Keenan Pepper's explanation of vals: Difference between revisions

Line 13:

A specific equal temperament (as opposed to an EDO) is represented by a single val. There are infinitely many vals, and therefore technically speaking infinitely many equal temperaments, corresponding to each EDO, but usually only a handful are anywhere near good enough to make sense musically. For example, <12 18 27| is a different 5-limit temperament with 12 equal steps to the octave, which maps 3/2 to 600 cents, and 5/4 to 300 cents, so it's musically useless. But in some cases there are different equal temperaments with the same number of steps that are about equally good. For example, 17edo could be used as two different equal temperaments in the 5-limit: one corresponds to the patent val <17 27 39|, in which 5/4 is very flat at 353 cents, and one corresponds to the <17 27 40| val (sometimes called "17c"), in which 5/4 is very sharp at 424 cents. Although these could be realized musically as exactly the same notes (17 equal divisions of 2/1), they are nevertheless different temperaments, because a 4:5:6 chord, for example, would be represented differently.

A val, v, is said to temper out a comma, c, whenever v(c) = 0. For example, <17 27 40| tempers out 81/80 (exercise for the reader), but the patent val for 17edo does not. Thus, if you played a meantone comma pump (which depends on 81/80 vanishing to return to the same pitch) in 17edo using the <17 27 40| val, it would work, whereas the version using the patent val would not return to the same pitch. We summarize this by saying "<17 27 40| ~~supports~~ meantone temperament" or "<17 27 40| is a meantone val".

A val, v, is said to temper out a comma, c, whenever v(c) = 0. For example, <17 27 40| tempers out 81/80 (exercise for the reader), but the patent val for 17edo does not. Thus, if you played a meantone comma pump (which depends on 81/80 vanishing to return to the same pitch) in 17edo using the <17 27 40| val, it would work, whereas the version using the patent val would not return to the same pitch. We summarize this by saying "<17 27 40| [[support]]s meantone temperament" or "<17 27 40| is a meantone val".

Temperaments other than equal temperaments (that is, rank 2 and higher) can be constructed out of vals. This operation can be denoted "v1 ^ v2" or "v1 & v2". One of the many possible ways to think about this operation is that the resulting temperament tempers out only those commas common to both vals.

@@ Line 13: / Line 13: @@
 A specific equal temperament (as opposed to an EDO) is represented by a single val. There are infinitely many vals, and therefore technically speaking infinitely many equal temperaments, corresponding to each EDO, but usually only a handful are anywhere near good enough to make sense musically. For example, &lt;12 18 27| is a different 5-limit temperament with 12 equal steps to the octave, which maps 3/2 to 600 cents, and 5/4 to 300 cents, so it's musically useless. But in some cases there are different equal temperaments with the same number of steps that are about equally good. For example, 17edo could be used as two different equal temperaments in the 5-limit: one corresponds to the patent val &lt;17 27 39|, in which 5/4 is very flat at 353 cents, and one corresponds to the &lt;17 27 40| val (sometimes called "17c"), in which 5/4 is very sharp at 424 cents. Although these could be realized musically as exactly the same notes (17 equal divisions of 2/1), they are nevertheless different temperaments, because a 4:5:6 chord, for example, would be represented differently.
-A val, v, is said to temper out a comma, c, whenever v(c) = 0. For example, &lt;17 27 40| tempers out 81/80 (exercise for the reader), but the patent val for 17edo does not. Thus, if you played a meantone comma pump (which depends on 81/80 vanishing to return to the same pitch) in 17edo using the &lt;17 27 40| val, it would work, whereas the version using the patent val would not return to the same pitch. We summarize this by saying "&lt;17 27 40| supports meantone temperament" or "&lt;17 27 40| is a meantone val".
+A val, v, is said to temper out a comma, c, whenever v(c) = 0. For example, &lt;17 27 40| tempers out 81/80 (exercise for the reader), but the patent val for 17edo does not. Thus, if you played a meantone comma pump (which depends on 81/80 vanishing to return to the same pitch) in 17edo using the &lt;17 27 40| val, it would work, whereas the version using the patent val would not return to the same pitch. We summarize this by saying "&lt;17 27 40| [[support]]s meantone temperament" or "&lt;17 27 40| is a meantone val".
 Temperaments other than equal temperaments (that is, rank 2 and higher) can be constructed out of vals. This operation can be denoted "v1 ^ v2" or "v1 &amp; v2". One of the many possible ways to think about this operation is that the resulting temperament tempers out only those commas common to both vals.