[Joe Monzo]

A "**pro**jective **mo**nzo" (ratio), which represents an entire infinite set of monzos, which are all integer multiples of the monzo used to define the promo, and which may be characterized geometrically as a line in prime-space. The promo's GCD = 1.

If the interval which is used to represent the promo is *not* tempered out, then it is simply a unison-vector and not a promo: the essential fact about a promo is that it physically vanishes.

The pair of monzos closest to the origin-point on the lattice (monzo [ 0 0, 0 > and ratio 1:1) lie at equal distances from it, on either side of it, along the line. It is always the case that one of this pair of intervals will have a positive pitch-height and the other a negative pitch-height (measured, for example, in cents). By convention, tuning theorists generally use the positive one as the promo. For example:

The most typical promo is probably [ -4 4, -1 >, which is the monzo of the syntonic-comma of ratio 81/80, which vanishes in all temperaments belonging to the meantone family. This "vanishing promo" represents all of the following "vanishing monzos": 2,3,5-monzo ratio ~cents etc. [ -4 4, -1 > * 3 = [-12 12, -3 > 531441/512000 64.51886879 [ -4 4, -1 > * 2 = [ -8 8, -2 > 6561/6400 43.01257919 [ -4 4, -1 > * 1 = [ -4 4, -1 > 81/80 21.5062896 [ -4 4, -1 > * 0 = [ 0 0, 0 > 1/1 0 [ -4 4, -1 > * -1 = [ 4 -4, 1 > 80/81 -21.5062896 [ -4 4, -1 > * -2 = [ 8 -8, 2 > 6400/6561 -43.01257919 [ -4 4, -1 > * -3 = [ 12 -12, -3 > 512000/531441 -64.51886879 etc. On a prime-space lattice, all of these coordinates would be points which lie on a single line, both ends of which extend into infinity. The two unison-vectors which lie closest to the origin are those with coefficients of +1 and -1.
The coeffecient of 1 produces [ -4 4, -1 > = 81/80 = ~ 21.5062896 cents. The latter interval's pitch-height is negative, so [ -4 4, -1 > = 81/80, whose pitch-height is positive, is used to designate the promo. All temperaments in which any of these intervals vanishes belong to the meantone family; thus, since the [ -4 4, -1 > promo represents all of them, it is part of the definition of all meantone tunings, and of the entire meantone family. (for further discussion of this particular promo, see "tempering out the syntonic-comma".) |

For sets of more than one promo (for example, temperaments which temper out two or more promos), a prefix is added before "promo" to denote the dimensionality of the set.

Thus, a bipromo represents a 2-dimensional lattice of monzos; a tripromo represents a 3-dimensional lattice of monzos; etc.

. . . . . . . . .

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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)