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I was looking into microtonal series, and started reading a bit on the 31-ET series, and came across something that doesn't quite make sense to me.

I noticed this chart on the wiki page for 31-ET that listed all of the notes in the chromatic scale:

31-ET Chromatic Notes

Source

According to this chart, C♭ is situated lower than B♯, and similarly, F♭ is also lower than E♯:

31-ET FE

Is this chart correct? If so, can anyone explain why it's notated like this, instead of the natural progression being: ..., B, B♯, C♭, C, ...?

  • 2
    Please feel free to change the tags as needed - I didn't see a 31-ET tag, and wasn't comfortable making a new one. – Siyual Apr 30 at 15:18
  • The difference is 39 cents. Also, A# is 39 cents below Bb, and C# is 39 cents below Db. In fact, every step on the scale is 39 cents above the previous. – GalacticCowboy Apr 30 at 17:49
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    Found this by randomly browsing, and first I'd ever heard of 31-ET... but to understand better, it seems to me that the note whose primary name is "C-flat" could also be called "B half-sharp", and similarly "B sharp" could be called "C half-flat"... similar to how on a 12-note scale the same note is used for C sharp and D flat, or for C flat and B etc. – Steve May 1 at 9:41
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    Per the answers, make sure the chart shows E# > Fb (not visible in your post) – Carl Witthoft May 1 at 13:27
  • I don‘t understand the 31-ET, but as Cb and Fb are enharmonic B and E, and at the other hand B# and E# are corresponding to C and F it is seams logic and obvious to me that B# and E# are higher than Cb and Eb. – Albrecht Hügli May 2 at 5:06
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This question seems to arise from a “linear” mental model of notes.

C♭ C C♯ D♭ D D♯ E♭ E E♯ F♭ F F♯ G♭ G G♯ A♭ A A♯ B♭ B B♯ C♭ C C♯

Like a piano keyboard, but somehow with 31 notes per octave instead of 12. (Building or playing such an instrument is left as an exercise for the reader.)

But instead, look at the notes in Circle of Fifths order.

F♭ C♭ G♭ D♭ A♭ E♭ B♭ F C G D A E B F♯ C♯ G♯ D♯ A♯ E♯ B♯

It may be helpful to rearrange alternating notes in this series into two staggered rows, like this:

C♭ D♭ E♭ F G A B C♯ D♯ E♯

F♭ G♭ A♭ B♭ C D E F♯ G♯ A♯ B♯

This produces an isomorphic note layout (specifically, two rows of Wicki-Hayden), in which:

  • Each move in the ↗ direction increases pitch by a perfect fifth (P5).
  • Each move in the ↘ direction decreases pitch by a perfect fourth (P4).
  • Each move in the → direction increases pitch by P5 - P4 = M2.

Note that, if you pick a single frequency ratio (call it x) for the P5, then you get the P4 (2/x, the octave inversion of P5, taking as axiomatic that an octave is a double of frequency) and M2 (x2/2) for free, and from any base pitch (e.g., A4 = 440 Hz), you can calculate the fundamental frequency of any note. If a reasonable choice is made for x, this gives you a syntonic temperament.

(Relaxing the requirement that P5 be a consistent pitch ratio allows other tunings, such as 5-limit just intonation, in which M2 intervals alternate between 9/8 and 10/9 ratios, so that M3 can be a “nice” 5/4, at the expense of making P5 sometimes be 40/27 instead of 3/2. But this over-complicates my analysis here, so ask another question if you wish to discuss the idea further.)

Specifically, the bounds on x are:

  • Minimum: x = 24/7 ≈ 1.48599428914. If it were lower, then sharps would lower pitches and flats would raise them.
  • Maximum: x = 23/5 ≈ 1.51571656651. If it were higher, then B would be higher than C, and E higher than F.

Some possible choices for x are:

  • x = 3/2 = 1.5. This is Pythagorean tuning, with pure P5 (3/2), P4 (4/3), and M2 (9/8).
  • x = 51/4 ≈ 1.49534878122. This is Quarter-comma meantone, with pure M3 (5/4), but irrational P4 and P5.
  • x = 27/12 ≈ 1.49830707688. This is the familiar 12-tone equal temperament.
  • x = 218/31 ≈ 1.49551788235. This the 31-ET under discussion. It is very close to Quarter-comma meantone.

