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## Math and Music

I was a guest on Michel Meiffren’s Tuesday morning classical music show on KMUN this past Tuesday, and we had an opportunity to talk about Math and Music.  We weren’t able to get a usable recording, but there is a transcript available…

Segment 1

Michel:    As promised, we now start our conversation with our guest, Good Morning.

Rich:    Good morning.

M:    Would you introduce yourself?

R:    Sure.  My name is Rich Beveridge and I teach math up at Clatsop Community College.  I’ve been there for about five years.  A couple of years ago I started getting interested in the relationship between math and music and I turned that into a series of two talks about it…

M:    And also an article in The Bandit, which is the newspaper from the Clatsop Community College and I thought it would be very interesting to our listening audience to dive into the connection between, fundamental connection between math and music.  So let’s start at the beginning.  What is the difference between noise and a note of music?  And what is sound?

R:    Mm, interesting.  Well the second part, I think, is easier to answer first.  Sound is generally produced by the vibration of whatever medium it’s traveling through.  So, if it’s traveling through air, the air itself is vibrating.  And how quickly it vibrates determines the pitch of the music.  If it vibrates more quickly, then the pitch is higher and if it vibrates more slowly, then the pitch is lower.

M:    So what’s the difference between noise and a note of music?

R:    OK, so then the difference between noise and a note of music is that a note of music will produce what is essentially a perfect sine wave.  A very smooth, single wave, very regular without really any breaks.  Although that’s actually different for different musical instruments.  For instance, on a tuning fork, the sine wave is a perfect wave.  Now, a musical note produced by a musical instrument has a slightly more complicated wave and the wave is different for each instrument.  Noise itself is going to be very chaotic and even though the human voice is not considered noise that wave is also very uh, essentially chaotic.  That’s why different individuals have different voice prints.

M:    Chaotic…do you mean that many waves overlap?

R:    Exactly.   They’re all interacting in various different ways and that’s also what produces the distinctive wave for each musical instrument is the way that these different sounds – the way these different waves are interacting with each other.  The instruments themselves actually produce a number of different tones at any given time and then the way those tones interact is what produces the signature wave for the instrument.

M:    So you’re saying that the wave generated by a clarinet is different compared to the wave generated by the violin for example.  How so?  Can you describe a little more the difference…what the difference would be?

R:    Well, a lot of it, I think, has to do…well for instance in the clarinet versus the violin, you’ve got a woodwind instrument versus a stringed instrument.  And so in the woodwind instrument the sound is produced by a vibrating column of air inside the instrument.  And then the string…the sound is produced by the vibrating string.  But even within two different stringed instruments or two different woodwinds, it essentially, to the best of my understanding, comes down to the difference between the way the overtones, or harmonic tones, are produced by each instrument.  They’re produced a little bit differently in each instrument and then all the different tones, the way that they interact with each other produces the signature wave for the instrument.

M:    And the richness of the orchestra….where there are many different “flavors.”

R:    Uh-huh.

M:    Many different, uh, would it be the amplitude?

R:    Um, well yeah.  It’s both the amplitude and the frequency, and the different frequencies have different amplitudes.  For instance, one instrument might produce a particular frequency with a particular amplitude.  Another instrument might produce the same frequency, but with a different amplitude.  So it’s a little bit stronger in that instrument than the other one.

M:    And what is pitch?

R:    Well, essentially the pitch is the frequency of vibration.  Something that vibrates very quickly will produce a higher pitch.

M:    OK

R:    And then something that vibrates more slowly produces a lower pitch.

M:    Well, this is the first wave of information and we’ll be back for more after this song by Claudio Monteverdi.

Segment 2

M:    And we’re back with Rich Beveridge, discussing the relationship between math and music.  Very fundamental relationship.  So, I was surprised to learn that the scale that we’re most familiar with was standardized only at the beginning of the century.  So, the history of the scale is a fascinating one and Rich, if you could take us through that then.

R:    There’s a lot in there and I know that I’m going to skip over a good amount of it.  But it’s true that concert pitch was only standardized, I think, uh, …I was just doing a little more research last week and one of the latest things I found was sometime in the thirties.  It was progressing throughout the 1800’s and finally accepted in the early 20th century.  The chromatic scale, the twelve tone scale essentially came about in, I think the mid-1300’s was what I was looking at.  It grew out of the medieval music, the Gregorian chants and whatnot and it was in the mid-1300’s when the first keyboard organ was built in Germany.  And that was in Halberstadt, Germany in the mid-1300’s.  And they had a keyboard with three octaves and twelve tones within each octave.

M:    OK, that’s the birth of the twelve tone scale?

