Douglas Blumeyer's RTT How-To: Difference between revisions
Cmloegcmluin (talk | contribs) →multicommas: monospace the table contents |
Cmloegcmluin (talk | contribs) remove double spaces |
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| Line 461: | Line 461: | ||
<math> | <math> | ||
\left[ \begin{array} {rrr} | \left[ \begin{array} {rrr} | ||
-4 & -10 \\ | |||
4 & -1 \\ | |||
-1 & 5 | -1 & 5 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 507: | Line 507: | ||
<math> | <math> | ||
\left[ \begin{array} {rrr} | \left[ \begin{array} {rrr} | ||
-4 & 4 & -1 \\ | |||
-10 & -1 & 5 | |||
\end{array} \right] | |||
</math> | </math> | ||
| Line 516: | Line 516: | ||
<math> | <math> | ||
\left[ \begin{array} {rrr} | \left[ \begin{array} {rrr} | ||
-1 & 4 & -4 \\ | |||
5 & -1 & -10 | |||
\end{array} \right] | |||
</math> | </math> | ||
| Line 525: | Line 525: | ||
<math> | <math> | ||
\left[ \begin{array} {rrr} | \left[ \begin{array} {rrr} | ||
-1 & 4 & -4 \\ | |||
5 & -1 & -10 \\ | |||
\hline | \hline | ||
1 & 0 & 0 \\ | 1 & 0 & 0 \\ | ||
0 & 1 & 0 \\ | 0 & 1 & 0 \\ | ||
0 & 0 & 1 | 0 & 0 & 1 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 538: | Line 538: | ||
<math> | <math> | ||
\left[ \begin{array} {rrr} | \left[ \begin{array} {rrr} | ||
-1 & 4 & 0 \\ | |||
5 & -1 & -30 \\ | |||
\hline | \hline | ||
1 & 0 & -4 \\ | 1 & 0 & -4 \\ | ||
0 & 1 & 0 \\ | 0 & 1 & 0 \\ | ||
0 & 0 & 1 | 0 & 0 & 1 | ||
\end{array} \right] | |||
→ | → | ||
\left[ \begin{array} {rrr} | \left[ \begin{array} {rrr} | ||
-1 & 0 & 0 \\ | |||
5 & 19 & -30 \\ | |||
\hline | \hline | ||
1 & 4 & -4 \\ | 1 & 4 & -4 \\ | ||
0 & 1 & 0 \\ | 0 & 1 & 0 \\ | ||
0 & 0 & 1 | 0 & 0 & 1 | ||
\end{array} \right] | |||
→ | → | ||
\left[ \begin{array} {rrr} | \left[ \begin{array} {rrr} | ||
-1 & 0 & 0 \\ | |||
5 & 19 & -570 \\ | |||
\hline | \hline | ||
1 & 4 & -76 \\ | 1 & 4 & -76 \\ | ||
0 & 1 & 0 \\ | 0 & 1 & 0 \\ | ||
0 & 0 & 19 | 0 & 0 & 19 | ||
\end{array} \right] | |||
→ | → | ||
\left[ \begin{array} {rrr} | \left[ \begin{array} {rrr} | ||
-1 & 0 & \color{lime}0 \\ | |||
5 & 19 & \color{lime}0 \\ | |||
\hline | \hline | ||
1 & 4 & \color{green}44 \\ | 1 & 4 & \color{green}44 \\ | ||
0 & 1 & \color{green}30 \\ | 0 & 1 & \color{green}30 \\ | ||
0 & 0 & \color{green}19 | 0 & 0 & \color{green}19 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 587: | Line 587: | ||
\color{green}30 \\ | \color{green}30 \\ | ||
\color{green}19 | \color{green}19 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 595: | Line 595: | ||
\left[ \begin{array} {rrr} | \left[ \begin{array} {rrr} | ||
44 & 30 & 19 | 44 & 30 & 19 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 603: | Line 603: | ||
\left[ \begin{array} {rrr} | \left[ \begin{array} {rrr} | ||
19 & 30 & 44 | 19 & 30 & 44 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 628: | Line 628: | ||
0 & 1 & 0 \\ | 0 & 1 & 0 \\ | ||
0 & 0 & 1 | 0 & 0 & 1 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 640: | Line 640: | ||
0 & 1 & 0 \\ | 0 & 1 & 0 \\ | ||
0 & 0 & 19 | 0 & 0 & 19 | ||
\end{array} \right] | |||
→ | → | ||
| Line 650: | Line 650: | ||
0 & 1 & 0 \\ | 0 & 1 & 0 \\ | ||
0 & 0 & 19 | 0 & 0 & 19 | ||
\end{array} \right] | |||
→ | → | ||
| Line 660: | Line 660: | ||
0 & 19 & \color{green}0 \\ | 0 & 19 & \color{green}0 \\ | ||
