Math and Music

If a tree falls in the forest and no one is around to hear it, does it make a sound? As is elucidated by the riddle, there are two related but distinct phenomena encompassed by the term sound. Sound could refer to the qualitative experience of hearing a sound of which the explanation would be neurological with appeals to electrochemical mechanisms. In this case no sound was made by the tree falling. Meanwhile an explanation of the vibration of air molecules which is then decoded by various parts of our ear/brain to create the qualitative experience would appeal to the laws of physics and math. Maybe they’d both appeal to physics one day if you believe reductionism, but regardless, in this case the tree most certainly made a sound. In this presentation I seek to connect these two realms in order to explain how laws of sound are reflected in mechanisms of brain and later in various musical systems.

I would like to start by defining some of the terms people who work with vibrating air molecules use to talk about their subject matter.

Waveforms represent sounds! They are composed of

1)       Amplitude = volume/loudness

The amount that air molecules that are displaced

Measured in decibels

The lower waveform is louder because it has larger amplitude; everything else about the two tones is the same.

2)      Frequency = pitch

Measured in hertz (hz), or periodicity…

Or compressions/expansions per second

 

Imagine that in the lower waveform the oscillation is occurring twice for every time it occurs once in the upper waveform. This would means that if the first waveform had a frequency of 100 hz, then the second one would be 200 hz, a 2:1 ration, or in music theory terms, an octave. 

To hear what this sounds like go to http://en.wikipedia.org/wiki/Octave (it plays first two sounds together, then the lower one, then the higher one)

3      Complexity = timbre                                                 

   Created by naturally occurring overtones.

Very few things in the world create pretty waveforms like the ones above. A tuning fork is the classic example; the sound you get when you press a button on a telephone is a more modern one. For just about everything else we get multiple waves appear simultaneously, the fundamental frequency and its integer multiples each occurring with a lower amplitude than the fundamental. The diagram bellow illustrates how this happens using a string.

•       440 Hz          (440*1)          

1:1 ratio – note = A

•  880 Hz          (440*2)          

2:1 ration – note = A

•  1320 Hz       (440*3)          

3:2 ratio – note = E

•  1760 Hz       (440*4)          

4:3 ratio – note = D

•  2200 Hz       (440*5)

5:4 ratio – note = C#

Interestingly, our brain doesn’t consciously perceive all these tones, although you can train yourself to hear them. The illusion of the missing fundamental demonstrates this property; you can remove the fundamental frequency but you still hear the vibrating air molecules as the pitch corresponding to the fundamental frequency. To try it yourself click the bellow. http://sites.sinauer.com/wolfe3e/chap10/missingfundF.htm

Something that may not be obvious from the above is what the 2:1 ratio refers to the real world. Take a guitar and pluck the string, then take a ruler and measure the string (or just imagine you’re doing it). Cut the distance in half and you’ll find your finger on the twelfth fret, the only one with two dots on it. Play it and it will be twice the hz as compared to the same string played without your finger on it. You might notice this interval (distance between two notes) sound pleasant or consonant. It’s also the same interval you listened to if you followed the Wikipedia link above.

Those in psycho-acoustics argue that it sounds pleasant due to what Helmholtz termed the coincidence of partials, which translates in laymen’s terms to overlapping overtones (indeed consonance literally means overlapping sounds). 100 hz and 200 hz tones (a 2:1 ratio) will have overlapping overtones at 400 hz 600 hz and 800 hz. As you can see this includes the majority of the overtones propagated when these notes are struck. For this reason notes that are related to one another by a 2:1 ratio not only sound consonant but in fact sound like the same note just in a different octave. Most people would agree that something about these two notes is fundamentally the same despite different pitches and qualia. This is then reflected in music systems around the world. If men and woman start singing together, they will naturally sing an octave apart.

Meanwhile, the other notes with overtones overlapping with the overtones of the 100 hz fundamental frequency will also sound consonant but not as starkly similar.  For example, a 150 hz fundamental will overlap with 100 hz at 300 hz, 600hz etc, but not 450 hz. The ratio of 150:100 is the same as that of 3:2, or a perfect fifth in music theory terms. The octave and fifth are present in the majority of music systems around the world. A pattern emerges already, the smaller ratio of frequencies, the more overlap in overtones. 

