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Topic: Fundamentals of music
Replies: 1   Last Post: Jun 14, 1996 9:46 PM

 Vincent R. Johns Posts: 131 Registered: 12/6/04
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
> 12-note 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 12-tone 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
good-sounding 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 equally-spaced 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
base-2 logarithmic scale; doubling the pitch adds exactly 1 to its
base-2 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 base-2 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 sub-dominant
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 continued-fraction 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 5-tone 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, non-trivial 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 whole-tone scales (6 per octave, skipping alternate notes of a
12-tone chromatice scale) and diminished (4 evenly-spaced pitches
per octave) and augmented (3 evenly-spaced pitches per octave)
chords. Unfortunately, the 6-, 4-, and 3-tone 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 17-tone
scale might have some possibilities, but it also misses landing on
the 3/2 ratio (step 179 of a 306-tone 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 2-tone 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 continued-fraction
expansions and drawing your own conclusions, I am including a QBasic
program and its output. QBasic is part of MS-DOS 5.0 and later
(including Windows 95).

-- Vincent Johns

===================================================
PRINT " Continued-fraction 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) 'well-tempered 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
===================================================
Continued-fraction expansion program
Original value = .5849625007211562
0 + 0 / 1 0.000000000000000 -1.00D+00
1 + 1 / 1 1.000000000000000 7.10D-01
1 + 1 / 2 0.500000000000000 -1.45D-01
2 + 3 / 5 0.600000000000000 2.57D-02
2 + 7 / 12 0.583333333333333 -2.79D-03
3 + 24 / 41 0.585365853658537 6.90D-04
1 + 31 / 53 0.584905660377359 -9.72D-05
5 + 179 / 306 0.584967320261438 8.24D-06
2 + 389 / 665 0.584962406015038 -1.62D-07
23 + 9126 / 15601 0.584962502403692 2.88D-09
2 + 18641 / 31867 0.584962500392255 -5.62D-10
2 + 46408 / 79335 0.584962500787799 1.14D-10
1 + 65049 / 111202 0.584962500674448 -7.98D-11
1 + 111457 / 190537 0.584962500721645 8.35D-13
55 + 6195184 / 10590737 0.584962500721149 -1.21D-14
1 + 6306641 / 10781274 0.584962500721158 2.85D-15
4 + 31421748 / 53715833 0.584962500721156 -1.90D-16
2 + 69150137 / 118212940 0.584962500721156 1.90D-16
1 + 100571885 / 171928773 0.584962500721156 0.00D+00