Monzo: Difference between revisions

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 group="note">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>.

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 {{monzo| 1 -2 1 }} for 14/9 or 2.3.13/5 {{monzo| 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 {{monzo| 1 -2 … 1 }} for 14/9.  

{{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 tempered monzo for the major third in meantone (or more generally, the diatonic major third in a fifth-generated temperament) is ~2.~3 {{monzo| -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).  

{{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 and stacks thereof. For example, {{monzo| -1/2 1/2 }} is a monzo representing a neutral third equal to half of a perfect fifth, and {{monzo| 1/12 }} is a monzo representing a 12edo semitone. {{monzo| 1/12 1/13 }} is a monzo representing 1\12edo stacked with 1\13edt. (Numerically, this is the 156th root of 2¹³ × 3¹².) Note that we write the fractional monzo entries with forward slashes, as they represent fractions, despite writing edosteps with backslashes.

* [[Extended bra–ket notation]]