[Joe Monzo]

the recognition of two pitches or intervals, which are separated by a small pitch distance, as the same musical gestalt. This occurs mainly as a result of musical context.

As a trivial example, in the 12-edo scale, since the interval between all degrees is exactly 1.00 Semitone [= 100 cents], any interval plus or minus 0.50 Semitone [= 50 cents] from one of the 12-edo degrees, will be perceived as the closest 12-edo degree. This example is for explanatory purposes only - in most actual music the situation is considerably more complex.

. . . . . . . . .

(added 2001.1.2, adapted from YahooGroup Tuning posts:)

... What I meant was that I've seen Beethoven scores which have notated pitches with flats which are then tied to their enharmonically equivalent sharp, or vice versa, and that Paul [Erlich] has pointed out to me that Mozart wrote a few things like this also.

But Paul's point is that Mozart (and Beethoven, and Wagner) must have had in mind when they wrote these, some form of temperament in which what he calls a "commatic" unison-vector disappears (now referred to as "vapro"), and I agree.

I put "commatic" in quotes because in this 5-limit case the only actual comma which might be involved is the pythagorean comma. The diaschisma is another possibility and it is also comma-sized. But other unison-vectors which might vanish have very different sizes, both larger and smaller than a comma; these could be the skhisma, diesis, or possibly a few others.

Here's a section of the 5-limit rectangular lattice in which I've notated only the D#'s and Eb's. Observe that notes with the same notation are a syntonic comma apart. We're already assuming that *that* vanishes, because the notation of the composers in the standard late-romantic repertoire never distinguishes it.

4 D# . . . . . . . . . . . . . . . . 3 . . . . D# . . . . . . . . . . . . 2 . . . . . . . . D# . . . . . . . . 1 Eb . . . . . . . . . . . D# . . . . 5^y 0 . . . . Eb . . n^0 . . . . . . . . D# -1 . . . . . . . . Eb . . . . . . . . -2 . . . . . . . . . . . . Eb . . . . -3 . . . . . . . . . . . . . . . . Eb -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 3^x

In this example, we're considering unison-vectors which connect any of the D#'s with any of the Eb's.

I decided to add numbers to each of the notes on my lattice, so that we can formulate an "algebra of enharmonicity" for the 5-limit:

4 D#1. . . . . . . . . . . . . . . . 3 . . . . D#2. . . . . . . . . . . . 2 . . . . . . . . D#3. . . . . . . . 1 Eb1 . . . . . . . . . . D#4. . . . 5^y 0 . . . . Eb2. . n^0 . . . . . . . . D#5 -1 . . . . . . . . Eb3. . . . . . . . -2 . . . . . . . . . . . . Eb4. . . . -3 . . . . . . . . . . . . . . . . Eb5 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 3^x

All the D#-Eb pairs on this lattice are: (note the last two columns)

x,y higher lower ratio coordinates ~cents note subtract note Pyth.comma 531441:528244 (12, 0) 23.46 D#(x+3) - Eb(x) skhisma 32805:32768 ( 8, 1) 1.95 D#(x+2) - Eb(x) diaschisma 2048:2025 (-4,-2) 19.55 Eb(x) - D#(x+1) diesis 128:125 ( 0,-3) 41.06 Eb(x) - D#(x) 648:625 ( 4,-4) 62.57 Eb(x+1) - D#(x) 6561:6250 ( 8,-5) 84.07 Eb(x+2) - D#(x) 531441:500000 (12,-6) 105.6 Eb(x+3) - D#(x) (16,-7) 127.1 Eb(x+4) - D#(x)

(Observe that D# is the higher note of the pair for the skhisma and Pythagorean comma, whereas it is Eb for the rest.)

. . . . . . . . .

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)