Monzo: Difference between revisions
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| de = Monzo-Notation | |||
| en = Monzo | |||
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| ja = モンゾ | |||
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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]]. | |||
Monzos can be thought of as counterparts to [[vals]]. When notating just intonation, they only permit integers as their entries. | |||
Monzos can be | == History and terminology == | ||
Monzos are named in honor of [[Joseph Monzo]], given by [[Gene Ward Smith]] in July 2003. These were also previously called ''factorads'' by [[John Chalmers]] in ''[[Xenharmonikôn]] 1'', although the basic idea goes back at least as far as [[Adriaan Fokker]] and probably further back, so that the entire naming situation can be viewed as an example of [[Wikipedia: Stigler%27s law of eponymy|Stigler's law]] many times over. More descriptive but longer terms include '''prime-count vector'''<ref>Used by [[Douglas Blumeyer]] and [[Dave Keenan]] on this wiki, notably in [[Dave Keenan & Douglas Blumeyer's guide to RTT]]</ref>, '''prime-exponent vector'''<ref>[http://tonalsoft.com/enc/m/monzo.aspx Tonalsoft | ''Monzo'']</ref>, and in the context of just intonation, '''harmonic space coordinates'''<ref>[https://www.plainsound.org/HEJI/ Plainsound Music Edition | ''Plainsound Harmonic Space Calculator'']</ref>. | |||
For | == Examples == | ||
For example, the interval [[15/8]] can be thought of as having <math>5 \cdot 3</math> in the numerator, and <math>2 \cdot 2 \cdot 2</math> in the denominator. This can be compactly represented by the expression <math>2^{-3} \cdot 3^1 \cdot 5^1</math>, which is exactly equal to 15/8. We construct the monzo by taking the exponent from each prime, in order, and placing them within the {{monzo| … }} brackets, hence yielding {{monzo| -3 1 1 }}. | |||
Here are some common [[5-limit]] monzos, along with their factorizations to show how to derive them: | |||
{| class="wikitable center-1" | |||
| | |- | ||
| | ! Ratio | ||
| | ! Factors | ||
| | ! Monzo | ||
| | |- | ||
| [[3/2]] | |||
| <math>2^{-1} \cdot 3</math> | |||
| {{monzo| -1 1 0 }} | |||
|- | |||
| [[5/4]] | |||
| <math>2^{-2} \cdot 5</math> | |||
| {{monzo| -2 0 1 }} | |||
|- | |||
| [[9/8]] | |||
| <math>2^{-3} \cdot 3^2</math> | |||
| {{monzo| -3 2 0 }} | |||
|- | |||
| [[81/80]] | |||
| <math>2^{-4} \cdot 3^4 \cdot 5^{-1}</math> | |||
| {{monzo| -4 4 -1 }} | |||
|} | |||
Here are a few 7-limit monzos: | Here are a few [[7-limit]] monzos: | ||
= | {| class="wikitable center-1" | ||
|- | |||
! Ratio | |||
! Factors | |||
! Monzo | |||
|- | |||
| [[7/4]] | |||
| <math>2^{-2} \cdot 7</math> | |||
| {{monzo| -2 0 0 1 }} | |||
|- | |||
| [[7/6]] | |||
| <math>2^{-1} \cdot 3^{-1} \cdot 7</math> | |||
| {{monzo| -1 -1 0 1 }} | |||
|- | |||
| [[7/5]] | |||
| <math>5^{-1} \cdot 7</math> | |||
| {{monzo| 0 0 -1 1 }} | |||
|} | |||
:'''Practical hint:''' On the wiki, the monzo template helps you getting correct brackets ([[Template:Monzo|read more…]]). | |||
== Relationship with vals == | |||
( | : ''See also: [[Val]], [[Keenan's explanation of vals]], [[Vals and tuning space]] (more mathematical)'' | ||
Monzos are important because they enable us to see how any JI interval "maps" onto a val. This mapping is expressed by writing the val and the monzo together, such as ⟨ 12 19 28 | -4 4 -1 ⟩. The mapping is extremely easily to calculate: simply multiply together each component in the same position on both sides of the line, and add the results together. This is perhaps best demonstrated by example: | |||
<math> | |||
\left\langle \begin{matrix} 12 & 19 & 28 \end{matrix} \mid \begin{matrix} -4 & 4 & -1 \end{matrix} \right\rangle \\ | |||
