A Diatonic Scale
[ The task of the reader of this detailed screed will be lighter if (s)he
has previously taken a look at the screed on Intonation, available
on Alan McConnell's web site ]
We create a diatonic scale according to the principle: we want a seven-
note scale whose notes are chosen to be as close to the bottom of the
harmonic overtone series as possible/feasible.
We recall the harmonic overtone series, familiar to all string players
who have played around with harmonics on their instruments. We shall
assume that our fundamental tone is C, and to avoid cumbersome notation
we shall not put primes in for octaves, relying on the reader's understanding
of the range of the notes.
The first ten harmonics over the bottom C are:
1 2 3 4 5 6 7 8 9 10
C C G C E G Bb C D E
Explicitly: we mean that e.g. the fifth note, E, has a frequency five
times greater than the fundamental C; the ninth note, D, has a frequency
nine times greater than the fundamental, etc.
A couple of remarks:
1. We have denoted the seventh harmonic as a Bb; but it really sounds
perceptibly lower than an "ordinary" Bb.
2. We remark on the multiplicative consistency of this series; just
as the second tone is an octave above the first, so is the fourth tone
an octave over the second and the sixth tone an octave over the third;
just as the third tone is an octave and a fifth over the first, so is
the ninth tone an octave and a fifth over the third tone. We shall
rely on these kinds of relationships implicitly in our further development.
It will be convenient to have a notation for the amount one has to multiply
a frequency by in order to obtain the frequency of the next note. Thus,
from our harmonics given above, we can write:
C - 3/2 - G
to indicate that the first G above -- the one with '3' written over it --
has a frequency obtained from the second C by multiplying by 3/2.
Armed with this notation, we can write the first three notes of our scale:
C - 9/8 - D - 10/9 - E
It is worth pausing here to observe that we have derived this hunk of
scale from the right end of our harmonic series above. And we note(pun
intended, as always!) that 9/8 times 10/9 equals 10/8 = 5/4, which is
exactly the ratio between a C and an E. So everything is consistent!
Observe however that all our relations are _multiplicative_, which can
lead us to counter-intuitive, indeed almost contradictory results, as
we shall see.
Our next note -- let's skip the F for a moment -- shall be G. Very
pleasantly, and consistently, we can obtain a G by saying that it
must be a (minor) third over the E, as specified in the basic series:
C - 9/8 - D - 10/9 - E - 6/5 - G
And, checking, we multiply 9/8 times 10/9 time 6/5 to get 3/2, which is
indeed the note that is a fifth above our starting C.
What about A? Here is where the going gets tough! for we have two choices.
1. The A should be a fifth above D. This means that it should be
9/8 times 3/2 = 27/16 times the frequency of C. And hence we
would have: G - 9/8 - A.
2. The A should be a minor third below the upper C. We do a little
algebra: if 'x' represents the ratio of the lower C to the A we are looking
for, we must have:
x times 6/5 equals 2
which means: x = 10/6 = 5/3.
Method 1 yields C - 27/16 - A; method 2 yields C - 5/3 - A. What is
the difference? The ratio of 27/16 to 5/3 is 81/80. This ratio is
called the "Musical Comma" in the literature; we'll run into it again.
Two tones in the ratio of 81/80 do sound distinct to the careful
listener; it is the difference between the two 'E's that I invite
string players to play in my article on Intonation; see my web site.
We must choose. And we choose the A with the ratio 5/3, because the
numbers 5 and 3 are smaller than 27 and 16.
What about F? F should certainly be a fourth above C, so we have
C - 4/3 - F. A little arithmetic -- by now you are used to doing these
multiplicative calculations -- shows that
E - 16/15 - F
It remains to do B. The following choice can be justified, but we
simply assert: let B be such that we have
A - 9/8 - B - 16/15 - C
Note that 9/8 times 16/15 is 6/5, which compares nicely with the
E - 6/5 - G obtained earlier.
The Fundamental Diatonic Scale(FDS) that we have created, following the
principle that smallest ratios get preference, is:
C - 9/8 - D - 10/9 - E - 16/15 - F - 9/8 - G - 10/9 - A - 9/8 - B - 16/15 - C
[ NB: There is nothing original to AMcC in the above; John Redfield gives
the above diatonic scale in his 70 year old book, and it is sure that this
goes back to Helmholtz at least. Indeed Bach undoubtedly knew the above
ratios. ]
It is now time to sit back and give this sequence of notes a good long
stare. For there is much to check, much to observe in the above ratios;
here, in no particular order of importance, are some comments.
1. The product of all these ratios -- 9/8 * 10/9 * . . .* 16/15 is 2;
so the FDS does indeed span an octave.
2. We have two different kinds of "whole steps", a big one where the jump
is by a factor of 9/8, and a smaller one where the jump is by a factor of
10/9. This corresponds to the difference between the first finger 'E's
mentioned in AMcC's screed on Intonation. Note also that the ratio of
of 9/8 to 10/9 is our already mentioned Musical Comma; a musical comma
is what a string player, checking the E on the D string that "groks" with
the G below, vs the E on the D string that "groks" with the A above, must
adjust by!
3. We have, pleasantly, several fifths -- C - 3/2 - G and F - 3/2 C and
D - 3/2 - A and E - 3/2 - B, as can easily be checked. Correspondingly,
we have several good fourths; we leave as exercises in arithmetic to see
which intervals give fourths.
4. We also have three good major thirds: C - 5/4 - E, F - 5/4 - A,
G - 5/4 -B. But our minor third situation is a bit chaotic; we recommend
to the interested reader to check the variety that can be found.
But now the following distressing observation must be made: this FDS
is extremely _rigid_, insusceptible to modulation. Let's try modulating
to the key of F; we have, as a start:
F - 9/8 - G - 10/9 - A - 16/15 - Bb(**) - 9/8 - C ; we of course have to
introduce the new note Bb, as we all know.
But the trouble comes as we try to continue: C - 9/8 - D - 10/9 - E -
16/15 - F, and this is not right! It should be -- compare the FDS as above --
C - 10/9 - D' - 9/8 - E
where the two D's are not the same! So in modulating a fifth down one must
not only put in a flat -- we're all used to that -- but one must introduce
a much more subtle change in one of the notes that we all rely on!
I conclude this tentative essay with the personal observation: for me,
a _just_ intonation, based, as it seemingly must be, on our FDS derived
above, puts us into a straitjacket. We have these seven notes, and their
octave transposes, to play with(pun intended), and modulations require
new notes to be introduced, or, equivalently, changes in the FDS.
Historically, we introduce extra notes -- five of them -- and fudge them
all up a tiny bit to achieve our chromatic scale as exemplified on the
piano. We string players can, however, do better; we can, if we are
sharp-eared and nimble-fingered, play, occasionally, in an "appropriate"
FDS. The price: to be controversial, to make subjective judgments about
what key we are in, and whether e.g. a big whole step or a small whole
step is appropriate. It is a difficult business, this "playing in tune"!!