Okay to understand this, need to understand how our tuning systems developed historically in the West.
This answer is about "tuning colour".
There are many other "key colour" effects of course. You get many orchestral colour differences depending on the tuning used as well, of course, and on particular instruments, different keys may have a fair bit of timbre variation, for instance you get resonances with the body of the instrument, eccentricities of instruments such as register shifts and timbre changes - and on some instruments each note has its own unique character, differences between open and stopped strings, preferred emphasis of some of the partials and so on
Many instruments also have limited range - so that for instance if the lowest pitch on your instrument is say C and it's range is just a little over a couple of octaves, - then you can't play a two octave scale starting from a B, but can do so on a C or D. So the things you can do in B on that instrument - the types of melodies - the chord patterns if it is polyphonic - and so on - may be noticeably different from what you can do in C.
But - these things are widely understood, so just mentioning them to move on - other answers here have talked about them. "Tuning colour" however is much less widely understood amongst musicians.
It's because of the way different keys were tuned from Bach's time onwards.
But to understand why that happened, it helps to go back a bit further.
First - all this is through choice, and not through any difficulty they had in tuning to twelve equal. The maths needed to tune to twelve equal is simply twelfth roots, i.e. cube roots of square roots of square roots.
Certainly was no problem at all by the time of Bach. First reference in writing, seems to be Aristoxenus around 350 BC (I haven't found the original text yet, just various statements without a citation to back it up e.g. here BBC - h2g2 - Sol-Fa - The Key to Temperament - A1339076)
And - they did in fact use twelve equal. They used it for lutes, viols, and the early guitars. That was for reasons of practicality - because these are all fretted instruments - and if you use uneven frets - then that works on one string, but not on the next string - unless you also have broken frets that only cover one string at a time - or sliding frets or other complex systems.
Also - guitars and lutes didn't sound too bad in twelve equal - and viols were also tolerable - so there was less incentive to do something about it. They just tuned them to twelve equal.
Players could adjust the pitches of individual notes on these instruments also by changing position of their fingers on the frets, as a kind of "adaptive intonation".
So - what was the problem they had with twelve equal - why didn't they use it for everything like we do?
Well - the reason is - that how we hear and understand the music, and the objectives of the musicians tuning instruments, have both changed.
First - we had the pythagorean system in early medieval times. This involved use of pure fifths, and the major and minor thirds were considered to be dissonances (they are very sharp / flat respectively if you tune to pure fifths).
You can try this yourself, if you have a suitable stringed instrument to hand - try tuning the fifths C G D A E as pure fifths. If they are all pure, the C to E interval will be a sharp third even sharper than twelve equal.
In this early medieval system of harmonies, then instead of chord progressions as we understand them - they have the idea of resolution by step-wise movement of individual voices (upwards or downwards) from a dissonant chord to either fourths or fifths. The resulting chord, after this resolution - the point of rest in their system of harmony - has only open fourths and fifths in it.
(This is a vast subject, which I have hugely simplified in this description - if you want to know more about it, see Margo Schulter's Pythagorean Tuning and Medieval Polyphony)
Then - later on they began to use harmonies that resolved to major and minor third chords as a consonance. When you do that, then you need to make the major third a fair bit flatter to keep it beatless and as harmonious as you can make it.
You can try this for yourself, again if you have a suitable instrument to hand which you can tune. Try playing two strings a third apart and tune the higher one to be as consonant as possible with the lower one. You will find that you tune it a tiny smidgen flatter than the twelve equal major third.
So - one of the early systems they used was quarter comma meantone - this starts with the major third, usually the E of C major, as a pure harmonic 5/4, which can be played beatless with the C.
Then - they adjust the size of the fifth so that in the chain C G D A E then the interval C to E is a pure 5/4.
To achieve that, all the fifths have to be flatter. The name for the musical interval between the pythagorean major third and the harmonic major third is the "syntonic comma".
