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Pythagorean intervals (3-limit) are strong and clear and go with everything, but don't have much character. Let's use white to describe them. Major intervals (otonal 5-limit) are warm and sunny and also somewhat bright. Let's use yellow. Minor (utonal 5-limit) is more subdued than yellow. Let's use green. Subminor (otonal septimal) is dark and bluesy. Let's use blue. Supermajor (utonal septimal) is sharper than nearby major intervals and more dissonant. It reminds me of something inflamed or swollen. Let's use red.
Yellow and blue are otonal colors, whereas green and red are utonal. Here are all the 7-limit thirds with small-number ratios, in descending order, with their shorthand abbreviations:
9/7
435¢
red 3rd
r3
5/4
386¢
yellow 3rd
y3
6/5
316¢
green 3rd
g3
7/6
267¢
blue 3rd
b3
They make a nice red-yellow-green-blue rainbow. The four “bands” of the rainbow are about 50-70¢ wide. This rainbow shows up not only for 3rds, but for most degrees of the scale, reminding us of the relative sizes of the intervals, and justifying the somewhat arbitrary color choices. (One disadvantage: The “blue third” played in jazz and blues is actually a slightly sharpened minor 3rd. We'll just have to call that a “blues third” from now on.)
Every 7-limit interval can be described by these five colors. Here's how it works: any two intervals an octave or a 5th apart are the same color. Thus 9/8 is white and 10/9 is yellow. Yellow and green, when added together, cancel out to make white (y3 + g3 = w5), as do blue and red. Blue and green combine to make blue-green, or bluish for short (but not bluish-green, see the next chapter). For example b3 +g3 = bg5 = 7/5, the bluish 5th. Likewise red & yellow make reddish. Intervals add up logically, so 10/7 = 617¢ is a 4th, not a 5th, because 10/7 = 9/7 x 10/9 = r3 + y2 = ry4. Bluish and reddish also cancel out, bg5 + ry4 = w8.
In the harmonic lattice, each row is a different color:
Figure 3.1 – The 7-limit harmonic lattice with colors, qualities, degrees and centslattice31.png
The following table is (more or less) the 9-odd-limit ratios, along with their counterparts up and down a 5th. The intervals are grouped by scale degree into six rainbows, each with five bands. The rainbow of 4ths runs blue-white-green-yellow-reddish, and the overlapping rainbow of 5ths runs bluish-green-yellow-white-red.
Table 3.2 – 7-limit JI intervals
ratio
cents
keyspan in semitones
deviation from 12-ET
quality & degree
color & degree
shorthand notation
1/1

0

perf unison
white unison
w1
(or more simply, unison)
21/20
84
1
-16¢
min 2nd
bluish 2nd
bg2

16/15
112
1
+12¢
min 2nd
green 2nd
g2

10/9
182
2
-18¢
maj 2nd
yellow 2nd
y2

9/8
204
2
+4¢
maj 2nd
white 2nd
w2

8/7
231
2
+31¢
maj 2nd
red 2nd
r2

7/6
267
3
-33¢
min 3rd
blue 3rd
b3

32/27
294
3
-6¢
min 3rd
white 3rd
w3

6/5
316
3
+16¢
min 3rd
green 3rd
g3

5/4
386
4
-14¢
maj 3rd
yellow 3rd
y3

9/7
435
4
+35¢
maj 3rd
red 3rd
r3

21/16
471
5
-29¢
perf 4th
blue 4th
b4

4/3
498
5
-2¢
perf 4th
white 4th
w4
(or simply 4th)
27/20
520
5
+20¢
perf 4th
green 4th
g4

7/5
583
6
-17¢
dim 5th
bluish 5th
bg5

45/32
590
6
-10¢
aug 4th
yellow 4th
y4

64/45
610
6
+10¢
dim 5th
green 5th
g5

10/7
617
6
+17¢
aug 4th
reddish 4th
ry4

40/27
680
7
-20¢
perf 5th
yellow 5th
y5

3/2
702
7
+2¢
perf 5th
white 5th
w5
(or simply 5th)
32/21
729
7
+29¢
perf 5th
red 5th
r5

