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Re: Fundamentals of music
Posted:
Jun 14, 1996 9:46 PM


Martin Bright wrote: > > In article <4pkg6o$5gm@news.abs.net>, Bill Vanyo <vanyo@ezaccess.net> wrote: > >Which is why, if you've ever wondered, most musical instruments have a > >12 note octave. Notes combine well (to the ear) when the ratios of > >their frquencies are expressable as small whole numbers. > > ? > I would question this. > > For a start, the 12 note octave is very much a Western thing. > > Also, all 12 notes are only needed once you have the idea of playing > in different keys; before a certain point (not sure when, about 500 years > ago maybe?) music was simply played in different modes. > > Of course a lot of the intervals found, particularly perfect 5ths and > 4ths, which are the intervals which have a very simple rational > representation, are those which are found in the harmonic series produced > by, say, a string or a tube, and so these naturally find their way into > music through the design of different instruments (early brass instruments, > in particular, had no valves, so the only notes you could get from them > were harmonics). If you try to match the harmonic series with a modern > musical scale, a lot of the notes are quite a long way from the ones used > in the scale. > > As a last point, as somebody has already pointed out, the equally tempered > 12note octave, which makes all the notes an equal frequency ratio apart, > doesn't actually give any of these ratios given above exactly, but just > gets fairly close to all of them, rather than tuning perfectly in one key > at the expense of tuning badly in another key. > > I can't believe that somebody sat down one day and decided that 12 notes > was a sensible number to have in a scale, and from then on all musical > instruments were designed thus. > > Martin > >  > > Martin Bright _ _ _ ___ > Clare College, Cambridge  V  _o_) > mjb47@cam.ac.uk _V(____o_)
Possibly the 12tone scale evolved through experimentation with the tuning of strings, boring of holes in flutes, etc., with some experiments sounding better than others. You can imagine that a goodsounding flute would be more likely to be imitated than a poor sounding one.
I present here a suggestion of how one might derive 12 as a reasonable number of equallyspaced tones for a scale.
Suppose that we know that doubling the frequency of a constant tone raises its pitch by one octave, as the ancient Greeks knew. They also knew that the most pleasant harmonies involved pitches with low common multiples.
Converting frequencies to octaves can most simply be done with a base2 logarithmic scale; doubling the pitch adds exactly 1 to its base2 logarithm.
The simplest ratio not yielding an integer would be 3/2 (later called Sol, or a fifth interval, or a dominant tone, or V, or G on a C scale, etc.).
Expressed as a base2 logarithm, the number we would like to approximate (sorry, there's no way to do it exactly) via rational numbers is
log (3/2) , or about 0.585 . 2
Alternatively, moving down instead of up, we can get a subdominant interval (instead of dominant; also called Fa, a fourth, IV, F, etc.). This gives us a ratio of 2/3 or, to put it into the current octave, 4/3, in which case the number of interest is
log (4/3) , or about 0.415 . 2
A continuedfraction expansion (see computer program at the end of this message) of either of these numbers shows that the good approximations have denominators in the set {1, 2, 5, 12, 41, 53, 306, 665, ...}. (190537 looks terrific, but probably way higher than the number of organs of Corti in any ordinary mortal's ear.)
A scale with only 1 tone per octave is not too interesting, but I have heard percussion music that employs 2 tones (roughly, octave and fifth), and there is plenty of 5tone music around, although the scale is not always evenly spaced. 12 you already know about. 53 would be usable, and is certainly easy to produce with electronic components, but gets into the realm of fine distinctions in pitch that probably exceed most people's capabilities. So, of low, nontrivial numbers, 12 seems to have been a pretty good choice for a basis for interesting yet accessible harmonies. It also has several divisors, giving rise to wholetone scales (6 per octave, skipping alternate notes of a 12tone chromatice scale) and diminished (4 evenlyspaced pitches per octave) and augmented (3 evenlyspaced pitches per octave) chords. Unfortunately, the 6, 4, and 3tone sets all skip over step 7, which gives the (close to) 3/2 ratio that we wanted to include.
Although 41 and 53 are prime, 306 = 2 * 3^2 * 17, so a 17tone scale might have some possibilities, but it also misses landing on the 3/2 ratio (step 179 of a 306tone scale).
