Monzo: Difference between revisions
m remove stray asterisk |
No edit summary |
||
| Line 89: | Line 89: | ||
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}}. | 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}}. | ||
--> | --> | ||
== Subgroup monzos == | |||
{{Main|Smonzos and svals}} | |||
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|Tmonzos and tvals}} | |||
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 == | == See also == | ||
* [[Extended bra-ket notation]] | * [[Extended bra-ket notation]] | ||
== External links == | == External links == | ||