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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.

3 Answers 3

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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).

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  • Downvote -- why?
    – phoog
    Feb 10 at 23:13
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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.

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    "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.
    – Eary Chow
    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.
    – Aaron
    Feb 10 at 1:21
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    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
    – Eary Chow
    Feb 10 at 1:28
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    And here is 31 et dot com 31et.com/interval/3
    – Eary Chow
    Feb 10 at 1:31
  • After 1:25 of this YT video, they called it "Chromatone" and "Diatone", I think they just combinef chromatic and semitone into chromatone, same for diatone. youtu.be/FtPp5QvEops?si=bmmG4UxGQKwxYvMT
    – Eary Chow
    Feb 10 at 1:46
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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.

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    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.
    – Eary Chow
    Feb 10 at 13:21
  • @EaryChow I'm having a hard time understanding what you're trying to say.
    – 000
    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.
    – Eary Chow
    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.
    – 000
    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)?
    – Eary Chow
    Feb 10 at 13:54

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