Keenan Pepper's explanation of vals
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A val is a function that assigns a whole number to a JI interval in a **consistent** way. Here's an example of what "consistent" means: if v is a val and x and y are JI intervals, then v(x+y) = v(x) + v(y), where x+y means the composition of the intervals x and y (so you add their cents values, which means you **multiply** their ratios). For example, if v is any val, then v(3/2) + v(4/3) = v(2/1). The mathematical term for this is a "homomorphism". Because of the consistency property, if you want to write down a specific val, you don't need to give its value on every possible JI interval you might want to apply it to. Instead, you only need to give the value for primes, because every other interval can be decomposed into primes. For example, if I tell you that v(2/1) = 12, v(3/1) = 19, and v(5/1) = 28, then you can figure out the value of v applied to any other interval of 5-limit JI. For example, 25/24 = 5^2/(2^3*3), so v(25/24) = 2*v(5) - 3*v(2) - v(3) = 2*28 - 3*12 - 19 = 1, so 25/24 must be mapped to 1 by this val. The standard way to write this val in symbols is < 12 19 28 | The three whole numbers are the values of the val applied to the first three primes (that is, 2, 3, and 5). This is enough information to figure out the value of the val applied to any 5-limit JI interval. For higher-limit JI intervals, its value is undefined. Every EDO has a particular val associated to it called the "patent val". This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo is <12 19 28|. This is **not** the same thing as the function you'd get by rounding every interval independently to the nearest number of steps - that kind of rounding is not consistent, so it isn't a val at all. Consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the mapping of 5/4, it must be represented as 4 steps, even though that's farther away from the JI interval 625/512. In other words, the result of tempering using a patent val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for non-prime intervals. ==Vals supporting temperaments== 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". 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. For example, consider the statement "5-limit meantone is 12p & 19p". Here's a list of the simplest commas tempered out of those two 5-limit equal temperaments: * 12p: 81/80, 128/125, 648/625, 2048/2025, 6561/6400... * 19p: 81/80, 3125/3072, 6561/6400, 15625/15552... In the 12p equal temperament, all of the commas in the first list vanish (are mapped to 0). In 19p, all of the commas in the second list vanish. In the temperament you get from combining them, "12p & 19p", only the commas common to both lists are tempered out. These are 81/80, 6561/6400 = (81/80)^2, (81/80)^3, (81/80)^4... In other words, it works out that 81/80 is the only basic, fundamental comma that vanishes in both 12p and 19p - all the other commas are powers of 81/80, so they automatically vanish if 81/80 vanishes. So we say that the 5-limit temperament "12p & 19p" is the same thing as "the 81/80 temperament" or "meantone". In practice, the easy way to find information about temperaments like this is to go to Graham Breed's temperament finder, http://x31eq.com/temper/net.html , type "12p 19p" into the equal temperaments field, and type "5" into the limit field. It tells you that the resulting temperament is called "meantone", it has 81/80 as its only "unison vector" (aka comma), and other information you might find useful.
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<html><head><title>Keenan's explanation of vals</title></head><body>A val is a function that assigns a whole number to a JI interval in a <strong>consistent</strong> way. Here's an example of what "consistent" means: if v is a val and x and y are JI intervals, then v(x+y) = v(x) + v(y), where x+y means the composition of the intervals x and y (so you add their cents values, which means you <strong>multiply</strong> their ratios). For example, if v is any val, then v(3/2) + v(4/3) = v(2/1). The mathematical term for this is a "homomorphism".<br /> <br /> Because of the consistency property, if you want to write down a specific val, you don't need to give its value on every possible JI interval you might want to apply it to. Instead, you only need to give the value for primes, because every other interval can be decomposed into primes. For example, if I tell you that v(2/1) = 12, v(3/1) = 19, and v(5/1) = 28, then you can figure out the value of v applied to any other interval of 5-limit JI. For example, 25/24 = 5^2/(2^3*3), so v(25/24) = 2*v(5) - 3*v(2) - v(3) = 2*28 - 3*12 - 19 = 1, so 25/24 must be mapped to 1 by this val. The standard way to write this val in symbols is<br /> <br /> < 12 19 28 |<br /> <br /> The three whole numbers are the values of the val applied to the first three primes (that is, 2, 3, and 5). This is enough information to figure out the value of the val applied to any 5-limit JI interval. For higher-limit JI intervals, its value is undefined.<br /> <br /> Every EDO has a particular val associated to it called the "patent val". This is the val where you simply round each prime to the nearest number of steps of the EDO. For example, the 5-limit patent val of 12edo is <12 19 28|. This is <strong>not</strong> the same thing as the function you'd get by rounding every interval independently to the nearest number of steps - that kind of rounding is not consistent, so it isn't a val at all. Consider the interval 625/512. This JI interval is about 345 cents, so it's closer to 3 steps of 12edo than to 4 steps. However, 625/512 = (5/4)^4/(2/1), so in order to be consistent with the mapping of 5/4, it must be represented as 4 steps, even though that's farther away from the JI interval 625/512.<br /> <br /> In other words, the result of tempering using a patent val is that primes are consistently represented as the best possible approximations, at the cost of sometimes using a worse approximation for non-prime intervals.<br /> <br /> <!-- ws:start:WikiTextHeadingRule:0:<h2> --><h2 id="toc0"><a name="x-Vals supporting temperaments"></a><!-- ws:end:WikiTextHeadingRule:0 -->Vals supporting temperaments</h2> <br /> 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.<br /> <br /> 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".<br /> <br /> 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.<br /> <br /> For example, consider the statement "5-limit meantone is 12p & 19p". Here's a list of the simplest commas tempered out of those two 5-limit equal temperaments:<br /> <br /> <ul><li>12p: 81/80, 128/125, 648/625, 2048/2025, 6561/6400...</li><li>19p: 81/80, 3125/3072, 6561/6400, 15625/15552...</li></ul><br /> In the 12p equal temperament, all of the commas in the first list vanish (are mapped to 0). In 19p, all of the commas in the second list vanish. In the temperament you get from combining them, "12p & 19p", only the commas common to both lists are tempered out. These are 81/80, 6561/6400 = (81/80)^2, (81/80)^3, (81/80)^4... In other words, it works out that 81/80 is the only basic, fundamental comma that vanishes in both 12p and 19p - all the other commas are powers of 81/80, so they automatically vanish if 81/80 vanishes. So we say that the 5-limit temperament "12p & 19p" is the same thing as "the 81/80 temperament" or "meantone".<br /> <br /> In practice, the easy way to find information about temperaments like this is to go to Graham Breed's temperament finder, <!-- ws:start:WikiTextUrlRule:32:http://x31eq.com/temper/net.html --><a class="wiki_link_ext" href="http://x31eq.com/temper/net.html" rel="nofollow">http://x31eq.com/temper/net.html</a><!-- ws:end:WikiTextUrlRule:32 --> , type "12p 19p" into the equal temperaments field, and type "5" into the limit field. It tells you that the resulting temperament is called "meantone", it has 81/80 as its only "unison vector" (aka comma), and other information you might find useful.</body></html>