From: Ken.Pledger@vuw.ac.nz (Ken Pledger)
Subject: Re: Mathematics behind western music : the number 12
Date: Thu, 14 Jun 2001 09:46:51 +1200
Newsgroups: sci.math
Summary: [missing]
In article , Karl Forsberg
wrote:
> mareg@primrose.csv.warwick.ac.uk wrote:
> > In article ,
> > bobcarls@aol.com (Bob Carlson) writes:
> > >....
> > >Is there something special about the number 12 that made this choice
> > >of frequencies in western music ?
> > >If we divide the octave by another number let's say 16 to get an exotic
> > >keyboard , will it sound less "harmonic" to our ears? ....
> > ....
> > IIRC, then there is no number below 40 something that is definitely
> > superior to 12 in this respect, but I am sure somebody will correct
> > me if I am wrong!
> >
> > Derek Holt.
> >
>
> The successive approximations obtained from a continued fraction
> expansion of log_2(3/2) is:
>
> {1/2,3/5,7/12,24/41,31/53....}
>
> This implies that if your main motivation is good approximations to a
> perfect fifth the next alternatives would be 41 or 53 tones.
Yes, and next 306. There's a very good explanation of all this in
Sir James Jeans, "Science and Music," Chapter V. On p.188 he mentions the
above result (not the details) of the continued fraction method for
finding increasingly good approximations to log_2(3/2). That's fine for
perfect fifths, but what about major thirds? He points out that the 5/4
ratio is equivalent to 3.86 equal-tempered semitones (i.e. 12ths of an
octave), to 13.20 41sts of an octave, and to 17.06 53rds of an octave. So
to improve upon our division of the octave into 12ths, you need to move up
to 53rds.
Jeans says that "two harmoniums with 53 notes to the octave were
built" in the mid-19th century. They may have been wonderful for harmony;
but what about melody, I wonder?
Ken Pledger.
==============================================================================
From: Brandon Hombs
Subject: Re: Mathematics behind western music : the number 12
Date: Thu, 7 Jun 2001 09:56:53 -0500
Newsgroups: sci.math
I read a book by Hindemith that addressed this topic nicely. He was very
clear and I found it quite interesting. I'm sorry I can't remember the
name of the book. I would guess that if you check a library for books by
Paul Hindemith you would not have any problems determining to which book
I'm refering.
Hindemith went in to adequate detail about different tonal systems and
describes attempts to use semitones and higher precision systems. He
gives good mathematical and musical reasons for choosing the 12 tone
system. I highly recommend the book.
Brandon Hombs
==============================================================================
From: Victor Eijkhout
Subject: Re: Mathematics behind western music : the number 12
Date: 07 Jun 2001 11:12:51 -0400
Newsgroups: sci.math
bobcarls@aol.com (Bob Carlson) writes:
> I guess that the answer depeneds both on mathematics and the
> perception of the human brain
I don't know why the brain perceives two notes an octave apart as "the same"
and doesn't do this with ratio of 3, but if you take this for given ...
The scale comes from filling in the other harmonics.
Basis frequency x (C), 2nd harmonic is 3x (G), bring this down to your basis
octave and you get 3/2 x. Do the same with other fractions, (9/8 (D), 5/4 (E))
and pretty soon you have 12 notes. However, if you would do this starting
at, say, 9/8 x (D), you get slightly different notes. Using root(12,2)
is a pretty good approximation, and it makes transposing possible, as well
as playing in different keys without retuning your keyboard.
Start reading Helmholtz for a good physics & psycho-acoustics combo.
Eh, "Sensations of Tone as the basis for a theory of music"? I'm too lazy
too look up the exact title. It's old, but a classic. Reprinted by Dover.
--
Victor Eijkhout
"One of the great things about books is sometimes there are some
fantastic pictures." [G.W. Bush]
==============================================================================
From: Ioannis
Subject: Re: Mathematics behind western music : the number 12
Date: Thu, 07 Jun 2001 20:26:37 +0200
Newsgroups: sci.math
Bob Carlson wrote:
[snip]
> Is there something special about the number 12 that made this choice
> of frequencies in the western music ?
Yes. It has to do with the way the intervals of fifths are perceived
(subjectively), but as another poster suggested it was not a purely
calculated choice to this extent. It was a matter of evolution and
although its use was exemplified best by JSB's Well-Tempered Clavier,
there ARE examples of Eastern music which use larger increments (thus
lesser number of intervals).
