Diatonic, chromatic, enharmonic, and subchromatic steps

The concept of diatonic and chromatic movements are well-known in common practice music theory. In particular, the distinction between the diatonic and chromatic semitone (C-Db vs C-C#) has been known for centuries, and before the advent of 12-EDO, the difference was necessary to preserve as these are tuned differently in meantone temperament.

This concept can be generalized considerably when working with microtonal tunings, where it has applications not only higher-rank temperament design, but also the modeling of hierarchical, layered "interval classes." This can likely be tied to categorical perception in an as-of-yet undiscovered way, and possibly to tonal interval "identities" as well.

Furthermore, all of this can use the methods of straightforward regular temperament theory -- by simply looking at a different mapping matrix than you are used to.

Introduction: Mapping Matrices

Suppose that we are working in 5-limit meantone temperament. The mapping matrix for meantone, represented by generators of 2/1 and 3/2, is

[1  1  0]
[0  1  4]

Reading the columns, this tells us that in meantone:

• 2/1 maps to |1 0> (up an octave)
• 3/1 maps to |1 1> (up an octave and a fifth)
• 5/1 maps to |0 4> (up four fifths)

where |x y> are tempered monzos reflecting the |octave, fifth> basis, so that x is the number of octaves and y is the number of fifths.

As you know, there is more than one mapping matrix that is valid for meantone. For instance, if we wanted to go with an P8 and a P4 for a generator, if you do the math, you instead get the following matrix:

[1  2  4]
[0 -1 -4]

Now our |x y> basis represents |octaves, fourths>. This now tells us that:

• 2/1 maps to |1 0> (an octave)
• 3/1 maps to |2 -1> (up two octaves, down a fourth)
• 5/1 maps to |4 -4> (up four octaves, down four fourths).

In general, there are an infinite set of mapping matrices for meantone, all of which simply give you different generators. You are not even restricted to having the first generator be an octave, although it is usually useful to do so.

However, the mapping matrix below is also a valid matrix for meantone:

[ 7 11 16]
[12 19 28]

There are three ways to think about this matrix:

1. The meantone mapping matrix where the generators are, oddly, |d2 A1>, or the diesis (C-B#) and the chromatic semitone (C-C#) -- (yes, this does generate the whole lattice)
2. The matrix you get by simply stacking the 5-limit patent vals for 7-EDO and 12-EDO (which define meantone), pre-reducing to the usual set of generators
3. A way to measure the "diatonic interval class" and "chromatic interval class" for any meantone interval

These are all equivalent, thanks to linear algebra. However, it is the third interpretation which is most interesting for the purposes of diving into a hierarchical, layered representation of interval classes, and hence understanding how to generalize chromaticism to the microtonal setting.

Chromatic and Diatonic Interval Classes

Again, we are looking at the following somewhat unusual meantone mapping matrix:

[ 7 11 16]
[12 19 28]

And of course, we can read this matrix the usual way, with the columns giving the mapping for 2/1, 3/1, and 5/1, just in this weird |d2 A1> or |diesis chromatic-semitone> basis. Just for thoroughness, if we do so, we get:

• 2/1 maps to |7 12> (7 dieses plus 12 chromatic semitones)
• 3/1 maps to |11 19> (11 dieses plus 19 chromatic semitones)
• 5/1 maps to |16 28> (16 diesis plus 28 chromatic semitones)

You may wonder: why would anybody ever care about using this basis?

Well, somewhat mysteriously, this basis tells us the "meantone diatonic interval category" and "meantone chromatic interval category" for any interval. So for 2/1, above, that |7 12> tells us that 2/1 maps to 7 diatonic steps (counting from 0, so an "eighth" if you count from 1), and maps to 12 chromatic steps. 3/1 maps to |11 19>, so 11 diatonic steps (or a "twelfth" if you count from 1), and 19 chromatic steps. And so forth.

How does this work? Is it mere coincidence?

No! It's simply a nice property of linear algebra, and one which we can make great use of in generalizing chromaticism. In this case, it turns out that all of these things become much simpler to understand if you totally forget about the |d2 A1> interpretation. Instead, we will develop a dual, complementary interpretation, one which involves us looking at the rows rather than the columns.

