Note names
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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: gu D, wa 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 wa. Thus in F, the yo 3rd is yo A, or yA. The gu 3rd is gu A-flat, gA♭. There is no gu A or yo A♭ in the key of F (unless you count large & small notes). If the color is omitted, the note is assumed to be wa.
For example, “Fur Elise” in E might be intoned:
| B – yA# – B – yA# – B – wF# – A – gG – E |
(Upper-case G is a note, lower-case g is a color.)
Notice the difference between note color and interval color. The first melodic step from wa B down to yo A# is a gu interval, even though neither of the notes is gu.
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 + z7 = zy9. It's a 9th, so it's an F, but is it F natural or F sharp? In conventional terms, 35/16 = y3 + z7 = maj3 + min7 = maj9, so it's major, and thus a zoyo 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 = zyM2. For the more remote colors, it's often very helpful to include the quality. But remember that the quality is not an independent variable; zym2 is meaningless.
Another example: 49/40 = 351¢ in A. 49/40 = 7/6 x 21/20 = zm3 + zgm2 = zzgd4 = zozogu D♭.
Magnitude (large vs small) is indicated by sharps and flats. Magnitude is different than size, an interval's width in cents. In E, a central (not large or small) wa 3rd is G, so a wa G# must be large.
In general, yoyo adds a sharp and gugu adds a flat. Large adds a sharp, and small adds a flat. Ruyo adds a sharp and zogu 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 yoyo semitone |
| 135/128 | 92¢ | Ly1 | the large yo semitone |
| 37/ 211 | 114¢ | Lw1 | the large wa semitone |
| 15/14 | 119¢ | ry1 | the ruyo semitone |
In addition to the large wa semitone Lw1 = 114¢, there is a small wa 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 wa semitone, and not the large wa chromatic semitone, because there is no large wa 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 wa F# is always exactly g1 sharper than a yo F#. A wa anything is always g1 sharper than the yo 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 wa version will always be a gu comma wider than the yo 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 zo to wa? There is no zo perfect unison, but there is one of the inverse color, ru. The wa version of any note is alwys a ru comma r1 sharper than the zo version. These two commas are sufficient to compare any two colors in 7-limit JI. For example to get from a zo F to a gu F, use a ru comma to change the zo to wa, and a gu comma to change wa to gu: zF + r1 + g1 = gF. To get from zgD to rD, add two ru commas and subtract a gu one: zgD + r1 + r1 - g1 = rD.
This next (printable) 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 wa, and that whole row as wa. 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 yaza JI Note Lattice
