[Joe Monzo]

©2001 by Joe Monzo

In this webpage, I present several different lattice diagrams which plot various fraction-of-a-comma meantone-like systems on a JI lattice with prime-factor axes of 3 and 5.

The meantone chain forms its own linear axis on the lattice, and makes it easy to visualize which JI ratios are acoustically the most closely implied by the meantone pitches. The meantone pitch-classes are called by their generator number. These lattices are "8ve"-invariant, that is, prime-factor 2 is ignored.

*
CAVEAT: Compositional practice may insist that a meantone pitch
implies a different JI ratio than the one which is acoustically
the closest.
*

These lattices are given with the reference pitch of C =
*n*^{0}.

In every case, the list of implied JI ratios includes both of the two nearest, except for instances where the meantone tuning gives one exact ratio (as, for example, the +3, +6, +9 generators A, F#, D# of 1/3-comma; the +7 generator C# of 2/7-comma; the +4, +8, +12 generators E, G#, B# of 1/4-comma; the +5, +10 generators B, A# of 1/5-comma; etc.).

The lattice diagrams, however, only give two ratios in cases where the meantone is exactly at the midpoint between them (as with the +2 generator "D" in 1/4-comma meantone, and with the +3 generator "A" in 1/6-comma meantone); otherwise only the nearest JI implied ratio is given.

The meantone chains could be extended beyond my diagrams; I chose a 27-note chain of +/- 13 generators (Gbb to Fx) as an arbitrary limit in every example.

In the 1/3-comma quasi-meantone, the generator is 1/3-comma
narrower than 3:2. Thus, the +1 generator is 1/3-comma flatter than
G 3:2 (= 3^{1}) and 2/3-comma sharper than
G 40:27 (= 3^{-3}5^{1}).

The next note in the chain (the +2 generator)
is 2/3-comma flatter than D 9:8 (= 3^{2}).
Since 2/3-comma is more than 1/2-comma,
it is evident that the +2 generator
will be closer to the note a comma flatter than 9:8 (= 10:9 ratio
= 3^{-2}5^{1}), namely, 1/3-comma sharper than D 10:9.

The +3 generator is exactly the JI ratio A 5:3 (= 3^{-1}5^{1}).

In the 2/7-comma quasi-meantone, the generator is 2/7-comma
narrower than 3:2. Thus, the +1 generator is 2/7-comma flatter than
G 3:2 (= 3^{1}) and 5/7-comma sharper than
G 40:27 (= 3^{-3}5^{1}).

The next note in the chain (the +2 generator)
is 4/7-comma flatter than 9:8 (= 3^{2}).
Since 4/7-comma is more than 1/2-comma,
it is evident that the +2 generator
will be closer to the note a comma flatter than 9:8 (= 10:9 ratio
= (= 3^{-2}5^{1}), namely, 3/7-comma sharper.

The +3 generator is 6/7-comma flatter than A 27:16 (= 3^{3})
and only 1/7-comma sharper than A 5:3 (= 3^{-1}5^{1}).

The +4 generator is 1/7-comma flatter than E 5:4 (= 5^{1})and
6/7-comma sharper than E 100:81 (= 3^{-4}5^{2}).

The +5 generator is 3/7-comma flatter than B 15:8 (= 3^{1}5^{1}) and
4/7-comma sharper than B 50:27 (= 3^{-3}5^{2}).

The +6 generator is 5/7-comma flatter than F# 45:32 (= 3^{2}5^{1})
and 2/7-comma sharper than F# 25:18 (= 3^{-2}5^{2}).

The +7 generator is exactly the JI ratio C# 25:24 (= 3^{-1}5^{2}).

In the 1/4-comma meantone, the generator is 1/4-comma
narrower than 3:2. Thus, the +1 generator is 1/4-comma flatter than
G 3:2 (= 3^{1}) and 3/4-comma sharper than
G 40:27 (= 3^{-3}5^{1}).

The next note in the chain (the +2 generator)
is 1/2-comma flatter than D 9:8 (= 3^{2})
and 1/2-comma sharper than D 10:9 (= 3^{-2}5^{1}).
Thus, this is exactly the "mean tone" between the two JI "whole-tones".

The +3 generator is 3/4-comma flatter than A 27:16 (= 3^{3})
and 1/4-comma sharper than A 5:3 (= 3^{-1}5^{1}) .

The +4 generator is exactly the JI ratio E 5:4 (= 5^{1}).

In the 1/5-comma quasi-meantone, the generator is 1/5-comma
narrower than 3:2. Thus, the +1 generator is 1/5-comma flatter than
G 3:2 (= 3^{1}) and 4/5-comma sharper than
G 40:27 (= 3^{-3}5^{1}).

The next note in the chain (the +2 generator)
is 2/5-comma flatter than 9:8 (= 3^{2}).
and 3/5-comma sharper than 10:9 (= 3^{-2}5^{1}).