Greek Eastern and Byzantine church music uses just 7 tones (Pa, Vu Ga,
The, Ke, Zo, Ni) and although appropriate approximate conversions can be
made, this particular music cannot be played on a well-tempered
instrument.
> If we divide the octave by another number lets say 16 to get an exotic
> keyboard , will it sound less "harmonic" to our ears ?
I have no idea. Define "Less Harmonic" :*)
> I guess that the answer depeneds both on mathematics and the
> perception of the human brain , I asked some classical musicians
> about this and found that they know very little about frequencies and
> mathematics so I am trying to ask here .
Not all classical musicians know very little about this. If you want to
see some extensive analyses about the well-tempered issue, do a google
search on the newsgroup alt.music.j-s-bach. There were several
discussions there recently and one particular poster, mr Anton Kellner,
has posted detailed analyses of this.
Check:
> Thanks in advance,
> Bob
--
Ioannis Galidakis
_______________________________________________
"The traits of the good scientist: good command
of logic and _excellent_ command of insanity."
==============================================================================
From: hrubin@odds.stat.purdue.edu (Herman Rubin)
Subject: Re: Mathematics behind western music : the number 12
Date: 7 Jun 2001 20:27:45 -0500
Newsgroups: sci.math
In article ,
Bob Carlson wrote:
>The western music uses 12 different keys (modulo a ahift by an octave)
>.
>The frequebcy ratio between a key and the same key shifted by an
>octave above is 1:2 , so the ratio between 2 consecutive keys on the
>piano keyboard is 2^(1/12) .
>Is there something special about the number 12 that made this choice
>of frequencies in the western music ?
>If we divide the octave by another number lets say 16 to get an exotic
>keyboard , will it sound less "harmonic" to our ears ?
>I guess that the answer depeneds both on mathematics and the
>perception of the human brain , I asked some classical musicians
>about this and found that they know very little about frequencies and
>mathematics so I am trying to ask here .
The original scale was based on simple rational numbers, and
was used by the Pythagoreans.
The basic scale (C) has the relative frequencies
1 9/8 5/4 4/3 3/2 5/3 15/8 2
with only three ratios between successive notes, 9/8, 10/9, and 16/15.
This makes it difficult to transcribe to different starting
points, even with interpolations, and the idea of making
the larger intervals into a full step and the smaller
intervals to a half step led to the tempered scale with the
ratios all 2^(1/12).
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Dept. of Statistics, Purdue Univ., West Lafayette IN47907-1399
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
==============================================================================
From: Kevin Foltinek
Subject: Re: Mathematics behind western music : the number 12
Date: 09 Jun 2001 01:02:22 -0500
Newsgroups: sci.math
bobcarls@aol.com (Bob Carlson) writes:
> Is there something special about the number 12 that made this choice
> of frequencies in the western music ?
> If we divide the octave by another number lets say 16 to get an exotic
> keyboard , will it sound less "harmonic" to our ears ?
As others have mentioned or implied, the notion of "harmonic" is of
some importance here.
The mathematical harmonics arise from a Fourier series: any periodic
function [technical hypotheses deleted] is a linear combination of
sine and cosine functions, specifically, sin(2 n pi t/T) and cos(2 n
pi t/T) where t is time, T is the period, and n is a non-negative
integer. Why a Fourier series? Well, our perception (and perhaps the
universe itself) is invariant under time-shift, so the measure on
[0,1] should be uniform; hence the relevant function space is the
usual L^2[0,1]. I can only speculate why the trigonometric functions
are more appropriatae than, say, the Legendre polynomials (with
appropriate periodic extensions), other than the observation that the
trig functions satisfy x''=-x and whatever mechanisms are involved in
sound perception might tend towards a linearization-type effect; there
may be other considerations, such as smoothness of the periodic
extensions of the basis elements. (Question: characterize the
orthonormal bases of L^2[0,1] whose periodic extensions are smooth or
analytic functions.)
Once you accept the observation that the second harmonic (n=2) "sounds
the same" as the first harmonic, it is natural to work modulo these
octaves. (See below for the importance of the octave.) An immediate
consequence of this is the importance of the ratios 3/2, 4/3, 5/3,
5/4, etc. Turns out that 2^(k/12) gives good approximations to some
of these; we (or at least, musicians) are perhaps fortunate that such
a small number happens to work so well. I have heard of the use of 19
tones in some other musical systems; someone else gave a table of
numbers in which 19 was a candidate.
