Talk:Diatonic, Chromatic, Enharmonic, Subchromatic

Quartertones and Diatonic-Chromatic Distinctions

Hey, I've been using quartertones with a kind of diatonic-chromatic distinction between the for a while. Specifically, two "parachromatic" quartertones add up to a chromatic semitone, but one "parachromatic" quartertone and one "paradiatonic" quartertone add up to a diatonic semitone. Specifically, I define this by breaking up a 9/8 whole tone into four parts, with three of the parts each being the same interval, in this case 33/32, while the last part is different, in this case 4096/3993. From what I've gathered, this seems to be the best way of actually breaking up the whole tone into quartertones, even in JI, as compared to other JI subgroups, the 2.3.11 subgroup accomplishes this division with the lowest odd-limit possible, with 4096/3993 only having four digits in either the numerator or the denominator, just like with the apotome. I bring this up here because I wonder what the people editing the page would think of this. --Aura (talk) 01:22, 21 April 2021 (UTC)

I think this is a good idea. It is definitely useful to have something corresponding to |0 1/2⟩, which you call parachromatic, and |1 1/2⟩, which you call paradiatonic, being two different ways to generalize a quarter tone. The first would keep the diatonic class the same but move to a chromatic class intermediate between two chromatic steps; in simple terms |0 1/2⟩ would be the interval between C and C‡, and also between C‡ and C#, etc. The interval |1 1/2⟩ would instead be the interval between C and Ddb, or between C‡ and Db, which also subtends half of a chromatic step but now also increments the diatonic category. The two sum to |1 1⟩ which would be a diatonic scale step. So these are two different kinds of "quarter tones" and incorporating them into the theory would generalize this to e.g. porcupine, mavila, whatever, so that each has a coherent notion of generalized "quarter tones" in this way.

There is also an easy and important generalization to 1/nth-sharps. Third-sharps are also important and exist in 22-EDO, as are fourth-sharps, which exist in 27-EDO. This is easy to formalize for arbitrary 1/n sharps as the intervals |0 1/n⟩ and |1 1/n⟩. The first is the difference between C and C-1/nth-sharp, and the second is the difference between C-(n-1)/nth-sharp and D flat.

I also think it's somewhat useful to look at intervals which split the diatonic semitone in half, such as you get in 19-EDO. These intervals would be |1/2 1/2⟩. You can also round it out and split the enharmonic step in half, if you want (which for meantone is equivalent to splitting the octave in half, giving you a perfect semi-octave interval between F# and Gb, or half an enharmonic step between F# and Gb).

I would perhaps suggest "semichromatic" instead of "parachromatic" as it seems like the correct prefix...

Also: If |0 1/2⟩ is parachromatic or semichromatic then what is |1/2 1/2⟩? That is literally half of a diatonic step so I would expect it to be paradiatonic or semidiatonic, but that is what you are calling |1 1/2⟩. Thoughts? Mike Battaglia (talk) 20:59, 21 December 2021 (UTC)

Okay, first things first. The term "parachroma" in particular has connections to MOS scales in general- specifically, it's the interval that forms the difference between a neutral scale degree and the nearest major or minor form of the same scale degree on the other, thus, it is an interval that can be easily tempered to equal half of a MOS-chroma. On the page detailing my ideas on functional harmony, I go into more of the reasons why I chose the prefix "para-" (meaning "resembling" and or "alongside") to denote the quartertones. Specifically, it has a lot to do with how, for example, 11/8 and 128/99 act almost like a major-minor pair in a Pythagorean diatonic context except for firstly not being located on the diatonic scale, and secondly for the interval between 11/8 and 128/99 not being the same as the apotome in JI, thus, the designation of "paramajor fourth" for 11/8 and "paraminor fourth" for 128/99.

Therefore, while in basic diatonic tunings, the parachroma is a type of quartertone, it is something else in other tunings, and the interval that remains after subtracting as many parachromas from a Major MOS-step as possible without resulting in a negative interval is called the "parastep", or in diatonic contexts, the "paralimma". Thus, when it comes to different tunings, we should consider also the ratio between the Minor MOS-Step and the parastep, and the ratio between the parachroma and the parastep. So, I would think that terms like "semichromatic", "semidiatonic", "triendiatonic" and "trienchromatic" can be used alongside the term "parachromatic", but one must keep the significance of each in mind, since parachromas can exist even in the absence of the ability to split the MOS-chroma perfectly in two. --Aura (talk) 01:01, 22 December 2021 (UTC)

While I'm thinking about it, we need to have further discussion on the nature of |1/2 1/2⟩ intervals and I need to see how this relates to the traditional diatonic and chromatic classes. Something tells me we also need to think more about ambiguous intervals- intervals that can either be diatonic or chromatic- but thankfully Flora and I seem to have a few ideas on hand. However, we may need to discuss this idea of ambiguous intervals in further detail on Discord. --Aura (talk) 01:21, 22 December 2021 (UTC)

That makes sense. Ok, so the idea is that semichromatic or trienchromatic refer to |0 1/2> and |0 1/3>, and parachromatic refers in general to |0 1/n> for some n? And then subchromatic is |0 0 1>. From a meantone standpoint, |1/2 1/2> would be like, you've split the fourth in half to get a semaphore-ish thing, and now there's an interval right between D and Eb which is half of a diatonic semitone sharp of D. I have no time to get involved in Discord right now but curious what you folks think up. I'm not sure if the ambiguous intervals you are talking about exist unless you're in some lower-rank temperament though, like an EDO, in which there are now enharmonic equivalences that don't exist in the original higher-rank temperament... Mike Battaglia (talk) 09:14, 22 December 2021 (UTC)