You can see that the rows of this matrix are just the patent vals for 7-EDO and 12-EDO stacked on top of each other. Basically, you can completely forget that these happen to define any sort of meantone basis at all, and just look at how monzos map onto these two vals independently. If you do so, then you get the following:

• In 7-EDO (<7 11 16|):
  • 2/1 is a type of "7-step interval" (7 steps to 2/1, or a type of "eighth")
  • 3/1 is a type of "11-step interval" (11 steps to 3/1, or a type of "twelfth")
  • 5/1 is a type of "16-step interval" (16 steps to 5/1, or a type of "seventeenth")

This is literally what <7 11 16| means. Note that if you simply add one to these, you get the usual diatonic interval classes for each of these intervals. The only reason you need to add one is because we conventionally count interval classes starting at "one" for the unison, whereas with mappings we start at "zero."

Implied from above, by adding and subtracting mappings linearly, we get:

• • 3/2 is a type of "4-step interval" (4 steps to 3/2, or a type of "fifth")
  • 5/4 is a type of "2-step interval" (2 steps to 5/4, or a type of "third")
  • 6/5 is a type of "2-step interval" (2 steps to 6/5, or a type of "third")
  • 5/3 is a type of "5-step interval" (5 steps to 5/3, or a type of "sixth")

Again you will note that this wondrously gives us the correct diatonic interval category for each 5-limit rational. This is simply due to the way that mappings work: the 7-EDO is literally an equidiatonic scale, so once you specify that 2/1 is an octave (7 steps), 3/1 is a twelfth (11 steps), and 5/1 is a seventeenth (16 steps), everything else falls into place in the perfect lattice-like fashion.

You can also see, that using this metric alone, we get the same thing for 5/4 and 6/5: a mapping of two steps, indicating that we have a "third." This shows us that while the 7-EDO patent val gives us the correct "diatonic" interval category for each rational, and nothing more: it does not distinguish between the major, minor, augmented, or diminished versions of intervals.

Although we do not have as nice a terminology for 12-EDO, we can the same type of thing:

• In 12-EDO (<12 19 28|):
  • 2/1 is a type of "12-step interval" (a type of "12\12")
  • 3/1 is a type of "19-step interval" (a type of "19\12")
  • 5/1 is a type of "28-step interval" (a type of "28\12")

Likewise, for more complex rationals, we get

• • 3/2 is a type of "7-step interval" (a type of "7\12")
  • 5/4 is a type of "4-step interval" (a type of "4\12")
  • 6/5 is a type of "3-step interval" (a type of "3\12")
  • 5/3 is a type of "9-step interval" (a type of "9\12")

Likewise, we can see that this val gives us the correct "chromatic" interval category for any 5-limit interval, expressed as a number of 12-EDO steps. By "chromatic", in this case, I mean it gives you the correct number of "generic steps" in the chromatic scale, even though the meantone chromatic scale is unequal and in fact an MOS, containing two different specific interval sizes for each generic interval class.

We can literally just concatenate these two valuations to get the tempered monzo. This is the same exact thing that we did in the other interpretation, but rather than starting with the columns representing prime mappings and adding them, we instead did each row independently, and then concatenated them. So for example, 2/1 maps to |7 12>, 3/1 maps to |11 19>, 5/1 maps to |16 28>, 3/2 maps to |4 7>, 5/4 maps to |2 4>, 6/5 maps to |2 3>, and 5/3 maps to |5 9>.

To make this much clearer, here is an example. The interval 75/64 is a 5-limit "detempering" of the meantone augmented second. Let's try seeing how that maps, and compare with 6/5:

• In 7-EDO (<7 11 16|):
  • 75/64 is a type of "1-step interval" (a type of second)
  • 6/5 is a type of "2-step" interval (a type of third)

• In 12-EDO (<12 19 28|):
  • 75/64 is a type of "3-step interval" (a type of "3\12")
  • 6/5 is a type of "3-step interval" (a type of "3\12")

Or, in tempered monzo form, we get

• Concatenating 7-EDO and 12-EDO:
  • 75/64 maps to |1 3>
  • 6/5 maps to |2 3>

And now we see what is going on! Note that 75/64 and 6/5 map to the same exact position chromatically, represented by the second coordinate: they are both, generically speaking, a type of "3\12", or represented by 3 generic steps in the chromatic scale. However, they map to different positions diatonically, represented by the first coordinate: one is a type of "second" (represented by a mapping of "one step"), and the other is a type of "third" (represented by a mapping of "two steps").

