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(This page is part of a series on Kite's color notation)

For certain intervals and chords, slightly adjusting the intonation will increase the prime limit up to the next prime and greatly decrease the odd limit. I think of these chords as "deal-breakers";

IFthey are perceived as consonant (or "interesting", see the limits discussion in chapter 1), they will force us to use a higher prime limit. In the Renaissance, when the major third became accepted as a consonance, and the major triad became used more, it "broke" 3-limit, as 81/64 was effectively overshadowed by 5/4. Likewise as the dom7 chord and the dim chord come to be accepted as consonant, they break 5-limit. Diminished 5ths like 36/25 and 64/45 are overshadowed by 7/5. Among 7-limit's deal-breakers are the neutral 3rd (60/49 or 49/40 overshadowed by 11/9) and the half-augmented 4th (135/98 overshadowed by 11/8). If these intervals are considered consonant, 11-limit JI becomes desirable. However, the concept of deal-breakers itself breaks down when one tries to argue thence for the desirability of JI systems above 11-limit because several would-be deal-breakers are close enough (at least, as they say, "for government work") to intervals which can already be considered particularly consonant by lower prime limits; i. e. they fall within +\- 16 cents of these intervals. Among the breakdowns are the major, neutral and minor thirds (overshadowed by or only just noticeably different from 11-limit intervals [28/33*13/11=364/363=4.763 cents]), the minor second (17/16 overshadowed by 16/15; also 256/255=6.776 cents, making either a passable representation of the other), the hemisixth (13/10 overshadowing 35/27 [351/350=4.939 cents]), the superminor and submajor thirds (17/14 overshadowing 39/32 or 63/52 and 21/17 overshadowing 26/21 [442/441=3.921 cents, 273/272=6.353 cents]), the neutral sixth (13/8 overshadowing 18/11 [144/143=12.064 cents] and the superminor second (13/12 overshadowing 88/81 [352\351=4.925 cents]).The color notation is easily expanded to higher primes. Pure undecimal (11) and tridecimal (13) ratios are clumped tightly together midway between yellow and green, with the otonal being closer to green than yellow. The main difficulty is finding enough color names in that part of the spectrum. For undecimal ratios I use

jadeandamber. For tridecimal,emeraldandochre. The otonal names are mineral, and the utonal names are mineral/vegetative. The midpoints are 11/6, 12/11, 13/12 and 24/13, which gives us these 7 central intervals on each row:Table 15.1 – Jade, amber, emerald and ochre central intervals

Lower-case a means amber; upper-case A means augmented, or the note A. All the 7-limit JI concepts can be applied to higher primes: there's large amber, deep emerald, pure undecimal, jade triads, etc. There are compound colors, although sometimes without the "-ish" suffix: 11/10 is jade-green, 14/13 is ochre-blue, 56/55 is amber-bluish. Compound means the ratio contains at least two primes other than 2 or 3. The augmented triad (which could be considered one of 7-limit's deal-breakers) finds a lower-odd-limit representation as w1 – r3 – jr5, or 7:9:11. The 4:5:6:7:9:11:13 chord (which could be considered [weakly] one of 11-limit's deal-breakers) is written yb,9,j11,e13. Remoteness is calculated with the same (p-1)/2 formula.The quality-chain for all these colors is the same as for purple, ambiguous (i.e. neutral, half-aug, etc).

Each prime adds a dimension to the lattice. White, yellow and blue create a 3-D lattice of tetrahedrons. Where to put these new ratios in the harmonic lattice? If you can visualize 4-D, you can put them anywhere. Otherwise, the layout of higher primes' rungs should reflect small commas of deep jade or deep emerald. This allows you to "collapse" the lattice down to 3-D. The jade 4th might be between w2 & w6 because 2 jade fourths are about equal to 5 white fifths (missing by Lw7 - j4 - j4 = aa1 = 7¢), and thus one j4 is two and a half w5's. Since two j4's are also about equal to y7 (missing by jjg1 = 14¢), the jade rung could also be set to half a y7, with rungX = 150 and rungY = 86. Laying out higher rungs this way preserves an important property of lattices: if three notes line up, the center one is midway between the other two melodically as well (allowing for octave transposition).

