A small unit of interval measurement, about the same size as, or a bit smaller than, the interval size-range denoted by the strict definition of the word "comma". There are historically two types of moria.
A term used by Cleonides in discussing the work of the important ancient Greek theorist Aristoxenos, to designate the small interval describing 1/30th part of the "perfect 4th".
It must be kept in mind that Aristoxenus himself never gave an exact measurement for the "perfect 4th", calling it simply a "concord". His method of "tuning by concords" results in what appears to be 12-edo, in which case the moria described by Cleonides would in fact refer to the 72-edo-morion described below. Cleonides refers to the "4th" simply as the "diatessaron", the usual Greek term for the interval; thus no exact measure can be applied.
Let us assume for the purpose of this definition that the "perfect 4th" is the ratio 4:3. This type of morion is calculated as the 30th root of 4:3, or (4/3)(1/30), thus having a ratio itself of approximately 1:1.009635528. It is an irrational number. The width of this morion interval is ~16.60149997 (pretty close to 16 & 3/5) cents.
This interval therefore divides the "octave", which is assumed to have the ratio 2:1, into ~72.28262519 equal parts. Thus this type of morion represents one degree in 72.28262519-edo "non-octave" tuning.
There are just over 6 of these moria (a more exact figure is ~6.023552099, about 6 & 1/42) in a Semitone.
The formula for calculating this moria-value of any ratio r is: moria = log10r / log10[ (4/3)(1/30) ]. (Thanks to Paul Erlich for help in simplifying that formula.)
Because it is so close to the size of 1 degree of 72-edo, the term "morion" is also used to designate that interval.
This type of morion is calculated as the 72nd root of the "octave" ratio 2:1, or 2(1/72), thus with a ratio itself of approximately 1:1.009673533. It is an irrational number, and the width of this morion interval is exactly 16 & 2/3 cents.
This interval therefore divides the "octave", which is assumed to have the ratio 2:1, into exactly 72 equal parts. Thus this type of morion represents one degree in 72-edo tuning.
There are thus exactly 6 of these moria in a Semitone, and (as in Cleonides's description) 30 of them in a 12-edo "perfect 4th" of 500 cents.
The formula for calculating this moria-value of any ratio r is: moria = log10r * [ 72 / log10(2) ] or moria = log2r * 72.
The difference in size between the two different types of moria is exactly 2 temperament-units. Proof:
2,3-monzo
2 3
2(1/72) [ 1/72 0 ]
÷ (4/3)(1/30) - [ 2/30 -1/30]
-------------- = -----------------
[-19/360 1/30] = [2(-19/720) * 3(12/720)]2
(For an explanation of the vector subtraction used in the middle column of this formula, see Monzo, JustMusic Prime-Factor Notation.)
Cleonides. c 100 AD. Eisagoge.
[English translation in Strunk 1950.]
Strunk, Oliver. 1950. Source Readings in Music History.
Selected and annotated [and translated].
W. W. Norton. New York.
[English translation of Cleonides on p 34-46.]
The tonalsoft.com website is almost entirely the work of one person: me, Joe Monzo. Please reward me for my knowledge and effort by choosing your preferred level of financial support. Thank you.