The +3 generator is 3/5-comma flatter than A 27:16 (= 3^{3})
and 2/5-comma sharper than A 5:3 (= 3^{-1}5^{1}) .

The +4 generator is 4/5-comma flatter than E 81:64 (= 3^{4})and
1/5-comma sharper than E 5:4 (= 5^{1}).

The +5 generator is exactly the JI ratio B 15:8 (= 3^{1}5^{1}).

In the 1/6-comma quasi-meantone, the generator is 1/6-comma
narrower than 3:2. Thus, the +1 generator is 1/6-comma flatter than
G 3:2 (= 3^{1}) and 5/6-comma sharper than
G 40:27 (= 3^{-3}5^{1}).

The next note in the chain (the +2 generator)
is 1/3-comma flatter than 9:8 (= 3^{2})
and 2/3-comma sharper than 10:9 (= 3^{-2}5^{1}).

The +3 generator is 1/2-comma flatter than A 27:16 (= 3^{3})
and 1/2-comma sharper than A 5:3 (= 3^{-1}5^{1}).
Thus, it is precisely the mean "6th" between the two JI ratios.

The +4 generator is 2/3-comma flatter than E 81:64 (= 3^{4})and
1/3-comma sharper than E 5:4 (= 5^{1}).

The +5 generator is 5/6-comma flatter than B 243:128 (= 3^{5})and
1/6-comma sharper than B 15:8 (= 3^{1}5^{1}).

The +6 generator is exactly the JI ratio F# 45:32 (= 3^{2}5^{1}).

And here is a more accurate lattice of the above, showing a closed 55-tone 1/6-comma meantone chain and its implied pitches, all enclosed within a complete periodicity-block defined by the two unison-vectors 81:80 = [-4 4 -1] (the syntonic comma, the shorter boundary extending from south-west to north-east on this diagram) and [-51 19 9] (the long nearly vertical boundary), portrayed here as the white area.

For the bounding corners of the periodicity-block, I arbitrarily chose
the lattice coordinates [-7.5 -5] for the north-west corner,
[-11.5 -4] for north-east, [11.5 4] for south-west, and [7.5 5]
for south-east. This produces a 55-tone system centered on
**n**^{0}.

The grey area represents the part of the JI lattice outside the defined periodicity-block (and thus, with each of those pitch-classes in its own periodicity-block), and the lattice should be imagined as extending infinitely in all four directions. The other periodicity-blocks, all identical to this one, can be tiled against it to cover the entire space.

In the 1/11-comma quasi-meantone, which is audibly indistinguishable
from the usual 12-EDO tuning,
the generator is 1/11-comma
narrower than 3:2. Thus, the +1 generator is 1/11-comma flatter than
G 3:2 (= 3^{1}) and 10/11-comma sharper than
G 40:27 (= 3^{-3}5^{1}).

The +2 generator is 2/11-comma flatter than 9:8 (= 3^{2})
and 9/11-comma sharper than 10:9 (= 3^{-2}5^{1}).

The +3 generator is 3/11-comma flatter than A 27:16 (= 3^{3})
and only 8/11-comma sharper than A 5:3 (= 3^{-1}5^{1}).

The +4 generator is 4/11-comma flatter than E 81:64 (= 3^{4})and
7/11-comma sharper than E 5:4 (= 5^{1}).

The +5 generator is 5/11-comma flatter than B 243:128 (= 3^{5}) and
6/11-comma sharper than B 15:8 (= 3^{1}5^{1}).

The +6 generator is 6/11-comma flatter than F# 729:512 (= 3^{6})
and 5/11-comma sharper than F# 45:32 (= 3^{2}5^{1}).

The +7 generator is 7/11-comma flatter than C# 2187:2048 (= 3^{7})
and 4/11-comma sharper than C# 135:128 (= 3^{3}5^{1}).

The +8 generator is 8/11-comma flatter than G# 6561:4096 (= 3^{8})
and 3/11-comma sharper than G# 405:256 (= 3^{4}5^{1}).

The +9 generator is 9/11-comma flatter than D# 19683:16384 (= 3^{9})
and 2/11-comma sharper than D# 1215:1024 (= 3^{5}5^{1}).

The +10 generator is 10/11-comma flatter than A# 59049:32768 (= 3^{10})
and 1/11-comma sharper than A# 3645:2048 (= 3^{6}5^{1}).

The +11 generator is exactly the JI ratio E# 10935:8192 (= 3^{7}5^{1}).

Note that because the JI lattice implied by 1/11-comma meantone includes the skhisma, the chain effectively closes at 12 tones; a typical version could be -3 Eb to +8 G#.

See also: Ellis's Duodene and a "best-fit" meantone.

updated:

2000.12.09

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)

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)