Syntonic comma
It's about 21.5 cents or a bit larger than a fifth of a semitone. (the 5/4 is also about a seventh of a semitone flatter than the 400.0 cents of 12 equal).
So - you have four of those fifths and in total about 20 cents flat compared with the sharp Pythagorean E, so if you spread it out evenly - each one will be about 5 cents flat, flatter by a quarter of a syntonic comma.
That's quarter comma meantone.
All it's fifths are a little bit flat, except for one, the "wolf fifth" which is very sharp. It also has many pure major thirds, but they can't all be, for instance in C E Ab C', one of the major thirds there has to be very sharp.
It's called "meantone" because of the way it tunes the D in C G D A E.
If you tune E as 5/4, and D as a 9/8 - the pythagorean whole tone - then the interval from D to E is 10/9, another harmonious but rather flatter whole tone. Quarter comma meantone tunes the D to a note half way between the 9/8 and the 10/9, so that's why it is called "meantone",
So that's a tuning system that has a fair number of decent keys - but some that sound quite a bit different, and one that is totally unusable (the one with the wolf fifth, which also happens to have a sharp major third as well to make it even worse) - at least for the type of music they were making at the time.
It also has some other things about it that aren't so good - although the major thirds are great, it has somewhat flat fifths, and the occasional very sharp major thirds - two thirds of the major chords work just fine, but one third of them have these very sharp major thirds.
That worked quite well while music was relatively unadventurous, just doing things like maybe one modulation of key per movement or not much more, changing between a few keys occasionally.
But as composers got more adventurous and wanted to wander through tuning systems more than that - then it just became too confining for them.
So - tuners started to experiment with "well temperaments".
Basically, they are compromises. You can't have all the fifths pure, because in a twelve tone system twelve fifths stack to make nineteen octaves which is mathematically impossible with pure fifths.
You can't have all the major thirds pure either because three major thirds have to stack to make an octave and if you use pure thirds, it will be flat by the Diesis - a very noticeable 41 cents flat.
So - they spread the notes out a bit so some of the intervals are close to harmonious - you might even have a pure 5/4 or 3/2 somewhere in the tuning (perhaps in C major) - and others are sharper or flatter than the harmonic series intervals.
It's a compromise - and they found ways to make all the keys tolerable so you could write music in any of them - but some had far more of the harmonious major or minor thirds than others.
Basically it's a compromise. And not just one system of compromises.
There are many tuning systems of this type, by a huge number of temperament designers - each making different approximations and adjustments.
They are far more complex, in a way, than twelve equal, and designed in intricate detail to minute fractions of a cent.
This site gives technical instructions on how to tune to various historical well temperaments - I've seleted the page for Werckmeister III - one of the earliest of the Well Temperaments. CBH Carey Beebe Harpsichords Australia - Tuning to Werckmeister III
Then Bach came along - he didn't invent "well temperament". And he didn't use twelve equal at all - except of course that he continued to use it for viol, guitar and lute, where it was already in use.
He was a champion, instead, of these new "Well temperaments" - these intricate compromise tunings that let you play music in any of the twelve keys - but with a different "tuning flavour" for each key.
So, what he did do is to write music for all the different keys of the twelve tone system, and showed that the result sounded good, and was effective on the new well temperaments the tuners had invented.
This showed clearly that you could use all the keys - and move from one to another in the same piece also - with no wolf key at all, and create good music in the process.
It opened the gates for composers to be more adventurous and use any of the twelve keys of twelve equal, without worrying about sharp major thirds or wolf fifths or any such thing.
So - that's the origin of the various tunings and key colour.
A just intonation system is one that uses pure harmonic series based harmonies. Intervals between pitches in the harmonic series can be shown as ratios. E.g interval from the second to the third harmonic is 3/2 which is a pure fifth. The interval from the fourth to the fifth is 5/4, the pure, or "just intonation" major third.
