Xenharmonic Wiki

Diatonic, chromatic, enharmonic, and subchromatic steps

Revision as of 20:43, 12 October 2018 by Mike Battaglia (talk | contribs) (write up stuff about chromaticism)

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" (or "diminished second"). The diesis 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 coordinate, representing something like a "comma adjustment" from our diatonic/chromatic scale position? Doing so will introduce a new notion, that of a "subchromatic" (or "commatic") movement, and thus bring us to a 3-dimensional system.


Subchromatic Movements: Rank-3

Suppose that we want to now add a new type of interval adjustment chroma to our system: one which is a small "comma" that you can adjust the other intervals by.

This adjustment does not alter our diatonic or chromatic interval category, nor does it change our "coarse position" on the chain of fifths. It simply shifts things in a new direction, totally independent of the first two.

In this case, suppose we want to stay in meantone, but add a new 7-limit "comma" that we can change things by to turn the meantone major third into a "supermajor third" of 9/7, and the meantone minor third into a "subminor third" of 7/6. This comma would be a tempered version of 36/35.

To represent this system, we first establish how we want to establish 9/7 and 7/6 in terms of diatonic and chromatic interval mappings. For instance, is 7/6 a type of "third," or a type of "second?" There is no right answer to this; these are simply two different systems, with two different types of hearing, corresponding to two different mappings. For the purposes of this example, we will lay out a system whereby 9/7 and 7/6 are considered types of altered "third."

Doing so means that we extend our meantone mapping matrix to the 7-limit as follows, by adding one more column: 

[ 7 11 16 20]
[12 19 28 34]

This new column tells us that 7/1 maps to 20 steps of 7-EDO, or is considered a type of "twenty-first"; if you reduce within the octave this means it is a type of "seventh." Likewise, 7/ maps to 34 steps of 12-EDO, or if you reduce to the octave, is a type of "10-step interval."

To add our new comma, then, we simply add the following row to the above: 

[ 7 11 16 20]
[12 19 28 34]
[ 0  0  0 -1]

This last entry coordinate tells us that 7/1 maps to a type of "twenty-first", maps to a "10-step dodecaphonic interval," and also maps to an adjustment level of "minus one" of these new comma-sized adjustment intervals. If you reduce to the octave, this means that 7/4 is a type of "seventh," fits into the 10-step dodecaphonic bucket, and has an adjustment level of minus one.

Note also that 5/1, in the above mapping, has an adjustment level of 0. This means that the way we designed the mapping hasn't broken our 5/4 off the main chain of fifths, so we are still in a meantone temperament where 81/80 vanishes.

The mapping above can literally be read as saying that 5/1 maps to |16 28 0>, or when reduced to the octave, 5/4 maps to |2 4 0>. Read using the interpretation given previously, this says that 5/4 is a heptatonic "2-step" interval (or a type of "third"), that it is a dodecaphonic "4-step" interval, and that 5/4 does not have any adjustment by this new comma-sized interval we've introduced.

(Of course, we did not have to map it this way, but we did it to keep things consistent for pedagogical reasons with the meantone example from before.)

This gives us a new type of interval adjustment, in a totally independent way of the ones discussed before. Revisiting the old ones, and adding the new one, we now have the following:

  • |1 1 0>: "Diatonic" adjustment (diatonic and chromatic interval classes go up by one)
  • |0 1 0>: "Chromatic" adjustment (only the chromatic interval class goes up by one)
  • |1 0 0>: "Enharmonic" adjustment (only the diatonic interval class goes up by one)
  • |0 0 1>: "Subchromatic" adjustment (diatonic and chromatic classes preserved, things only change on this new layer)

Now, a note about terminology. Some people may think it more intuitive to use the term "commatic" adjustment for the last one, rather than "subchromatic." However, I deliberately use the term "subchromatic" for the following reasons:

  1. The term "commatic" is also often used to refer to things that are tempered out.
  2. We are now deliberately talking about something that is not being tempered.
  3. This new interval really does function like a type of "chroma," in that it leads to a lattice of possible adjustments.
  4. However, since this chroma is operating on a new layer that is "underneath" the usual diatonic and chromatic one, I will call it a "subchroma."
  5. Such motions, being distinct from chromatic or enharmonic ones, are called "subchromatic."

Here are how the following "thirds" map according to this system:

  • |2 3 -1>: 7/6
  • |2 3 0>: 6/5, 32/27
  • |2 4 0>: 5/4, 81/64
  • |2 4 1>: 9/7

Note how descriptive the notation is: the first coordinate in all these cases is "2", representing a "2-step" or "third" diatonic mapping. The second coordinate changes from 3 to 4, giving you a 3\12 mapping for the first 3, and a 4\12 for the second 3, telling you the first two are a kind of minor third and the second two are a kind of major third. The last coordinate gives you the subchromatic adjustment.

Putting that all together, this shows you that in this particular system, while 7/6, 6/5, 32/27 are all equated diatonically (2-step) and chromatically (3-step), that 7/6 is distinguished from the other two subchromatically (-1 subchroma).

