Changing Keys and Just Intonation
Since we are unable to derive the following from anything stated previously, we
state it as an axiom.
Key Variety Axiom: It is very important in tonal music to be able to have different portions of the composition in different (but usually related) keys.
How do we generate a new key? We have the JI major scale array [ 1 9/8 5/4 4/3 3/2 5/3 15/8 ] and we can multiply it by a frequency to build the major scale at that frequency. The first idea might be to take every note of the C scale and build a new major scale from it. There are also many other ways, and unfortunately, they do not yield the same results. The standard way seems to be to follow the progression of 5ths upward for sharps and downward for flats. The progression of 5ths is taking the 5th scale step (that is, the 3rd harmonic) repeatedly:
Key of G major: (3/2) [ 1 9/8 5/4 4/3 3/2 5/3 15/8 ] = [ 3/2 27/16 15/8 2/1 9/4 15/6 45/16 ]
Key of D major: (3/2)(3/2) [ 1 9/8 5/4 4/3 3/2 5/3 15/8 ] = [ 9/4 81/32 45/16 3/1 27/8 45/12 135/32 ]
And so forth. See the ascending 5th sequence calculated by the applet below.
Let's look more closely at the key of G we just derived. Its fractions correspond
to the notes
( G A(3/2) B C D E
Note the special notes A(3/2) and F#(3/2) where the 3/2 in parenthesis means
the version of the note that occurs in the scale with tonic frequency 3/2f.
Observe A(1) = 5/3 but A(3/2) = 27/16; they are not quite
the same note! (see the drawing by the applet)
Indeed, the ratio A(3/2)/A(1) = 81/80 is the infamous syntonic comma [Benson p.116], [Backus 1969 p.122], which comes up again and again.
What is the definition of F#?
What should it be?
The point of this discussion is that as we move away from C, generating new scales from the JI array, the new notes do not coincide with old notes, so we can't re-use the old ones. This is the primary problem with Just Intonation; it gives harmonically perfect chords, but requires an unpractical number of distinct physical keys on the piano keyboard. (And even before we start changing keys, the D-A scale 5th is dissonant in C major). It should be noted that [Sethares 1997] has proposed a clever fix for electronic instruments: dynamically re-tuning. The standard fix, however, is 12 tone equal temperment.
In the diagram drawn by the following applet, the vertical axis is the pitch in an octave. Each column represents a different key or scale.
view the source code If you save it to disk, don't forget to change the .txt to .java before you try to javac it. Indeed, this program also computes the exact frequency ratios (as fractions) of all the various keys, but the printing of that to stdout that was commented out when it was applet-ized.
However, the standard way seems to be to assign names to notes based on the progression of 5ths and the historical Pythagorean scale.
ASIDE: The Pythagorean scale is derived from starting with C = f and going below to F = 2/3f and then progressing upward, taking 3rd harmonics: G = 3/2f, D = 9/4f, A = 27/16f, E = 81/64f, B = 243/128f, and rescaling into a single octave, starting at C. With these note names, we can continue the progression of 5ths; going up from B we generate a pitch between F and G which we name "F#". Going down from F, we generate a pitch between A and B we name "B-flat" and so on. [Benson 2002 ch 5] gives a good discussion of the Pythagorean and many other scales.
Therefore, the convention seems to be that F# is defined to be the tonic pitch of the key of F#, what we would write as F#(729/512 ). Perhaps the real conclusion to draw from this is that it is a mistake to think in terms of a single definition for F# (or any other sharps or flats) unless we are refering to equal temperment. Otherwise, the frequencies of the sharps, flats, and naturals all vary and are not tied to our origin key of C; they are determined only by the tonic frequency and the JI array.
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What is the definition of F#? What should it be?In the above diagram, look at all the horizontal lines midway between the F and G of the C scale. We have multiple choices for F# and G-flat! And we'd get even more if we continued in the progression of 5ths; we have no logical reason to stop, since it never exactly closes on itself. So what is the real F#? One idea would be to define it to be the first possibility we come to, F#(3/2) in the key of G, if we want the idea that the key of C, derived a priori from harmonics, is the most "pure".