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 are either 9/8 or 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♭.