In 31-ET, an octave is divided logarithmically into 31 equal units, and a P5 interval is taken to be 18 of these units. Starting from an arbitrary “origin” note C, and doing all arithmetic modulo 31, the circle of fifths can be labelled as:

  • -144 = 11 = F♭
  • -126 = 29 = C♭
  • -108 = 16 = G♭
  • -90 = 3 = D♭
  • -72 = 21 = A♭
  • -54 = 8 = E♭
  • -36 = 26 = B♭
  • -18 = 13 = F
  • 0 = 0 = C
  • 18 = 18 = G
  • 36 = 5 = D
  • 54 = 23 = A
  • 72 = 10 = E
  • 90 = 28 = B
  • 108 = 15 = F♯
  • 126 = 2 = C♯
  • 144 = 20 = G♯
  • 162 = 7 = D♯
  • 180 = 25 = A♯
  • 198 = 12 = E♯
  • 216 = 30 = B♯

Or, sorting by the modulo-31 pitch class:

  • 0 = C
  • 2 = C♯
  • 3 = D♭
  • 5 = D
  • 7 = D♯
  • 8 = E♭
  • 10 = E
  • 11 = F♭
  • 12 = E♯
  • 13 = F
  • 15 = F♯
  • 16 = G♭
  • 18 = G
  • 20 = G♯
  • 21 = A♭
  • 23 = A
  • 25 = A♯
  • 26 = B♭
  • 28 = B
  • 29 = C♭
  • 30 = B♯

So yes, B♯ is higher than C♭, just like E♯ is higher than F♭. What's important is the consistency of intervals. A perfect fifth is always 18 units, and thus:

  • A whole tone (M2), like from C to D, is always 5 units.
  • A diatonic semitone (m2), like from E to F, is always 3 units.
  • A chromatic semitone (A1), represented by a ♯ or ♭ sign, is always 2 units.

Swapping around B♯/C♭ or E♯/F♭ would break this consistency. And by having ♯ or ♭ always be 2 units, you can similarly construct double sharps/flats of 4 units, or half-sharps/flats of 1 unit, thus filling in the rest of the chart with note names.

In fact, 31-ET is far from the only system that has B♯ higher than C♭. If the note C is assigned a frequency of 1, then B♯ = x12 / 27 and C♭ = 24 / x7, and a bit of algebra reveals that the situation occurs when x > 211/19 ≈ 1.49375896165. The threshold is 19-ET, which makes B♯ is the same as C♭.

  • 4
    The more I read, the more questions I had, but then the more I kept reading, the more those questions were answered. I think this answered every question I had, and every question I didn't know I had. I'm not the greatest when it comes to music theory, but the maths used here made everything fit into place. Thank you so much! – Siyual May 1 at 11:10
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It is because B and C are closer together than the difference between B and B♯ and the difference between C and C♭. That is, they are all some sort of semitone apart.

but this doesn't appear anywhere else in the chromatic series

Actually, F♭ is also lower than E♯.

By contrast, when the natural notes are a (some sort of) whole step apart, as with C and D, the flattened higher note is higher than the sharpened lower note. That is, D♭ is higher than C♯. The same is true of E♭ and D♯, G♭ and F♯, A♭ and G♯, and B♭ and A♯.

To put it in terms of the 31-tone temperament, there are two pairs of neighboring unadorned letter names that are separated by 3/31 of an octave, those being B to C and E to F. The other pairs (A to B, C to D, D to E, F to G, and G to A) are separated by 5/31 of an octave.

Altering any unadorned letter note with a sharp or flat means raising or lowering it by 2/31 of an octave. Therefore, with B to C and E to F, the altered pitches will "cross" because 2 is more than half of 3. With the other pairs of notes, the altered pitches will not cross, because 2 is less than half of 5.

As the Wikipedia article notes, 31-tone equal temperament grew out of attempts to derive an equal temperament that allows (approximate) just major thirds (having a 5:4 frequency ratio). If you derive C♭ and B♯ using just intervals, you also arrive at a C♭ that is lower than B♯. For example, B♯ could be three major thirds above C, while C♭ could be three minor thirds above D:

    C  =    1     D  =   9/8
    E  =   5/4    F  =  27/20
    G♯ =  25/16   A♭ =  81/50
    B♯ = 125/64   C♭ = 243/125
    B♯ = 1.953125 C♭ = 1.944

Here, B♯ is 1158.94 cents, while C♭ is 1150.83 cents.

Finally, in 12-tone equal temperament, of course B♯ is also higher than C♭, since B♭ = C = 1200 cents, while Cb = B = 1100 cents.

  • This makes sense as far as theory is concerned, but why not alter the naming convention for the notes themselves so that it doesn't jump around? The portion that I can't wrap my head around is that we have a note named B# that is higher in pitch than a note named Cb. The logical sequence would be the opposite. – Siyual Apr 30 at 16:14
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    @Siyual presumably because the point of 31-tet is not to invent an entirely new musical system, but to extend the existing one. 31-tet grew out of just intonation and quarter-comma meantone temperament, as noted in the wikipedia article. I'll add another paragraph. – phoog Apr 30 at 16:16
  • Ah, that makes a lot more sense - thank you! – Siyual Apr 30 at 16:33
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    @phoog It would also (most importantly imo) make all of the chords go out of tune. It is a historical convention, but by no means an arbitrary one, it serves a functional purpose. – Some_Guy Apr 30 at 18:23
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    @siyual I think the most important take away here is that this has nothing to do with 31-ET. Even in 12-ET (or just intonation, etc.) you have the same situation because there is only a single semitone between B and C (and E and F). We can't simply change the notation because we have centuries of notation and tradition using this notation. – S. Burt Apr 30 at 19:44
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This isn't quite an answer, but is also too long for a comment, and I think it will point you in the right direction.