You do get "just intonation" twelve tone systems - but they are rather restricting if you try to write using pure harmonies only (rather than experiment with other intervals as well). Even more so than quarter comma meantone.
You will probably get only a few harmonious triads and tetrads throughout all the notes in your twelve tone system. That is - unless you can change the tuning of notes flexibly as you play - adaptive just intonation.
So - systems like that are useless for Bach and later composers and they didn't use them. Except of course that unaccompanied "a capella" choirs might well have sung naturally in adaptive just intonation, whenever they try to sing the most harmonious possible chords they can.
As time went on -composers wrote music that moved between distant keys more and more - and used distant keys frequently.
Listeners also got accustomed to the sound of the rather sharp major thirds in some of the keys.
So - from Bach onwards- the temperaments got more and more even.
By the early twentieth centuries - the tunings Chopin used were almost identical to twelve equal. But not quite exactly. Still some variation of tuning "colour" from key to key.
Nowadays of course in twelve equal, tuning colour no longer exists at all in this sense. Except that is for those who do informed historical performance, or microtonal composers who use these tunings and many others intentionally for effect.
The effects may also linger on in the pieces themselves - the composers were influenced by the tuning of the music - and this perhaps influenced the characteristics of the music they wrote, so perhaps some of the characteristics of the "tuning colour" of the various keys. communicates to us even when we hear their music in twelve equal.
This answer is about "tuning colour".
OTHER TYPES OF "KEY COLOUR"
There are many other "key colour" effects of course. You get many orchestral colour differences depending on the tuning used as well, of course, and on particular instruments, different keys may have a fair bit of timbre variation, for instance you get resonances with the body of the instrument, eccentricities of instruments such as register shifts and timbre changes - and on some instruments each note has its own unique character, differences between open and stopped strings, preferred emphasis of some of the partials and so on
Many instruments also have limited range - so that for instance if the lowest pitch on your instrument is say C and it's range is just a little over a couple of octaves, - then you can't play a two octave scale starting from a B, but can do so on a C or D. So the things you can do in B on that instrument - the types of melodies - the chord patterns if it is polyphonic - and so on - may be noticeably different from what you can do in C.
TUNING COLOUR NOT SO WIDELY UNDERSTOOD
But - these things are widely understood, so just mentioning them to move on - other answers here have talked about them. "Tuning colour" however is much less widely understood amongst musicians.
It's because of the way different keys were tuned from Bach's time onwards.
But to understand why that happened, it helps to go back a bit further.
TUNING IN NON TWELVE EQUAL WAS THROUGH CHOICE, NOT TECHNICAL DIFFICULTIES
First - all this is through choice, and not through any difficulty they had in tuning to twelve equal. The maths needed to tune to twelve equal is simply twelfth roots, i.e. cube roots of square roots of square roots.
Certainly was no problem at all by the time of Bach. First reference in writing, seems to be Aristoxenus around 350 BC (I haven't found the original text yet, just various statements without a citation to back it up e.g. here BBC - h2g2 - Sol-Fa - The Key to Temperament - A1339076)
And - they did in fact use twelve equal. They used it for lutes, viols, and the early guitars. That was for reasons of practicality - because these are all fretted instruments - and if you use uneven frets - then that works on one string, but not on the next string - unless you also have broken frets that only cover one string at a time - or sliding frets or other complex systems.
Also - guitars and lutes didn't sound too bad in twelve equal - and viols were also tolerable - so there was less incentive to do something about it. They just tuned them to twelve equal.
Players could adjust the pitches of individual notes on these instruments also by changing position of their fingers on the frets, as a kind of "adaptive intonation".
SO WHY NOT USE IT FOR EVERYTHING AS WE DO?
So - what was the problem they had with twelve equal - why didn't they use it for everything like we do?
Well - the reason is - that how we hear and understand the music, and the objectives of the musicians tuning instruments, have both changed.
