Definition of Chromatic and Diatonic Semitone in 31 TET

I am trying to learn about music theories in 31 TET, I learned that 2 fifthtones are a chromatic semitone, 3 fifthtones are a diatonic semitone, 4 fifthtones are a neutral semitone, five fifthtones are a whole tone.

The problem is, when I google the def of chromatic and diatonic semitone, the result tells me chromatic semitones are semitones that share the same letter name, so they are written on the same line in a sheet music, a diatonic semitones are semitones with different letter names, written on different lines on a sheet music.

Then here comes the question, C and C half sharp are 1 fifthtone apart in 31 TET, but they share the same letter name, are they a chromatic semitone? But they are not 2 fifthtones apart! C and C sharp and a half are three fifthtones apart in 31 TET, so they should be a diatonic semitone, but they share the same letter name still, all three notes are still written on the same C line in sheet music, are they all a chromatic semitone? Why not? The definitions are conflicting themselves.

• I've expanded my answer a bit, and I'd be grateful if you would let me know what you think of it. Thanks. Commented Feb 28 at 16:14

One use of 31-TET is to approximate meantone temperaments of the sort that were used for several centuries from the early Renaissance to the mid Baroque. In so doing, one maps a subset of 31-TET to the chromatic 12-tone system. When you do this, you find the mapping described in the question: a chromatic semitone is 2/31 of an octave and a diatonic semitone is 3/31 of an octave. The composition of these two intervals yields a tone of 5/31 of an octave.

For this purpose, you don't have much use for C half sharp, and, if you're using 31-TET in a context where you do want C half sharp, you won't necessarily have a use for the concept of chromatic and diatonic semitones (or you'll need to add another class of semitones or commas or limmas or something, so you can include the 1/31-octave intervals in the system of chromatic and diatonic semitones).

In other words, this definition of chromatic and diatonic semitones belongs to the more limited tonal system in which "sharing the same letter name" implies "and being modified by some integer number of sharps and flats such that they are a semitone apart." G♯, after all, is not a chromatic semitone above G♭. The definition is expressed in a context where half-flats and half-sharp do not exist.

Since half flats and half sharps don't exist in that system, you have to rethink the definitions when you introduce them. After giving it some thought, it has occurred to me that the pitch midway between C and C♯ isn't even necessarily C half sharp. For example, if you are in fact representing quarter-comma meantone, 1/31 of an octave is the enharmonic limma (or comma), which is the distance between B♯ and C. This arises from extending the circle of fifths, where a fifth is 18/31 of an octave. If you do this, you find that the pitch halfway between C and C♯ is D♭♭.

One context in which you might in fact want a C-half-sharp is 7-limit just intonation, and 31-TET is also a pretty good approximation for that. You could certainly use C-half-sharp as the just 9:7 third above A, or as the 7:4 seventh above D♯. But here we have once again ventured beyond the boundaries of the traditional system, which has no way to distinguish, for example, 7:4 sevenths from 9:5 sevenths or 16:9 sevenths. If you invented a system that added harmonic sevenths to just intonation, you would also in that case need to come up with a names for for the interval between C and C-half-sharp. You would also need a name for the interval between C-half-sharp and C♯ and for the interval between C half-sharp and D. None of these would be a chromatic or diatonic semitone.

Another thing to consider is that the definition of chromatic semitone and diatonic semitone holds regardless of your tuning system provided you don't venture into seven-limit tuning or into systems that support fractional accidentals. For example, an ascending diatonic semitone can be defined as the interval you get by going up three fifths and a major third, then down two octaves. That works in Pythagorean tuning, 5-limit just intonation, and quarter-comma meantone. Similarly, in all of those systems an ascending diatonic semitone may be found by going up a fourth and down a major third. This is one reason why the concept of chromatic and diatonic semitones is useful independent of the tuning system in use.

(When you start to look at enharmonic equivalents such as B♯ and C, things start to get a little weird, because the interval between B♯ and C doesn't go in the same direction in all systems. In Pythagorean tuning, B♯ is slightly higher than C, while in 5-limit and meantone, it is slightly lower. This means that you can't establish a general definition of "enharmonic limma or comma" unless you are prepared to accept that its direction will be different in different systems.)

