Thursday, September 25, 2008

Octaves and other pure intervals

The octave is a ratio of 1:2. The two notes in an octave give the effect of being one and same note.

There are no natural numbers between one and two. However, 1:2=2:4.

If you spell the octave 2:4, it falls apart into two intervals, 2:3 and 3:4. These are the perfect or major fifth, the perfect or minor fourth. Since the higher of the one is an octave above the lower of the other, since the lower of the one is the higher of the other, they are considered equivalent: a minor fourth is the inversion of a major fifth.

We continue to subdivide the fifth: 2:3=4:6. 4:6 subdivides into 4:5, major third, and 5:6, minor third. Inversions: 5:8, minor sixth; 3:5, major sixth.

The major third, 4:5=8:10, subdivides into 8:9, major tone or (major) major second and 9:10, minor tone or (minor) major second. Inversions: 9:16, 5:9, minor sevenths.

The minor or perfect fourth, take away a major third:

3:-:4
4:5:-
12:15:16

15:16 is the major or diatonic semitone or the minor second. Inversion: 8:15, major seventh.

The minor or perfect fourth, take away a major tone:

3:-:4
8:9:-
24:27:32

27:32 is also a minor third, though not the best ratio. Inversion: 16:27, major sixth, also not the best ratio. 27:32 are less apart, a smaller interval, than 5:6, approaching 6:7, which is horrible, according to what J J Rousseau observed. I do not think I had the displeasure to hear it.

Or a minor tone:

3:-:4
9:10:-
9:10:12

10:12=5:6, the standard ratio for a minor third.

Minor third, 27:32, take away a minor tone:

27:-:32
9:10:-
27:30:32

30:32=15:16, minor second.

Minor third, 5:6, take away a major tone:

5:-:6
8:9:-
40:45:48

45:48=15:16.

Of the intervals so far, all except the seconds (tones and semitones) and the sevenths are good in simultaneous relations between voices, all except the sevenths are good in consecution in the same voice. Note that these intervals can be exactly applied only in monoscalar compositions.

The most basic intervals for ancient Greek music theory were minor fourth, major fifth, octave. 6:8:9:12. That parted the octave into two halves, a tone apart, each comprising a fourth. It is called the fourth (diatesseron) because the lowest is the fourth scale tone from highest to lowest or reverse. Between 6 and 8 you do not put 7, since 6:7 sounds bad. You put two different tones. And since they are smaller, more apt for steps than the fourth that is clearly a leap and a harmonic event, they are to begin with not fixed as the outer tones. Nearly all of ancient Greek music theory went into dividing the fourth in different ways. As well as rhythmic and versification theory - the main musical instrument was the song, thereafter came lyre and aulos. There were three kinds of division of the fourth: diatonic, chromatic, enharmonic. Diatonic counted, in its purest theoretical form, the Pythagorean, only fourths and fifths as pure and built all the scale on them. The major third was thus:

8 :9: -
- :8: 9
64:- :81

64:81 > 64:80=4:5.

And the semitone was not 3:4 - 4:5=15:16 but 243:256 -

3 :- :4
64 :81 :-
192:243:256

The minor third was, of course, 27:32.

64:81 is four major fifths up, and cut the superfluous octaves. 27:32 is three minor fourths up, also cutting superfluous octaves. When older theoreticians called fifth=2:3 and fourth=3:4 major and minor instead of perfect, they observed the fact that the major fifth is the basis of all major intervals and the minor fourth for all minor ones.

On the other hand, the ratios 4:5 and 5:6 were known - they were the standard bigger intervals in the enharmonic and chromatic genera. Then the chromatic kind had two intervals dividing 9:10 and enharmonic had two intervals dividing 15:16. Arabic and Hindustani musics still have intervals resembling some of these. In the Greco-Roman space, there was first a great vogue for the chromatic, then for the enharmonic kinds, and then a great revulsion aganst the enharmonic music. It has not survived at all among us. Chromatic was suppressed by the Church from Church music, and in the West Church music became as determining for popular music as ever Ecclesiastic Latin for the languages now spoken in Western Europe. In theory, only diatonic=Pythagorean was left.

That meant, all major fifths 2:3, all minor fourths 3:4, all major seconds 8:9, all minor seconds 243:256. Each fourth was divided into two major and one minor second. But this does not tell us where they were placed.

The diatonic kind had eight modes surviving in the West. Actually, more like four by two. I will give their by now traditional, but later names, and I will design them by enumerating their notes from grave to shrill, it being understood I name only what is on the piano white touches, and that the semitones, the minor seconds, are between B and C, then again between E and F. All other letters following each other immediately in alphabetic order, as well as G and A are major seconds:

First tone, doric authentic:
DEFGABCD, final note the first D, dominant note A.
DE, DA, DB are its major intervals (from/to final - which is understood in the following). ED etc invert them.
DF, DG, DC are its minor intervals. FD etc invert them.
Final is in middle of fifth series:
FCG-D-AEB

Second tone, doric plagal:
ABCDEFGA, final note D, dominant note F
CD, DE, DA major intervals.
AD, BD, DF and DG are its minor intervals.

Third tone, phrygian authentic:
EFGABCDE, final note E, dominant note B or C
EB only major interval, inverts to BE.
EF, EG, EA, EC, ED minor intervals inverting to FE et c.
Final has only one dominant in fifth series:
FCGDA-E-B

Fourth tone, phrygian plagal:
BCDEFGAB, final note E, dominant note A.
CE, DE, EB major intervals.
BE, EF, EG, EA, minor intervals.