How to fit undecimal and tridecimal ratios onto a standard keyboard?

If the jade 4th 11/8 were augmented,

j3 = 11/9 = 347¢ would be major (j3 = j4 - w2 = A4 - P2 = M3),

and a3 = 27/22 = 355¢ would be minor (a3 = w6 - j4 = M6 - A4 = m3).

But then we would have major flatter than minor,

in other words aa1 = am3 - jM3 = 243/242 = 7¢ would be a diminished prime.

But if the jade 4th were perfect,

then jg2 = 11/10 = 165¢ would be minor (jg2 = j4 - y3 = P4 - M3 = m2)

and a2 = 12/11 = 151¢ would be major (a2 = w5 - j4 = P5 - P4 = M2),

again, major is flatter than minor,

and gjj1 = jgm2 – aM2 = 121/120 = 14¢ would be a diminished prime.

What's wrong with diminished primes? They're upside-down. Let's put it in more musical terms. In C, the 11/8 is either F or F#. Supposing it's F#, consider the chord wDj = wD – jF# – wA. The lower 3rd is narrower than the upper one, but spans more semitones. If 11/8 = F, in the melody wD – yE – jF – wG – wA, the yE – jF step is wider, but spans fewer semitones, than jF – wG. So the correlation between an interval's size and its keyspan, already weakened by purple intervals, is further eroded.

The same problem arises with tridecimal intervals:

If the emerald 6th 13/8 were major,

then e3 = 39/32 = 343¢ = e6 - w4 = M3,

and o3 = 16/13 = 359¢ = w8 - e6 = m3.

But then eM3 < om3 and oo1 = 512/507 = 16¢ = dim1.

Also sgrm6 = 512/315 = 841¢,

thus eM6 < sgrm6 and sgro1 = dim1.

But if e6 were minor,

then e2 = 13/12 = 139¢ is minor (e2 = e6 - w5 = m6 - P5 = m2)

and ob2 = 14/13 = 128¢ would be major (ob2 = b7 - e6 = m7 - m6 = M2),

and em2 > obM2 and eer1 = 169/168 = 11¢ = dim1.

Musically: In C, 13/8 is either A♭ or A.

If A: in the chord Fe = F – eA – C, F – A is narrower than A – C.

If A♭: in the melody F – G – eA♭– bB♭– C, G – A♭ is wider than A♭– B♭.

These diminished primes are small, but they are not as remote as the ones in chapter 12, and the examples are not at all contrived.

There is no obvious way to notate these intervals. Notation and keyspans for jade and/or emerald must be determined on a case-by-case basis, depending on the scale. There will usually be contradictions; we must take the lesser-of-the-two-evils approach.

For example, take the harmonic-series scale w1, w2, y3, j4, w5, e6, b7, y7, w8. We look at the intervals between all the notes: jr5 = 11/7 = 782¢ is a fifth that's too big to be a perfect fifth. It should be augmented, so j4 = jrA5 - rM2 must be augmented too. That takes care of jade, what about emerald? The ea3 = 13/11 = 289¢ is too small to be a major 3rd, so it must be minor. Likewise ob2 = 14/13 = 128¢ should also be minor, not major. This leads to e6 = 13/8 being major. That gives us wP1, wM2, yM3, jA4, wP5, eM6, bm7, yM7, wP8. In C, that would be wC, wD, yE, jF#, wG, eA, bB♭, yB, wC.

There are still contradictions. For example eM2 = 13/12 = 139¢ and am2 = 12/11 = 151¢, thus the major 2nd is flatter than the minor 2nd by ao1 = 144/143 = 12¢ = dim prime. However as e2 and a2 both sound fairly neutral, I find e2 being major and a2 being minor less problematic than jrP5 or ead3 or obM2 would be. The aod1 also shows up as jM3 < om3 and jA4 < oP4.