= 12 \cdot (-4) + 19 \cdot 4 + 28 \cdot (-1) \\ | |||
= 0 | |||
</math> | |||
In this case, the val {{val| 12 19 28 }} is the [[patent val]] for [[12-equal]], and {{monzo| -4 4 -1 }} is 81/80, or the [[syntonic comma]]. The fact that ⟨ 12 19 28 | -4 4 -1 ⟩ tells us that 81/80 is mapped to 0 steps in 12-equal—in other words, it is tempered out—which tells us that 12-equal is a [[meantone]] temperament. It is noteworthy that almost the entirety of Western music composed in the [[Historical temperaments|Renaissance]] and from the sixteenth century onwards, particularly Western music composed for 12-tone circulating temperaments ([[12edo|12 equal]] and unequal [[well temperament]]s), is made possible by the tempering out of 81/80, and that almost all aspects of modern common practice Western music theory (chords and scales) in both classical and non-classical music genres are based exclusively on meantone. | |||
In general: | |||
& | <math> | ||
\left\langle \begin{matrix} a_1 & a_2 & \ldots & a_n \end{matrix} \mid \begin{matrix} b_1 & b_2 & \ldots & b_n \end{matrix} \right\rangle \\ | |||
= 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: | |||
& | ⟨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 === | |||
{{Main|Subgroup monzos and vals}} | |||
A subgroup monzo is like a standard monzo, except that it is in a just intonation [[subgroup]] that is not a prime limit. It is usually provided along with the subgroup it is in, such as 2.3.7 [1 -2 1⟩ for 14/9 or 2.3.13/5 [1 -1 1⟩ for 26/15. For subgroups with primes as basis elements (such as 2.3.7), a subgroup monzo can be seen as a standard monzo abbreviated to hide zeroes. To make this more clear, the abbreviated sections may be replaced with ellipses, such as 2.3.7 [1 -2 ... 1⟩ for 14/9. | |||
=== Tempered monzos === | |||
{{Main|Tempered monzos and vals}} | |||
A tempered monzo represents a tempered interval, or an interval in a regular temperament. It functions similarly to a subgroup monzo, except each entry represents not a JI interval but a generator for that regular temperament. For example, the tmonzo for the major third in meantone (or more generally, the diatonic major third in a fifth-generated temperament) is ~2.~3 [-6 4⟩. (Note that we write the generators with tildes to indicate that they are tempered intervals). This means that to find the major third, you go up four perfect twelfths (~3/1) and down six octaves (~2/1). | |||
More generally, tempered monzos are applicable to any regular tuning, regardless of JI mapping, so corresponding intervals in two different regular temperaments that are tuned the same way have the same tempered monzo. | |||
=== Fractional monzos === | |||
{{Main|Fractional monzos}} | |||
Any of the previous categories of monzo can also be a "fractional monzo", allowing entries to be fractions or non-integer rational numbers as opposed to just integers. This allows monzos to express equal divisions of just intervals (or stacks thereof). For example, [-1/2 1/2⟩ is a monzo representing a neutral third equal to half of a perfect fifth, and [1/12⟩ is a monzo representing a 12edo semitone. [1/12 1/13⟩ is a monzo representing 1\12edo stacked with 1\13edt. (Numerically, this is the 156th root of 2<sup>13</sup>*3<sup>12</sup>.) Note that we write the fractional monzo entries with forward slashes (as they represent fractions), despite writing edosteps with backslashes. | |||
== See also == | |||
* [[Extended bra-ket notation]] | |||
== External links == | |||
* [http://tonalsoft.com/enc/m/monzo.aspx Tonalsoft | ''Monzo''] | |||
== Notes == | |||
<references /> | |||
[[Category:Regular temperament theory]] | |||
[[Category:Just intonation]] | |||
[[Category:Terms]] | |||
[[Category:Notation]] | |||
[[Category:Math]] | |||
[[Category:Monzo]] | |||