7-limit interval names
top next (this page is part of a series on Kite's color notation)
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 (5-over) are warm and sunny and also somewhat bright. Let's use yellow.
Minor (5-under) is more subdued than yellow. Let's use green.
Subminor (7-over) is dark and bluesy. Let's use blue.
Supermajor (7-under) is sharper than nearby major intervals and more dissonant. It reminds me of something inflamed or swollen. Let's use red.
To avoid confusing synesthetes, and to avoid certain connotations (e.g. a blue 3rd is neutral, not subminor), new words are coined for these colors: white becomes wa, yellow becomes yo, green becomes gu ("goo"), etc. The vowel (-o or -u) indicates otonal vs. utonal, or over vs. under. The -a in wa indicates all 3-limit ratios, both over and under. These new color names are more concise, and easier for non-English-speakers to learn, spell and pronounce.
The colors have one-letter abbreviations w, y, g, etc. The problem with using blue is that the flat sign is often written with a b. Therefore 7-over is represented by the Spanish and Portuguese word for blue, azul, or by the English word azure. Its new name is zo. (Additional mnemonic: Z looks like 7 with a line on the bottom.) Red becomes ru.
Here are all the thirds of odd-limit 9 or less, in descending order, with a one-letter shorthand:
| 9/7 | 7-under | 435¢ | supermajor | sharpest | hot | red | ru 3rd | r3 |
| 5/4 | 5-over | 386¢ | major | sharper | warm | yellow | yo 3rd | y3 |
| 6/5 | 5-under | 316¢ | minor | flatter | cool | green | gu 3rd | g3 |
| 7/6 | 7-over | 267¢ | subminor | flattest | cold | blue/azure | zo 3rd | z3 |
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.
Just as wa means 3-all or 3-limit, ya means 5-all, i.e. 5-limit. Za refers to the 2.3.7 subgroup, and yaza means 7-limit.
Every yaza 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 because 3/2 is wa, so is 9/8. Because 5/3 is yo, so is 10/9. Yo and gu, when added together, cancel out to make wa (y3 + g3 = w5), as do zo and ru (z3 + r2 = w4). Zo and gu combine to make zogu. (Not guzo, because the higher prime always comes first.) For example, z3 +g3 = zg5 = 7/5, the zogu 5th. Likewise ru & yo make ruyo. Intervals add up logically, so 10/7 = 617¢ is a 4th, not a 5th, because 10/7 = 9/7 · 10/9 = r3 + y2 = ry4. Zogu and ruyo also cancel out: zg5 + 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 zo-wa-gu-yo-ruyo, and the overlapping rainbow of 5ths runs zogu-gu-yo-wa-ru.
Table 3.2 – 7-limit JI intervals
| ratio | cents | 12-edo keyspan | deviation from 12-ET | quality & degree | color & degree | shorthand notation | |
|---|---|---|---|---|---|---|---|
| 1/1 | 0¢ | 0 | 0¢ | perf unison | wa unison | w1 | (or more simply, unison) |
| 21/20 | 84 | 1 | -16¢ | min 2nd | zogu 2nd | zg2 | |
| 16/15 | 112 | 1 | +12¢ | min 2nd | gu 2nd | g2 | |
| 10/9 | 182 | 2 | -18¢ | maj 2nd | yo 2nd | y2 | |
| 9/8 | 204 | 2 | +4¢ | maj 2nd | wa 2nd | w2 | |
| 8/7 | 231 | 2 | +31¢ | maj 2nd | ru 2nd | r2 | |
| 7/6 | 267 | 3 | -33¢ | min 3rd | zo 3rd | z3 | |
| 32/27 | 294 | 3 | -6¢ | min 3rd | wa 3rd | w3 | |
| 6/5 | 316 | 3 | +16¢ | min 3rd | gu 3rd | g3 | |
| 5/4 | 386 | 4 | -14¢ | maj 3rd | yo 3rd | y3 | |
| 9/7 | 435 | 4 | +35¢ | maj 3rd | ru 3rd | r3 | |
| 21/16 | 471 | 5 | -29¢ | perf 4th | zo 4th | z4 | |
| 4/3 | 498 | 5 | -2¢ | perf 4th | wa 4th | w4 | (or simply 4th) |
| 27/20 | 520 | 5 | +20¢ | perf 4th | gu 4th | g4 | |
| 7/5 | 583 | 6 | -17¢ | dim 5th | zogu 5th | zg5 | |
| 45/32 | 590 | 6 | -10¢ | aug 4th | yo 4th | y4 | |
| 64/45 | 610 | 6 | +10¢ | dim 5th | gu 5th | g5 | |
| 10/7 | 617 | 6 | +17¢ | aug 4th | ruyo 4th | ry4 | |
| 40/27 | 680 | 7 | -20¢ | perf 5th | yo 5th | y5 | |
| 3/2 | 702 | 7 | +2¢ | perf 5th | wa 5th | w5 | (or simply 5th) |
| 32/21 | 729 | 7 | +29¢ | perf 5th | ru 5th | r5 | |
| 14/9 | 765 | 8 | -35¢ | min 6th | zo 6th | z6 | |
| 8/5 | 814 | 8 | +14¢ | min 6th | gu 6th | g6 | |
| 5/3 | 884 | 9 | -16¢ | maj 6th | yo 6th | y6 | |
| 27/16 | 906 | 9 | +6¢ | maj 6th | wa 6th | w6 | |
| 12/7 | 933 | 9 | +33¢ | maj 6th | ru 6th | r6 | |
| 7/4 | 969 | 10 | -31¢ | min 7th | zo 7th | z7 | |
| 16/9 | 996 | 10 | -4¢ | min 7th | wa 7th | w7 | |
| 9/5 | 1018 | 10 | +18¢ | min 7th | gu 7th | g7 | |
| 15/8 | 1088 | 11 | -12¢ | maj 7th | yo 7th | y7 | |
| 40/21 | 1116 | 11 | +16¢ | maj 7th | ruyo 7th | ry7 | |
| 2/1 | 1200 | 12 | 0¢ | octave | wa 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 |
|---|---|---|
| ru | +27¢ to +39¢ | mostly major, with aug 4th & perf 5th |
| ruyo or fifthward wa | +2¢ to +21¢ | " |
| yo | -22¢ to -10¢ | " |
| ------- | ||
| gu | +10¢ to +22¢ | mostly minor, with perf 4th & dim 5th |
| zogu or fourthward wa | -21¢ to -2¢ | " |
| zo | -39¢ to -27¢ | " |
Note the large gap between yo and gu. The missing band will be filled in later.