By the same logic, the subharmonic-series scale becomes wP1, gm2, rM2, om3, wP4, ad5, gm6, wm7, wP8, or in C, wC, gD♭, rD, oE♭, wF, aG♭, gA♭, wB♭,wC.

Another example is the arabic maqam Rast, which runs P1, M2, n3, P4, P5, M6, n7, P8. It might be tuned w1, w2, j3, w4, w5, w6, j7, w8. In this case j3 and j7 could be either major or minor intervals, as long as they both have the same quality, so that j3 to j7 makes a perfect 5th. However the descending scale uses a minor 7th, forcing both j3 and j7 to be major: wP1, wM2, jM3, wP4, wP5, wM6, wm7, jM7, wP8. In C, that would be wC, wD, jE, wF, wG, wA, wB♭, jB, wC. Note that jM3 to wP5 is am3, and am3 > jM3.

Maqam Sikah runs P1, n2, n3, hA4, P5, n6, n7, P8 ascending, with a hd5 in the descending form. It might be tuned w1, j2, j3, j4, sj5, w5, j6, j7, w8. The w5 is of course perfect, so the sj5 must be diminished, which means the j4 can't be augmented and must be perfect. The j7 is a 4/3 above j4, so it must be minor; in shorthand, j7 = jP4 + wP4 = jm7. Likewise j3 = jm7 - wP5 = jm3, j6 = jm3 + wP4 = jm6, j2 = jm6 - wP5 = jm2, and sj5 = jm2 + P4 = sjd5. Thus we have wP1, jm2, jm3, jP4, sjd5, wP5, jm6, jm7, wP8. In C: wC, jD♭, jE♭, jF, sjG♭, wG, jA♭, jB♭, wC.

This tuning actually has no upside-down intervals, although some minor 2nds & 3rds are only aa1 = 7¢ narrower than some major ones. It can be extended with white intervals to 12 chromatic notes:

wP1, jm2, wM2, jm3, LwM3, jP4, sjd5, wP5, jm6, wM6, jm7, LwM7, wP8

In C: wC, jD♭, wD, jE♭, wE, jF, sjG♭, wG, jA♭, wA, jB♭, wB, wC

Because the quality of a jade interval can change depending on context, the magnitude can't be inferred from the quality, as it can with 7-limit colors. For example, in the C scale above, we know the wE is large, because it's a major 3rd. We can't assume that the jG♭ is small just because it's diminished, and in staff notation we must write "s" next to the note.

A full 12-note jade tuning without contradictions! But if we change Lw3 to y3, contradictions result. It seems that to avoid upside-down intervals, jade must be minor / perfect, and yellow must not be present. Green can be used; we could reduce the odd limit by replacing sj5 = 352/243 and j2 = 88/81 with g5 and g2.

Let's explore this tuning further. Here are some of its modes:

In G: wG, jA♭, wA, jB♭, wB, wC, sjD♭, wD, jE♭, wE, jF, sjG♭, wG

wP1, jm2, wM2, jm3, LwM3, wP4, sjd5, wP5, jm6, wM6, jm7, sjd8, wP8

In D: wD, jE♭, wE, jF, sjG♭, wG, sjA♭, wA, jB♭, wB, wC, sjD♭, wD

wP1, jm2, wM2, jm3, sjd4, wP4, sjd5, wP5, jm6, wM6, wm7, sjd8, wP8

When you invite 11 and 13 to the party, it gets rather crowded. Every ambiguous interval contains a miniature rainbow, with bands only 4-6¢ wide. We are like astronomers, finding a large gap between Mars and Jupiter, looking for a planet there and instead finding the asteroid belt. Shown here high to low, omitting the more remote intervals:

Table 15.2 Neutral, half-augmented and half-diminished intervals in 13-limit JI

pLp157¢p2147¢p3351¢p4555¢p5645¢p6849¢p71053¢sp81143¢The neutral 3rds e3, gr3, j3, p3, a3, Lyb3 and o3 all have roughly the same remoteness, class 6 or 7. A purple triad is class 6, jade and emerald triads are class 7.

