[Joe Monzo]

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.

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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 = log _{10}r / log_{10}[ (4/3)^{(1/30)} ]`. (Thanks to Paul Erlich for help in simplifying that formula.)

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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 = log _{10}r * [ 72 / log_{10}(2) ]` or

The difference in size between the two different types of moria is exactly 2 temperament-units. Proof:

2,3-monzo2 32^{(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.]

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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.

a&b temperament [a&b are numbers]

55-edo (comma) (Mozart's tuning)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

a&b temperament [a&b are numbers]

apotome (Greek interval)

aristoxenean (temperament family)

atomic (temperament family)

augmented / diesic (temperament family)

augmented-2nd / aug-2 / #2 (interval)

augmented-4th / aug-4 / #4 (interval)

augmented-5th / aug-5 / #5 (interval)

augmented-6th / aug-6 / #6 (interval)

augmented-9th / aug-9 / #9 (interval)

blackjack (tuning)

cent / ¢ (unit of interval measurement)

centitone / iring (unit of interval measurement)

chromatic-semitone / augmented-prime (interval)

daseian (musical notation)

dekamu / 10mu (MIDI-unit)

diapason (Greek interval)

diapente (Greek interval)

diatessaron (Greek interval)

diatonic semitone (minor-2nd) (interval)

diesic (temperament family)

diezeugmenon (Greek tetrachord)

diminished-5th / dim5 / -5 / b5 (interval)

diminished-7th / dim7 / o7 (interval)

doamu / 2mu (MIDI-unit)

dodekamu / 12mu (MIDI-unit)

dominant-7th (dom-7, x7) (chord)

dorian (mode)

eleventh / 11th (interval)

enamu / 1mu (MIDI-unit)

endekamu / 11mu (MIDI-unit)

enharmonic semitone (interval)

ennealimmal (temperament family)

enneamu / 9mu (MIDI-unit)

farab (unit of interval measurement)

fifth / 5th (interval)

flu (unit of interval measurement)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

fourth / 4th (interval)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

grad (unit of interval measurement)

hexamu / 6mu (MIDI-unit)

Hurrian Hymn (Monzo reconstruction)

hypate (Greek note)

hypaton (Greek tetrachord)

hyperbolaion / hyperboleon (Greek tetrachord)

hypophrygian (Greek mode)

imperfect (interval quality)

iring / centitone (unit of interval measurement)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

jot (unit of interval measurement)

JustMusic: A New Harmony [Monzo's book]

JustMusic prime-factor notation [Monzo essay]

kwazy (temperament family)

leimma / limma (Greek interval)

lichanos (Greek note)

limma / leimma (Greek interval)

locrian (mode)

lydian (mode)

magic (temperament family)

Mahler 7th/1 [Monzo score and analysis]

marvel (temperament family)

meantone (temperament family)

mem (unit of interval measurement)

meride (unit of interval measurement)

mese (Greek note)

meson (Greek tetrachord)

millioctave / m8ve (unit of interval measurement)

mina (unit of interval measurement)

minerva (temperament family)

miracle (temperament family)

mixolydian (mode)

monzo (prime-exponent vector)

Monzo, Joe (music-theorist)

morion / moria (unit of interval measurement)

mutt (temperament family)

mystery (temperament family)

octamu / oktamu / 8mu (MIDI-unit)

octave (interval)

oktamu / octamu / 8mu (MIDI-unit)

orwell (temperament family)

p4, perfect 4th, perfect fourth (interval)

p5, perfect 5th, perfect fifth (interval)

pantonality of Schoenberg [Monzo essay]

paramese (Greek note)

paranete (Greek note)

parhypate (Greek note)

pentamu / 5mu (MIDI-unit)

prime-factor notation (JustMusic) [Monzo essay]

proslambanomenos (Greek note)

savart (unit of interval measurement)

schismic / skhismic (temperament family)

Schoenberg's pantonality [Monzo essay]

second / 2nd (interval)

semisixths (temperament family)

semitone (unit of interval measurement)

seventh / 7th (interval)

sixth / 6th (interval)

sk (unit of interval measurement)

skhismic / schismic (temperament family)

sruti tuning [Monzo essay]

studloco (tuning)

subminor 3rd (interval)

Sumerian tuning [speculations by Monzo]

synemmenon (Greek tetrachord)

temperament-unit / tu (unit of interval measurement)

tenth / 10th (interval)

tetrachord-theory tutorial [by Monzo]

tetradekamu / 14mu (MIDI-unit)

tetramu / 4mu (MIDI-unit)

third / 3rd (interval)

thirteenth / 13th (interval)

tina (unit of interval measurement)

tone (interval, and other definitions)

tredek (unit of interval measurement)

triamu / 3mu (MIDI-unit)

tridekamu / 13mu (MIDI-unit)

trihemitone (Greek interval)

trite (Greek note)

tu / temperament-unit (unit of interval measurement)

Türk sent (unit of interval measurement)

twelfth / 12th (interval)

whole-tone (interval)

woolhouse-unit (unit of interval measurement)