0 & 0 & \color{green}19 | 0 & 0 & \color{green}19 | ||
\end{array} \right] | |||
→ | → | ||
| Line 670: | Line 670: | ||
0 & \color{green}19 & \color{green}0 \\ | 0 & \color{green}19 & \color{green}0 \\ | ||
0 & \color{green}0 & \color{green}19 | 0 & \color{green}0 & \color{green}19 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 680: | Line 680: | ||
\color{green}19 & \color{green}0 \\ | \color{green}19 & \color{green}0 \\ | ||
\color{green}0 & \color{green}19 | \color{green}0 & \color{green}19 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 708: | Line 708: | ||
5 & 8 & 12 \\ | 5 & 8 & 12 \\ | ||
7 & 11 & 16 | 7 & 11 & 16 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 727: | Line 727: | ||
0 & 1 & 0 \\ | 0 & 1 & 0 \\ | ||
0 & 0 & 1 | 0 & 0 & 1 | ||
\end{array} \right] | |||
→ | → | ||
| Line 738: | Line 738: | ||
0 & 1 & 0 \\ | 0 & 1 & 0 \\ | ||
0 & 0 & 5 | 0 & 0 & 5 | ||
\end{array} \right] | |||
→ | → | ||
| Line 749: | Line 749: | ||
0 & 1 & 0 \\ | 0 & 1 & 0 \\ | ||
0 & 0 & 5 | 0 & 0 & 5 | ||
\end{array} \right] | |||
→ | → | ||
| Line 760: | Line 760: | ||
0 & 5 & 0 \\ | 0 & 5 & 0 \\ | ||
0 & 0 & 5 | 0 & 0 & 5 | ||
\end{array} \right] | |||
→ | → | ||
| Line 771: | Line 771: | ||
0 & 5 & 0 \\ | 0 & 5 & 0 \\ | ||
0 & 0 & 5 | 0 & 0 & 5 | ||
\end{array} \right] | |||
→ | → | ||
| Line 782: | Line 782: | ||
0 & 5 & -20 \\ | 0 & 5 & -20 \\ | ||
0 & 0 & 5 | 0 & 0 & 5 | ||
\end{array} \right] | |||
→ | → | ||
| Line 793: | Line 793: | ||
0 & 5 & -4 \\ | 0 & 5 & -4 \\ | ||
0 & 0 & 1 | 0 & 0 & 1 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 815: | Line 815: | ||
5 & 8 & 12 \\ | 5 & 8 & 12 \\ | ||
7 & 11 & 16 | 7 & 11 & 16 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 836: | Line 836: | ||
5 & 8 & 12 \\ | 5 & 8 & 12 \\ | ||
7 & 11 & 16 | 7 & 11 & 16 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 871: | Line 871: | ||
5 & 8 & 12 \\ | 5 & 8 & 12 \\ | ||
7 & 11 & 16 | 7 & 11 & 16 | ||
\end{array} \right] | |||
→ | → | ||
| Line 878: | Line 878: | ||
5 & 8 & 12 \\ | 5 & 8 & 12 \\ | ||
2 & 3 & 4 | 2 & 3 & 4 | ||
\end{array} \right] | |||
→ | → | ||
| Line 885: | Line 885: | ||
1 & 2 & 4 \\ | 1 & 2 & 4 \\ | ||
2 & 3 & 4 | 2 & 3 & 4 | ||
\end{array} \right] | |||
→ | → | ||
| Line 892: | Line 892: | ||
1 & 2 & 4 \\ | 1 & 2 & 4 \\ | ||
1 & 1 & 0 | 1 & 1 & 0 | ||
\end{array} \right] | |||
→ | → | ||
| Line 899: | Line 899: | ||
0 & 1 & 4 \\ | 0 & 1 & 4 \\ | ||
1 & 1 & 0 | 1 & 1 & 0 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 908: | Line 908: | ||
1 & 1 & 0 \\ | 1 & 1 & 0 \\ | ||
0 & 1 & 4 | 0 & 1 & 4 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 927: | Line 927: | ||
Two points make a line. By the same logic, three points make a plane. Does this carry any weight in RTT? Yes it does. | Two points make a line. By the same logic, three points make a plane. Does this carry any weight in RTT? Yes it does. | ||