Many people agree the 5:4 and 6:5 ratios are the most beautiful. I personally believe that this is due to the Goldilocks Effect of similarity; their overtones overlap just enough but not too much. Pythagoras didn’t know about overtones, but he thought that all these small integer ratios (3:2, 4:3 etc) sounded really pleasant, which led him to claim to have discovered divinity in sound. Using the hand out on page 5 you should be able to play any one of these intervals on piano or guitar.

So you take the fundamental and the overtones and you perform this complex mathematical procedure called Fourier Analysis and you get the complex waveform. Like many interesting things this makes a lot more sense as a visual (left). The inverse is also possible, aka we can decompose the complex wave into its component frequencies. Cool right?

But what about in the real world? Well it just so happens that depending on    what physical thing is producing the sound (your voice box, a violin, a piano, etc), different overtones will naturally occur with different amplitudes. In the same vein, whether you strike the guitar string with a pick or with your finger will also create differences in the amplitude of different overtones, even if the amplitude of the fundamental is the same. This is a large part of what causes a guitar to sound like a guitar, and your voice to sound like your voice; it’s called timbre!

Helmholtz figured this out and build the worlds first sythesizer using multiple tuning forks and resonator boxes!

•       So the big question: Why, when creating scales and chords, melodies and harmonies, do humans employ a relatively small number of the infinite possibilities?

•       Well we already determined that what we hear is biased by the overtone series. Studies have shown that these intervals are then represented in language prosody (the tones we use in speech to display different emotions) and musical composition (“A Biological Rationale for Musical Scales”,  Gill and  Purves“Musical intervals in speech”, Ross and Choi). This leads to accurate decoding of emotions from prosody and music across languages and cultures (“Universal Recognition of Three Basic Emotions in Music”, Fritz, Thomas, Jentschke, Gosselin, Sammler, Peretz, Turner, Friederici, and Koelsch “Recognition of emotion in Japanese, Western, and Hindustani music by Japanese listeners”, Balkwill, Thompson, and Matsunaga, “Communication of Emotions in vocal expression and music performance: different channels same code?”, Juslin, and Laukka).

•       It doesn’t matter what tuning system you use, whether you have 12 notes, 19 notes, or 24 notes dividing the octave. Those are cultural variations, overtones are naturally occurring. Even in those alternate tuning systems, scales are generally based on 5 or 7 tones that correspond in some predictable way with the overtone series  e.g. demarcate the octave and fifth as the most stable notes in the scale

•       This isn’t to say dissonant intervals are not used. Rather, when they are used is dependent upon norms specified by cultural practices. Furthermore, cross-cultural variation comes to effect how we perceive and respond to different musical stimuli in important and documentable ways. One that comes to mind is the studies of interpolated notes. Imagine we take someone who has been raised in a western musical system and someone raised in an eastern music tradition. Let’s say the most common eastern scale shares 5/7 notes with the most common western scale.

Play each of these people a melody with just the 5 common notes, and then afterwords play them a note which was not included in the melody. If that note occurs in the western scale that encompasses the other 5 notes that were played, the westerner will say they heard it in the passage, but easterner will not, and vica-versa. We interpolate notes based on the musical system which we have grown up in. This in turn effects expectations, the heart of musical understand according to most theorists, and even modulates what you are actually hearing due to top down processing.

Pitch	Frequency (Hz)	Half-steps from C	Interval with C	Ratio with C
C	262	0	Unison	1:1
C sharp/D flat	262(2^1/12) = 278	1	Minor 2nd	16:15
D	262(2^2/12) = 294	2	Major 2nd	9:8
D sharp/E flat	262(2^3/12) = 312	3	Minor 3rd	6:5
E	262(2^4/12) = 330	4	Major 3rd	5:4
F	262(2^5/12) = 350	5	Perfect 4th	4:3
F sharp/G flat	262(2^6/12) = 371	6	Tritone	45:32
G	262(2^7/12) = 393	7	Perfect 5th	3:2
G sharp/A flat	262(2^8/12) = 416	8	Minor 6th	8:5
A	262(2^9/12) = 440	9	Major 6th	5:3
A sharp/ B flat	262(2^10/12) = 467	10	Minor 7th	16:9
B	262(2^11/12) = 495	11	Major 7th	15:8
C	262(2) = 524	12	octave	2:1