PYTHAGOREAN PURE FIFTHS, THIRDS AS A DISSONANCE
First - we had the pythagorean system in early medieval times. This involved use of pure fifths, and the major and minor thirds were considered to be dissonances (they are very sharp / flat respectively if you tune to pure fifths).
You can try this yourself, if you have a suitable stringed instrument to hand - try tuning the fifths C G D A E as pure fifths. If they are all pure, the C to E interval will be a sharp third even sharper than twelve equal.
In this early medieval system of harmonies, then instead of chord progressions as we understand them - they have the idea of resolution by step-wise movement of individual voices (upwards or downwards) from a dissonant chord to either fourths or fifths. The resulting chord, after this resolution - the point of rest in their system of harmony - has only open fourths and fifths in it.
(This is a vast subject, which I have hugely simplified in this description - if you want to know more about it, see Margo Schulter's Pythagorean Tuning and Medieval Polyphony)
MAJOR AND MINOR CHORDS A LATER DEVELOPMENT
Then - later on they began to use harmonies that resolved to major and minor third chords as a consonance. When you do that, then you need to make the major third a fair bit flatter to keep it beatless and as harmonious as you can make it.
You can try this for yourself, again if you have a suitable instrument to hand which you can tune. Try playing two strings a third apart and tune the higher one to be as consonant as possible with the lower one. You will find that you tune it a tiny smidgen flatter than the twelve equal major third.
QUARTER COMMA MEANTONE, REASONABLE CHOICE FOR MAJOR / MINOR TYPE HARMONY WITH FEW KEY CHANGES
So - one of the early systems they used was quarter comma meantone - this starts with the major third, usually the E of C major, as a pure harmonic 5/4, which can be played beatless with the C.
Then - they adjust the size of the fifth so that in the chain C G D A E then the interval C to E is a pure 5/4.
WHY "QUARTER COMMA"?
To achieve that, all the fifths have to be flatter. The name for the musical interval between the pythagorean major third and the harmonic major third is the "syntonic comma".
Syntonic comma
It's about 21.5 cents or a bit larger than a fifth of a semitone. (the 5/4 is also about a seventh of a semitone flatter than the 400.0 cents of 12 equal).
So - you have four of those fifths and in total about 20 cents flat compared with the sharp Pythagorean E, so if you spread it out evenly - each one will be about 5 cents flat, flatter by a quarter of a syntonic comma.
That's quarter comma meantone.
All it's fifths are a little bit flat, except for one, the "wolf fifth" which is very sharp. It also has many pure major thirds, but they can't all be, for instance in C E Ab C', one of the major thirds there has to be very sharp.
WHY "MEANTONE"?
It's called "meantone" because of the way it tunes the D in C G D A E.
If you tune E as 5/4, and D as a 9/8 - the pythagorean whole tone - then the interval from D to E is 10/9, another harmonious but rather flatter whole tone. Quarter comma meantone tunes the D to a note half way between the 9/8 and the 10/9, so that's why it is called "meantone",
ONE UNUSABLE KEY AND MANY "FAILED" MAJOR CHORDS
So that's a tuning system that has a fair number of decent keys - but some that sound quite a bit different, and one that is totally unusable (the one with the wolf fifth, which also happens to have a sharp major third as well to make it even worse) - at least for the type of music they were making at the time.
It also has some other things about it that aren't so good - although the major thirds are great, it has somewhat flat fifths, and the occasional very sharp major thirds - two thirds of the major chords work just fine, but one third of them have these very sharp major thirds.
That worked quite well while music was relatively unadventurous, just doing things like maybe one modulation of key per movement or not much more, changing between a few keys occasionally.
But as composers got more adventurous and wanted to wander through tuning systems more than that - then it just became too confining for them.
"WELL TEMPERAMENTS" - A HAPPY COMPROMISE
So - tuners started to experiment with "well temperaments".
