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Encyclopedia of Microtonal Music Theory

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[Joe Monzo, with Paul Erlich]

A term used by Fokker to describe a musical interval which is small enough that it can be disregarded (possibly due to tempering), thereby setting up an equivalence relationship in the tonespace.

The unison-vector most frequently invoked by Fokker was the variety of the septimal kleisma with ratio 225/224, allowing 7/4 and 225/128 to be perceived as being interchangeable with each other. This particular unison-vector is also an example of a xenharmonic-bridge.

In Fokker's lattice theory, the unison vectors divide the lattice into regions called periodicity blocks.

In the case of temperaments, where a certain unison-vector and all of its multiples are tempered-out ("vanish"), the set of monzos that vanish are vapros.


Tanaka, Shohé. 1890.

"Studien im Gebiete der reinen Stimmung"
Vierteljahrsschrift für Musikwissenschaft vol. 6 no. 1,
Friedrich Chrysander, Philipp Spitta, Guido Adler (eds.),
Breitkopf und Härtel, Leipzig, pp. 1-90.

English translation of pages 8 to 18 by Daniel J. Wolf,
"Studies in the Realm of Just Intonation",
Xenharmonikôn vol. 16, autumn 1995, pp. 118-125.

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[Joe Monzo, Yahoo tuning group, message 55499 (Sat Aug 14, 2004 3:54 am)]

A unison-vector in the "pure" sense is an interval which is not an actual unison. It is an interval with a real pitch difference by however small or large an amount. at least as often as not, it is a rational interval.

The essential concepts embodied in the term "unison-vector", are these:

Now, venturing into my own terminology ...

There is also a specific type of unison-vector called a "xenharmonic-bridge", which links a note within the periodicity-block of a certain prime-space, to one which lies not only in another "tiling" of the periodicity-block, but also in a different prime-space.

I coined this term, but Fokker often invoked the septimal-kleisma in precisely the way i describe here -- to make singers become familiar with the 5-limit just intonation "augmented-6th", then have them use it as a "target pitch" when he notates a harmonic-7th in his compositions. This was at least as early as his 1949 English book on singing JI.

The unison-vector has, within the past couple of years, also been referred to as a "comma", a usage which i personally do not encourage, because "comma" has already had a millennia-long established meaning, as an interval of about 12 to 35 cents.

A unison-vector certainly may be a "real" comma ... for example, the syntonic-comma (ratio 81:80 = 2,3,5-monzo [-4 4, -1> = ~ 21.5 cents) definitely does exist in theory in 5-limit just intonation.

In theory and in practice, a unison-vector may be tempered-out ("vanish"), ignored, or deliberately used as a valid scale degree (or "step"), either to play around with a listener's expectations, or to secure exact just intonation with commatic drift.

A unison-vector's multiples, both positive and negative, are all equivalent to it. thus they may be modeled geometrically as a line cutting across the lattice.

The monzo describing the one, of the pair of unison-vectors closest to the origin-point of the lattice, which has a positive pitch-height (the other will always be negative), is chosen to represent the whole line, and is a "promo" (projective monzo).

If a promo vanishes in a temperament, i propose to call that a "vapro".

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