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[Joe Monzo]

In musical tuning theory, a "**va**nishing **pro**mo" (or vanishing unison-vector), which represents an entire infinite set of vanishing monzos, which are all integer multiples of the promo, and which may be characterized geometrically as a line in prime-space. The prime-space must include prime-factor 2 in order for the vapro (or promo) to appear as a line; in the fairly common usage of lattices which do not show 2 as a dimension, all the vapros/promos would appear as a single point. The vapro's GCD, like that of all promos, is 1.

If the interval which may be used to represent the vapro is *not* tempered out, then it is simply a unison-vector and not a representation of a vapro: the essential fact about a vapro is that it physically disappears.

The pair of unison-vectors closest to the origin-point (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 and vapro.

For example: The most common and familiar vapro is [ -4 4, -1 >, which is the monzo of the syntonic-comma of ratio 81/80. This vapro represents all of the following unison-vectors: 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 (which includes 2 as a dimension), 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 the monzo [ -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 vapro. All temperaments in which any of these intervals vanishes belong to the meantone family; thus, since the [ -4 4, -1 > vapro represents all of them, it is part of the definition of all meantone tunings. |

For temperaments which temper out two or more vapros, a prefix is added before "vapro" to denote the dimensionality of the set of vapros. Thus, a bivapro represents a 2-dimensional lattice (plane) of vanishing unison-vectors; a trivapro represents a 3-dimensional lattice (cube) of vanishing unison-vectors; 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)

méride (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)