Fifth tone, lydian authentic:
FGABCDEF, final note F, dominant C.
FG, FA, FB!, FC, FD, FE major intervals, invertible as GF et c.
minor intervals - none except the inversions
Final has no subdominants in fifth series:
F-CGDAEB

Sixth tone, lydian plagal:
CDEFGABC, final note F, dominant A.
FG, FA, FB!, FC major intervals
CF, DF, EF minor intervals

Seventh tone, mixolydian authentic:
GABCDEFG, final note G, dominant D
GA, GB, GD, GE major intrevals, inverted as AG et c
GC, GF minor intervals, invertible as CG et c
Final is one further to subdominants than doric in fifth series:
FC-G-DAEB

Eighth tone, mixolydian plagal:
DEFGABCD (like first tone, but) final note G, dominant C.
FG, GA, GB, GD major intervals
DG, EG, GC minor intervals.

Note one FB - this is NOT the minor fourth of 3:4, but a major fourth of - well, in Pythagorean it is very complex, in just intonation it may be 32:45. Its inversion, 45:64 is the minor fifth. Consecutively they sound not good. Simultaneously they sound not good either, by themselves. Add a third, above, below or between, it softens the badness. But even such a chord wants to dissolve into one without that interval. Given that the major fifth and the minor fourth are the standards of major and minor intervals, you may surmise that the reverse qualities are far off. Modern theory, calling minor fourths and major fifths perfect, call the reverse augmented fourth/diminished fifth. It is the only pair of intervals in a strictly Pythagorean scale, where the augmented and the diminished correspond to no other intervals. Since it is not singable, singers want to avoid it. What about the lydian mode? FB=32:45. How do you deal with it? Flatten the B when it runs toward F, since FBflat=3:4. And BF=45:64, make it BflatF=2:3.

FBflatB=
96:128:135
96:128=3:4

BflatBF=
128:135:192
128:192=2:3

ABflatB=
120:128:135
120:128=15:16
120:135=8:9

128:135=8:9 - 15:16

Note that the difference (interval) between Bflat and B is smaller than between Bflat and A. If 15:16 is major semitone, 128:135 is minor semitone. The major semitone is the minor second. The minor semitone is "augmented unison", but then 128:135 is outside the strictly Pythagorean. On the piano Bflat is a black. Its inversion is the "diminished octave".

Nough said for this here lesson.

Hans Lundahl
Avignon,
14/27 Sept 2008
OC Elevation of the Cross.

7 comments:

Old Fashioned Liberal said...

Your analysis of tuning is far better than mine, as it is much more mathematically complete, while mine only uses approximations. You also include a great deal of history that I didn't know. But there's one thing I don't understand:

"Of the intervals so far, all except the seconds (tones and semitones) and the sevenths are good in simultaneous relations between voices, all except the sevenths are good in consecution in the same voice. Note that these intervals can be exactly applied only in monoscalar compositions"

What sorts of ratios do you define as "good in simultaneous relations between voices?" I see ratios and ratios, and some are more complex than others, but I don't see why you should draw the line at any one spot, as it is a difference of degree of complexity that separates one ratio from the other. Granted, you can call one consonant and one disnnonant, as we actually do, this does not seem sufficient to call them good or not good. (Schoenberg, to his credit but to our eternal displeasure, also saw this difficulty, by the way. Why he cared, as he didn't care about good or evil anyway, is beyond me.)

Perhaps something was lost in the translation (the original, even if this original be just in your head, is in French, right?). Or perhaps I am assuming that you, like me, want to prove Schoenberg wrong when I shouldn't be making such assumptions.

Hans Lundahl said...

if you play two tones simultaneously, fifths, fourths, thirds and sixths will sound good (major third 4:5)

not even so if you tune thirds and sixths strictly pythagoreanly (major third 64:81)

simultaneous seconds are ok when leading to or from unison from or to third or fourth

otherwise seconds and sevenths are ok only when mitigated by thirds

sevenths are usually not sung

Old Fashioned Liberal said...

So you judge by the evidence of your ears. Schoenberg did not, so he would love to dispute the statement. That was part of the reason I wrote "The Book, Part III," because I needed to prove that there was no way around the ancient mathematics. Merely the statement that one ratio was simpler than another was not good enough for me, although it was for the Greeks and the Baroquists and the Romantics etc. Well, I am foolish and have a scupulous Descartean (or is it Thomistic?) streak, and some of them are probably wise.

Hans Lundahl said...

The ratio 6:7 was tested on monochord and found faulty by judgement of the ear: I have not tested it in overtone tables, but I suspect there are bad spacings.

As for the discussion on senses, it is continued under your post.

Hans Lundahl said...

p s:

I do not consider your Adornoesque streak anything close to Thomistic. I was discussing the Thomistic definition of beauty - what pleases when perceived - with a lady who is a common aquaintance to me and the then Cardinal Ratzinger.

He - or at least she, but I think she referred to him - thought St Thomas simply must have meant this allegorically for a purely spiritual beauty.

I disagreed, pointed out the obvious, and her standard answer was that is not defined. I do not need and ex cathedral definition that 2 * 2 = 4 before believing it.

Hans-Georg Lundahl said...

This is part of my writings on Musical theory and history with Schenkerian analyses and syntheses.

Hans-Georg Lundahl said...

On the octaves of the Nychthemeron, Octaves and other pure intervals, Why isthe tritone bad?, Game of avoiding the tritone: from modes via early twelve tone to major and minor, Between major and minor