14/9
765
8
-35¢
min 6th
blue 6th
b6

8/5
814
8
+14¢
min 6th
green 6th
g6

5/3
884
9
-16¢
maj 6th
yellow 6th
y6

27/16
906
9
+6¢
maj 6th
white 6th
w6

12/7
933
9
+33¢
maj 6th
red 6th
r6

7/4
969
10
-31¢
min 7th
blue 7th
b7

16/9
996
10
-4¢
min 7th
white 7th
w7

9/5
1018
10
+18¢
min 7th
green 7th
g7

15/8
1088
11
-12¢
maj 7th
yellow 7th
y7

40/21
1116
11
+16¢
maj 7th
reddish 7th
ry7

2/1
1200
12

octave
white octave
w8
(or simply octave)

There's no need to memorize this table, because every interval's name can be deduced from its ratio, and vice versa.
Two handy terms are fourthward and fifthward, which means leftwards or rightwards on the harmonic lattice. Table 3.2 has six rainbows of five bands each. Looking farther fourthward and fifthward yields 6-band rainbows. They all follow the same general form, shown here high to low:
Table 3.3 The 6-band rainbow
color
cents from 12-ET
quality
red
+27¢ to +39¢
mostly major, with aug 4th & perf 5th
reddish or fifthward white
+2¢ to +21¢
"
yellow
-22¢ to -10¢
"

