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In Pythagorean and Just tuning, according to Wikipedia, there are different frequencies for flat and sharp notes. Most notably, there is a tritone of which there seem to be 2 quite different versions.

In equal temperament, there are just 12 semitones.

In meantone temperaments, like Werckmeister's, is there a difference in frequency between A♯ and B♭? Or do they, like equal temperament, just have 12 notes?

I know a violin may be playing a different frequency for A♯ and B♭, and it may sound better, but if they do, are they technically outside the bounds of both equal and meantone temperaments?

EDIT

As per comments below, Werckmeister's is NOT a 'meantone temperament' but a 'well temperament'. Meantone happened early 17th century, well temperament late 17th century (says https://www.albany.edu/piporg-l/tmprment.html)

So I now have two questions:

  • Are A♯ and B♭ similar in meantone temperaments?
  • Are A♯ and B♭ similar in well temperaments?
  • I don’t have time to put in my answer but, yes, in something like 1/4 meantone, in theory, they’ll be different, audibly so; one is about 20 cents below 800, one 20 cents higher. Using one in place of the other is what gives you wold intervals. Well temperament involves shifting the notes to avoid wolf intervals. – Dave Feb 18 at 18:34
  • Dave's answer and MattPutnam's answer are correct. A temperament is a system for tuning a keyboard instrument, so its specification must take into account the keys available on the instrument. Any keyboard instrument that has the same key for A♯ and B♭ will necessarily be tuned in a temperament that specifies the same pitch for A♯ and B♭. – phoog Mar 20 at 17:03
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    But that doesn't answer the question. The question is whether when using such a tuning, the tuner needs to choose which note to tune such a key to. Does a meantone temperament or e.g. Werckmeister III prescribe a single pitch for the A#/Bb key, to be used for both, or does it give two different pitches out of which the tuner must choose? That is the question. I think the answer is: the former for well temperaments, the latter for meantone temperaments (which simply assumes notes such as A# do not occur). – reinierpost Jun 15 at 9:28
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No. "Meantone" means the tuning has been adjusted so the pitch produced by either Bb or A# is the average (or the 'mean') of the two pitches. You end up with just 12 pitches per octave, just as you do in equal temperament.

EDIT: The OP has requested I elaborate on my answer, which I see has gotten a number of downvotes. Perhaps that’s deserved – as a music teacher I shade answers to the perceived level of what a student knows (e.g. in teaching a time signature “the top number is the number of beats per measure” is sufficient, and a discussion of simple vs compound meters waits until later).

With that out of the way…

Keyboard instruments prior to about 1500 had as many as 19 keys per octave, with split keys for all of the modern black keys, plus a B# and E# that were different from C and F. (See the first photograph in http://www.harpsichord.org.uk/wp-content/uploads/2015/04/archicembalo.pdf for a reproduction of an early 16th century archicembalo with 19 keys per octave in the lower manual.)

The large number of keys was required by Pythagorean tuning: as you go up the harmonic series you get different pitches for F#/Gb etc.

Manufacturing technology was improving quickly in the 16th century, with new techniques for pulling wire (the strings in most of these instruments were drawn brass). As a result, we were now capable of making instruments with a larger range. Even in Pythagorean tuning that means bumping up against the Pythagorean comma: three pure 5:4 thirds do not add up to a 2:1 octave. The larger the range of the instrument, the more pronounced the discrepancy.

Concurrent with this was a desire to make instruments simpler, reducing the number of keys. This would make them both easier to manufacture and easier to play.

Solving the Pythagorean comma at a practical level was needed. The earliest solution that I know of was that of Pietro Aron in his book Toscanello in musica published in 1539: he proposed dividing the comma into four parts, and lowering the tuning of four consecutive fifths by that amount to create a perfect octave while retaining most of the pure thirds. His system resulted in eight pure thirds and four that were terrible (wide by about 40 cents). The fifths were narrow by about six cents, except for one – that was wide by a whopping 35 cents. In case you’re interested in looking into it in depth, the tuning portion of his book is online in translation at http://www.tonalsoft.com/monzo/aron/toscanello/aron_toscanello.htm

Using Aron’s tuning made it possible to create an instrument with 12 keys per octave that sounded at least acceptable in a number of keys. Keys that required the bad notes, known as “wolf tones” were avoided.

Aron’s method came to be known as meantone tuning. Tempering for two consecutive fifths meant that the major second interval was halfway (the mean) to the major third. Give or take 1/10th of a cent, which is certainly close enough for the purpose. Any tuning that results in equal whole steps can properly be called meantone, and meantone tunings can be created by using 1/3 of the comma, or 1/5th, or any other fraction less than the number of discrete pitches per octave (our 12TET system is one variety of meantone).