But in any event this system certainly breaks down when you get to 7-limit and above, or when you get to accidentals that change the pitch by less than a semitone. And indeed, this brings us back to probably the core point: the quoted definition does not apply to the interval between C and C-half-sharp:

chromatic semitones are semitones that share the same letter name, so they are written on the same line in a sheet music, a diatonic semitones are semitones with different letter names, written on different lines on a sheet music.

This definition does not apply because the half-sharp isn't a semitone. It is smaller.

• Downvote -- why? Commented Feb 10 at 23:13

The meaning that chromatic and diatonic have in standard major/minor tuning systems doesn't apply outside of them. In 31-TET, those terms have an analogous but distinctly different meaning. In X-TETs, and 31-TET specifically, they are defined by the number of steps that comprise the interval. In general, chromatic intervals will have a smaller number of steps, and diatonic intervals will have larger ones.

In major/minor tonality, chromatic and diatonic intervals can be identified by their letter names: C-C# is a chromatic interval, while C-Db is a diatonic one. However, outside of tonality, those letter-based definitions don't necessarily apply. In 31-TET they do not.

• "In 31-TET, those terms aren't defined" But I have seen a lot of sources about 31 TET using these two terms for 2 and 3 steps intervals. Commented Feb 10 at 1:08
• @EaryChow You'll have to post those sources. In order for diatonic and chromatic to have meaning, there has to be a conception of "key". So the definitions would depend on how those sources choose to define "key" in 31-TET. Commented Feb 10 at 1:21
• Wikipedia is one of them, they listed a bunch of interval names, and 2 & 3 steps intervals are called chromatic and diatonic semitone. en.m.wikipedia.org/wiki/31_equal_temperament#interval_size Commented Feb 10 at 1:28
• And here is 31 et dot com 31et.com/interval/3 Commented Feb 10 at 1:31
• "But once you leave that system — which one does when dealing with 31-TET": not necessarily. One reason for 31-TET's popularity is that it can be used as a quarter comma meantone temperament. To do that, you map (a subset of) the 31 tones to the tonal system of standard music notation. It is in the context of this mapping that 2/31 of an octave corresponds to a chromatic semitone and 3/31 of an octave to a diatonic semitone, Thus C sharp is 2/31 of an octave above C, D flat is 1/31 of an octave above that, and D is 2/31 of an octave higher still. Commented Feb 28 at 11:32

The problem is, when I google the def of chromatic and diatonic semitone, the result tells me chromatic semitones are semitones that share the same letter name, so they are written on the same line in a sheet music, a diatonic semitones are semitones with different letter names, written on different lines on a sheet music.

A chromatic semitone is the amount a sharp raises by and a flat lowers by. A diatonic semitone is the smaller of the two intervals that you'll find in the natural modes of the diatonic scale. Two notes a chromatic semitone apart make an augmented unison, while two notes a diatonic semitone apart form a minor second.

Then here comes the question, C and C half sharp are 1 fifthtone apart in 31 TET, but they share the same letter name, are they a chromatic semitone?

Not really, they are one diesis apart. In 31-TET, sharps raise by two steps and flats lower by two steps, so a half-sharp raises by one step and half-flat lowers by one step. Since one step is also the size of a diesis (diminished second, such as C♯–D♭), half-sharps and half-flats effectively raise and lower, respectively, by one diesis. More generally, any tuning where the sharp raises by an even number of steps will enable true half-sharps and half-flats.

But they are not 2 fifthtones apart! C and C sharp and a half are three fifthtones apart in 31 TET, so they should be a diatonic semitone, but they share the same letter name still, all three notes are still written on the same C line in sheet music, are they all a chromatic semitone? Why not?

C to C sesquisharp (C sharp and a half) is a sesquiaugmented unison, which in 31-TET is 3 steps, making it enharmonically equivalent to a minor second or diatonic semitone.