R:    Mm-hmm.  And that grew out of…what the monks were doing during the medieval period originally came out of what had been built by the Greeks.  The Greek scale got started with just four tones.  You have the root tone itself and then the tone an octave higher.  And then In between those two, you had what are called the fourth and the fifth.  The fundamental relationship of the fourth and the fifth to the root tone is that the frequencies of those tones…The frequency of the fifth is one and a half times the frequency of the root tone and the frequency of the fourth is one and a third times the frequency of the root tone.

M:    A very specific ratio.

R:    Exactly.  The Greek actually saw that, uh…they were focused a lot on the length of a vibrating string.  The Greek lyre was an instrument that they often used to produce the tones and those tones were based on the length of the strings, assuming that the other qualities of the string were the same…the cross section and the thickness and whatnot.

M:    The same thickness?

R:    Uh-huh.

M:    What would be the ratio of length that would create the octave?

R:    That’s always the inverse of the ratio of the frequencies.  For instance, the frequency of the octave is twice the frequency of the root tone and so the length of the string would be half the length of the root tone.  And so the Greeks developed the four tone system – the root, fourth, fifth and octave.  And they used a four stringed lyre to work with those four tones.  From that, you can continue, for instance, if you take the root tone and multiply it by one and a half, you get the fifth.  Now, if you take the fifth and you multiply that by one and a half, then you get another tone, and by continuing to raise those tones to next fifth, multiplying by one and a half each time, you can fill in more tones between the root and the octave.  They eventually ended up with seven tones, what is called the diatonic scale, which the DO, RE, MI, FA, SO and so on.

M:    The A, B, C, D…?

R:    It is, it is, but without the accidentals.

M:    Right, right, so tell me more about the twelve tone scale.  Is it a good way to represent it as twelve tones?

R:    It is, it is a good way.  There really is no good way to do it, but that’s kind of what I love about trying to build a musical scale is that there is always an approximation that you need to make if you’re not going to – I mean, if you just have the four tones that the Greeks started with, then everything is exact and everything’s fine.  But if you need, if you want to have more tones and you want them to be in a fixed relationship to each other, then you kind of have to, kind of have to – well, in making calendars they call it intercalating.  You have to kind of slide them around a little bit to get them to fit just right.  And the twelve tone scale is a good way of doing it.  Actually, if you look at the mathematics of trying to make the approximation with the fixed relationships of each tone to each other, there are a number of different ways that are better than others.  And the five tone scale works well, the twelve tone works well, then there’s also a  19 and a 41 tone scale.

M:    Right, there are exotic scales from our point of view from different cultures – from China and India.

R:    Exactly, exactly, Chinese music often uses the five tone pentatonic scale and the Indian music is a 24 tone scale.  Similar to the 12 tone scale but with other tones in between – halfway in between.

M:    OK, there again, a lot of information and to help us digest it – more music.  More music by Claudio Monteverdi.

Segment 3

M:    We’re back with Rich Beveridge and so, Rich, from you have said so far can we conclude that, in fact,  music is not an exact science.

R:    Yes, exactly.  Composers and musicians often run into this with what are called wolf tones.  These are tones that, ideally, should be harmonic with each other, they should in harmony, but because of the approximations that are made in creating a standardized scale, the chromatic twelve tone scale, they actually don’t sound as they should with each other because they’ve been approximated in a way that, essentially, destroys the harmony that should have been there.  The approximations that I keep talking about is – if you’ve got a root tone and an octave they wanted a standardized increase in pitch from one note to the next, because in the Greek diatonic system the difference between Do and Re is not necessarily the same as the difference between La and Ti.

[note: Actually the difference between Do and Re is the same as the difference between La and Ti, but the difference between Mi and Fa is not the same as the others.]

So what they wanted was they wanted a standardized, regulated – the same difference in pitch between every note.

M:    Are you saying that you can have dissonance or discordance or impossible chords?

R:    Yes, yes and I was thinking about it because I was thinking about in terms of an approximation and a student of mine actually asked me about the wolf tones and whether or not there was any way he could avoid them.  My initial reaction was, well, since it’s an approximation, if you shift one of the notes a little bit if you’re fingering it on a fretboard if you just shift it a little bit that should put it back into a perfect harmonic relationship with the notes that you’re using.  But, what creates the wolf tones in the first place is that even if you shift it so that it vibrates harmonically with one tone, you’ve shifted it away from another tone.  And that’s what creates the wolf tone is that, as you’re shifting it around, if you put in a good relationship with one tone, you’ve then destroyed its relationship with another tone.

M:    Mm-hm.  Is there any advantage to fretless instruments from that point of view.

R:    I think so.  But you still run into the problem that the composers ran into with the wolf tones – is that you then have to limit yourself to certain notes and avoid the notes that produce the wolf tones.