My (limited) understanding of Oriental music, which seems to lean more heavily toward 5 tones per octave, is that melody is relatively (compared to the Greek tradition) more important than harmony. If this be true, could the relative richness of 12 tones over 5 have contributed to that? For that matter, in the case of a society using a 2tone scale (e.g., drums), would there likely be a greater emphasis on complex rhythms? I guess questions like that are addressed in another newsgroup. If we choose to build a scale on 5/4 (Mi) instead of 3/2 (Sol), we get numbers like {3, 28, 59, ...}. 28 is only a bit more than twice the 12 in common use, so many people can hear the pitches; and it is composite. The closest that comes to 3/2 is step 16 (of 28), which is about 1% too low; step 17 is about 1.5% too high. A version of Bach's "Little Harmonic Labyrinth" adapted to the 28 keys of a 28 tone scale would be an interesting composition, but perhaps a bit difficult to appreciate on first hearing. :)
Anyway, if you are interested in looking at the continuedfraction expansions and drawing your own conclusions, I am including a QBasic program and its output. QBasic is part of MSDOS 5.0 and later (including Windows 95).
 Vincent Johns =================================================== PRINT " Continuedfraction expansion program" ' ' Copyright (c) 1996 Vincent R. Johns  This program is freeware. ' You may freely use this software for any application, but I ' would appreciate your giving me credit if you use it. ' ' Let f be a fraction to be converted to rational form ' ' Continued fraction = r(1) + 1 / ( r(2) + 1 / ( r(3) + 1 / ... )) ' ' r(1) + 1 / ( r(2) ) = u(2) / v(2) ' ON ERROR GOTO ExceptionHandler
N = 20 'Number of iterations DIM r(N) AS LONG 'Integer part of ratio DIM s(N) AS DOUBLE 'Remainder DIM u(N) AS LONG 'Numerator of ratio DIM v(N) AS LONG 'Denominator of ratio
f# = ATN(1) * 4 'pi f# = (SQR(5) + 1) / 2 'golden ratio (converges poorly) f# = EXP(1) 'e f# = LOG(3 / 2) / LOG(2) 'welltempered scale ratio for Sol
PRINT "Original value = "; f# s#(1) = f# FOR i = 1 TO N  1 r&(i) = INT(s#(i)) PRINT r&(i); "+", u&(i) = r&(i) v&(i) = 1 IF i > 1 THEN FOR j = i  1 TO 1 STEP 1 u&(j) = r&(j) * u&(j + 1) + v&(j + 1) v&(j) = u&(j + 1) NEXT j END IF g# = u&(1) / v&(1) PRINT u&(1); "/"; v&(1), 'Numerator & denominator PRINT USING "#.############### "; g#; 'Decimal expansion PRINT USING "##.##^^^^"; (g# / f#)  1 s#(i + 1) = 1 / (s#(i)  r&(i)) NEXT i END
ExceptionHandler: SELECT CASE ERR CASE 6, 11 'Overflow or division by zero PRINT "We appear to be done." END CASE ELSE PRINT PRINT "Unexpected error "; ERR; " on line "; ERL END END SELECT =================================================== Continuedfraction expansion program Original value = .5849625007211562 0 + 0 / 1 0.000000000000000 1.00D+00 1 + 1 / 1 1.000000000000000 7.10D01 1 + 1 / 2 0.500000000000000 1.45D01 2 + 3 / 5 0.600000000000000 2.57D02 2 + 7 / 12 0.583333333333333 2.79D03 3 + 24 / 41 0.585365853658537 6.90D04 1 + 31 / 53 0.584905660377359 9.72D05 5 + 179 / 306 0.584967320261438 8.24D06 2 + 389 / 665 0.584962406015038 1.62D07 23 + 9126 / 15601 0.584962502403692 2.88D09 2 + 18641 / 31867 0.584962500392255 5.62D10 2 + 46408 / 79335 0.584962500787799 1.14D10 1 + 65049 / 111202 0.584962500674448 7.98D11 1 + 111457 / 190537 0.584962500721645 8.35D13 55 + 6195184 / 10590737 0.584962500721149 1.21D14 1 + 6306641 / 10781274 0.584962500721158 2.85D15 4 + 31421748 / 53715833 0.584962500721156 1.90D16 2 + 69150137 / 118212940 0.584962500721156 1.90D16 1 + 100571885 / 171928773 0.584962500721156 0.00D+00