Regarding why the second harmonic sounds the same but the third sounds
different, an explanation is given in Benade's book "Fundamentals of
Musical Acoustics". To quickly paraphrase, look at the numbers:
200 400 600 800 1000 1200 ... [harmonics of 200]
400 800 1600 ... [harmonics of 400]
600 1200 1800 ... [harmonics of 600]
In most "real tones", many harmonics, not just the fundamental, are
present; however, only the first few appear with significant
amplitude. (The higher frequencies "carry more energy" and are
therefore more difficult to excite.) In the 400/200 case, these first
few overlap at 400 and 800; in the 600/200 case, these first few
overlap only at 600. (I just defined "few"=5.)
If the higher tone is not exactly at 400 or 600, but slightly off,
then there are "beat frequencies" present, and because of
nonlinearities in the ear, there are additional beat-like frequencies;
these beat frequencies are low but also appear near the overlapping
frequencies which I mentioned above. The more overlapping
frequencies, the more "clues" we have with which to compare the tones;
maybe one overlap (at 600) is not enough for us to say that the two
tones "sound the same". Experiments with sinusoid generators show
that the effect is still present, though perhaps to a lesser extent,
for single-harmonic tones, so the nonlinearities are evidently an
important factor.
About those harmonics, and the periodic functions which they so well
represent: Many musical instruments do not generate periodic tones.
For example, pianos and bells decay. Also in Benade's book is a list
of frequencies produced by some bells (the experiment was done by Lord
Raleigh). One such set of these frequencies (converted so that the
bell sounded a C4, or 261.6Hz) is
278 467 620 786 1046 .
What happened to the harmonics
261.6 523.2 784.8 1046.6
of the pitch of the bell? Well, they show up eventually, but the
first three are rather far off, and - there are three where there
should have been two!
Notice that 620/467 =~= 4/3, and 467/278 =~= 5/3 =~= 786/620. The
perceived pitch comes from the upper harmonics. Now consider the
"fundamental pitch" of the (278,467,620) sequence: it's about 31Hz.
(31*9=279, 31*15=465, 31*20=620.) This is very far below the
perceived pitch, and near the limit of our perception anyway. So what
we have is a "phantom" 31 Hz (approximately B0, the note which would
be to the left of a piano keyboard) sitting underneath the harmonics
of C4. You can sort of imagine, but not quite hear, a nice "bass"
component to a church bell.
Notice I said "=~=" (approximately equal). The beat frequencies
arising from the discrepancies are on the order of 1Hz, i.e., one
cycle per second. You can hear these beats as oscillations of
amplitude. Perhaps more significant is the observation that a real
bell decays; the decay rates of each component need not be the same,
but in any case the "half-life" is comparable to the beat period, and
the result is an interaction between decay rates and beats which gives
a real bell a rather interesting dynamic nature.
What about a piano string? The frequency components of a piano
string are much closer to mathematical harmonics, but not exact.
There are two (competing) forces which perturb the frequencies; one of
these is a linear effect and the other is nonlinear. The first is the
damping effect. If you look at the solutions of the damped wave
equation
u_tt + k u_t = u_xx , u(0)=0, u(1)=0 ,
you will find that the frequencies are all lowered from the integer
harmonics. On the other hand, if you think about the nonlinearity of
the physical string (the length of the string increases with a
deflection, causing an increase in the tension in the string), you
will see that the frequencies are increased from the integer
harmonics. In real pianos, the nonlinear increasing effect
dominates. (This makes tuning a piano rather difficult - the
harmonics of the notes separated by an octave do not line up!) You
can hear the beat frequencies if you listen closely to a piano; once
again, they interact with the decay rates to give a nice dynamic
character to the instrument. (Of course the body of the piano
contributes to the sound as well.)
Kevin.
==============================================================================
From: Kevin Foltinek
Subject: Re: Mathematics behind western music : the number 12
Date: 11 Jun 2001 23:46:25 -0500
Newsgroups: sci.math
Randy Poe writes:
> ...But I'm not sure I get it. So the reason why it sounds
> "the same" is more overlap of harmonics? Or does it
> have to do with the fact that the harmonics overlapped
> are strong ones?
