Outside of Equal Temperament, what decides the spelling of notes in a major scale?

Question

I've been reading up on the history of temperament, and how enharmonic notes are more of a limitation of the modern piano (only one black key), and also mathematically they are the same if you use equal temperament, but the situation is not ideal harmonically.

For example, F-major has a B-flat. I can also see that the A natural is already in use, so having A-sharp wouldn't be feasible (to write down). But my understanding is that historically at least, sharps and flats would differ by a comma?

So is using B-flat in F-major a compromise given the design of the modern piano and the temperament, or is there a more theoretical reason why it was chosen?

(Apologies in advance if this is a confused question, I am somewhat new to this!)

Answer 1

Why is Bb preferred over A# in F major?

In F major, Bb is the fourth scale degree. Thus, the "flat" doesn't really mean anything since Bb is naturally part of the scale. In terms of traditional Just Intonation tuning systems, this is simply the 4:3 perfect fourth frequency ratio.

Think of it in terms of C major, where there are fewer enharmonic spellings to worry about, and it will be much clearer. Obviously, with the tonic as C, F is the 4th scale degree at the perfect fourth ratio of 4:3 above the tonic. If the spelling E# was encountered for some reason, say as the major third above the note C#, it would then be calculated with the major third ratio (typically 5:4) above that C# note. This is where the difference between E# and F would come into play, since the two note names represent the different calculation methods that determined the notes' frequencies.

Basically, the confusion here is coming from the name Bb in F major. The flat doesn't actually refer to anything; in F major, Bb isn't a flattened B♮ anymore, it's a diatonic note in its own right. Since enharmonic spellings are no longer equivalent, it is even more important to understand how each note's spelling reflects its position in the music.

7-note scales are spelled to set each diatonic note in the scale with its own letter name, and since F major already has an A as its major third, Bb is its own note to represent the fourth.

Well, then why is the fourth note represented with a flat when we could just call it B?

This is done to represent the fact that there is only one semitone between the 3rd and 4th scale degrees. Technically, there can be many different sizes of semitones/minor seconds outside of equal temperament, but it is clear that the distance between the 3rd and 4th must be narrower than the distance between the 4th and 5th. Since B is nominally closer to C than A (there's no note between B♮ and C♮, since only seven letters represent all twelve chromatic notes), the flat in the name is an indicator that this note must be the normal perfect fourth.

Also consider that during any kind of co-occurrence of C major and F major (modulation, comparison, secondary dominants, etc.), the "B" note in C major is much sharper than the "B" note of F major; clearly it would be nonsensical to call them both B! Thus the flat symbol also makes sense in reconciling the letter names to their positions relative to each other.

Starting with the original 7 letter-named notes CDEFGAB, the major scale on C can be constructed. Any other notes outside of that scale can use accidentals in their spelling, since all notes are constructed as a series of specific intervals relative to the tonic. Mapping that same major-scale frequency pattern onto the note F gives a "B" note that is much flatter than the "B" leading tone of C, so it was assigned a flat. Similarly, starting on G, the "F" note is much sharper than the fourth of C major, so it grew a sharp symbol. On either side, this pattern can be extended ad infinitum to fill in all of the diatonic keys of modern-day 12-tone equal temperament, creating the vestigial sharps and flats in diatonic keys.

Answer 2

I think the matter wasn't about intonation, but is historically related to the fact that the flat sign was developed first and Medieval music didn't have a sense of fix pitches for the staff.

Long before the notion of keys developed, during the Medieval period, only the flat symbol was used. It was only used on B to avoid the tritone between F and B.

So, at least a partial answer to your question is, it wasn't a matter of sharps versus flats and intonation, because there wasn't a choice, there was only the flat sign.

Also, the staff did not represent fixed pitches like in later times. They didn't have a concept like middle C was 261Hz. The notation was more about setting the mode and the singers would sing in whatever range was comfortable. That mitigates some of the enharmonic issues. For example, in the modern system you can questions whether to use key signature F# major or Gb major. But that problem didn't exist in the Medieval period.