Basically, they are compromises. You can't have all the fifths pure, because in a twelve tone system twelve fifths stack to make nineteen octaves which is mathematically impossible with pure fifths.
You can't have all the major thirds pure either because three major thirds have to stack to make an octave and if you use pure thirds, it will be flat by the Diesis - a very noticeable 41 cents flat.
So - they spread the notes out a bit so some of the intervals are close to harmonious - you might even have a pure 5/4 or 3/2 somewhere in the tuning (perhaps in C major) - and others are sharper or flatter than the harmonic series intervals.
It's a compromise - and they found ways to make all the keys tolerable so you could write music in any of them - but some had far more of the harmonious major or minor thirds than others.
Basically it's a compromise. And not just one system of compromises.
There are many tuning systems of this type, by a huge number of temperament designers - each making different approximations and adjustments.
They are far more complex, in a way, than twelve equal, and designed in intricate detail to minute fractions of a cent.
This site gives technical instructions on how to tune to various historical well temperaments - I've seleted the page for Werckmeister III - one of the earliest of the Well Temperaments. CBH Carey Beebe Harpsichords Australia - Tuning to Werckmeister III
WHERE BACH COMES IN - CHAMPION OF WELL TEMPERAMENTS
Then Bach came along - he didn't invent "well temperament". And he didn't use twelve equal at all - except of course that he continued to use it for viol, guitar and lute, where it was already in use.
He was a champion, instead, of these new "Well temperaments" - these intricate compromise tunings that let you play music in any of the twelve keys - but with a different "tuning flavour" for each key.
So, what he did do is to write music for all the different keys of the twelve tone system, and showed that the result sounded good, and was effective on the new well temperaments the tuners had invented.
This showed clearly that you could use all the keys - and move from one to another in the same piece also - with no wolf key at all, and create good music in the process.
It opened the gates for composers to be more adventurous and use any of the twelve keys of twelve equal, without worrying about sharp major thirds or wolf fifths or any such thing.
So - that's the origin of the various tunings and key colour.
THESE ARE NOT "JUST INTONATION" SYSTEMS
A just intonation system is one that uses pure harmonic series based harmonies. Intervals between pitches in the harmonic series can be shown as ratios. E.g interval from the second to the third harmonic is 3/2 which is a pure fifth. The interval from the fourth to the fifth is 5/4, the pure, or "just intonation" major third.
You do get "just intonation" twelve tone systems - but they are rather restricting if you try to write using pure harmonies only (rather than experiment with other intervals as well). Even more so than quarter comma meantone.
You will probably get only a few harmonious triads and tetrads throughout all the notes in your twelve tone system. That is - unless you can change the tuning of notes flexibly as you play - adaptive just intonation.
So - systems like that are useless for Bach and later composers and they didn't use them. Except of course that unaccompanied "a capella" choirs might well have sung naturally in adaptive just intonation, whenever they try to sing the most harmonious possible chords they can.
HOW THIS DEVELOPED LATER
As time went on -composers wrote music that moved between distant keys more and more - and used distant keys frequently.
Listeners also got accustomed to the sound of the rather sharp major thirds in some of the keys.
So - from Bach onwards- the temperaments got more and more even.
By the early twentieth centuries - the tunings Chopin used were almost identical to twelve equal. But not quite exactly. Still some variation of tuning "colour" from key to key.
Nowadays of course in twelve equal, tuning colour no longer exists at all in this sense. Except that is for those who do informed historical performance, or microtonal composers who use these tunings and many others intentionally for effect.
The effects may also linger on in the pieces themselves - the composers were influenced by the tuning of the music - and this perhaps influenced the characteristics of the music they wrote, so perhaps some of the characteristics of the "tuning colour" of the various keys. communicates to us even when we hear their music in twelve equal.
About the Author
Robert Walker
Writer of articles on Mars and Space issues - Software Developer of Tune Smithy, Bounce Metronome etc.Studied at Wolfson College, Oxford
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