We can, of course, use different mappings for the above than extending meantone. For example, suppose we really do want to break 5/4 off the main chain of fifths, and specify that it is instead one subchroma lower from the Pythagorean major third, just like 9/7 is one adjustment higher. Then we would instead get the following mapping matrix: 

[ 7 11 16 20]
[12 19 28 34]
[ 0  0 -1 -1]

where the only difference is that the 5-coordinate for the last row has been changed to -1. Now the mapping for 5/1 would be |2 4 -1>, specifying that it is a type of 2-step interval diatonically (or a "third"), a type of 4-step interval chromatically, and is adjusted down by one subchroma. We then get the following nice sequence of thirds:

  • |2 3 -1>: 7/6
  • |2 3 0>: 32/27
  • |2 3 1>: 6/5
  • |2 4 -1>: 5/4
  • |2 4 0>: 81/64
  • |2 4 1>: 9/7

So you can now see that while 7/6, 32/27, and 6/5, are all equivalent diatonically and chromatically, they are all subchromatically distinct.

This is an example of the hemifamity temperament, and now our subchroma represents both 64/63 and 81/80.

We could likewise decide to go in a "superpyth" direction: put 9/7 on the main chain of fifths, and break 5/4 off from it. We would then get 

[ 7 11 16 20]
[12 19 28 34]
[ 0  0 -1  0]

And now we get the following mappings for the thirds:

  • |2 3 0>: 7/6, 32/27
  • |2 3 1>: 6/5
  • |2 4 -1>: 5/4
  • |2 4 0>: 9/7, 81/64

So you can see that now it is 6/5 that is distinguished subchromatically from the other two.

This concept doesn't just apply to the 7-limit, but can even apply just in the 5-limit. For instance, suppose we go with the following 5-limit mapping: 

[ 7 11 16]
[12 19 28]
[ 0  0 -1]

Then we have the following mapping for these 5-limit thirds:

  • |2 3 0>: 32/27
  • |2 3 1>: 6/5
  • |2 4 -1>: 5/4
  • |2 4 0>: 81/64

It so happens that the above system is equal to 5-limit JI, but now given a hierarchical interpretation and a diatonic, chromatic, and subchromatic structure, represented by this coordinate system. Using the above, we can see that we can view 5-limit JI as being generated by a coarse, 7-based diatonic mapping, a coarse, 12-based chromatic mapping, and then one last coordinate representing this new subchroma, which happens to be a just 81/80.

Since in that last row, the 3-coordinate is 0, this means that in this particular interpretation, we consider all 3-limit intervals to have no subchromatic adjustment whatsoever. They are subchromatically "clean," and it is the 5-limit intervals which we consider adjusted. We could change this to get totally different diatonic/chromatic/subchromatic hierarchies for the same exact temperament -- 5-limit JI -- but which represent different modes of hearing it. After all, 5-limit JI is just as much a "comma-adjusted porcupine" as it is a "comma-adjusted meantone". We will return to this later.


A Quick Tangent: Rank-3 "Subchromatic" or "Commatic" Scales

In the above exposition, we started by showing that, rather than going with the usual choice of meantone generators - octave and fifth - we can go with the pair of vals representing the "diatonic" and "chromatic" interval mappings, which is 7-EDO and 12-EDO, to obtain a different (but equivalent) representation for every meantone interval.

The two vals 7-EDO and 12-EDO are sensitive to changes in diatonic and chromatic scale position, respectively, but are not sensitive to changes in 81/80. If we add a third val to our matrix that is sensitive to changes in 81/80, this would give us a third coordinate, and we would have a rank-3 lattice.

Now, if we were going to strictly continue with the paradigm from earlier, we would do so by finding a third val that represents something like a "subchromatic" (or "commatic") scale, within which fit multiple subchroma-altered (or "comma-altered") versions of the chromatic scale. This would be directly analogous to how the chromatic scale fits multiple chroma-altered versions of the diatonic scale.

This would be a very interesting model to develop! For example, we could add a third 15-tone, 17-tone, or 22-tone background scale, for which 81/80 is a single step. Then, assuming we are in the 5-limit, the combination of these three mappings would uniquely specify any 5-limit interval, but do so in a way that specifies a tertiary hierarchical, layered representation of interval mappings, building on the diatonic/chromatic one from before. In this model, there would be simultaneously three levels of hearing, with the third one specifying the interval's mapping along this new "subchromatic" scale.

The relevant mapping matrix would look like this (in the case of 22-EDO): 

[ 7 11 16]
[12 19 28]
[22 35 51]

In the interpretation from before, the first two coordinates still give you the diatonic and chromatic mappings for any 5-limit interval, and correspond roughly to the 7-based and 12-based "rounded off categories" for every 5-limit interval. However, by also adjoining a coordinate that represents a hypothetical "22-based," rounded off categorical perception, we can represent every interval in the 5-limit.

Put differently, in our new interpretation of regular mappings, the combination of these three "modes of hearing" enable us to uniquely understand any 5-limit interval.

We could also use the 17-EDO patent val, using the variant where 5/4 and 6/5 both map to 5 steps, or exactly half of 3/2, as types of middle third: 

[ 7 11 16]
[12 19 28]
[17 27 39]

In this approach, we would map a 5/4 as a "third" diatonically, a "4-step" interval chromatically, and a "6-step" interval in this new "subchromatic" scale. In contrast, 6/5 would be a "third" diatonically, a "3-step interval" dodecaphonically, and a "6-step" interval heptadecatonically. But, 32/27 would be a "third" diatonically, a "3-step interval" dodecaphonically, and a "5-step" interval heptadecatonically, so that it only moves "subchromatically" on that last layer.

This is one very neat interpretation of rank-3 lattices: rather than three "generators", we think of three layers of perception. A better description of this model would involve diving into Fokker blocks, which we will leave for future research.

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)

Retrieved from "https://en.xen.wiki/index.php?title=Diatonic,_chromatic,_enharmonic,_and_subchromatic_steps&oldid=36705"