In the modern system, you choose a tonic then apply accidentals. For example, for G minor your tonic is G and you then need to add a flat to B, not to avoid a tritone, but to set the basic mode to minor. During the Baroque period, you would stop there. In was only later that an E flat was added to the key signature. But, in the Medieval period, if you wanted a minor mode, you used one of the dorian or phrygian modes, the final (tonic) would be set to staff D or E. You didn't need a key signature to alter intervals to create the mode, rather you changed the location of the final on the staff. And remember, you didn't have a concern about whether the D was, for example, D3 146Hz, because the singer sang whatever range was comfortable.

Well after the Medieval period, when the major/minor key system developed and staff pitches became fixed, the choice of sharps or flats becomes a matter of simply raising or lowering letters from the gamut A B C D E F G. So, if the tonic is G and you want a major key, you need the F to be a half step away. Raising F with a sharp does that with only one change. Lowering the G with a flat doesn't make sense. For one it changes the tonic, and also you would apply flats to six letters total! If the question is whether F# or Gb for G major, there are two problems: the basic concept of the staff is to represent the gamut A B C D E F G where the scale has seven degree, each with a separate letter, separate staff lines and spaces. G A B C D E Gb uses one letter twice. That brings us to the second problem: how to write a key signature if you simultaneously use the G line or space for two pitches?

The choice of sharps of flats is not about intonation. It's about maintaining a gamut of seven letters on the staff.

Another way you can think of it, a way that sort of (hopefully) reconciles modern key signatures with the Medieval system, is to think of sharps and flats in key signatures not as pitch alterations, but as a sign of transposing. In other words, a key signature of one sharp, is sort of saying: treat G as if it were C, but change your actual pitch up a P5. Or, probably more in line with Medieval thinking, the key signature of one sharp means: treat G as DO, therefore if F is TI you must raise it up. Raise it how high? The same as the distance between C down to B which requires no sharps or flats. If the staff tells you the distance of a half step between C and B, and brings up no questions about intonation, then G and F# mean the same half step size... just transposed up a P5.

But my understanding is that historically at least, sharps and flats would differ by a comma?

I think this brings up an issue different that key signatures.

Instruments like the violin or voice that don't have fixed pitch can perform half steps smaller than those on an equally tempered instrument like the piano. Regardless of that the notes would be notated the same. The choice of sharp or flat relating back to the stuff about the gamut of letters and key signatures.

Answer 3

In a comment you ask

if a singer were to sing an F-major scale using just intonation, would they sing an A♯ or a B♭ ?

The answer is B♭, for reasons well explained in other answers. But the question seems to assume a strict relationship between note names and tuning that does not exist.

If a singer were to sing an F major scale in just intonation, the B-flat would be about 2 cents lower than equal-tempered, a negligible difference in almost every context. The A, however, would be approximately 14 cents lower than equal-tempered. However, the medieval system is far more likely to have been developed using Pythagorean tuning than just intonation. One clue to this is that they tended to treat the major third as a dissonance. (Another clue is medieval writings about tuning.) The Pythagorean major third is about 8 cents higher than equal (and 8+14 cents gets you a syntonic comma). Whichever A we use, however, wherever it is between 386 cents above F and 408 cents above, we still call it A.

In other words, you seem to assume that the "black" notes are the most flexible in their tuning, but that isn't necessarily the case. In the major scale, the third, sixth, seventh, and second scale degrees are the most unstable for use in harmonic music.

But my understanding is that historically at least, sharps and flats would differ by a comma?

"Comma" is a whole class of intervals (the very small ones), so yes, this is true in the sense that bread is made from a plant. But normally, the word "comma" by itself, in the context of just intonation, denotes the syntonic comma, and that's not the difference between two enharmonically equivalent pitches; rather, it's the difference between the A that is a major third above F and the A that is a perfect fifth above D (specifically, above the D that is a perfect fifth above G). More generally, it's the difference between four perfect fifths and a major third, after adjusting to the same octave. Four perfect fifths is (3:2)⁴, or 81:16, while two octaves plus a major third is 5:1 or 80:16, giving 81:80 for the syntonic comma.

