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The previous section is all relative notation; no actual frequency or pitch is specified. Absolute notation requires the use of standard note names A, B, C#, etc. A note is named by its color and its letter name: green D, white E, etc. Of course individual notes don't have color, only intervals have color. But we can assign colors to the notes relative to the tonic, which is always white. Thus in F, the yellow 3rd is yellow A, or yA. The green 3rd is green A-flat, gA. There is no green A or yellow A in the key of F (unless you count large & small notes, see chapter 11). If a note is the tonic or a perfect interval from it (4th, 5th, octave, twelfth, etc.) it is assumed to be white and we can omit the “w”.
For example, “Fur Elise” in E might be intoned:
B – yA# – B – yA# – B – wF# – A – gG – E
(Upper-case B & G are notes, lower-case b & g are colors.)
Notice the difference between note color and interval color. The first melodic step from white B down to yellow A# is a green interval, even though neither of the notes is green.
Unfortunately, we can no longer use lower-case b for the flat sign, as it now means blue. We have to use a real flat sign (B), or a superscripted b (Bb), or else spell it out (B-flat), or possibly use "f" for flat (Bf).
Given a key and a remote interval, say, 35/16 = 1355¢ in E, exactly which note name do we use, and which keyboard key do we assign it to? First find the color & degree: 35/16 = 5/4 x 7/4 = y3 + b7 = yb9. It's a 9th, so it's an F, but is it F natural or F sharp? In conventional terms, 35/16 = y3 + b7 = maj3 + min7 = maj9, so it's major, and thus a yellowish F#. (This interval could well be used, in the relative minor it's the V7 chord's seventh.)
We can combine color, quality & degree into one term: 35/32 = ybM2. For the more remote colors, it's often very helpful to include the quality. But remember that the quality is not an independent variable; ybm2 is meaningless.
Another example: 49/40 = 351¢ in A. 49/40 = 7/6 x 21/20 = bm3 + bgm2 = bbgd4 = bluish-blue D.
Magnitude (large vs small) is indicated by sharps and flats. Magnitude is different than size, defined in chapter 1 as an interval's width in cents. In E, a central (not large or small) white 3rd is G, so a white G# must be large.
In general, deep yellow adds a sharp and deep green adds a flat. Large adds a sharp, and small adds a flat. Reddish adds a sharp and bluish adds a flat. Thus in G, w4 is C, but y4, yy4, Lw4 and ry4 are all C#, because each one is an augmented 4th. The interval the w4 is raised by is conventionally known as an augmented prime, an augmented unison, or a chromatic semitone (as opposed to a diatonic semitone, which is a minor second).
Table 5.1 – Augmented primes, aka chromatic semitones
25/24
71¢
yy1
the deep yellow semitone
135/128
92¢
Ly1
the large yellow semitone
37/ 211
114¢
Lw1
the large white semitone
15/14
119¢
ry1
the reddish semitone
In addition to the large white semitone Lw1 = 114¢, there is a small white semitone sw2 = 256/243 = 90¢, which is a diatonic semitone. By a happy accident, magnitude corresponds to size. We can safely refer to Lw1 as the large white semitone, and not the large white chromatic semitone, because there is no large white diatonic semitone.
It's generally easier to compare two "versions" of the same note using absolute notation than relative notation. For example, if we compare y2 and w2, the w2 is a green comma g1 wider. But the w7 isn't g1 wider than y7, in fact it's narrower. However a white F# is always exactly g1 sharper than a yellow F#. A white anything is always g1 sharper than the yellow version, regardless of the tonic. The reason for this difference is that absolute notation always represents keyspan (via sharps and flats), whereas relative notation only sometimes represent keyspan (via quality). Adding qualities to the relative notation comparision, it becomes wM2 - yM2 = g1, but wm7 - yM7 = descending LyA1. Major and minor intervals of the same degree, or perfect and augmented ones, will differ by an augmented prime. But for any given degree and quality, the white version will always be a green comma wider than the yellow version.
So why not have the keyspan and quality a mandatory feature of relative notation, and let the magnitude be optional? Because the magnitude of a ratio is always known, but with higher primes, sometimes the keyspan can't be specified exactly.
To generalize the above to other colors, look for an interval that will get you from one row (color) to the other without changing either the keyspan or degree. In other words, look for a perfect unison of a given color or its inverse. There will only be one such interval. How about comparing blue to white? There is no blue perfect unison, but there is one of the inverse color, red. The white version of any note is alwys a red comma r1 sharper than the blue version. These two commas are sufficient to compare any two colors in 7-limit JI. For example to get from a blue F to a green F, use a red comma to change the blue to white, and a green comma to change white to green: bF + r1 + g1 = gF. To get from bgD to rD, add two red commas and subtract a green one: bgD + r1 + r1 - g1 = rD.
This next (printer-friendly) diagram shows a large lattice that contains every key. It extends about as far as possible without encountering triple-sharps and triple-flats. Start by picking an instance of your keynote somewhere in the middle of the chart. Use the notes on the lines, not inside the triangles. For example, for E, use the third row from the top, 6th note from the left. Think of that note as white, and that whole row as white. Assign colors to nearby rows based on that. You can see all possible scales and chord progressions using that E note. Comma pumps will take you to a different E note. Try to visualize the "Fur Elise" melody!

Figure 5.1 – The 7-limit JI Note Lattice
lattice51.png

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