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Introduction


So, the "just intonation" system introduces a scale where every note is aimed to be a more harmonious low-integer ratio (harmonic) relative to the fundamental. In the case of C, the scale is characterised by the following ratios (taken from Science & Music, Sir James Jean, 1937):

Note Ratio of frequency to C
C 1/1
D 9/8
E 5/4
F 4/3
G 3/2
A 5/3
B 15/8
C 2/1

A problem that arises from this system is that if we want to modulate to another note such as G, the A note will no longer have 5/3 times the frequency of C, but 9/8 times the frequency of G, and so 27/16 times the frequency of C, meaning that A would need to change its pitch. So pitches would need to vary depending on the key in which we happen to be playing at the moment. I understand this idea.

Differences in key


So, if we want to play in different keys, we need new notes, different in frequency from what we were using in a previous key.

The aforementioned book takes this fact to conclude that different scales have therefore different, special and unique characteristic qualities, that differ from just a change in absolute pitch.

It even quotes this testimony from Helmholtz:

There is decidedly different character in different keys on pianofortes and bowed instruments. C major and the adjacent Db major have different aspects. The difference is not caused by a difference of absolute pitch, as can easily be verified by comparing two instruments which are tuned to different pitches. If Db on one instrument has the same pitch as C on the other, the C major still retains its brighter and stronger character on both, and the Db its soft, veiled harmonious quality.

Googling around I found sources which also assign different qualities to different keys (when playing in "just intonation"), such as this one, where it agrees that there is a difference between C and Db (aside from the difference in pitch).

C Major Completely Pure. Its character is: innocence, simplicity, naïvety, children's talk.

Db Major A leering key, degenerating into grief and rapture. It cannot laugh, but it can smile; it cannot howl, but it can at least grimace its crying.--Consequently only unusual characters and feelings can be brought out in this key.

Question


Where does these differences among scales which are not in pitch come from?

Even if when modulating from C to G you need different notes, both scales will consist of the same ratios. As far as I understand, in both C and G the scales are going to be:

1/1 , 9/8 , 5/4 , 4/3 , 3/2 , 5/3 , 15/8 , 2/1

So the only difference I could see between C and G is pitch.

If Db on one instrument has the same pitch as C on the other, the C major still retains its brighter and stronger character on both, and the Db its soft, veiled harmonious quality.

Why? Where are these differences coming from if it's not pitch? I'm guessing the ratios DO change from one note to another but I'm not seeing it.

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5 Answers 5

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It doesn't. This myth arises from a confusion between just intonation, a tuning system in which no pitch is fixed, and various temperaments, which are different ways of assigning fixed pitches to the keys on a keyboard. Those temperaments, except for one, cause different keys to have different frequency ratios between pitches that are nominally separated by the same interval. The one exception is called "equal temperament."

Even if when modulating from C to G you need different notes, both scales will consist of the same ratios.

Both scales consist of the same ratios only if you can retune your instrument as you modulate. Most instruments can adjust their tuning in performance, but traditional keyboard instruments cannot (there's no pitch-bend function on a piano).

Consider tuning a C-major scale on a keyboard using the frequency ratios given above. Then consider the ratios of some of the perfect fifths. Between the first and fifth degrees of the scale, or pitches ^1 and ^5, the ratio is 3:2. The same ratio exists between ^6 and ^3. The ratio between ^2 and ^6, however, is 40:27.

But those pitches, in the G major scale, are ^4 and ^1, ^2 and ^6, and ^5 and ^2. So the position of the out-of-tune fifth is different. In C major, the ii chord is out of tune, but in G major, the V chord is out of tune.

In fact, because 40:27 is significantly out of tune, nobody ever actually tuned a keyboard to "just intonation" for practical use. Instead, they used "meantone temperament," which is a class of temperaments that prefer just major thirds while having most of the perfect fifths slightly out of tune and one or two rarely-used fifths very much out of tune. These temperaments also give different characters to different keys.

By the 19th century, temperaments were close to equal or in fact equal. The difference between a piano and a string instrument, however, can still be attributed to tuning: string instruments can alter pitches to achieve just intonation, but not if the pitch is an open string. In C major, many of the pitches will correspond to open strings. In D-flat major, very few will.