The development of meantone tuning was concurrent with the elimination of split accidentals, or at least as concurrent as the speed of information traveling in the 16th century would allow. This meant it also had the effect of adjusting the black keys to fall between the Pythagorean sharp and flat pitches.

In the context of the question, my original answer – that meantone split the difference between sharps and flats – is accurate. The fact that you can have meantone tunings with divisions of the octave other than twelve is true… but I judged it irrelevant to the question.

The next problem was how to handle the wolf tones. Many tuners devised adjustments to meantone tuning to reduce or eliminate the wolves by raising or lowering specific pitches. Werkmeister was one of them, and devised a number of possible tunings. These tunings are called well temperament, because it allowed the instrument to play well in all keys.

Because well temperament results from the adjustment of individual pitches you end up with intervals that are not all the same size. Your minor third in C will be different from your minor third in D or Ab. Some found this a feature rather than a bug: each key now had its own unique character, distinct from any other key. Various well temperaments stuck around through the 1800s, and are still used by “period” instrumentalists.

12TET began to replace well temperament in the mid 19th century, and is considered the standard today.

  • Do you have a reference that lays out the math behind the idea that the pitch is the mean of the two enharmonic notes? – Dave Feb 16 at 16:01
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    @Dave there's different versions of meantone temperament. here are 4 of them: en.wikipedia.org/wiki/Werckmeister_temperament – commonpike Feb 16 at 19:15
  • @Dave math wasn't possible, as meantone tuning developed several hundred years before we were able to measure frequency. The "mean" was the guess of the tuner. – Tom Serb Feb 16 at 21:08
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    @dekkadeci - as a new user I don't yet have enough "reputation" to comment directly on your edit to the question, but it is incorrect. Werkmeister tunings are well temperaments, not meantone tunings. Meantone tunings take the average of the Pythagorean split accidentals; they are a "regular" or "linear" temperament, evenly dividing the Pythagorean comma. Well temperaments make adjustments to meantone to reduce the wolf tones, and are "irregular" temperaments, which means not all steps are the same size. – Tom Serb Feb 17 at 9:15
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    Wrong. In just intonation, do re mi have frequencies 8:9:10, making intervals of a greater tone & a lesser tone. "X is a meantone temperament" means that X makes these tones the same size, i.e. makes 4 perfect 5ths = 2 octaves + major 3rd. Nothing to do with Bb/A#. One possible meaning of Bb=A# is that 12 perfect 5ths = 7 octaves. In this case 12ET is the only temperament with both properties; in quarter-comma meantone, 31ET and other meantones Bb and A# are different. – Rosie F Feb 17 at 10:31
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Most temperaments that you'll come across in practice do not differentiate between enharmonic notes, however there are many temperaments that do; they're rarely used, or "just theoretical" because there are not many fixed pitch instruments that provide more than 12 notes per octave so there aren't instruments that can be tuned to them. An important thing to keep in mind is that a temperament is a highly practical construct -- how do I tune this thing so that it sounds good?

Take the Werkmeister temperaments: they're primarily used for organs, which almost always have 12 note octave keyboards, and thus you need to assign a single pitch to the pairs of enharmonic notes.

However, there were some instruments built with split key keyboards that provide the mechanism to differentiate between some enharmonic notes. Indeed, people have gone pretty far with this idea, e.g. this instrument with 84 notes per octave tuned in 53-ET(?).

Another example is the English concertina -- it has 14 buttons/octave and differentiates between a♭/g♯, and e♭/d♯. These can be tuned in 1/5 comma meantone to take advantage of this fact.

Finally, yes, continuous pitch instruments don't always (try to) create notes right at any particular tuning standard instead they can, especially in solo/lead situations, apply expressive intonation.

  • You are not addressing the possibility of retuning instruments between pieces. Conceivably a portative organ or other keyboard instrument can be retuned between pieces in different keys; such instruments must be tuned before performances anyway. So conceivably, temperaments may have been used that require such retuning. – reinierpost Jun 19 at 14:26
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In any 12-tone system, A♯ and B♭ are always the same. If they're different, then you have more than 12 tones.

When using a ratio-based tuning, there are always multiple ways to construct a particular interval. If it's a 12-tone temperament, you have to pick just one. The different ways to pick intervals constitute unique 12-tone temperaments.

  • Indeed, even most split-key keyboard with more than 12 keys per octave will have the same pitch for A♯ and B♭ unless the A♯/B♭ key is one of the split ones (which it usually isn't). – phoog Mar 20 at 17:10
  • So the question is: when using meantone or well temperaments, were only 12 tones used, or did players retune certain keys depending on the key a piece was in? Clearly, they didn't on church organs, but on home instruments, frequent retuning is necessary anyway, so it would not be too much of a problem. – reinierpost Jun 15 at 9:33

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