Projection: Difference between revisions
m →Examples: typos |
Cmloegcmluin (talk | contribs) →Examples: great work Kite, thanks for the extra examples. I just made this section stylistically consistent with the rest of the article and some conventions Dave and I prefer |
||
| Line 40: | Line 40: | ||
==Examples== | ==Examples== | ||
The generator of meantone temperament is the fifth. A justly intoned fifth is the interval <math>\frac32</math> at about 701.955 ¢, which as a 5-limit prime-count vector looks like {{vector|-1 1 0}}. But in the [[quarter comma meantone| | The generator of meantone temperament is the fifth. A justly intoned fifth is the interval <math>\frac32</math> at about 701.955 ¢, which as a 5-limit prime-count vector looks like {{vector|-1 1 0}}. But in the [[quarter comma meantone|quarter-comma tuning of meantone]], the fifth is flattened. Since 1 fifth is a quarter comma flat, 4 fifths are a full comma flat. 4 just fifths equals <math>\frac{81}{16}</math>, and 4 fifths minus a comma works out to exactly <math>\frac51</math>. Thus the tuning of the fifth is one-quarter of <math>\frac51</math>, which is <math>5^\frac14 = \sqrt[4]5</math> at about 696.578 ¢, which as a prime-count vector looks like {{vector|0 0 1/4}}. JI ratios have prime-counts that contain only integers, but this one has fractions in it; <math>\sqrt[4]5</math> is an irrational number, so it is not JI. | ||
So, by combining this vector for the tuned fifth with the vector {{vector|1 0 0}} for a purely-tuned octave <math>\frac21</math> as the period, we produce the full generator embedding G for | So, by combining this vector for the tuned fifth with the vector {{vector|1 0 0}} for a purely-tuned octave <math>\frac21</math> as the period, we produce the full generator embedding <math>G</math> for quarter-comma meantone as {{rbra|{{vector|1 0 0}} {{vector|0 0 1/4}}}}: | ||
| Line 55: | Line 55: | ||
And so here is the projection matrix for quarter-comma meantone, shown as the product of that generator embedding with the meantone mapping: | And so here is the projection matrix for quarter-comma meantone, shown as the product of that generator embedding with the meantone mapping: | ||
<math> | <math> | ||
| Line 89: | Line 90: | ||
The columns of P are prime-count vectors, one for each prime. The | The columns of <math>P</math> are prime-count vectors, one for each prime. The 1<sup>st</sup> column of <math>P</math> tells us that prime 2 is projected to {{vector|1 0 0}} = <math>\frac21</math>. Thus <math>\frac21</math> is projected to itself, and is an unchanged-interval. The 2<sup>nd</sup> column tells us that prime 3 is projected to {{vector|1 0 1/4}} = an octave plus the tempered fifth. The 3<sup>rd</sup> column tells us that prime 5 is projected to {{vector|0 0 1}} = <math>\frac51</math>. Thus <math>\frac51</math> is also an unchanged-interval, as is any combination of our two unchanged-intervals <math>\frac21</math> and <math>\frac21</math>, such as <math>\frac54</math>, <math>\frac85</math>, <math>\frac{25}{16}</math>, etc. | ||
We can use this matrix to determine what a JI interval <math>\textbf{i}</math> is projected to. Multiply <math>P</math> by <math>\textbf{i}</math> to get <math>P\textbf{i}</math>. Let's start with <math>\textbf{i}</math> = <math>\frac43</math>: | |||
<math> | <math> | ||
| Line 105: | Line 107: | ||
\begin{array} {c} | \begin{array} {c} | ||
\textbf{i} \\ | |||
\left[ \begin{array} {r} | \left[ \begin{array} {r} | ||
2 \\ | 2 \\ | ||
| Line 116: | Line 118: | ||
\begin{array} {c} | \begin{array} {c} | ||
P\textbf{i} \\ | |||
\left[ \begin{array} {r} | \left[ \begin{array} {r} | ||
1 \\ | 1 \\ | ||
| Line 127: | Line 129: | ||
Thus | Thus <math>\frac43</math> is projected to {{vector|1 0 -1/4}}. Note that {{vector|1 0 -1/4}} is one quarter of an exact untempered JI ratio, <math>\frac{16}{5}</math>, and thus four tempered fourths equals a pure 5-limit minor thirteenth. | ||
Now let <math>\textbf{i}</math> = <math>\frac65</math>. | |||
<math> | <math> | ||
| Line 143: | Line 146: | ||
\begin{array} {c} | \begin{array} {c} | ||
\textbf{i} \\ | |||
\left[ \begin{array} {r} | \left[ \begin{array} {r} | ||
1 \\ | 1 \\ | ||
| Line 154: | Line 157: | ||
\begin{array} {c} | \begin{array} {c} | ||
P\textbf{i} \\ | |||
\left[ \begin{array} {r} | \left[ \begin{array} {r} | ||
2 \\ | 2 \\ | ||
| Line 165: | Line 168: | ||
Thus | Thus <math>\frac65</math> becomes {{vector|2 0 -3/4}}, and four of them equals an untempered <math>\frac{256}{125}</math>. The reader can do similar calculations to verify that <math>\frac21</math> is projected to <math>\frac21</math>, <math>\frac54</math> is projected to <math>\frac54</math>, and <math>\frac32</math> is projected to {{vector|0 0 1/4}}. | ||
For ''third''-comma meantone, however, three fifths (<math>\frac{27}{8}</math>) are a full comma flat. That works out to <math>\frac{10}{3}</math>. Thus the generator is {{vector|1/3 -1/3 1/3}}. Again, the period is a pure octave. This gives us our generator embedding <math>G</math>. Multiply it by the same mapping <math>M</math> to find the projection matrix <math>P</math>: | |||
<math> | <math> | ||
| Line 202: | Line 206: | ||
Let's use P to find out what | Let's use <math>P</math> to find out what <math>\frac65</math> gets projected to: | ||
<math> | <math> | ||
| Line 216: | Line 221: | ||
\begin{array} {c} | \begin{array} {c} | ||
\textbf{i} \\ | |||
\left[ \begin{array} {r} | \left[ \begin{array} {r} | ||
1 \\ | 1 \\ | ||
| Line 227: | Line 232: | ||
\begin{array} {c} | \begin{array} {c} | ||
P\textbf{i} \\ | |||
\left[ \begin{array} {r} | \left[ \begin{array} {r} | ||
1 \\ | 1 \\ | ||
| Line 238: | Line 243: | ||
Thus | Thus <math>\frac65</math> is an unchanged-interval of third-comma meantone, instead of <math>\frac54</math>. The reader can verify that <math>\frac21</math> is still an unchanged-interval. | ||
==Generator embedding== | ==Generator embedding== | ||