The close proximity of purple, jade and emerald suggest another approach, using septimal approximations. The purple 4th is a good approximation of the jade fourth j4 = 11/8 = 551¢. They differ by only the purple-amber subcomma, pa1 = (9√3) / (11√2) = 3.57¢. The purple 5th matches 16/11 just as well. Other undecimal ratios can be derived from p4 & p5:

jg2 = 11/10 = 11/8 ÷ 5/4 ≌ p4 - y3 = pg2

a2 = 12/11 = 16/11÷ 4/3 ≌ p5 - w4 = p2

The 11-limit intervals can be made even better by tempering out pa1, for example by flattening the fifths by about 1¢.

Furthermore the purple 6th approximates the emerald 6th 13/8 = 841¢ by the purple-ochre subcomma, (16√2) / (13√3) = 8.49¢. Every ratio of 11 or 13 is closely approximated by a purple interval. Here's several versions of a 4:5:6:7:9:11:13 chord in 7-limit JI using purple intervals:

The greenish 4th 48/35 = 547¢ approximates 11/8 almost as well, missing by only the yellowish-jade subcomma ybj1 = 385/384 = 4.50¢. And the small greenish 6th sgr6 = 512/315 = 841.0¢ is

veryclose to 13/8. The difference is the greenish-ochre microcomma sgro1 = 4096/4095 = 0.42¢. This is fortunate because we are up to ten colors now, plus deep colors and compound colors, and that's a lot. So we have the option of fudging the microcomma and calling emerald and ochre intervals greenish and yellowish. Another way of thinking of this is that over in the small greenish and the large yellowish parts of the harmonic lattice, the numbers in the ratios suddenly get much smaller, although the remoteness is hardly changed.Using the greenish approximations for both 11/8 & 13/8 on a y3 root gives us the chord yIIIyb,9,gr11,gr13:

The point is, rather than adding another dimension or two to your harmonic lattice, you might want to just add another row.

The nearest 11-limit negative 2nd that I know of is jrr-2 = 99/98 = 18¢ = class 9.

The nearest 11-limit upside-down interval depends on how keyspan is calculated. If j4 = A4, it's the dim prime aa1 = 243/242 = 7¢ = class 13. If j4 = P4, it's the dim prime gjj1 = 121/120 = 14¢ = class 10.

The nearest 13-limit negative 2nd that I know of is ojryy-2 = 275/273 = 13¢ = class 11.

The nearest 13-limit upside-down interval depends on how keyspan is calculated. If e6 = M6, it's the dim prime oo1 = 512/507 = 16¢ = class 13, or else the dim prime sgro1 = 4096/4095 = 1¢ = class 13. If e6 = m6, it's the dim prime eer1 = 169/168 = 11¢ = class 12.

Even higher primes are used in JI. For example, 19/16 has a high prime limit but not too large an odd limit, widens well, and is extremely otonal, so it gives 6/5 and 32/27 some competition as a consonant minor third.

For higher primes, we could make up new colors, but that would just create more arbitrary terms to memorize. Plus there's more primes than letters in the alphabet. Instead we'll use the suffixes

-ishand-esquefor otonal and utonal, as in these colors:17ish = 17 on the top, "seventeenish"

19esque = 19 underneath, "nineteen-esque"

19ish-g = 19 above, 5 underneath = "nineteenish-green"

The ratio is shown by the usual color / degree combo. In the shorthand, -esque can be abbreviated to -esq:

17/16 = 17ish2 = "seventeenish second"

17/15 = 17ish-g3 = "seventeenish-green third"

19/14 = 19ish-r4 = "nineteenish-red fourth"

20/19 = 19esq-yA1 = "nineteen-esque-yellow augmented prime"

19/17 = 19ish17esqM2 = "nineteenish-seventeen-esque major second"

323/320 = 17ish19ish-gd2 = 16¢ = "seventeenish-nineteenish-green comma"

