Monzo: Difference between revisions

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We can generalize the concept of monzos and vals from the ''p''-limit to other [[JI subgroup]]s. This can be useful when considering different edo tunings of [[subgroup temperaments]].

Proposed notation: To write a JI ratio as a monzo in a JI subgroup, we choose a [[basis]] for the subgroup and factor an interval into the basis elements as we factor an interval in the ''p''-limit into primes at most ''p''. Then we write the monzo so as to explicitly state what basis elements we factor the intervals into and how many of each basis element the interval has in the factorization. For example, we can write [[81/80]] = 92/(24 51) in the 2.9.5 subgroup as {{monzo|2: -4, 9: 2, 5: -1}}. (We reserve the notation {{monzo|a b c ...}} and {{val|a b c ...}} for the ''p''-limit.)

Proposed notation: To write a JI ratio as a monzo in a JI subgroup, we choose a [[basis]] for the subgroup and factor an interval into the basis elements as we factor an interval in the ''p''-limit into primes at most ''p''. Then we write the monzo so as to explicitly state what basis elements we factor the intervals into and how many of each basis element the interval has in the factorization. For example, we can write [[81/80]] = 92/(24 51) in the 2.9.5 subgroup as {{monzo|2^-4, 9^2, 5^-1}}. (We reserve the notation {{monzo|a b c ...}} and {{val|a b c ...}} for the ''p''-limit.)

Vals can be defined the same way in other subgroups as well; they represent how a subgroup is (viewed as being) tuned in terms of that edo's steps. For example, [[13edo]]'s "2.9.5 [[patent val]]" can be written as {{val|2: 13, 9: 41, 5: 30}}, since [[13edo]]'s best approximation to the 9th harmonic is 41\13 (reduces to 2\13) and its best approximation to the 5th harmonic is 30\13 (reduces to 4\13). To see that this val "tempers out [[81/80]]", we do the same operation (of matching up and multiplying the components and summing the products) as described in the previous section:

Vals can be defined the same way in other subgroups as well; they represent how a subgroup is (viewed as being) tuned in terms of that edo's steps, but the basis element and the entry are separated by ~ instead of ^. For example, [[13edo]]'s "2.9.5 [[patent val]]" can be written as {{val|2~13, 9~41, 5~30}} (think "2 is approximately 13 steps, ..."), since [[13edo]]'s best approximation to the 9th harmonic is 41\13 (reduces to 2\13) and its best approximation to the 5th harmonic is 30\13 (reduces to 4\13). To see that this val "tempers out [[81/80]]", we do the same operation (of matching up and multiplying the components and summing the products) as described in the previous section:

⟨2: 13, 9: 41, 5: 30][2: -4, 9: 2, 5: -1⟩ = 13*-4 + 41*2 + 30*-1 = 0.

⟨2~13, 9~41, 5~30][2^-4, 9^2, 5^-1⟩ = 13*-4 + 41*2 + 30*-1 = 0.

== Monzos in regular temperaments ==

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 We can generalize the concept of monzos and vals from the ''p''-limit to other [[JI subgroup]]s. This can be useful when considering different edo tunings of [[subgroup temperaments]].
-Proposed notation: To write a JI ratio as a monzo in a JI subgroup, we choose a [[basis]] for the subgroup and factor an interval into the basis elements as we factor an interval in the ''p''-limit into primes at most ''p''. Then we write the monzo so as to explicitly state what basis elements we factor the intervals into and how many of each basis element the interval has in the factorization. For example, we can write [[81/80]] = 9<sup>2</sup>/(2<sup>4</sup> 5<sup>1</sup>) in the 2.9.5 subgroup as {{monzo|2: -4, 9: 2, 5: -1}}. (We reserve the notation {{monzo|a b c ...}} and {{val|a b c ...}} for the ''p''-limit.)
+Proposed notation: To write a JI ratio as a monzo in a JI subgroup, we choose a [[basis]] for the subgroup and factor an interval into the basis elements as we factor an interval in the ''p''-limit into primes at most ''p''. Then we write the monzo so as to explicitly state what basis elements we factor the intervals into and how many of each basis element the interval has in the factorization. For example, we can write [[81/80]] = 9<sup>2</sup>/(2<sup>4</sup> 5<sup>1</sup>) in the 2.9.5 subgroup as {{monzo|2^-4, 9^2, 5^-1}}. (We reserve the notation {{monzo|a b c ...}} and {{val|a b c ...}} for the ''p''-limit.)
-Vals can be defined the same way in other subgroups as well; they represent how a subgroup is (viewed as being) tuned in terms of that edo's steps. For example, [[13edo]]'s "2.9.5 [[patent val]]" can be written as {{val|2: 13, 9: 41, 5: 30}}, since [[13edo]]'s best approximation to the 9th harmonic is 41\13 (reduces to 2\13) and its best approximation to the 5th harmonic is 30\13 (reduces to 4\13). To see that this val "tempers out [[81/80]]", we do the same operation (of matching up and multiplying the components and summing the products) as described in the previous section:
+Vals can be defined the same way in other subgroups as well; they represent how a subgroup is (viewed as being) tuned in terms of that edo's steps, but the basis element and the entry are separated by ~ instead of ^. For example, [[13edo]]'s "2.9.5 [[patent val]]" can be written as {{val|2~13, 9~41, 5~30}} (think "2 is approximately 13 steps, ..."), since [[13edo]]'s best approximation to the 9th harmonic is 41\13 (reduces to 2\13) and its best approximation to the 5th harmonic is 30\13 (reduces to 4\13). To see that this val "tempers out [[81/80]]", we do the same operation (of matching up and multiplying the components and summing the products) as described in the previous section:
-  &#x27E8;2: 13, 9: 41, 5: 30&#93;&#91;2: -4, 9: 2, 5: -1&#x27E9; = 13*-4 + 41*2 + 30*-1 = 0.
+  &#x27E8;2~13, 9~41, 5~30&#93;&#91;2^-4, 9^2, 5^-1&#x27E9; = 13*-4 + 41*2 + 30*-1 = 0.
 == Monzos in regular temperaments ==