M:    Lots to ponder.  And lets do that while listening to Antonin Dvorak.

Segment 4

M:    Continuing our conversation with Rich Beveridge about the connection between mathematics and music.  So what were the contributions of the new technology that allowed the measurement of the frequencies.

R:    Hm.  That’s an interesting question and it was one that I pondered over for a while, because nowadays you have, you know, computer software and electronic technology, oscilloscopes.  So I kept wondering to myself, well, before you have all these technologies, how in the world do you know that something is vibrating 400 times per second?  Because it’s far too fast to count.  So, I had to dig through a lot of material to find my way to this and essentially it comes down to – in math what’s known as a system of equations.  Two equations with two unknowns will allow you to solve for those two unknowns.  Where these two equations and two unknowns come from is that these things that are observable.  Things that we do know about.  I was talking earlier about the length of the vibrating string and its frequency and people knew very clearly that the length of the string was intimately related to the vibrational frequency of the string.

M:    They knew the ratio.

R:    That was exactly it was that if you knew the ratio of the lengths of the strings then the ratio of the frequencies would be the reciprocal or the inverse.  We were talking about the octave, where the frequency of the octave is twice the frequency of the root tone and the
length of the string would be half.  And so if you know the lengths of the strings, you just invert them and then you have the ratio of the frequencies to each other.  The other thing that you can use is the difference between the two frequencies, if they’re very close to each other, they can’t be very far apart.  But, for instance if you’re tuning a guitar often you will hear what are called beats.  And this is a sound – [makes sound] and it’ll do that and actually, if you count the number of beats that you hear per second, that is actually the difference between the two frequencies.  There are equations that show that that’s true.  So then if you have the difference between the two frequencies and you have the ratio of the two frequencies to each other, then you can use those to solve one for the other and you can actually solve for the frequencies.

M:    Did measurement provide a benchmark?

R:    It did, but there were different benchmarks in different countries and even in different concert halls in the same country.  This continued through the 16, 17 and 1800’s.  In the 1800’s it started to be standardized.  But the concert pitch of, you know, the 18th century was actually a little bit flat compared to the concert pitch of the 20th century.

M:    And we will return to our muscial program now with Prelude de l’apres- midi d’un faune by Claude Debussy.

Segment 5

M:    And we are here with Rich Beveridge, mathematics teacher at Clatsop Community College discussing the relationship between mathematics and music.  And we have come to the end of this hour, Rich, and so are there thoughts about mathematics or music that you would like to leave us with?

R:    Well, actually, just listening to that Debussy piece I thought that it’s nice that the mathematics can allow us to kind of analyze and make sense of something as complex as music.  And then, if we use those mathematical ideas to make sense of something so complex, that allows us to then work with it.  And that’s what the composers were doing, that’s what the instrument makers were doing, that’s what the symphony orchestras were doing, but it’s also interesting in that that’s not the only way to do it.  You can also – different cultures made sense of music in different ways.  So then, you also get beautiful Asian music, Chinese and Japanese, as well as the Indian musical scale produces a different kind of beautiful way to make sense of something so complex.

M:    If people are interested in continuing a conversation with you abut music or mathematics or something entirely different, how do they get a hold of you?

R:    Well, I have an e-mail address at the college, and it’s rbeveridge@clatsopcc.edu and I also have a blog now at wordpress at richbeveridge.wordpress.com

M:    OK, well Rich, thanks so much for coming this morning it is much appreciated.

### 2 Responses

1. I know the frequencies of the notes in a pentatonic scale are the same, are their sound waves the same as well? Meaning do the waves meet up in the same way that the waves do in a western scale.

2. on March 15, 2010 at 11:55 AM | Reply richbeveridge

If, when you say “meet up,” you are referring to harmony, then the answer would be “sort of.”

The pentatonic scale is built around the musical fourth and fifth the same way that the Western equal tempered scale is. Because of this, the harmonies that exist in the root, fourth, fifth and octave are the same in the pentatonic scale as they are in the Western scale.

But, other harmonies that exist in the 12-tone Western scale might not appear in the pentatonic scale because the pentatonic scale might not have those notes.

For example, any harmony involving the third tone in the Western scale (major or minor 3rd) wouldn’t exist in the same way in the pentatonic scale because there is no note in the pentatonic scale that corresponds to the third note in the western scale.

Hope this helps!

UPDATED: I’ve been looking more closely at the pentatonic scale, and the last note in the scale before the octave is very nearly 1.75 times the frequency of the root tone. The equal-tempered scale has no note that corresponds to this tone so that the harmony produced by this tone with the root tone doesn’t exist in the equal-tempered scale.