-------

green
+10¢ to +22¢
mostly minor, with perf 4th & dim 5th
bluish or fourthward white
-21¢ to -2¢
"
blue
-39¢ to -27¢
"
Note the large gap between yellow and green. The missing band will be filled in later.
I play 7-limit JI music on a retuned conventional keyboard. I like knowing that if I spread my hand to a fifth, I'll play something that sounds more like a fifth than a fourth or a sixth. This means having a consistent method of determining the keyspan (width in semitones) of an interval. Therefore each 7-limit interval has not only a degree (3rd, 5th, etc.) but also a quality (major, perfect, augmented, etc.) which together determine its keyspan. As an added benefit, this approach allows the use of standard staff notation. Those who play fretless instruments, array keyboards, densely fretted guitars, or other instruments with more than 12 tones per octave will find the keyspan concept less useful.
Interval quality is redundant (if a third is yellow, it must be major), it's not unique (there are other major thirds available), and its main purpose is to indicate keyspan (all major thirds are 4 semitones wide on a standard keyboard). Subminor and supermajor are not needed to determine keyspan, and are cumbersome, so they are not used.
It may seem odd to see the dissonant yellow 5th 40/27 called a perfect 5th. Of course it's not the perfect 5th = 3/2, but it's played as one on a retuned keyboard and written as one in staff notation. Perfect refers only to keyspan; it merely means not augmented or diminished. In conventional music theory, "perfect" implies consonance. While perfect white intervals are highly consonant, non-white perfect intervals are generally quite dissonant.
Let's expand our lattice a bit. The augmented 5th 25/16 = yy5 is double yellow, or deep yellow for short. Diminished is deep green, 36/25 = gg5. Likewise ratios involving 49 are deep blue (bb) or deep red (rr). Yellow & blue can be combined to make yellowish (35/24 = yb5), and green & red make greenish (gr). Reddish & yellow make reddish-yellow (ryy), and bluish & green make bluish-green (bgg).
There is no deep white; remote pythagorean intervals are large or small white. 32/27 = w3 is a white 3rd and 81/64 = Lw3 is a large white 3rd. It's inverse 128/81 = sw6 is a small white sixth. Large and small are also used for other colors, see chapter 11.
Figure 4.1 – The Harmonic Lattice with Deep Colors
lattice41.png
Colors with an “-ish” in them are compound colors, representing ratios with both 5 and 7 factors. The other colors have either 5 or 7, or neither, and are primary colors.
Short descriptive names for those frequently-discussed smaller intervals:
Table 4.1 – Commas
225/224
7.7¢
desc dim 2nd
ryy-2
the reddish-yellow subcomma (a.k.a. the sub)
81/80
22¢
perf unison
g1
the green comma
312 / 219
23¢
desc dim 2nd
LLw-2
the white comma (full name: double large white comma)
64/63
27¢
perf unison
r1
the red comma
50/49
35¢
desc dim 2nd
rryy-2
the deep reddish comma
49/48
36¢
min 2nd
bb2
the deep blue comma
36/35
49¢
perf unison
gr1
the greenish comma, a.k.a. the double comma, so-called
because 36/35 = g1 + r1
LLw-2, ryy-2 and rryy-2 have a degree of minus 2 because they are actually descending 2nds. For more on these negative 2nds, see “Paradoxical Intervals”.
A subcomma is any comma smaller than 10¢. Most subcommas are quite remote (many steps away on the harmonic lattice). Because 225/224 is by far the least remote one, and because “reddish-yellow subcomma” is such a mouthful, it gets a nickname, the sub. The sub shows up quite often in 7-limit JI as the difference between two intervals. For example y4 = 590¢ is a sub sharper than bg5 = 583¢. The bg5 is said to be a subflat y4, and y4 is a subsharp bg5. In the harmonic minor scale, the interval from g6 to y7 is a subsharp b3.
Table 3.2 has six rainbows. There is a seventh rainbow, the rainbow of octaves. It overlaps the neighboring rainbows and its colors run out of order: bluish – blue – yellow – white – green – red – reddish:
Table 4.2 – The Rainbow of Octaves
28/15
1081¢
dim 8ve
bluish octave
bg8
63/32
1173¢
perf 8ve
blue octave
b8
160/81
1178¢
perf 8ve
yellow octave
y8
2/1
1200¢
perf 8ve
white octave
w8
81/40
1222¢
perf 8ve
green octave
g8
128/63
1227¢
perf 8ve
red octave
r8
15/7
1319¢
aug 8ve
reddish octave
ry8
This curious rainbow is the result of taking two rainbows, sb8 – bg8 – sg8 – y8 – w8 – r8 and b8 – w8 – g8 – Ly8 – ry8 – Lr8, and discarding the large and small ratios. The most consonant one after the white octave is probably the reddish octave, which sounds like a minor 9th. Again, note the difference between perfect and perfect white.
The harmonic lattice groups all notes separated by a white octave into one node. One can think of the lattice as having an "invisible rung" of length zero, the octave rung. Occasionally the octave rung needs consideration, for example when stretching octaves in alt-tuner. 2-limit ratios, which are ratios with no factors other than 2, are clear, abbreviated c. Technically 1/1 and 2/1 should be called c1 and c8, but for simplicity's sake they're called w1 and w8.



Large and small intervals are defined relative to the midpoint of each row. Midpoints are any ratio such that if r = 2^a * 3^b * 5^c * 7^d, b + c + d = 0, like 5/3 or 7/5. Each row has only one midpoint ratio, which is one of seven central (not large or small) ratios. Central ratios are at most 3 steps along the row away from the midpoint. Thus for large ratios, b + c + d ranges from 4 to 10. Defining large and small this way means that central intervals will usually be simpler (smaller odd limit) than the corresponding large or small interval. (Although unfortunately yy6 = 400/243 has a slightly higher odd limit than Lyy6 = 225/128. The yyb and yybb rows are also problematic.)

To extend the rows further, use double large LL and double small ss. If needed, there's triple large, etc. Triple large can be written with an exponent, L^3, but this wiki's superscript is buggy, so I'll write LLL instead here.To extend the number of rows, use triple yellow (y^3 or yyy) for ratios involving 125, etc.

The word plain means non-deep. Plain and central are useful words when comparing intervals. For example, the deep yellow 4th = 25/18 is flatter than the plain yellow 4th = 45/32, and the large white 3rd = 81/64 is sharper than the central white 3rd = 32/27. We can even compare large deep green to large plain green, or small deep yellow to central deep yellow.