Our hypothesis might be: this represents the entirety of 5-limit JI. If two rank-1 temperaments — | Our hypothesis might be: this represents the entirety of 5-limit JI. If two rank-1 temperaments — each of which can be described as tempering out 2 commas — when unioned result in a rank-2 temperament — which is defined as tempering out 1 comma — then when we union three rank-1 temperaments, we should expect to get a rank-3 temperament, which tempers out 0 commas. The rank-1 temperaments appear as 0D points in PTS but are understood to be a 1D line coming straight at us; the rank-2 temperaments appear as 1D points in PTS but are understood to be 2D planes coming straight at us; the rank-3 temperament appear as the 2D plane of the entire PTS diagram but is understood to be the entire 3D space. | ||
Let’s check our hypothesis using the PTS navigation techniques and matrix math we’ve learned. | Let’s check our hypothesis using the PTS navigation techniques and matrix math we’ve learned. | ||
| Line 940: | Line 940: | ||
15 & 24 & 35 \\ | 15 & 24 & 35 \\ | ||
22 & 35 & 51 | 22 & 35 & 51 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 950: | Line 950: | ||
0 & 1 & 0 \\ | 0 & 1 & 0 \\ | ||
0 & 0 & 1 | 0 & 0 & 1 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 987: | Line 987: | ||
1 & 0 \\ | 1 & 0 \\ | ||
0 & 1 | 0 & 1 | ||
\end{array} \right] | |||
→ | → | ||
| Line 996: | Line 996: | ||
1 & 0 \\ | 1 & 0 \\ | ||
0 & 12 | 0 & 12 | ||
\end{array} \right] | |||
→ | → | ||
| Line 1,005: | Line 1,005: | ||
1 & -19 \\ | 1 & -19 \\ | ||
0 & 12 | 0 & 12 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 1,057: | Line 1,057: | ||
5 & 8 & 12 \\ | 5 & 8 & 12 \\ | ||
7 & 11 & 16 \\ | 7 & 11 & 16 \\ | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 1,066: | Line 1,066: | ||
1 & 1 & 0 \\ | 1 & 1 & 0 \\ | ||
0 & 1 & 4 | 0 & 1 & 4 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 1,077: | Line 1,077: | ||
1 & 2 & 4 \\ | 1 & 2 & 4 \\ | ||
0 & -1 & -4 | 0 & -1 & -4 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 1,094: | Line 1,094: | ||
1 & 0 & -4 \\ | 1 & 0 & -4 \\ | ||
0 & 1 & 4 | 0 & 1 & 4 | ||
\end{array} \right] | |||
</math> | </math> | ||
| Line 1,160: | Line 1,160: | ||
<math> | <math> | ||
\begin{array}{ccc} | \begin{array}{ccc} | ||
\text{(2,3)} & \text{(2,5)} & \text{(3,5)} \\ | |||
\begin{bmatrix}\color{red}1 & \color{lime}0 \\ \color{red}0 & \color{lime}1 \end{bmatrix} & \begin{bmatrix}\color{red}1 & \color{blue}-4 \\ \color{red}0 & \color{blue}4 \end{bmatrix} & \begin{bmatrix}\color{lime}0 & \color{blue}-4 \\ \color{lime}1 & \color{blue}4 \end{bmatrix} | |||
\end{array} | \end{array} | ||
</math> | </math> | ||
| Line 1,204: | Line 1,204: | ||
<math> | <math> | ||
\begin{array}{ccc} | \begin{array}{ccc} | ||
\text{(2,3,5)} & | |||
\text{(2,3,7)} & | |||
\text{(2,5,7)} & | |||
\text{(3,5,7)} \\ | |||
\begin{bmatrix}\color{red}1 & \color{lime}0 & \color{blue}1 \\ \color{red}0 & \color{lime}1 & \color{blue}1 \\ \color{red}0 & \color{lime}0 & \color{blue}-2 \end{bmatrix} & | |||
\begin{bmatrix}\color{red}1 & \color{lime}0 & \color{magenta}4 \\ \color{red}0 & \color{lime}1 & \color{magenta}-1 \\ \color{red}0 & \color{lime}0 & \color{magenta}3 \end{bmatrix} & | |||
\begin{bmatrix}\color{red}1 & \color{blue}1 & \color{magenta}4 \\ \color{red}0 & \color{blue}1 & \color{magenta}-1 \\ \color{red}0 & \color{blue}-2 & \color{magenta}3 \end{bmatrix} & | |||
\begin{bmatrix}\color{lime}0 & \color{blue}1 & \color{magenta}4 \\ \color{lime}1 & \color{blue}1 & \color{magenta}-1 \\ \color{lime}0 & \color{blue}-2 & \color{magenta}3 \end{bmatrix} \\ | |||
\end{array} | \end{array} | ||
</math> | </math> | ||