Monzo: Difference between revisions

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A '''monzo''' is a way of notating a [[JI]] [[interval]] that allows us to express directly how any "composite" interval is represented in terms of simpler [[prime]] intervals. They are typically written using the notation {{monzo| a ~~b c d e f~~ … }}, where ~~the columns~~ represent how the primes 2, 3, 5, 7, 11, 13, etc, in that order, contribute to the interval's prime factorization, up to some [[~~Harmonic Limit~~|prime limit]].

A '''monzo''' is a way of notating a [[JI]] [[interval]] that allows us to express directly how any composite interval is represented in terms of simpler [[prime]] intervals. They are typically written using the notation {{monzo| ''a''1 ''a''2 ''a''3 ''a''4 ''a''5 ''a''6 … }}, where ''a''''i'' are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, contribute to the interval's prime factorization, up to some [[harmonic limit|prime limit]].

Monzos can be thought of as counterparts to [[vals]]. When notating just intonation, they only permit integers as their entries.

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~~:'''Practical hint:'''~~ On the wiki, the monzo template helps you getting correct brackets ([[Template:Monzo|read more…]]).

{{Tip| On the wiki, the monzo template helps you getting correct brackets ([[Template: Monzo|read more…]]). }}

== Relationship with vals ==

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= a_1 b_1 + a_2 b_2 + \ldots + a_n b_n

</math>

~~<!--~~

~~== Monzos in JI subgroups ==~~

~~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.)

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.~~

~~== Monzos in regular temperaments ==~~

Proposed notation: We write a tempered interval (an interval in a [[regular temperament]]) as a (generalized) monzo by taking a set of [[generator]]s (for rank-2 temperaments, this will be the period and the generator), then writing what JI ratio each generator approximates (distinguished from pure JI intervals by putting it in quotes), followed by the number of that specified generator that the interval has. For example, the major third in [[meantone]] temperament can be written as {{monzo|"2"^-2, "3/2"^4}}, meaning "4 perfect fifths minus 2 octaves".

Similarly, edo tunings of a temperament can be given in terms of (a generalized version of) vals, by specifying how many edo steps are used for each generator of the temperament. For example, [[31edo]]'s tuning of meantone temperament can be written as {{val|"2"~31, "3/2"~18}}.

~~-->~~

== Generalizations ==

=== Subgroup monzos ===

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 }}
 {{Beginner|Monzos and interval space}}
-A '''monzo''' is a way of notating a [[JI]] [[interval]] that allows us to express directly how any "composite" interval is represented in terms of simpler [[prime]] intervals. They are typically written using the notation {{monzo| a b c d e f … }}, where the columns represent how the primes 2, 3, 5, 7, 11, 13, etc, in that order, contribute to the interval's prime factorization, up to some [[Harmonic Limit|prime limit]].
+A '''monzo''' is a way of notating a [[JI]] [[interval]] that allows us to express directly how any composite interval is represented in terms of simpler [[prime]] intervals. They are typically written using the notation {{monzo| ''a''<sub>1</sub> ''a''<sub>2</sub> ''a''<sub>3</sub> ''a''<sub>4</sub> ''a''<sub>5</sub> ''a''<sub>6</sub> … }}, where ''a''<sub>''i''</sub> are numbers that represent how the primes 2, 3, 5, 7, 11, 13, etc., in that order, contribute to the interval's prime factorization, up to some [[harmonic limit|prime limit]].
 Monzos can be thought of as counterparts to [[vals]]. When notating just intonation, they only permit integers as their entries.
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 |}
-:'''Practical hint:''' On the wiki, the monzo template helps you getting correct brackets ([[Template:Monzo|read more…]]).
+{{Tip| On the wiki, the monzo template helps you getting correct brackets ([[Template: Monzo|read more…]]). }}
 == Relationship with vals ==
@@ Line 85: / Line 85: @@
 = a_1 b_1 + a_2 b_2 + \ldots + a_n b_n
 </math>
-<!--
-== Monzos in JI subgroups ==
-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.)
-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.
-== Monzos in regular temperaments ==
-Proposed notation: We write a tempered interval (an interval in a [[regular temperament]]) as a (generalized) monzo by taking a set of [[generator]]s (for rank-2 temperaments, this will be the period and the generator), then writing what JI ratio each generator approximates (distinguished from pure JI intervals by putting it in quotes), followed by the number of that specified generator that the interval has. For example, the major third in [[meantone]] temperament can be written as {{monzo|"2"^-2, "3/2"^4}}, meaning "4 perfect fifths minus 2 octaves".
-Similarly, edo tunings of a temperament can be given in terms of (a generalized version of) vals, by specifying how many edo steps are used for each generator of the temperament. For example, [[31edo]]'s tuning of meantone temperament can be written as {{val|"2"~31, "3/2"~18}}.
--->
 == Generalizations ==
 === Subgroup monzos ===