This is just like how C♯ and D♭ are the same pitch in 12-TET. There are many cases where two or more intervals have the same size, or how two or more notes have the same pitch, but have to be spelled differently depending on the context. Unless you're composing a microtonal work, you'd spell 3 steps as a minor second.

(In 31-TET, half-sharps and half-flats can also be used in non-microtonal contexts to reduce the need for double sharps or double flats in key signatures. For example, the key of D half sharp is enharmonically equivalent to E𝄫, and F half flat major is enharmonically equivalent to E♯ major.)

The definitions are conflicting themselves.

Not really, you just need to take some time to understand them more.

• I think your definition of chromatic and diatonic semitones are different from the standard definition. The standard definition says as long as they have the same letter name and they write on the same staff position on the sheet music they are chromatic semitone. It doesn't quite matter how many steps between them there, it only cares about the letter name. This is the standard definition you can find everywhere on the internet. Commented Feb 10 at 13:21
• @EaryChow I'm having a hard time understanding what you're trying to say.
– user59346
Commented Feb 10 at 13:26
• Your answer includes your definition of the two semitones, it seems to discuss the relative interval size between them. But Googling the terms tells that the definition only has to do with the letter name. As long as the two notes starts with the same letter "C", and they are semitone, then it's a chromatic semitone. C and C sharp&half satisfy that definition, which you seem to disagree. Commented Feb 10 at 13:32
• @EaryChow Most of the definitions of musical terms on the Web are specific to 12-TET without ever mentioning this inconvenient fact, or they are heavily dumbed down and end up being too ambiguous. You have to actually put in some effort to figure out how things work yourself so you can verify what you see.
– user59346
Commented Feb 10 at 13:50
• Can you link me some sources that discuss chromatic and diatonic semitones in an non-tuning specific way (I.E. the one size fits all definition across every possible tunings)? Commented Feb 10 at 13:54

tl;dr

The way that we write the notes is arbitrary, and surprisingly disconnected from the actual interval sizes.

"Chromatic semitone" doesn't mean "any interval between that someone decides to represent with the same base note name". It means the difference between the diatonic semitone and a whole tone. But this will normally look like an interval between two notes represented with the same note name, because the notation was invented for a tuning where that makes sense.

"Diatonic semitone" doesn't mean "the interval between E and F", and "whole tone" doesn't mean "the interval between C and D". Instead, these terms mean intervals such that five tones plus two diatonic semitones add up to an octave. Their exact sizes depend on the tuning. But again, the intervals will normally look this way, because the notation was invented for a tuning where that makes sense.

A note name like "C sharp" means "a note that is a chromatic semitone sharper than C", and similarly for "C flat". But this is specific to the concept of "sharp" and "flat". If you invent some other accidental, then it doesn't describe the same pitch relationship. Historically, (many centuries ago, before the common practice era), you couldn't start the scale there - the note only conceptually existed to support diatonic scales. That's why pianos have a black key / white key distinction.

Similarly, in the theory, the distance between "C" and "D flat" is not a chromatic semitone, even though it happens to be an identical pitch ratio in 12TET. It's a diatonic semitone, but that fact is not because of how we named the notes - it's because of how they relate to each other when creating a diatonic scale.

In traditional Western music theory - including Western tonal music theory - we like to write the notes of a diatonic scale using a different letter for each note, and then use accidentals to indicate the places where the actual note needs to be higher or lower than the standard pitch for that letter. The sharp and flat symbols don't refer to specific amounts of pitch alteration - they mean the amounts needed to make the scale work, in the current tuning.

When we use terms like "half sharp" etc. to try to explain the pitch of notes in a radically different tuning, we are consciously stepping outside of all of that musical theory.

On the other hand, when we apply concepts like "chromatic semitone" to those tunings, the underlying idea is that we can create a diatonic scale using a subset of the notes in that tuning. And by doing so, we directly imply definitions for "tone", "diatonic semitone" and "chromatic semitone".

Some theory

The most common definition of "chromatic semitone" and "diatonic semitone" comes from Western tonal music theory. In this context, to make sense of those terms, we must first have a diatonic scale.

