[Joe Monzo]

A set of numbers notated in the form [a b, c d e, ... z > , which is the list of exponents of the prime-factors up to the prime-limit z, and which uniquely identifies an interval as a vector in prime-space. It is a specific cardinality referring to the 1-dimensional case of the more general term multimonzo.

A comma punctuation-mark is used after the exponent of 3, and then subsequently after every third exponent (i.e., after the exponents of prime-factors 11, 19, 31, 43, etc.). This makes it easy to identify whether prime-factor 2 is being recognized or ignored, and aids in recognition of higher primes.

A wildcard character * may also be used for any prime or combination of primes which are considered to form an equivalence-interval. Thus, for example, the monzo for the syntonic-comma (ratio 81:80) in the usual case where prime-factor 2 is ignored, is [ * 4, -1 > . (see Robert Walker, Yahoo tuning group, message 55056 ... which contains some errors.)

The standard monzo is defined as admitting only integer exponents; a *rational monzo* allows rational exponents as well (... many examples of the latter may be found in this Encyclopaedia's entries describing temperaments such as 1/4-comma meantone).

A monzo may be thought of as a set of "driving directions" thru prime-space along the taxicab metric of a lattice diagram. Geometrically, a monzo may usually be thought of as a Euclidean directed line-segment in tone-space (prime-space) unless it vanishes, in which case it becomes a point.

The term was coined in July 2003 by Gene Ward Smith in honor of Joe Monzo's advocacy of prime-factor vectors as part of the musical notation, in Monzo 1997, JustMusic Prime-Factor Notation.

A special case of the monzo is when it is tempered-out (or "vanishes"), in which case it is represented by, or is, a promo.

. . . . . . . . .

[Gene Ward Smith, xenharmony.org]

For `p` an odd prime, the intervals of the `p`-limit `Np` may be taken as the set of all frequency ratios which are positive rational numbers whose factorization involves only primes less than or equal to `p`.

If `q` is such a ratio, it may be written in factored form as

`q` = 2^{e2} 3^{e3} ... `p`^{ep}

where `e2`, `e3`, ... `ep` are integer exponents.

We may write this in factored form as a row vector of the exponents, or monzo:

[`e2`, `e3`, ..., `ep`]

The `p`-limit rational numbers `Np` form an abelian group, or Z-module, under multiplication, so that it acts on itself as a transformation group of a musical space; this becomes an additive group using vector addition when written additively as a monzo.

. . . . . . . . .

[Gene Ward Smith, Yahoo tuning-math group, message 7527 (Tue Nov 11, 2003 5:42 am); thanks to Dave Keenan (private correspondence) for adding the Wolfram "ExponentVector" link]

Physicists sometimes use bra and ket vectors; if you did that, the monzo for 81/80 would be a ket, [-4 4 -1> and the val for 5-limit 12-et would be a bra, <12 19 28]. Putting them together would give the bra-ket, angle bracket, or inner product: <12 19 28 | -4 4 -1> = 0. See:

- http://mathworld.wolfram.com/ExponentVector.html
- http://mathworld.wolfram.com/ContravariantVector.html
- http://mathworld.wolfram.com/CovariantVector.html
- http://mathworld.wolfram.com/Ket.html
- http://mathworld.wolfram.com/Bra.html
- http://mathworld.wolfram.com/One-Form.html
- http://mathworld.wolfram.com/AngleBracket.html

. . . . . . . . .

[Joe Monzo]

In Yahoo tuning-math, message 7435 (Thu Nov 6, 2003 3:48 pm), George Secor proposed using comma punctuation marks after the exponent of prime-factor 3 and then again after every third prime-factor. I accept both this idea and the ket notation proposed by Gene Ward Smith described above. Thus, labeling each exponent with its prime-factor, the standard i propose for the notation of monzos is:

[2 3, 5 7 11, 13 17 19, 23 29 31, 37 41 43, ... etc.>

If there are large gaps in the set of prime-factors, those with a zero-exponent may be omitted, and a comma used to delimit every member of the monzo; the prime-factors must be explicitly stated. (An example of this may be seen the Tuning Encyclopaedia description of the epimoric approximation of the Pythagorean comma.)

Monzos for many of the ratios that might be encountered in 11-limit just intonation, may be found in the "big list of 11-limit intervals" webpage.

. . . . . . . . .

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)