To see a contrasting example, let's compare 5/4 and 6/5. Skipping the individual mappings and just going straight to the concatenation, we get

• Concatenating 7-EDO and 12-EDO:
  • 5/4 maps to |2 4>
  • 6/5 maps to |2 3>

And now we have the reverse situation: 5/4 and 6/5 map to the same position diatonically, which is two steps (or a third, counting from one), but they map to different positions chromatically, in that 6/5 maps to 3\12 and 5/4 maps to 4\12.

We can put this all together to arrive at our main result, which is in some sense the primary theorem we will use when understanding layered interval perception:

Result #1: Any meantone interval can be uniquely specified by simply giving its generic diatonic interval position, and its generic chromatic interval position.

Furthermore:

Result #2: For any 5-limit JI interval, if you specify its mapping when tempered to 7-EDO, and its mapping when tempered to 12-EDO, this is equivalent to meantone-tempering it, and giving the above meantone-tempered representation.

Diatonic, Chromatic and Enharmonic Movements

Using the above concepts, we can rigorously define "diatonic," "chromatic," and "enharmonic" modulations. We will stick with meantone at first, but then generalize this to other temperaments.

Suppose we start with some note, and then we want to move upward by "step" in some direction. There are three possible ways to do this, which are:

• Movement by |1 1> (diatonic coordinate changes, chromatic coordinate changes)
• Movement by |0 1> (diatonic coordinate doesn't change, chromatic coordinate changes)
• Movement by |1 0> (diatonic coordinate changes, chromatic coordinate doesn't change)

The first one is called diatonic, and corresponds to motion by a diatonic semitone (or "minor second"). Note that |1 1> is simply the representation of the meantone diatonic semitone in this basis. That the diatonic semitone is |1 1> tells you that if you add this to any other interval, both the diatonic interval position and the chromatic interval position change by 1. So if you add this to a major third, which is |2 4> (so it maps to 2\7, or a type of "third", and 4\12 in the background), you will get |3 5> which is a perfect fourth.

The second is called chromatic, and corresponds to motion by a chromatic semitone (or "augmented unison"). |0 1> is simply the representation of the chromatic semitone in this basis, and it tells you that if you add this to any other interval, the diatonic interval position will be held constant while the chromatic interval position will change by 1. For example, if you add this to a minor third, which is |2 3>, you get |2 4> which is the major third. Again, note that |2 3> is a type of "third" that maps to 3\12 in the background, and |2 4> is a type of "third" that maps to 4\12 in the background, or the minor and major third respectively.

The last one is called enharmonic, and corresponds to motion by what in meantone has historically been called the "diesis," but which you could reasonably also call an "enharmonic semitone" (or "diminished second"). The enharmonic semitone maps to |1 0>. This tells you that if you add this to any other interval, the diatonic coordinate changes, but the chromatic coordinate stays constant.

We defined this for meantone, but using our "Result #2" above, we can easily see which 5-limit JI "semitones" map to which category. For example, 16/15 is easily shown to map to |1 1>, meaning it is both a type of "second" and one step in the background chromatic scale, and hence the combination of these two specifies that it is a "diatonic semitone." Likewise, 25/24 maps to |0 1>, and 128/125 maps to |1 0>.

Some intervals, however, such as 81/80, map to |0 0> and do not correspond to *any* movement in meantone, either diatonically, chromatically, enharmonically, or otherwise. These intervals are of no importance in strict meantone temperament, having been tempered with the unison.

However, remember that we formed this mapping matrix by starting with 7-EDO, which is sensitive to changes in "diatonic" scale position, and then adding 12-EDO, which is additionally sensitive to changes in "chromatic" scale position.

What if we add one more val, which is now sensitive to changes by 81/80, so that we have three coordinates? Doing so will introduce a new notion, that of a "subchromatic" movement, and thus bring us to a 3-dimensional system.

Subchromatic Movements: Rank-3

(Still to do, insert 5-limit JI, meantone+, superpyth+, and garibaldi examples here, showing how the chromatic/diatonic/enharmonic structure is preserved, but the subchromatic structure changes)

Generalizing to Other Temperaments

(Still to do, insert Porcupine 7&15 porcupine example here)

Which Chroma is a Subchroma? Submodulations

(Talk about how in rank-3 and rank-4, you can change which is a "chroma" and which is a "subchroma")

True Categorical Perception

(Talk about how real categorical perception doesn't use JI)