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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

whiteto describe them. Major intervals (otonal 5-limit) are warm and sunny and also somewhat bright. Let's useyellow. Minor (utonal 5-limit) is more subdued than yellow. Let's usegreen. Subminor (otonal septimal) is dark and bluesy. Let's useblue. Supermajor (utonal septimal) is sharper than nearby major intervals and more dissonant. It reminds me of something inflamed or swollen. Let's usered.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:

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

bluishfor short (butnotbluish-green, see the next chapter). For example b3 +g3 = bg5 = 7/5, the bluish 5th. Likewise red & yellow makereddish. 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 centsThe 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

ratiocentskeyspan in semitonesdeviation from 12-ETquality & degreecolor & degreeshorthand notationperf unisonmin 2ndmin 2ndmaj 2ndmaj 2ndmaj 2ndmin 3rdmin 3rdmin 3rdmaj 3rdmaj 3rdperf 4thperf 4thperf 4thdim 5thaug 4thdim 5thaug 4thperf 5thperf 5thperf 5thmin 6thmin 6thmaj 6thmaj 6thmaj 6thmin 7thmin 7thmin 7thmaj 7thmaj 7thoctaveThere'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

fourthwardandfifthward, 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

colorcents from 12-ETqualityI 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

theperfect 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

doubleyellow, ordeepyellow 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 makeyellowish(35/24 = yb5), and green & red makegreenish(gr). Reddish & yellow make reddish-yellow (ryy), and bluish & green make bluish-green (bgg).There is no deep white; remote pythagorean intervals are

largeorsmallwhite. 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 ColorsColors with an “-ish” in them are

compoundcolors, representing ratios with both 5 and 7 factors. The other colors have either 5 or 7, or neither, and areprimarycolors.Short descriptive names for those frequently-discussed smaller intervals:

Table 4.1 – Commas

desc dim 2ndsubcomma(a.k.a. thesub)perf unisondesc dim 2nddouble largewhite comma)perf unisondesc dim 2ndmin 2ndperf unisondouble comma, so-calledbecause 36/35 = g1 + r1

negative2nds, 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 asubflaty4, and y4 is asubsharpbg5. 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:

dim 8veperf 8veperf 8veperf 8veperf 8veperf 8veaug 8veThe 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

midpointof 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 sevencentral(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 largeLL anddouble smallss. If needed, there'striplelarge, 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, usetripleyellow(y^3 or yyy) for ratios involving 125, etc.The word

plainmeans 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

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

ratiocentsnamequalityclassderivationsdesc dim 2ndperf unisondim 2nddim 2ndmicrocommadouble-dim 3rdaug primeA 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

smalltriple 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

squaredgreen 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 thetripledgreen comma = g31 = 64.6¢ is different than thetriplegreen comma.top next