Note the use of a hyphen after -ish or -esq in the shorthand, when a lower case letter follows. Never use a clarifying hyphen before a number, because it'll look like a minus sign. For really long names, we can abbreviate -ish and -esq to i and q, and leave out the hyphens:

323/320 = 17ish19ish-gd2 = 17i19ig2

324/323 = 17q19q-2 = "seventeen-esque-nineteen-esque negative second"

All JI concepts apply to higher primes, so there is deep nineteenish, large seventeen-esque, etc. Chords are named similarly: 1/1 – 19/16 – 3/2 is a nineteenish triad. If the root is C, it's written as a C19ish or C19i chord. Deep colors are signified with repeated "i"s and "q"s:

361/360 = 19iig2 = "green-deep-nineteenish second"

384/289 = 17qq3 = "deep-seventeen-esque third"

If you want fewer colors to memorize, -ish and -esque can be applied to 11 and 13 as well:

11/9 = 11ish3 = "elevenish third"

27/22 = 11esq3 = "eleven-esque third"

13/8 = 13ish6 = "thirteenish sixth"

We're almost done extending color notation to cover every possible JI ratio. All that's left to do is to determine the degree and quality of each prime number, using a representative octave-reduced ratio of that prime over a power of two. By the way, these are called

prime rungs, we'll be using them in part IV. From the prime rungs, we can deduce the quality, degree and keyspan of any ratio. Here's the ones we've seen so far:2/1 = wP8 = 12 semitones

3/2 = wP5 = 7 semitones

5/4 = yM3 = 4 semitones

7/4 = bm7 = 10 semitones

11/8 = jhA4 = 5 or 6 semitones

13/8 = en6 = 8 or 9 semitones

Onto the higher primes. 17 and 19 fit in very well on a standard keyboard. For most of the other primes, the keyspan and/or the degree are somewhat arbitrary. The table below follows alt-tuner's defaults, defining the keyspan relative to 12-EDO and the degree relative to 7-EDO. See the end of chapter 4 in the alt-tuner manual for details.

17/16 = 105¢ = 17ish-m2 = 1 semitone

19/16 = 298¢ = 19ish-m3 = 3 semitones

23/16 = 628¢ = 23ish-d5 = 6 semitones

29/16 = 1030¢ = 29ish-m7 = 10 semitones

31/16 = 1145¢ = 31ish-d8 = 11 semitones

37/32 = 251¢ = 37ish-A2 = 3 semitones

41/32 = 429¢ = 41ish-d4 = 4 semitones

43/32 = 512¢ = 43ishP4 = 5 semitones

47/32 = 666¢ = 47ishP5 = 7 semitones

53/32 = 874¢ = 53ishM6 = 9 semitones

59/32 = 1059¢ = 59ish-M7 = 11 semitones

61/32 = 1117¢ = 61ish-d8 = 11 semitones

etc.

Alt-tuner allows one to manually override the rung's keyspan and degree. One approach is to avoid negative intervals when possible. 31/16 is a d8, not an A7, so that 31/30 is a d2 and the smaller 32/31 is an A1. This prevents their difference the green deep 31ish 2nd = 31iig2 = 961/960 = 1.8¢ from being a negative 2nd. In general, diminished is preferred over half-augmented.

Because 37/32 defaults to an A2, 37/36 is an A1 and the smaller 38/37 is a d2. To avoid the 37ii19q comma being negative, set the 37ish rung's degree to 3.

What do higher primes sound like? Seventeenish ones and nineteenish ones sound quite ordinary, being so close to 12-ET intervals. The other ones seem rather obscure to me personally. I doubt I could tune even the simplest intervals like 23/1 by ear in isolation. The easiest way to hear them is in the harmonic series scale:

1, 17i2, w2, 19i3, y3, b4, j4, 23i5, w5, yy5, e6, w6, b7, 29i7, y7, 318, w8

Tuning a 23i5 is easier when it's part of a scale like this, as you can tune melodicallly, not harmonically, and shoot for the midpoint between the j4 and the w5.

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