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."