Next Page: 4/2 – Meanings of All 53 Notes
*Advanced*
Diagramming Scales
To summarize the basics of the WWMW terms, then, let’s use the example of E^{1}b one circle inside of the central circle. The descriptors for that note surround the intersection of the two lines. Top left: the note in its harmonic context relative to C. Top right: D#, the note you strike on the fiverank keyboard to get the pitch you want. Bottom left: 6/5, the harmonic ratio from C to E^{1}b (ratios appear only on notes close enough to C to have harmonic meaning). Also, the number 15, which means its step number is 15 in the 53note series. Bottom right: D#^{+11}, which refers to the cents notation system: eleven cents higher than the 12eq D#. Also, 316, which is to say 316 cents above the central C (C is C^{6}, C to D# is 300 cents above C in 12eq, and E^{1}b is just under 11 cents above 12eq C; E’b is also 14 cents above the Eb in 12eq).
There are all kinds of patterns in the WWMW that you get used to as you work with it. For example, let’s see how easy it is to figure out how to change keys in any scale. We’ll use the example of the Just Mixolydian scale, seven notes which look like this in C:
A_{1} – E_{1}
 
F — C — G – D
. 
. B^{1}b
Let’s say you modulate from C Mixolydian to A_{1} Mixolydian. You would still have two notes on the top rank above the home note, four perfect fifths on the line of the new home key of A_{1}, and one note along the bottom rank. It would have exactly the same shape, like this:
F_{2}# — C_{2}#
 
D_{1} – A_{1} — E_{1} – B_{1}
. 
. G
Same shape on the WWMW. Easy! Also, notice that C and A_{1} Just Mixolydian have A_{1}, E_{1} and G in common. (Hint – those would be good notes to pivot around in such a modulation!). Knowing the shape of C Mixolydian on the WWMW, you can create that same scale in whatever key you are in by simply preserving that mode’s shape. Here’s A^{3}bb Mixolydian, for example:
F^{2}b – C^{2}b
 
D^{3}bb—A^{3}bb—E^{3}bb—B^{3}bb
. 
. G^{4}bb
On my honeycombed generalized keyboard, the hexagonal black and white keys are all the same shape and distance from one another, so, as it happens, every scale has the same “shape” on the keyboard as well, if the colors of the keys are ignored.
The C Mixolydian mode can also be expressed as a series of fifths, as Pythagoras and the Catholic Church defined it:
Bb – F – C – G –D –A — E
So the shape of a Pythagorean Mixolydian mode will always be the same straight line with a fifth and two fifths below the starting point of the scale. Whatever the 3limit or 5limit scale or mode, it is always defined by its shape on the WWMW.
The same is true of chords. Every chord is defined by its intervals. For example, the C Maj^{7} chord is always a square, looking like this no matter where you start:
E_{1} – B_{1 }
 
C — G
You can create unnamed tone clusters. Very generally, a tone cluster makes a kind of harmonic sense when the notes are somewhat close to one another, like this:
G2# — D2#
.  — 
.  — F_{1}#
. C
There are patterns everywhere. Find all the “Pythagorean” commas by going up from, say, C twelve fifths to B# (or 12 perfect fifths above any note in the circles and a different pitch in 53eq); start at C and go up four fifths and down a third to find C^{1}, the same distance to C^{2}, etc.
Using the WWMW, you can also discover how all 53 notes relate to one another harmonically. Starting from the note C, you can create what I call the “Miracle Mode,” first developed by Alain Danielou (see Figure 221 in Chapter Two). And yes, the “Miracle Mode” is modulatable, just by starting on any other note you want, and creating the same matchstick structure. Because we are dealing with such a close approximation of Just Intonation, it is for composers of the near future (like you) to discover how far you can take the audience’s ears on a journey through as many of these 53 relationships as newly attuned ears can hear.
Now you can begin to see how 12eq compares with just intonation. Since every perfect fifth in 12eq is 2 cents away from a pure perfect fifth, if you start at the note A^{0} (A440) and go down by fifths, you get D^{2}, G^{4}, C^{6,} etc. The major thirds are a little over 14 cents off in 12eq (that’s very bad!), so go up the spine from C and you get C^{6}, E^{21}, G^{#36}, B^{+49}, etc. (in cents notation, when you are more than fifty cents away you just go up or down to the next note and express it that way; there is no such thing as B#^{51}, for example).
What about the seventh harmonic relationships? To diagram these, you would need a threedimensional lattice: the horizontal dimension for 3limit, the vertical dimension for 5limit, and then a 53story skyscraper where each “floor” is a fivelimit lattice stacked septimally atop one another. In 53eq, the seventh harmonic is about 4.5 cents off from the nearest note in the 53eq system. Starting from C, the closest note in 53eq is A_{2}#. However, to create a seventh chord in common practice tradition, you need the notes CEGB (with flats, double flats and naturals as needed depending on the chord). So we need some kind of CEGBb. Take the pitch of A_{2}# (two diagonal ascents from C) and find its enharmonic equivalent a skhisma away and you get B_{1}b. So it would look something like this (and of course this can be moved to any other starting point in the circle):
. — — A_{2}#/B_{1}b

E_{1}

C — G
We will talk more about the seventh harmonic in a later chapter. For now, suffice it to say that the Helmholtz notational system, which works so well to describe the 3limit and 5limit notes, creates but an approximation of the true seventh harmonic notes. Since the WWMW is only twodimensional and you need a third dimension for the seventh harmonic relationships, the 5limit notational system we are borrowing from Helmholtz necessarily distorts the septimal notation. We have the notes C, D, E, F, G, A and B for onedimensional 3limit Pythagorean notation. Helmholtz adds the superscripts and subscripts 1, 2, 3 etc. for 5limit notation. For the septimal approximations in 53eq, I don’t add any special notation; I simply use the notations to tell the performer what pitch to play. This is a compromise; the Just Intonation purists do have special notation for septimally generated notes. And by the way, what you see above is a septimal seventh chord. It “looks” like a dominant seventh chord, and can function as a dominant seventh chord. But in fact, it is free of dissonance and sounds more stable than a major triad in 12eq! So it can stand alone with no real need to “resolve” to an F chord as it would in 12eq. This is probably the kind of flat seventh chord jazz musicians employ, the kind that needs no resolution. Here are the other three possible nonseptimal patterns of CEGBb in 53eq, all of which have overtonal tensions that need resolving:
. E_{1 } E_{1}
.  
Bb — — C – G C – G – Bb — — C — G — — — E (Pythagorean)
. 
. B^{1}b
For the higher harmonics that are being explored by Johnny Reinhard and many other Just Intonation pioneers of the twentieth and twentyfirst centuries, the 53eq approximations vary by overtone (see Chapter Two). To try to create these approximations in 53eq using the WWMW, you will once again end up with an enharmonic approximation both notationally and pitchwise.
But for 3limit, 5limit or 7limit music, you can see that 53eq offers many more options for you as a composer than the old 12eq system. Over and over you will find that you are presented with musical choices you never had in 12eq. And the WWMW is a diagram that actually makes it easy to navigate these waters. Pattern after pattern reveals itself as you go through the WWMW. What I love about this template is the ease with which any scale or mode from anywhere in the world can be understood. You can diagram any ancient scale, compare it to any other ancient scale from anywhere else on the planet, and let them dance together. You can modulate to 53 different keys per scale/mode at will. It would take thousands of composers hundreds of years to exhaust the musical possibilities of 53eq. So next let’s look more carefully at the pitches of 53eq and how to notate them in 53eq.
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