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7-limit JI offers a bewildering variety of chords with an incredible range of consonance and dissonance. Color notation gives us clear, concise names for them.
A triad is named after the color of its 3rd. The 5th is assumed to be white. There are four main triads. They're shown here in close position with examples of both written names and spoken names. (The roots are mostly white in these examples; the next chapter discusses root colors.)
Table 6.1 – Triads
chord name
chord type
chord structure
examples
blue chord
minor chord
1, b3, 5
1/1 – 7/6 – 3/2
Fb
“F blue”
wF, bA, wC
green chord
minor chord
1, g3, 5
1/1 – 6/5 – 3/2
Cg
“C green”
wC, gE, wG
yellow chord
major chord
1, y3, 5
1/1 – 5/4 – 3/2
Gy
“G yellow”
wG, yB, wD
red chord
major chord
1, r3, 5
1/1 – 9/7 – 3/2
B♭r
“B-flat red”
wB, rD, wF


lattice62.png

Yellow A is a note, whereas A yellow is a chord. Chords can be referred to by structure as, say, y chords or b chords. The chord type (major, minor, etc.) is analogous to interval quality, in that it's redundant (if it's yellow, it must be major), it's not unique (there are other major triads available), and its main purpose is to indicate keyspan (both yellow and red triads will in close position have two intervals of 4 and 3 semitones each).
Like ratios, a chord's dissonance mostly comes from its "bigness" and "utonalness". The bigness is the odd limit; the previous four triads all have an odd limit of 9 or less. Yellow and blue are otonal, green and red are utonal.
Table 6.2 – More triads, mostly dissonant
chord name
chord type
chord structure
examples
white chord
minor chord
1, w3, 5
1/1 – 32/27 – 3/2
Gw
wG, wB, wD
large white chord
major chord
1, Lw3, 5
1/1 – 81/64 – 3/2
BLw
wB, wD, wF
white yellow-five chord
minor chord
1, w3, y5
1/1 – 32/27 – 40/27
Dw,y5
wD, wF, yA
yellow yellow-five chord
major chord
1, y3, y5
1/1 – 5/4 – 40/27
Gy,y5
wG, yB, yD
four chord
four chord
1, 4, 5
1/1 – 4/3 – 3/2
C4
wC, wF, wG
blue four chord
four chord
1, b4, 5
1/1 – 21/16 – 3/2
Cb4
wC, bF, wG
All but one of these chords have a fairly large odd limit, as seen by the size of the numbers in the 4th column. The first two are white chords, the bane of 3-limit JI. The next two are yellow-5 chords, the bane of 5-limit JI. The b4 chord shows up as a suspension in a V7 – I cadence, if the 7th in V7 is blue: G – yB – D – bF to G – C – bF to G – C – yE.
Augmented and diminished triads are named after the color of the third and the fifth. However if the fifth has a deep color, the third's color is implied. The terms bluish and reddish always refer to the bluish 5th and the reddish 4th, unless otherwise specified.
Table 6.3 – Augmented and diminished triads
chord name
chord type
chord structure
examples
deep yellow chord
augmented
1, y3, yy5
1/1 – 5/4 – 25/16
Ay,yy5
gA, wC, yE
yellow reddish-five chord
augmented
1, y3, ry5
1/1 – 5/4 – 45/28
By,ry5
wB, yD, ryF#
red reddish-five chord
augmented
1, r3, ry5
1/1 – 9/7 – 45/28
Br,ry5
wB, rD, ryF#
deep green chord
diminished
1, g3, gg5
1/1 – 6/5 – 36/25
Eg,gg5
yE, wG, gB
green bluish chord
diminished
1, g3, bg5
1/1 – 6/5 – 7/5
Cg,bg5
wC, gE, bgG
blue bluish chord
diminished
1, b3, bg5
1/1 – 7/6 – 7/5
Cb,bg5
wC, bE, bgG
yellow reddish chord
maj dimin
1, y3, ry4
1/1 – 5/4 – 10/7
Cy,ry4
wC, yE, ryF#
red reddish chord
maj dimin
1, r3, ry4
1/1 – 9/7 – 10/7
Cr,ry4
wC, rE, ryF#
If an aug 4th or any non-white 5th is present, the white 5th is assumed to be absent: ry4 or bg5 or gg5 or y5 or yy5 or ry5 all imply a missing w5. If the white 5th is present too, it's an "add" chord: a chord with both w5 and bg5 is an "add bluish 5" chord.
Augmented chords always have a high odd limit. They, along with full diminished tetrads, have no obvious 7-limit JI intonation. Min-maj chords, which contain an augmented triad, also fall into this category.
Tetrads: We assume a white 5th and name the chord after the 3rd and the 6th/7th. We assume the 6th/7th lies in the “sweet spot” from 5/3 to 9/5 (I call this type of interval a subseventh, a name inspired by pentatonicism; see chapter 19.) We assume it is not white, which allows us to dispense with the terms 6th and 7th. Here are my favorite tetrads:
Table 6.4 – Some low odd limit tetrads
yellow yellow chord
maj6
1, y3, 5, y6
Cyy
wC, yE, wG, yA

green green chord
min7
1, g3, 5, g7
Cgg
wC, gE, wG, gB
(inversion of yellow yellow)
blue blue chord
min7
1, b3, 5, b7
Cbb
wC, bE, wG, bB

red red chord
maj6
1, r3, 5, r6
Crr
wC, rE, wG, rA
(inversion of blue blue)
blue yellow chord
min6
1, b3, 5, y6
Cby
wC, bE, wG, yA

green green bluish
half-dim
1, g3, bg5, g7
Cgg,bg5
wC, gE, bgG, gB
(inversion of blue yellow)
blue blue bluish
half-dim
1, b3, bg5, b7
Cbb,bg5
wC, bE, bgG, bB

green red chord
min6
1, g3, 5, r6
Cgr
wC, gE, wG, rA
(inversion of blue blue bluish)
yellow blue chord
dom7
1, y3, 5, b7
Cyb
wC, yE, wG, bB

red green chord
dom7
1, r3, 5, g7
Crg
wC, rE, wG, gB

yellow-7 chord
maj7
1, y3, 5, y7
Cy,y7
wC, yE, wG, yB




lattice63.png
To my ears, the y,y7 chord, whose odd limit is 15, is considerably more consonant than the gr chord, odd limit 7. The gr chord has smaller numbers, but the green and red make it far more utonal and hence more dissonant.