Table 11.1 – Deep colors

rryy = deep reddish
ryy = reddish-yellow
yy = deep yellow
yyb = yellowish-yellow
yybb = deep yellowish
rry = reddish-red
ry = reddish
y = yellow
yb = yellowish
ybb = yellowish-blue
rr = deep red
r = red
w = white
b = blue
bb = deep blue
grr = greenish-red
gr = greenish
g = green
bg = bluish
bbg = bluish-blue
ggrr = deep greenish
ggr = greenish-green
gg = deep green
bgg = bluish-green
bbgg = deep bluish

The next table lists some rather remote commas. The large yellow subcomma Ly-2 is simply called the yellow subcomma.
Table 11.4 – More commas
ratio
cents
name

quality
class
derivations

38·5 / 215
1.95¢
yellow subcomma
Ly-2
desc dim 2nd
10
Ly-2 = LLw-2 - g1
5·210 / 7·36
5.8¢
reddish subcomma
sry1
perf unison
9
sry1 = r1 - g1 = ry1 - Lw1
211 / 34·52
19.5¢
deep green comma
sgg2
dim 2nd
8
sgg2 = g1 - Ly-2 = r1 - ryy-2
27 / 53
41¢
triple green comma
ggg2
dim 2nd
6
g1 + g1 + g1 - LLw-2 = gr1 - ryy-2
74 / 25·3·52
0.72¢
deep bluish-blue microcomma
bbbbgg3
double-dim 3rd
12
bb2 - rryy-2
aka the deep purple microcomma
54·7 / 2·37
0.40¢
blue quadruple yellow microcomma
yyyyb1
aug prime
13
bb2 + sry1 - ggg2

A microcomma is any comma too small to hear, i.e. it flunks the ear test. I define it as a comma less than 1¢, although less than 2¢ would also be a useful definition.
It's convenient to omit the large or small modifiers as much as possible, as we did with the yellow subcomma. To do this, we need a consistent method of determining "the" comma or subcomma for any row. This leads to the question, how many commas and subcommas of a given color are there?
The sum of two commas is another comma, at least up to a point. In particular, you can add the white comma to any comma or subcomma and get another comma. Thus each row has a series of commas 12 steps apart. The yellow subcomma plus the white comma is the yellow comma, the LLLy-3 = 25.4¢. Adding another white comma makes the large yellow comma, LLLLLy-4 = 48.9¢. Beyond that the intervals become too wide to be called commas. Note the naming sequence: subcomma, comma and large comma.
On the deep green row, there is both the deep green comma sgg2 = 2048/2025 = 19.5¢ and the large deep green comma Lgg1 = 6561/6400 = 43¢. The first one is "the" deep green comma because it has a lower odd limit.
When the smallest comma on a given row is far fourthwards, the sequence runs subcomma, small comma, comma (or possibly just small comma, comma). For example, consider the triple green comma = ggg2 = 41.1¢. By the way, this comma arises when tuning an augmented chord as a stack of three yellow 3rds. The triple green comma minus a white comma is the small triple green comma = ssggg3 = 17.6¢.
There is a deep green subcomma, but it's impossibly remote. Every row has a series of subcommas, each pair separated by the white subcomma L8w-6 = 3.6¢. Adding and subtracting white subcommas yield many more commas on a row, all extremely remote.
The large deep green comma Lgg1 = 6561/6400 = 43¢ happens to be the sum of two green commas. Because ratios multiply when adding intervals, 6561/6400 = (81/80)2, so this is called the squared green comma. Squared, cubed, etc. are general terms for intervals that break down into two or more identical ratios. Squared usually refers to commas, but could also include larger intervals like the white ninth, the deep yellow aug 5th, the large white 3rd, etc. Squared intervals can also be called doubled, tripled, etc., but this is potentially confusing because the tripled green comma = g31 = 64.6¢ is different than the triple green comma.


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