The conventions for the spelling of 31TET are related to the conventions of meantone temperaments, and in a sense 31TET can be seen as a special case of a "completed" meantone temperament. And if you pick out a subset of 31TET the intervals are really, really close to quarter comma meantone

I was helped by an unnamed answerer to my question on this here, it really opened my eyes to how 31TET functions.

Is this music (flow my tears) here being played in 31 TET or a form of renaissance unequal temperament such as quarter comma meantone?

The question was about this absolutely beautiful recording of "flow my tears" which has the most deliciously warm "unequal" chords


With that plug and shoutout out of the way, let's try and answer your question:

All of this would also apply to meantone temperaments, most of which are unequal, in that it matters what note you begin on when you derive the other notes (by a chain of tempered fifths in either direction). 31TET is just a special case of this - usually with a meantone temperament you would derive all the notes you need and then stop (usually 12, although sometimes a couple more so you can have say both Ab and G#, or both A# and Bb). but in 31TET, if you keep looping through your fifths, eventually, you get back to where you started.

And even though it might seem weird at first, the nomenclature in use is the really the only logical one; it means when you play a chord spelled out as a certain set of intervals, that's the chord you hear. So the spelling of B# and Cb is related to the fact that B# should be a fifth away from E#, a major third away from G#, a minor third below D#, the M7 of C#, etc. And Cb should be a fifth away from Fb, a minor third away from Ab etc.

My head isn't quite clued into 31TET at the moment, but if you play around with it and build chords and harmonic progressions out of it, then the resulting chords come out like they "should" sound based on their spelling. It is intuitive in the more basic keys, once you start travelling further and further down the circle of fifths rabbit hole it gets a bit weirder but it all still ends up making sense, even when you get to the point of playing in "C half flat" or whatever.

It's been a while since I've been really locked into 31TET, so that's why I haven't gone through the exact details more precise;y, but I think if you look at it and start building chords, or chains of intervals etc., you'll see why the existing spelling system makes the most sense.

A good start is the circle of fifths, if you build 31TET in fifths then the spelling works out like this.

But you'll see that it's not just the fifths, if you pick any given interval (and therefore any chord) then the spellings are the most logical ones. It does result in the chromatic scale written in order looking a bit weird, but that's a small sacrifice I think.

If I get time, I'll edit in the 31 TET circle of fifths (using half flat/half sharp nomenclature not double flat double sharp nomenclature which I don't like) but for now I'll post this and hope it helps!

  • 2
    This is extremely helpful - thank you! I wasn't considering the circle of fifths much at all. – Siyual Apr 30 at 17:19
  • @Siyual indeed. I also did not think of that in my answer, since I was looking at it from the point of view of just intervals. But of course the point of a temperament is to be able to close the circle of fifths, so both approaches will yield the same result. I also note that in the 31-tone system B-half-sharp is enharmonically equivalent to C-flat and that C-half-flat is enharmonically equivalent to B-sharp. – phoog Apr 30 at 18:41
  • glad to help! A curiosity of 31TET too is that (because 31 is a prime number) every interval loops back to itself after exactly 31 repeats, not just the fifth, so you can build a full circle of all the pitches by using any arbitrary interval (although the fifth is the most important one of course). – Some_Guy Apr 30 at 18:44
  • (in 12 you can only do this with fifths (7), fourths (5), and semitones (1), everything else "closes early" because 2,3,4 & 6 are factors of 12) Conversely this also means that 31TET doesn't have any "symmetrical chords" built of repeated patterns (like the dim7 or aug chord in 12TET), so beloved by improvisers for the fact that each chord does multiple jobs. So with increased "pitch resolution" you also sacrifice the some ability to easily "fudge" things, or pivot. I guess "giant steps" wouldn't really work in 31 for example. – Some_Guy Apr 30 at 18:49
  • @phoog That's the fun thing with music theory, often a completely different analysis gets you to the same endpoint, and neither is more "correct" than the other! :) (although yeah, I agree with you that the potential for delicious juicy consonant chords in an EDO framework is the biggest appeal of 31—I'd love to hear some music that really explores its potential to have tiny melodic intervals in a consonant and functional harmonic framework) – Some_Guy Apr 30 at 18:55

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