morion (pl.: moria)

1.

A term used by Cleonides in discussing Aristoxenos's work, 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 12edo, in which case the moria described by Cleonides would in fact refer to the 72edo-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 is:

moria = log₁₀(ratio) / log₁₀[ (4/3)^(1/30) ]

(Thanks to Paul Erlich for helping me simplify that formula.)

2.

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 12edo "perfect 4th" of 500 cents.

The formula for calculating this moria-value of any ratio is:

moria = log₁₀(ratio) * [ 72 / log₁₀(2) ]


The difference in size between the two different types of moria is exactly 2 tuning units.

Proof:

                         prime-factor vector
                               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 my article JustMusic Prime-Factor Notation.)


REFERENCES

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.]


See also:

Manuel Op de Coul's Logarithmic Interval Measures,
my paper on Aristoxenos.


[from Joe Monzo, JustMusic: A New Harmony]

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