Projection: Difference between revisions

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==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|1/4-comma tuning of meantone]], ~~since~~ 1 fifth is a quarter comma flat, 4 fifths are a full comma flat. 4 just fifths equals 81/16, and 4 fifths minus a comma works out to exactly 5/1. Thus the tuning of the fifth 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}}. Usually JI ratios have prime-counts that contain only integers, but this one has fractions in it. Since <math>\sqrt[4]5</math> is an irrational number, we could say it's sort of an irrational ratio!

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|1/4-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 81/16, and 4 fifths minus a comma works out to exactly 5/1. Thus the tuning of the fifth is one-quarter of 5/1, 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}}. Usually JI ratios have prime-counts that contain only integers, but this one has fractions in it. Since <math>\sqrt[4]5</math> is an irrational number, we could say it's sort of an irrational ratio!

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 for 1/4-comma meantone as {{rbra|{{vector|1 0 0}} {{vector|0 0 1/4}}}}:

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 1/4-comma meantone as {{rbra|{{vector|1 0 0}} {{vector|0 0 1/4}}}}:

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And so here is the projection matrix for quarter-comma meantone, shown as the product of that generator embedding with the meantone mapping:

<math>

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</math>

The columns of P are prime-count vectors, one for each prime. The 1st column of P tells us that prime 2 is tempered to {{vector|1 0 0}} = 2/1. Thus 2/1 is tempered to itself, and is an eigenmonzo. The 2nd column tells us that prime 3 is tempered to {{vector|1 0 1/4}} = an octave plus the tempered fifth. The 3rd column tells us that prime 5 is tempered to {{vector|0 0 1}} = 5/1. Thus 5/1 is also an eigenmonzo. As is any combination of 2/1 and 5/1, such as 5/4, 8/5, 25/16, etc.

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</math>

Thus 4/3 is tempered to {{vector|1 0 -~~\frac14~~}}. Even though we are discussing a temperament, note that {{vector|1 0 -~~\frac14~~}} is one quarter of an exact untempered JI ratio, 16/5. Thus four tempered 4ths equals a pure 5-limit minor 13th. Now let J = 6/5.

Thus 4/3 is tempered to {{vector|1 0 -1/4}}. Even though we are discussing a temperament, note that {{vector|1 0 -1/4}} is one quarter of an exact untempered JI ratio, 16/5. Thus four tempered 4ths equals a pure 5-limit minor 13th.

Now let J = 6/5.

<math>

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</math>

Thus 6/5 becomes {{vector|2 0 -3/4}}, and four of them equals an untempered 256/125. The reader can do similar calculations to verify that 2/1 is tempered to 2/1, 5/4 is tempered to 5/4, and 3/2 is tempered to {{vector|0 0 1/4}}.

For third-comma meantone, three fifths (27/8) are a full comma flat. That works out to 10/3. Thus the generator is {{vector|1/3 -1/3 1/3}}. Again the period is a just octave. This gives us our generator embedding G. ~~We can use~~ the same mapping M to find the projection matrix P:

For third-comma meantone, three fifths (27/8) are a full comma flat. That works out to 10/3. Thus the generator is {{vector|1/3 -1/3 1/3}}. Again the period is a just octave. This gives us our generator embedding G. Multiply it by the same mapping M to find the projection matrix P:

<math>

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</math>

Let's use P to find out what 6/5 gets tempered to:

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</math>

Thus 6/5 is now an eigenmonzo. The reader can verify that 2/1 is too, and 5/4 is not.

==Generator embedding==

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 ==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|1/4-comma tuning of meantone]], since 1 fifth is a quarter comma flat, 4 fifths are a full comma flat. 4 just fifths equals 81/16, and 4 fifths minus a comma works out to exactly 5/1. Thus the tuning of the fifth 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}}. Usually JI ratios have prime-counts that contain only integers, but this one has fractions in it. Since <math>\sqrt[4]5</math> is an irrational number, we could say it's sort of an irrational ratio!
+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|1/4-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 81/16, and 4 fifths minus a comma works out to exactly 5/1. Thus the tuning of the fifth is one-quarter of 5/1, 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}}. Usually JI ratios have prime-counts that contain only integers, but this one has fractions in it. Since <math>\sqrt[4]5</math> is an irrational number, we could say it's sort of an irrational ratio!
-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 for 1/4-comma meantone as {{rbra|{{vector|1 0 0}} {{vector|0 0 1/4}}}}:
+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 1/4-comma meantone as {{rbra|{{vector|1 0 0}} {{vector|0 0 1/4}}}}:
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 And so here is the projection matrix for quarter-comma meantone, shown as the product of that generator embedding with the meantone mapping:
 <math>
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 </math>
 The columns of P are prime-count vectors, one for each prime. The 1st column of P tells us that prime 2 is tempered to {{vector|1 0 0}} = 2/1. Thus 2/1 is tempered to itself, and is an eigenmonzo. The 2nd column tells us that prime 3 is tempered to {{vector|1 0 1/4}} = an octave plus the tempered fifth. The 3rd column tells us that prime 5 is tempered to {{vector|0 0 1}} = 5/1. Thus 5/1 is also an eigenmonzo. As is any combination of 2/1 and 5/1, such as 5/4, 8/5, 25/16, etc.
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 </math>
-Thus 4/3 is tempered to {{vector|1 0 -\frac14}}. Even though we are discussing a temperament, note that {{vector|1 0 -\frac14}} is one quarter of an exact untempered JI ratio, 16/5. Thus four tempered 4ths equals a pure 5-limit minor 13th. Now let J = 6/5.
+Thus 4/3 is tempered to {{vector|1 0 -1/4}}. Even though we are discussing a temperament, note that {{vector|1 0 -1/4}} is one quarter of an exact untempered JI ratio, 16/5. Thus four tempered 4ths equals a pure 5-limit minor 13th.
+Now let J = 6/5.
 <math>
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 </math>
 Thus 6/5 becomes {{vector|2 0 -3/4}}, and four of them equals an untempered 256/125. The reader can do similar calculations to verify that 2/1 is tempered to 2/1, 5/4 is tempered to 5/4, and 3/2 is tempered to {{vector|0 0 1/4}}.
-For third-comma meantone, three fifths (27/8) are a full comma flat. That works out to 10/3. Thus the generator is {{vector|1/3 -1/3 1/3}}. Again the period is a just octave. This gives us our generator embedding G. We can use the same mapping M to find the projection matrix P:
+For third-comma meantone, three fifths (27/8) are a full comma flat. That works out to 10/3. Thus the generator is {{vector|1/3 -1/3 1/3}}. Again the period is a just octave. This gives us our generator embedding G. Multiply it by the same mapping M to find the projection matrix P:
 <math>
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 </math>
 Let's use P to find out what 6/5 gets tempered to:
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 </math>
 Thus 6/5 is now an eigenmonzo. The reader can verify that 2/1 is too, and 5/4 is not.
 ==Generator embedding==