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  • Good point that it's not just just intonation where different keys sound different - it's every tuning system except equal temperaments!
    – user45266
    Jan 27 at 3:50
  • Thanks for the reply! Okay, so these differences between keys arise when a justly-intoned instrument modulates to a key different than the one it was tuned for. But I can't help but think that this wouldn't be bearable at all, just dissonant nonsense, yet the sources I mentioned are not only satisfied with these keys, but assign distinctive positive qualities onto them. That seems weird to me. Jan 27 at 18:36
  • @user45266 but also in equal temperaments other than 12-tone, if you're using (a subset of) them to approximate some other temperament. For example, 31-tone equal temperament approximates quarter-comma meantone.
    – phoog
    Jan 27 at 21:27
  • @MatíasCerioni the apparent inconsistency that is troubling you arises from your assumption that there is such a thing as "a justly-intoned instrument." This assumption is incorrect: they don't exist (except perhaps in experimental or other specialized contexts, where "dissonant nonsense" may be the point). Similarly, flexible-pitch instruments that can adjust their tuning easily do so, but can't be said to be tuned in any temperament. And remember: you don't have to go as far as another key to find dissonant nonsense with fixed-pitch "just" intonation: did you follow the link in this answer?
    – phoog
    Jan 27 at 22:06
  • 1
    In a string/symphonic context, how much of the difference in character between D major (bright) vs. D-flat major (dark) is attributed to (non)-resonance of open strings? Jan 28 at 14:27
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The thing is, the theoretical just intonation ratios themselves aren't different between different keys. The differences between different keys only comes into play due to the fact that most instruments are fixed-pitch. The musician has to tune their instrument one way, and they usually can't change how it's tuned in real time. So the actual differences arise from the instruments being unable to perfectly follow the ideal tuning system outside their intended keys.

Suppose you tune your piano in some form of just intonation centered on the note C. If you then play music that stays in C major, all your notes will be correctly tuned for your Just Intonation piano. However, if you wanted to then play some music in D minor with that same C-tuned piano, now your intervals will be messed up since the piano was tuned for just intonation based around the C tonality. If playing music in a different key, now the intervals will be slightly different.


As an example, let's examine in detail:

Say we tune a piano to a kind of just intonation based off of the harmonic series of the note C. Then C would be 0 cents, and according to the harmonic series, the 9:8 major second corresponds to the note D +4 cents, the 5:4 major third E would be -14 cents, the 4:3 perfect 4th would be -2 cents, and the 3:2 perfect fifth G would be +2 cents each compared to their respective equal temperament intervals.

So if we then decide to play music centered in D minor using the C piano, our root note is already 4 cents sharper than normal. But since it's the root note, let's just call this 0c and then subtract 4 cents from every other note in order to make it all relative to that D note. So the note F is now -2 - 4, or 6 cents flat compared to the equal-tempered F. In other words, the F note we're hearing from our C-tuned piano is 6 cents flatter than an equal-tempered F above this D, taking into account the fact that the root note D already being sharpened slightly flattens the F note by comparison.

But we wanted to play in Just Intonation based on D, so what would the interval have been between a D and an F in that system? Well, the simplest harmonic series ratio that represents a minor third is the 6:5 minor third, which is 16 cents sharper than its equal-tempered counterpart. So according to our tuning logic, this note F should sound 16 cents sharper than the corresponding equal-tempered minor third.

Unfortunately, since we're still playing on a piano tuned to just intonation based on C, all the F notes on the piano are actually tuned -6c compared to the D notes. A D-tuned piano would have gotten us an F that's +16c instead! That is a major audible difference between the key of D minor and, say, the key of C minor, which would have obviously simply used the idealized Eb +16c tuning to create that minor third above the root.


This is why it was possible to say that certain keys had different sounds compared to others - as the music moves to different keys than what it was tuned for, the carefully-calculated intervals stack up in inconsistent ways, and thus intervals that ideally should be at a certain ratio will end up at slightly different ratios because of the predetermined tuning of the piano.

Even if when modulating from C to G you need different notes, both scales will consist of the same ratios.

Correct! But that's what gives the different keys their different characters: you need different notes, but you can't get them! Incidentally, this problem is why just intonation is completely impractical for pianos: as phoog's answer describes, the direct predecessors to equal tempered pianos used meantone temperaments to try to compromise; these tunings still had the audible differences between different keys. Equal Temperament did away with these differences by tempering the 12 pitches such that every specific interval was the same size regardless of which note it was built from.

