |||
\end{array} \right] | |||
+ | |||
\left[ \begin{array} {rrr} | |||
1 \\ | |||
-5 \\ | |||
3 \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
(4+1) \\ | |||
(-4+-5) \\ | |||
(1+3) \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
5 \\ | |||
-9 \\ | |||
4 \\ | |||
\end{array} \right]</math> | |||
<math>\left[ \begin{array} {rrr} | |||
4 \\ | |||
-4 \\ | |||
1 \\ | |||
\end{array} \right] | |||
- | |||
\left[ \begin{array} {rrr} | |||
1 \\ | |||
-5 \\ | |||
3 \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
(4-1) \\ | |||
(-4--5) \\ | |||
(1-3) \\ | |||
\end{array} \right] | |||
= | |||
\left[ \begin{array} {rrr} | |||
3 \\ | |||
1 \\ | |||
-2 \\ | |||
\end{array} \right]</math> | |||
We could write this in ratio form — replacing addition with multiplication and subtraction with division — as 80/81 × 250/243=20000/19683 and 80/81 ÷ 250/243=25/24, respectively. The similarity in these temperaments can be seen in how all of them are supported by 7-ET. | |||
Temperament arithmetic is simplest for temperaments which can be represented by single vectors such as demonstrated in these examples. In other words, it is simplest for temperaments that are either rank-1 ([[equal temperament]]s, or ETs for short) or nullity-1 (having only a single comma). Because [[grade]] <math>g</math> is the generic term for rank <math>r</math> and nullity <math>n</math>, we could define the minimum grade <math>g_{\text{min}}</math> of a temperament as the minimum of its rank and nullity <math>\min(r,n)</math>, and so for convenience in this article we will refer to <math>r=1</math> (read "rank-1") or <math>n=1</math> (read "nullity-1") temperaments as <math>g_{\text{min}}=1</math> (read "min-grade-1") temperaments. We'll also use <math>g_{\text{max}}</math> (read "max-grade"), which naturally is equal to <math>\max(r,n)</math>. | Temperament arithmetic is simplest for temperaments which can be represented by single vectors such as demonstrated in these examples. In other words, it is simplest for temperaments that are either rank-1 ([[equal temperament]]s, or ETs for short) or nullity-1 (having only a single comma). Because [[grade]] <math>g</math> is the generic term for rank <math>r</math> and nullity <math>n</math>, we could define the minimum grade <math>g_{\text{min}}</math> of a temperament as the minimum of its rank and nullity <math>\min(r,n)</math>, and so for convenience in this article we will refer to <math>r=1</math> (read "rank-1") or <math>n=1</math> (read "nullity-1") temperaments as <math>g_{\text{min}}=1</math> (read "min-grade-1") temperaments. We'll also use <math>g_{\text{max}}</math> (read "max-grade"), which naturally is equal to <math>\max(r,n)</math>. | ||
| Line 471: | Line 587: | ||
0 & 1 & 4 & 10 \\ | 0 & 1 & 4 & 10 \\ | ||
\end{array} \right] | \end{array} \right] | ||
+ | + | ||
\left[ \begin{array} {rrr} | |||
1 & 0 & -4 & 17 \\ | 1 & 0 & -4 & 17 \\ | ||
0 & 1 & 4 & -9 \\ | 0 & 1 & 4 & -9 \\ | ||
\end{array} \right] | \end{array} \right] | ||
= | = | ||
\left[ \begin{array} {rrr} | |||
2 & 0 & -8 & 4 \\ | 2 & 0 & -8 & 4 \\ | ||
| Line 496: | Line 612: | ||
1 & 0 & -4 & -13 \\ | 1 & 0 & -4 & -13 \\ | ||
\end{array} \right] | \end{array} \right] | ||
+ | + | ||
\left[ \begin{array} {rrr} | |||
\color{OliveGreen}19 & \color{OliveGreen}30 & \color{OliveGreen}44 & \color{OliveGreen}53 \\ | \color{OliveGreen}19 & \color{OliveGreen}30 & \color{OliveGreen}44 & \color{OliveGreen}53 \\ | ||
| Line 520: | Line 636: | ||
\color{BrickRed}7 & \color{BrickRed}11 & \color{BrickRed}16 & \color{BrickRed}19 \\ | \color{BrickRed}7 & \color{BrickRed}11 & \color{BrickRed}16 & \color{BrickRed}19 \\ | ||
\end{array} \right] | \end{array} \right] | ||
+ | + | ||
\left[ \begin{array} {rrr} | |||
\color{OliveGreen}19 & \color{OliveGreen}30 & \color{OliveGreen}44 & \color{OliveGreen}53 \\ | \color{OliveGreen}19 & \color{OliveGreen}30 & \color{OliveGreen}44 & \color{OliveGreen}53 \\ | ||
| Line 535: | Line 651: | ||
\color{BrickRed}7 & \color{BrickRed}11 & \color{BrickRed}16 & \color{BrickRed}19 \\ | \color{BrickRed}7 & \color{BrickRed}11 & \color{BrickRed}16 & \color{BrickRed}19 \\ | ||
\end{array} \right] | \end{array} \right] | ||
+ | + | ||
\left[ \begin{array} {rrr} | |||
\color{BrickRed}7 & \color{BrickRed}11 & \color{BrickRed}16 & \color{BrickRed}20 \\ | \color{BrickRed}7 & \color{BrickRed}11 & \color{BrickRed}16 & \color{BrickRed}20 \\ | ||
\end{array} \right] | \end{array} \right] | ||
= | = | ||
\left[ \begin{array} {rrr} | |||
\color{BrickRed}14 & \color{BrickRed}22 & \color{BrickRed}32 & \color{BrickRed}39 \\ | \color{BrickRed}14 & \color{BrickRed}22 & \color{BrickRed}32 & \color{BrickRed}39 \\ | ||
| Line 581: | Line 697: | ||
\color{BrickRed}7 & \color{BrickRed}11 & \color{BrickRed}16 & \color{BrickRed}20 \\ | \color{BrickRed}7 & \color{BrickRed}11 & \color{BrickRed}16 & \color{BrickRed}20 \\ | ||
\end{array} \right] | \end{array} \right] | ||
= | = | ||
\left[ \begin{array} {rrr} | |||
\color{BrickRed}0 & \color{BrickRed}0 & \color{BrickRed}0 & \color{BrickRed}-1 \\ | \color{BrickRed}0 & \color{BrickRed}0 & \color{BrickRed}0 & \color{BrickRed}-1 \\ | ||