Note that gg is a green green tetrad and g,gg5 is a deep green triad. Also note that the otonal colors yellow and blue go together, as do the utonal colors, green and red. Imagine the harmonic lattice rotated so that you're looking at the rows end-on; you can see which colors go with which.


Figure 6.1 – Cross section of the harmonic lattice

lattice61.png
Neighboring colors, colors connected by a line, go together. Mixing non-neighboring colors makes more dissonant intervals with large numbers like 25, 35 and 49 in them, as in the first two chords in the next table:

Table 6.5 – Examples of tetrads with a high odd limit
yellow green chord
dom7
1, y3, 5, g7
Cyg
wC, yE, wG, gB
odd limit = 25
green blue chord
min7
1, g3, 5, b7
Cgb
wC, gE, wG, yB
odd limit = 35
yellow white-6 chord
maj6
1, y3, 5, w6
Fy,w6
wF, yA, wC, wD
odd limit = 27
yellow white-7 chord
dom7
1, y3, 5, w7
Cy,w7
wC, yE, wG, wB
odd limit = 45
Even though the last two chords use only neighboring colors, they have high odd limits. Neighboring colors don't guarantee consonant chords.


Full diminished tetrads are mostly non-neighboring, and always have a high odd limit.
Table 6.6 – Examples of full diminished tetrads
green red bluish chord
dim7
1, g3, bg5, r6
Cgr,bg5
wC, gE, bgG, rA
odd limit = 49
blue yellow bluish chord
dim7
1, b3, bg5, y6
Cby,bg5
wC, bE, bgG, yA
odd limit = 25
white yellow green-five chord
dim7
1, w3, g5, y6
Cwy,g5
wC, wE, gG, yA
odd limit = 75
green yellow deep green chord
dim7
1, g3, gg5, y6
Cgy,gg5
wC, gE, ggG, yA
odd limit = 125

Many interesting chords are subsets of tetrads. The simplest ones contain only a fifth:
Table 6.7 – "Five" chords (dyads)
5 chord
five chord
1, 5
C5
wC, wG
(or white dyad, aka a power chord)
bluish-5 chord
dim five chord
1, bg5
Cbg5
wC, bgG
(or bluish dyad)

Table 6.8 – Chords without a 3rd, but with a sub7th
5 blue chord
dom7, no 3rd
1, 5, b7
C5b
wC, wG, bB

5 red chord
maj6, no 3rd
1, 5, r6
C5r
wC, wG, rA

5 yellow chord
maj6, no 3rd
1, 5, y6
C5y
wC, wG, yA

5 green chord
dom7, no 3rd
1, 5, g7
C5g
wC, wG, gB

bluish blue chord
half-dim, no 3rd
1, bg5, b7
Cbg5b
wC, bgG, bB
(inversion of yellow reddish)
(Don't confuse bluish blue with blue bluish, which is 1, b3, bg5.)


Table 6.9 – Some fifth-less chords
blue blue no-5 chord
min7, no 5
1, b3, b7
Cbb -5
wC, bE, bB

green red no-5 chord
min6, no 5
1, g3, r6
Cgr -5
wC, gE, rA
(inversion of blue bluish)
blue yellow no-5 chord
min6, no 5
1, b3, y6
Cby -5
wC, bE, yA
(inversion of green bluish)

In pentads and hexads, the 9th & 11th are assumed to be white. A 9th implies a sub7th, and an 11th implies a 9th. A white 9th goes well with many chords with a major 3rd, and a white 11th goes well with many minor-3rd chords.
Table 6.10 – Chords with 9ths and/or 11ths
yellow chord add 9
add 9
1, y3, 5, w9
Cy,9
wC, yE, wG, wD

yellow blue, 9 chord
9 chord
1, y3, 5, b7, w9
Cyb,9
wC, yE, wG, bB, wD

yellow-7, 9 chord
maj7 + 9
1, y3, 5, y7, w9
Cy,y7,9
wC, yE, wG, yB, wD

blue blue add 11
min7 add 11
1, b3, 5, b7, w11
Cbb,11
wC, bE, wG, bB, wF

5 blue, blue 11
11 chord, no 3
1, 5, b7, w9, b11
C5b,9,b11
wC, wG, bB, wD, bF
(inversion of
blue blue add 11)
red green 9 chord
9 chord
1, r3, 5, g7, w9
Crg,9
wC, rE, wG, gB, wD

green red add 11
min6 add 11
1, g3, 5, r6, w11
Cgr,11
wC, gE, wG, rA, wF
(inversion of red green 9)
red red 9 chord
maj6 + 9
1, r3, 5, r6, w9
Crr,9
wC, rE, wG, rA, wD


Chords can be classified by the number of colors they contain (including white) as a rough measure of their complexity. For example, the triads in table 6.1 are bicolored, but the aug & dim triads in table 6.3 are all tricolored.




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