Why is B♯ higher than C♭ in 31-ET?

Question

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:

Source

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

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, ...?

Answer 1

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 = 2^4/7 ≈ 1.48599428914. If it were lower, then sharps would lower pitches and flats would raise them.
• Maximum: x = 2^3/5 ≈ 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 = 5^1/4 ≈ 1.49534878122. This is Quarter-comma meantone, with pure M3 (5/4), but irrational P4 and P5.
• x = 2^7/12 ≈ 1.49830707688. This is the familiar 12-tone equal temperament.
• x = 2^18/31 ≈ 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 > 2^11/19 ≈ 1.49375896165. The threshold is 19-ET, which makes B♯ is the same as C♭.

Answer 2

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.

Answer 3

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

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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!