Interval quality is redundant (if a third is yo, 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 12edo 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 yo 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 yo, or yoyo for short. The diminished 5th is gugu, 36/25 = gg5. Likewise ratios involving 49 and its multiples are zozo (zz) or ruru (rr). Ruyo & yo make ruyoyo (ryy). Zozo and gugu make zozogugu, or double zogu (zzgg).
There is no deep wa; remote wa intervals are large or small. 32/27 = w3 is a wa 3rd and 81/64 = Lw3 is a large wa 3rd. It's inverse 128/81 = sw6 is a small wa sixth. Large and small are also used for other colors. Central means neither large nor small.
Figure 4.1 – The Harmonic Lattice with Double Colors
Compound colors such as zogu represent ratios with both 5 and 7 factors. Primary colors such as wa and yoyo have either 5 or 7, or neither.
Short descriptive names for those frequently-discussed smaller intervals:
Table 4.1 – Commas
| 225/224 | 7.7¢ | desc dim 2nd | ryy-2 | the ruyoyo minicomma |
| 81/80 | 22¢ | perf unison | g1 | the gu comma |
| 312 / 219 | 23¢ | desc dim 2nd | LLw-2 | the wa comma (full name: double large wa comma) |
| 64/63 | 27¢ | perf unison | r1 | the ru comma |
| 50/49 | 35¢ | desc dim 2nd | rryy-2 | the double ruyo comma |
| 49/48 | 36¢ | min 2nd | zz2 | the zozo comma |
| 36/35 | 49¢ | perf unison | rg1 | the rugu comma, a.k.a. the double comma, so-called
because 36/35 = g1 + r1 |
LLw-2, ryy-2 and rryy-2 are negative 2nds that have a degree of minus 2 because they are actually descending 2nds.
A comma is defined as any interval smaller than 50¢, a minicomma is any comma smaller than 10¢, and a microcomma is any interval smaller than 1¢.
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: zogu – zo – yo – white – green – red – reddish:
| Table 4.2 – The Rainbow of Octaves | ||||
| 28/15 | 1081¢ | dim 8ve | zogu octave | zg8 |
| 63/32 | 1173¢ | perf 8ve | zo octave | z8 |
| 160/81 | 1178¢ | perf 8ve | yo octave | y8 |
| 2/1 | 1200¢ | perf 8ve | wa octave | w8 |
| 81/40 | 1222¢ | perf 8ve | gu octave | g8 |
| 128/63 | 1227¢ | perf 8ve | ru octave | r8 |
| 15/7 | 1319¢ | aug 8ve | ruyo octave | ry8 |
This curious rainbow is the result of taking two rainbows, sb8 – zg8 – sg8 – y8 – w8 – r8 and z8 – w8 – g8 – Ly8 – ry8 – Lr8, and discarding the large and small ratios. The most consonant octave after the white one is probably the ruyo 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. 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 yyz and yyzz 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, L3. To extend the number of rows, use triple yellow (y3 or yyy) for ratios involving 125, etc.
Table 11.1 – Deep colors
| rryy = double ruyo | ryy = ruyoyo | yy = yoyo | zyy = zoyoyo | zzyy = double zoyo |
| rry = ruruyo | ry = ruyo | y = yo | zy = zoyo | zzy = zozoyo |
| rr = ruru | r = ru | w = wa | z = zo | zz = zozo |
| rrg = rurugu | rg = rugu | g = gu | zg = zogu | zzg = zozogu |
| rrgg = double rugu | rgg = rugugu | gg = gugu | zgg = zogugu | zzgg = double zogu |
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¢ | yo 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.