Probably both. The book to which I referred said something about more
clues. I suspect there's a lot more going on here, though, including
1) The physics of the cochlea;
2) The neural connections;
3) Neural training;
4) Musical training and other social conditioning.
> I think from what you're saying that the musical ear
> would in fact get the illusion of 200 Hz from
> 600+800+1000.
Yes. It seems to be more than an illusion, though; apparently the
nonlinearities actually construct the "missing harmonics" in a tone.
> I know that in some piano music a composer will ask
> for a chord to be held open (the keys held down)
> while other notes are struck. I think that something
> like this is what they are aiming for. The undamped
> strings are supposed to resonate not only at the
> harmonics, but at the fundamental. Or give the
> illusion to the ear that that is happening.
Again, probably not an illusion. The nonlinearities in the strings
and the piano itself (together with the slight anharmonicity of the
notes being played) will no doubt produce enough of a spectrum that
the open notes will resonate.
As I understand it, the nonlinearities introduce harmonics as
follows. Consider a nonlinear "black box", F, with input and output
related by y(t) = F(x(t)). A sinusoidal input x(t) will produce a
non-sinusoidal but still periodic output y(t); hence, the construction
of harmonics.
Now, write a series expansion: F(x) = a0 + a1*x + a2*x^2 + ... .
(Assume convergence, or if you don't want to do that, truncate and use
a bounded remainder term.) The artificial harmonics come from the
terms of order two and higher. For the case of acoustics, most F will
be approximately linear, so the added harmonics will be small and of
low order. However, you can see that if x is the sum of two
sinusoidal inputs of slightly anharmonic frequencies, the collection
of all the added harmonics will be quite rich, producing more of a
spectrum than a tone. (Note that for anharmonicity on the order of
1Hz, which is barely noticeable [except to serious musicians] in the
middle of the musical range, you will get the usual harmonics of the
two source tones plus or minus 1Hz, 2Hz, 3Hz, etc; note also that,
often, musical tones will change far more frequently than once per
second. In other words, more or less, the tone changes faster than
the introduced harmonics.)
Kevin.
==============================================================================
From: pwomack@engage.com (bugbear)
Subject: Re: Mathematics behind western music : the number 12
Date: 12 Jun 2001 01:35:02 -0700
Newsgroups: sci.math
Here's a seriously unconventional system
http://www.harmonics.com/lucy/
BugBear
==============================================================================
From: Victor Eijkhout
Subject: Re: Mathematics behind western music : the number 12
Date: 07 Jun 2001 14:13:01 -0400
Newsgroups: sci.math
wild_dj@mit.edu (Jake Wildstrom) writes:
> >IIRC, then there is no number below 40 something that is definitely
> >superior to 12 in this respect, but I am sure somebody will correct
> >me if I am wrong!
> Theoretically a 29-step scale might be at least as euphonic as the
> 12-step scale. The sums on 31 and 34 look promising, but the
> discrepancies on the first two columns are unacceptable.
No time to check your calculations, but here's a link:
http://www.xs4all.nl/~huygensf/english/fokker.html
(For those of you unwilling to explore link: two Dutch guys apparently
explored the 31 option: Huygens wrote about it, and Fokker
built an organ with it.)
--
Victor Eijkhout
"One of the great things about books is sometimes there are some
fantastic pictures." [G.W. Bush]
==============================================================================
From: Ioannis
Subject: Re: Mathematics behind western music : the number 12
Date: Thu, 14 Jun 2001 17:36:24 +0200
Newsgroups: sci.math
Ken Pledger wrote:
[snip]
>
> Jeans says that "two harmoniums with 53 notes to the octave were
> built" in the mid-19th century.
Prior to the time when JSB made the well-tempered intonation famous,
there were designs and actual builts of even harpsichords with many
notes per octave. One I have seen had separate keys for C, C#, key for
between C# and Bb, Bb, B, etc, with entire "staircase" patterns
repeating which included many inbetween tones for the entire octave.
They were a keyboardist's nightmare.
The mechanisms were such that certain keys (the upper ones) forced
various side keys to come down as well. They were later abandoned in
favour of the well-tempered scheme.
--
Ioannis Galidakis
_______________________________________________
"The traits of the good scientist: good command
of logic and _excellent_ command of insanity."