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  • If you have a string instrument you should be able to play just intonation with a different root as long as you center your root at a different position on the fingerboard? Since that would "shorten" each string the same amount (think barré). Right? Is there a reason they did not have some kind of way to "shorten" the piano strings in equal amounts?
    – Emil
    Jan 27 at 7:21
  • 1
    There's a neat video available here that demonstrates what happens when you try to play a C# major triad on a piano tuned with C-major just intonation, and vice versa. Jan 27 at 21:00
  • 1
    "If you then play music that stays in C major, all your notes will be correctly tuned for your Just Intonation piano": this is incorrect unless you avoid playing D and A at the same time. No ii chords, no V/V chords... @MichaelSeifert that video doesn't really tell you much, because there are lots of different possibilities for where to tune the black notes, especially G sharp, which one might tune instead as A flat. The D-to-A perfect fifth is a much more convincing demonstration of the impracticality of using JI to tune a keyboard.
    – phoog
    Jan 27 at 21:44
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    @user45266: sounds like a nice summer project for some future year.
    – Emil
    Jan 27 at 22:58
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    @Emil (& @user45266) "Ive never seen a piano..." See this! (Or google "fluid piano")
    – Rusi
    Jan 28 at 15:32
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Just intonation does not really have the concept of a key in the sense we tend to use it. Rather just intonation puts everything into relation to a root note. So if we have a change of harmony this quickly gets really complicated, and you no longer can restict yourself to 12 chromatic pitches (as each pitch would need to be different, depending on the root (whose pitch might then also depend on the harmonic progression).

So this system only really makes sense with vocal music or something like strings.

If you were to tune a piano justly intonated you can get somewhat

But if you were to tune let’s say a piano to be just in regards to one specifiy key like C then other scales on the same piano will have a different character, as the intervals would then be different.

More common would be something like a temperated tuning system, which were actually used on pianos and stuff and did have different characters to different keys.

In your tuning example: The interval from C to D is 9/8 (a big wholestep). The interval from G to A would be (5/3)/(3/2)=10/9 (a small wholestep).

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Except for equal temperament, it is not possible to say just that you tune with a particular temperament. This is because the temperament fundamentally is a pattern of frequency ratios; to apply it, you need a starting point. (you also need to pick an absolute pitch - that is, a characteristic frequency - for that note.)

In practice, just intonation is used to mean two things:

  1. the simple practice of trying to perform the music with each interval being justly intoned. This is not practical in the general case even when you can choose the exact frequency of each note, because of the comma pump phenomenon.

  2. A temperament in which the pattern of frequency ratios is chosen such that the other notes are justly intoned relative to the tonic.

In the case of C, the scale is characterised by the following ratios

Right, so those are the pitches relative to C that you get, on an instrument tuned in just intonation, in C. We choose a specific frequency for one specific note (for example, we might set C4 to 256 Hz), and determine all other pitches by that ratio scheme.

The rest of the text is explaining that it's neither because of the specific frequency that we chose, nor the name we give to the note, that the scales have their character. It's because of how the set of frequency ratios relates to where the tonic of the scale is.

If "C" is the name for the note you just-intonation tuned around, then the key of C sounds "pure", because it is in just intonation. Other keys are not, because of Math(TM).

if we want to modulate to another note such as G, the A note will no longer have 5/3 times the frequency of C, but 9/8 times the frequency of G, and so 27/16 times the frequency of C, meaning that A would need to change its pitch.

Better to think of it the other way around. You describe the desired A note. G has 3/2 the frequency of C, and A has 5/3 the frequency of C, because that is how the instrument was tuned. The actual A note has 10/9 the frequency of G, where it should be 9/8.

So pitches would need to vary depending on the key in which we happen to be playing at the moment. I understand this idea.

Okay, so...

Where does these differences among scales which are not in pitch come from?

It comes from the fact that, since we cannot simply re-tune a piano before each piece to be performed, we do not have access to pitches that "vary depending on the key".

