Game of avoiding the tritone: From modes via first twelvetone to major and minor

There are two problems in just intonation, as in "octaves and other pure intervals". One I leave aside, it being that to get all major and all but one minor thirds pure, one fifth in the scale needs to be reduced, thus approaching tritone. Which is exactly why pythagorean tuning was preferred, even though the thirds were impure (64:81 rather than 64:80=4:5) The other is so much less necessary, but was felt more keenly: how to avoid the tritone.

The tritone being between final and fourth in lydian, to get a pure (minor) fourth down to final, or up from it, you lower the B to a B flat. The alternative would have been to avoid approaching them: which is so much easier in modes where the tones concerned involve neither final nor dominant. The first accidental was then a B flat. And a lydian (with its original final, not transposed) with a flattened B, is the scale of F major. The major mode transposed from modified "lydian" to pure naturals, called ionic by Glareanus, is of course between lydian and mixolydian:

lydian:
F-CGDAEB
major/ionic:
F-C-GDAEB
mixolydian:
FC-G-DAEB

He also instituted the eolic mode, between doric and phrygian:

doric:
FCG-D-AEB
eolic:
FCGD-A-EB
phrygian:
FCGDA-E-B

Eolic looks like minor, right? Well, when melodic minor descends, it is simply eolic. Melodic ascending, as well as harmonic minors are not pure eolic. We are not quite there yet.

Why is the tritone bad?

As you saw in the post on Octaves and other pure intervals, each scale authentic contained a line of seven tones intervalled a major fifth between each. It was not repeated in the plagal scale, since it only served to place the final, identic for both scales that share a name, in the line of fifths between F and B. B is not a final, since it has no major fifth above it (in natural notes, I will return to accidentals), only the minor fifth F. And no minor fourth below it, only the major fourth, F. Why is that bad?

Between 64 and 96 Hz, never mind what we call them, there is a perfect/major fifth, a 2:3. Each tone has its multiples, 1 being itself, 2 being twice as fast, 3 being 3 times as fast.

64=64*1
96=96*1
128=64*2
192=64*3=96*2
256=64*4
288=96*3
320=64*5
384=64*6=96*4

et c
Every forth multiple is a double one, a coincidence between a multiple of the one and the other. 64 and 96 Hz or any other two tones related 2:3 support each other. The other multiples are well spaced and make no swings or bad vibrations.

Now, between F and B there are six 2:3. 6 times as good? Uh-uh! It is 512:729. As between 64 and 91 1/8 Hz.

64=64*1
91 1/8=91 1/8*1
128=64*2
182 1/4=91 1/8*2
192=64*3
256=64*4
273 3/8=91 1/8*3
320=64*5
364 1/2=91 1/8*4
384=64*6
448=64*7
455 5/8=91 1/8*5
512=64*8
546 3/4=91 1/8*6
576=64*9
637 7/8=91 1/8*7
640=64*10
704=64*11
729=91 1/8*8
768=64*12
820 1/8=91 1/8*9
832=64*13
896=64*14
911 1/4=91 1/8*10
960=64*15
1002 3/8=91 1/8*11
1024=64*16
1088=64*17
1093 1/2=91 1/8*12

et c

Not one coincidence, not one reenforcement, but several bad spacings between overtones, like between the last two, which are closer than the two basic tones to start with. To make it worse, overtones are not just in descending strength (2 being half, 3 being a third as strong as 1, ideally), but, since each overtone itself has overtones, any overtone that is a multiple of a multiple rather than a prime number, will be stronger than the multiples of far off prime numbers:

1002 3/8=91 1/8*11 and no more than an eleventh as strong as 91 1/8 itself, but
1024=64*16=128*8=256*4=512*2 and duly enforced.

Next pair of overtones:
1088=64*17 and no more than a seventeenth as strong as 64, whereas
1093 1/2=91 1/8*12=182 1/4*6=273 3/8*4=364 1/2*3=576 3/4*2 and duly enforced.

Thus overtones close enough to kill each other when they do not enforce tend to the one and to the other, so to say fighting for supremacy - which may be one cause why it sounds bad. The usual explanation of beats is the closeness of the overtones. See Wiki. I am just pointing out that there are more of them in the overtones of interval 64:91 1/8 than in those of 64:96.

I have made similar tables with a G for upper tone, ratios 3:4 (good), 5:7 (not so good, even bad), 32:45 (real bad, but maybe not quite as much as 512:729). But these tables only explain that, when you strike two whites with three whites between them, it sounds good, unless there is one fewer black than otherwise, i e between B and F, whose inverse, F and B have between them two whites and one more black than other two whites separated by two whites. Why explain what you can simply hear? In intervals, the ear is a good judge. It is quite as sure as these tables that the tritone is bad.

Hans Lundahl

Schenker meets Couperin in Hypodoris

Alas, I tried to get a correct scan in a cyber that was, ultimately, not helping me very efficiently to get it right. However, the owner said that although the picture as you click on it will be too big, pringing it out (the right way?) will restore the right size. Your try.