A diatonic scale is one which divides the octave into five tones and two diatonic semitones, where the interval of the diatonic semitones is greater than zero and less than the smallest tone. (Recall that the size of an interval is determined by the ratio of the two pitches.) We play eight notes - the bottom and top of the octave, stepping between them by taking five tone-sized steps and two diatonic-semitone-sized steps.

In general we expect the tones to be at least approximately equal in size, and the diatonic semitone to be roughly half that size.

If we force the tones to be equal in size to each other - let's call the interval T, and the corresponding pitch ratio t - and the diatonic semitones to be equal to each other - let's call the interval D, and the corresponding pitch ratio d - then we have 5t * 2d = 2. (If we additionally force the diatonic "semitone" to be exactly half a tone - i.e. d2 = t - we get 12TET. The corresponding tuning, 12edo, is the unique solution to the equations.)

By definition, a chromatic semitone is the difference between a tone and a diatonic semitone. As long as we have a specific T and D, this is easy: C is defined as T - D (equivalently, c = t / d). (In tunings such as just intonation, where we may have more than one tone size or more than one diatonic semitone size, it stands to reason that we have more than one chromatic semitone size as well.)

Why we have these semitone sizes in 31TET

When we embed a diatonic scale within some other edo tuning, naturally we choose a specific number of steps to represent a tone, and some other number of steps - at least one, and fewer than the number representing a tone - to represent a diatonic semitone. Then, of course, the chromatic semitone is the difference.

To define these concepts in 31TET, we must solve 5T + 2D = 31, for integer T and 0 < D < T. This has a unique solution T = 5, D = 3 (thus C = 2). So those are the definitions we use.

As it happens, this gives us a diatonic semitone which is larger than the chromatic semitone. The opposite is also possible. For example, in 17TET, we get a unique solution T = 3, D = 1, C = 2. (In this tuning, the diatonic semitone very closely approximates the pure interval 25/24.)

Then here comes the question, C and C half sharp are 1 fifthtone apart in 31 TET, but they share the same letter name, are they a chromatic semitone?

The problem with this entire line of questioning is that it is ill defined.

Sheet music is specifically designed for Western music theory. Of course if we try to apply it to a system that divides the octave more than twice as finely, we need to either invent new supplemental notation, or else embrace enharmonicity in a way that most musicians will find very difficult to understand.

Terms like "C" pertain specifically to the diatonic scale. Tunings don't have note names; scales do. It is the same as if you tried to ask why that physical key on a standard keyboard has both "C sharp" and "D flat" as names. Neither of those is a name for the frequency of the sound produced when you press that key on the piano. They are names for the idealized note that is part of an idealized scale, which does not specify exact frequencies but which is instead represented by frequencies in a tuning, which in turn are approximated by sounds produced by a physical instrument.

And notice: on a traditional keyboard, the same key represents both of those note names. But in 31TET, the note "a chromatic semitone above C" and the note "a chromatic semitone below D" - whether you want to call those notes "C sharp" and "D flat", or invent a completely new notation - are distinct.

The reason the same key plays "A flat" in a C minor scale but "G sharp" in an A major scale, is that this key represents the frequency that is desired within each context, while allowing us to use a different note name (possibly with accidentals) for each note of these diatonic scales. We say "G sharp", for example, because the desired note is the seventh in the scale but is noticeably sharper than a standard G.

The term "C half sharp" is an invention created to allow musicians and theorists to express scales (and melodies) in the context of a tuning that includes more notes than are required for a diatonic scale. It's a way to describe that pitch in a way that's intuitive to people familiar with Western tonal music theory, without having to extend the wheel of fifths until we actually land on that note.

The fact that "C" and "C half sharp" share a note name is a consequence of how the terms were constructed. It has nothing to do with how semitones are defined. The entire point of the "half" is to denote that an existing "semitone"-like interval is being further divided.

In other words, a diatonic major scale starting on that note isn't really "C half sharp major"; it could be "D double flat major", or possibly even "B triple sharp major". That's what's needed in order to begin and end there, but use a different name for each note, with the standard system of accidentals.