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  • "the key of C sounds "pure", because it is in just intonation": no it doesn't. The perfect fifth between D and A sounds like this (this is actually transposed to A and E, so think of the G major just scale if you have perfect pitch): en.xen.wiki/w/40/27
    – phoog
    Jan 27 at 22:02
  • ... I don't understand the objection. Of course there are wolf intervals. I went over that. OP, as I understood, went over that too. How often are you playing D and A together while in the key of C, and why? You would commonly play them together in the key of D, where they are the tonic and dominant; which is why the key of D would not sound good on an instrument tuned in that particular way. Jan 28 at 7:20
  • 1
    "How often are you playing D and A together while in the key of C, and why?": Very often, thanks to the common use of ii to precede V and to the most common chromatic alteration being the secondary dominant, which raises the third of ii to yield V/V. The only pieces that commonly lack the interval are 2-part counterpoint (compare the various movements of Bach's 5th French Suite for example). How many songs use "rhythm changes" (the third chord of which is ii)? I can't imagine that anyone repeating this myth has actually tried it.
    – phoog
    Jan 28 at 7:32
  • I forgot another class of pieces that commonly lacks the ^2-^6 perfect fifth, of course, which is songs with simpler harmony, such as folk songs that use only I and V and possibly IV, also including the classic blues progression. Taking these into consideration, the statement can be qualified to yield a true statement: "the key of C sounds "pure" if you avoid certain chords." But then that statement is also true of the key of F, for a keyboard tuned in C; you just have to avoid vi instead of avoiding ii.
    – phoog
    Jan 28 at 8:21
  • This is a balanced answer not being so keyboard centric as the others — "cannot return a piano" means we can retune a violin mid performance (or voice). Not being kbd-centric means allowing for more than 12 pitch classes : infinite in theory, and, in practice assuming modulation-less music, at the minimum at least 22
    – Rusi
    Jan 29 at 2:43
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When a keyboard instrument is justly tuned, it will sound good - but really only when every piece is played in that just key - and doesn't modulate. Each and every note played is tuned to that key, and musically fits better than if the instrument was tuned to 12tet. That just key could be any key you want (excepting too much tension on strings, obviously!) but let's say it's tuned to sound good in C major.

Playing a piece then in G major, for example, would put all of the relative pitches out, by small degrees. Even though there's only one 'foreign' note introduced - F♯.

That's where the strange proclamation emanates from, that subsequently each key will have its own 'odd feel'. Tune the whole instrument from c to C♯, and then the C♯ pieces will sound "completely pure", while pieces using key C will have the same attributes as key B had previously!

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  • This is incorrect. A justly-tuned keyboard does not sound good in one key. Have you ever tried it? If you had tuned it to G major, and you had played the progression G-Em-Am-D, you would have heard this perfect fifth in the third chord: en.xen.wiki/w/40/27. That's hardly "sounding good."
    – phoog
    Jan 27 at 21:56
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    But why would you play G-Em-Am-D on your justly-tuned instrument anyway? @phoog I think for the most part, when people here are saying "in the key of C, JI would work", they mean something more along the lines of "harmonies that center around the note C will follow their just intonation ratios very closely" rather than using the sense of "key of C" used in other contexts. They're contrasting the key of C with the distant keys that obviously would not work.
    – user45266
    Jan 27 at 22:47
  • @user45266 Why wouldn't you? What would you play on it instead? I'm only speaking of a keyboard justly-tuned in the key of G because the audio file I linked to contains an A and an E. If you tune in C, you get that interval between D and A instead, so you ruin the third chord of the progression C-Am-Dm-G. It's a common progression. It's entirely diatonic, using all seven diatonic pitches and only those seven pitches, so it's a good example to illustrate why just intonation doesn't sound good in any key.
    – phoog
    Jan 27 at 23:22
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    "I think for the most part, when people here are saying "in the key of C, JI would work", they mean something more along the lines of "harmonies that center around the note C will follow their just intonation ratios very closely" rather than using the sense of "key of C" used in other contexts." Echoing this. I think it is important here to distinguish tuning theory from music theory. Helmholtz' impression of the "character" of various keys (in a particular context) is a separate matter from the underlying arithmetic, which I think we all covered in more than adequate detail. Jan 28 at 7:29
  • @KarlKnechtel distinguishing tuning theory from music theory is all well and good, but statements such as "a keyboard tuned in C using just intonation sounds good when you play music in C" bring us into the realm of the practical. Speaking as someone who has actually played music on such a keyboard, I find it difficult to believe that anyone making such statements has done so. With delight, I began playing my lovely in-tune diatonic piece, and a few bars in I heard something that made me think I'd messed up the tuning. But no, it was just that wolf fifth between re and la.
    – phoog
    Jan 28 at 7:56

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