And this brings us to another point: the specific tuning of any pitch in pure just intonation depends on how you get there. Sharps are normally the major third of a chord, while flats are normally either a perfect fourth above something or the root of a major chord whose third has already been established.

So, tuning from C, your B♭ can be two perfect fourths, 16:9, or a perfect fifth plus a minor third, 9:5. That's already a syntonic comma difference without considering A♯.

For a more plausible example that is easier to calculate, as well as less ambiguous, let's consider, with C as our base note, C♯ as the leading tone to D, and D♭ as the major third below F:

Pitch   Ratio                   Cents
F       4:3                       498
D       9:8                       204
D-flat  (4:3)*(4:5) = 16:15       112
C-sharp (9:8)*(15:16) = 135:128    92

Here, the difference between C♯ and D♭ is slightly less than a syntonic comma. But if you figure C♯ as a major third above A (the one that's a third above F), you get a very different result:

Pitch   Ratio                   Cents
F       4:3                       498
D       9:8                       204
D-flat  (4:3)*(4:5) = 16:15       112
C-sharp (5:6)*(5:4) = 25:24        71

This difference between C♯ and D♭ is just under two syntonic commas (which is not surprising, since the A we based it on is a syntonic comma flatter than the perfect fifth above our D).

Basically, for most music, you run into this kind of confusion very quickly if you try to tune everything in pure just intonation. Some pieces are "comma pumps" in which the harmonic progression causes the pitch to change if you keep all the intervals pure.

This difficult problem becomes all the more difficult when trying to tune a keyboard. Singers can sing a different A depending on whether it's part of an F major chord or some sort of D chord. Unfretted string players can do the same. Wind instruments can bend the pitch up and down. Fretted strings can bend it up. But, aside from the clavichord (and modern electronic instruments!) keyboardists cannot do this.

Furthermore, the difficult problem becomes still more difficult when you consider playing different pieces on the same instrument, and intractable when you try to tune for different keys.

One solution to this problem was keyboards with split keys. But that's only a partial solution to the problem, because, as we've seen, even if you have separate keys for (e.g.) G♯ and A♭, you might need more than one pitch for G♯ or for A♭.

The other solution, the one that stuck in the long run, was to temper the just intervals so they're all a little bit out of tune but tolerable. There's a common myth that holds that formerly (at some unspecified time) everything was tuned in just intonation, but that only sounded good in one key, so eventually we discovered temperament in order to play in every key.

But as we've seen, you can easily run into tuning trouble in just one key, with no chromaticism at all. Tempered tunings must have been in use from the first time anyone tuned a keyboard with a just major third, because as soon as you do that, you mess up at least one of your fifths. Initially, these would have been meantone temperaments, which resolve the problem of the unstable A, and really do sound good in a small number of closely related keys while sounding bad in others. It was meantone temperament, not just intonation, that drove the development of split-key keyboards. As harmonic complexity increased, keyboard temperaments became gradually more equal, and split keys lost their purpose.

All of this is intended to complement the other answers by pointing out that tuning considerations are far more complicated than, and separate from, the question of whether to use A♯ or B♭ as the fourth degree of the F major scale. The choice between enharmonic equivalents is more about melodic logic, in a fairly abstract theoretical way. The choice of a particular tuning for a particular pitch is about what sounds best, in a concrete practical sense.

Answer 4

The basic reason is that we need one of each letter name, as you suggest. The spacings between notes in a major scale are TTSTTTS. So, when we start on a white key, for instance, each note played in that scale gets a note in alphabetical order, starting wherever, and continuing from G back to A.

An example - key A. A, B, C something, D, E Fsomething, G something.Those somethings will be sharps, as the pattern goes to black keys, and if they were labelled flats, there couldn't be one of each letter name. Key F would be F, G, A, B something, C, D, E. Here, the something would have to be flat, as that black key couldn't be A♯ - two A names wouldn't work on written music.

The historical story is that in other temperaments, true, A♯ and B♭ were different pitches, but 12tet came along as a compromise, but this hopefully is a simplified answer.