Honestly, the term "parachromatic" is mainly distinct form "semichromatic" in that the term "parachromatic" also includes detemperings of |0 1/2⟩ while "semichromatic" is a strict |0 1/2⟩, however, the chromatic and diatonic semitones (the latter of which are basically Minor MOS-Steps) can differ dramatically in size depending on the system in question, and this leads to the parachromas having a number of wildly different sizes relative to the Major MOS-Step. I should also point out that "diatonic" class intervals, when broken down into smaller intervals, break down into a single "diatonic" class interval and one or more "chromatic" class intervals. As to the ambiguous intervals, one class of these tends to show up when the same interval can act as either a "diatonic" or and "chromatic" intervals due to "enharmonic" considerations of any stripe- these tend to happen when you temper out intervals like the schisma or the nexus comma, the latter of which strangely splits the 4/3 in two equal parts of 1536/1331~1331/1152 in the 2.3.11 subgroup. However, there's another class of these intervals, that requires further discussion. --Aura (talk) 15:19, 22 December 2021 (UTC)

It sounds like your idea is somewhat different but still interesting, and it would at least be a good idea to have a theory of |1/2 1/2⟩ and so on, whatever you call it. Mike Battaglia (talk) 00:11, 23 December 2021 (UTC)

I've been thinking about this recently, and I do indeed have a theory about it. In the case of |1/2 1/2⟩, we're talking a subchromatically modified parachroma and a subchromatically modified paralimma- at least as long as we're using the diatonic scale as a frame of reference. Effectively, what this means from the mathematical standpoint is that parachromatic intervals are found in their own row of this matrix between the chromatic and subchromatic rows. Thus, if we start with this set of rows...

[ 7 11 16 20]

[12 19 28 34]

We can tack on another row for 24edo like this...

[ 7 11 16 20]

[12 19 28 34]

[24 38 56 67]

What's more, since prime 11 in heavily involved in parachromatic representation relative to the Pythagorean Diatonic Scale, we need to add the 11-prime's column to the matrix...

[ 7 11 16 20 24]

[12 19 28 34 42]

[24 38 56 67 83]

You follow me so far? If so, then now is the part where things get interesting...

If we're going to both continue with EDOs and see the syntonic comma as one of the relevant subchromas, that means we need to choose an EDO which represents it well, and 53edo presents a decently accurate for the syntonic comma as a subchroma. However, one thing we know from the matrix so far is that prime 11 is poorly represented by both 12edo and 53edo. Thus, we need an additional subchroma to help define the 11-prime better relative to the syntonic comma, and thankfully, there is one- the rastma, which is almost exactly one third the size of the syntonic comma in JI. What's more, when you multiply 53edo by three to get 159edo, not only do you get a fair representation of the rastma relative to the syntonic comma, but you also get better representations of both primes 7 and 11. Even more strangely, all four of these EDOs are united by a single temperament. Thus, since the syntonic comma and the rastma are both relevant subchromas and both are represented well by 159edo, we need to tack on a 159edo row to the matrix...

[ 7 11 16 20 24]

[ 12 19 28 34 42]

[ 24 38 56 67 83]

[159 252 369 446 550]

Now, in light of the concepts previously covered by the article, we see that using this new tempered monzo notation, we can rigorously define not only "diatonic," "chromatic," and "enharmonic" steps, but also "parachromatic", "paradiatonic", "subchromatic", and even "dietic" steps (the term "dietic" is related to "diesis")- come to think of it, a lot of the stuff in the section of the article dealing with subchromatic scales applies here, only, we're actually dealing with a Rank-4 lattice rather than a Rank-3 lattice. Anyway, we can start building our discussions of how parachromatic subchromatic and enharmonic intervals relate to each other based on this, and from there, go on to discuss the nature of semilimmas in greater detail. The premise that I assume with enharmonic motion is that the only coordinate that changes is the diatonic coordinate- the chromatic, parachromatic, and even subchromatic coordinates don't change. On the other hand, if the subchromatic coordinate changes in addition to the diatonic coordinate, we have a "dietic" motion. As for parachromatic motion, two of these do equal some type of chromatic motion, but the result isn't necessarily clean on the subchromatic level. Am I making sense so far? --Aura (talk) 17:36, 21 January 2022 (UTC)

Outstanding article

Seriously I think this is one of the best articles on here. If there was some kind of star I could nominate it for I'd do that

- Sintel (talk) 12:45, 21 December 2021 (UTC)

Seconded. I know I can dig up some early emails to Dave saying how this page was a huge help to me. --Cmloegcmluin (talk) 14:48, 21 December 2021 (UTC)

Thirded, since it helped me to ultimately define how to make the layers of syntax for Syntonic-Rastmic Subchroma Notation. As an example, building off of what has been stated here, I'd add another row of syntax based on 24edo to cover "parachromatic" and "paradiatonic". --Aura (talk) 17:00, 21 December 2021 (UTC)

Glad you all enjoyed it! Mike Battaglia (talk) 20:59, 21 December 2021 (UTC)