Music & Noise

Discussion

music

The distinction between music and noise is mathematical form.

The foundation of music is the musical note: a combination of pitch (the musical word for frequency) and duration.

Music in its simplest form is monotonic; that is, a sequence of single frequencies played one at a time. Monotonic music is boring. Real music is polytonic; a mixture of different frequencies played together in a manner that sounds harmonious. And don't think that you need more than one instrument or more than one voice to make real music. Pretty much every musical instrument and every person's voice is a source of polytonic sound.

graph
Spectral analysis of a female voice. Note that the peaks occur at multiples of about 270 Hz (C♯).

Fourier analysis


A flute is essentially a tube that is open at both ends. Air is blown across one end and sound comes out the other. A spectral analysis confirms this. The harmonics are all whole number multiples of the fundamental frequency (436 Hz, a slightly flat A4 — a bit lower in frequency than is normally acceptable). Note how the second harmonic is nearly as intense as the fundamental. This strong second harmonic is part of what makes a flute sound like a flute.


A recorder is also a tube with two open ends. It produces a sound similar to a flute, but not exactly the same. Again the harmonics are whole number multiples of the fundamental frequency (923 Hz, a very sharp A5 -- much higher in frequency than is normally acceptable), but for some reason the second harmonic is nearly non existent. This nearly missing second harmonic is part of what makes a recorder sound like a recorder and sound different from a flute.


A tuning fork is forked; that is to say, it splits from it's handle into two branches called tines. Each tine is fixed to the handle at one end, but is free to vibrate at the other. As a result, one would expect to find only those harmonics that were odd multiples of the fundamental in the spectrum of a tuning fork. This is what the spectral analysis shows. The even harmonics are present, but are they are extremely weak and are probably due to the sympathetic vibrations of something in the vicinity. This spectra was produced by striking a large, demonstration-size tuning fork (not the one pictured above) with an excessively heavy blow. Tuning forks should always be tapped lightly and on a resilient surface. Doing so reduces the intensity of the "ping" overtones, which is a desirable thing. An ideal tuning fork would vibrate at just one frequency. The tuning fork used in this experiment was rated at 256 Hz, but the spectral analysis software picked up 259 Hz for the fundamental. Is the tuning fork out of tune or is the software in error?

consonance & dissonance

Did I say music was based on notes? That's not true. Real music is based on intervals (the ratio of two notes) with high degrees of consonance (shared harmonics).


[slide]

Intervals are named by their size …

and character (perfect, major, minor, augmented, and diminished).

Pythagoras of Samos (582–496 BCE) Greece was the first to try and pin this all down with a mathematical system called the cycle of fifths (or circle of fifths). Start with the tonic. Multiply by a perfect fifth (3/2). Do it again. This puts you into the next octave. Bring it down an octave (multiply by 1/2) so you can keep building your scale. Well …

(3/2)*(1/2) = 3/4

is the inverse of 4/3, an interval with a great deal of consonance. When you completely build the scale, the ratio 4/3 turns out to be the fourth interval in the series of eight that make up an octave. Thus the name fourth. The fifth and the fourth are inversions of one another in an octave. They are the only intervals that work out this way. That makes them special, in my mind, but the adjective that was ascibed to them was perfect. Thus the intervals 4/3 and 3/2 are called the perfect fourth and perfect fifth, respectively.

So here's the plan again: start with the tonic, bring it up a perfect fifth, take it down a perfect fourth, and repeat until the ratio equals an (2/1).


[slide]

We'll start on C since that's the middle of the modern piano. Behold!

F : C = 
3 −1

1 0
 =  2  = 0.6666666666666…
2 2 3
C : C = 
3 0

1 0
 =  1  = 1 ← start here
2 2 1
G : C = 
3 1

1 0
 =  3  = 1.5
2 2 2
D : C = 
3 2

1 1
 =  9  = 1.125
2 2 8
A : C = 
3 3

1 1
 =  27  = 1.6875
2 2 16
E : C = 
3 4

1 2
 =  81  = 1.265625
2 2 64
B : C = 
3 5

1 2
 =  243  = 1.8984375
2 2 128
F♯ : C = 
3 6

1 3
 =  729  = 1.423828125
2 2 512
C♯ : C = 
3 7

1 4
 =  2187  = 1.06787109375
2 2 2048
G♯ : C = 
3 8

1 4
 =  6561  = 1.601806640625
2 2 4096
D♯ : C = 
3 9

1 5
 =  19683  = 1.2013549804688…
2 2 16384
A♯ : C = 
3 10

1 5
 =  59049  = 1.8020324707031…
2 2 32768
E♯ : C = 
3 11

1 6
 =  177147  = 1.3515243530273…
2 2 131072
B♯ : C = 
3 12

1 6
 =  531441  = 2.0272865295410…
2 2 262144

Oh oh. For those of you familiar with the piano, you will note that the errors occur at notes that do not exist on the keyboard.

interval pythagorean ratio interval name
C :  C 1 tonic
C♯ :  C 1.06787109375 minor second
D :  C 1.125 major second
D♯ :  C 1.2013549804688… minor third
E :  C 1.265625 major third
F :  C 1.3333333333333… perfect fourth
E♯ :  C 1.3515243530273… bigger than a perfect fourth
F♯ :  C 1.423828125 tritone*
G :  C 1.5 perfect fifth
G♯ :  C 1.601806640625 minor sixth
A :  C 1.6875 major sixth
A♯ :  C 1.8020324707031… minor seventh
B :  C 1.8984375 major seventh
B♯ :  C 2.0272865295410… bigger than an octave
The Pythagorean intervals sorted and named * Also known as an augmented fourth or diminished fifth

It should really be called the spiral of fifths since it never closes up. One complete lap around the circle should equal a whole number of octaves. That turns out to be 12 fifths and 7 octaves. But 12 fifths is a bit larger than seven octaves. This discrepancy is known as the pythagorean comma and is equal to …

B♯  : C = 
3 12

1 7
 =  531441  = 1.0136432647705…
2 2 524288

Thus B♯ is a bit higher than C by about one-quarter of a semitone. This is a small difference that would be audible to trained ears were the two notes to be played in succession. (Ordinary folks might not perceive the difference at all.) Play them together as a part of a chord and your ears would definitely not enjoy it.

ƒB♯  −  ƒC  =  ƒbeat
(256 Hz)(1.013643264771)  −  (256 Hz)  =  3.5 Hz

The dissonance would be audible as a 3.5 Hz beat for C = 256 Hz. No musician would ever want to be this far out of tune and no audience would want to listen to them.

just intonation

Just intonation is the basis of the Western scale. It is built on a series of intervals that results in a high degree of consonance between tones.

The scale shown below is a type of just intonation scale based on a middle C value of 256 Hz. This philosophical or scientific scale is not much used in music, but the numbers it generates are nice and simple.


Ratios to the Tonic in a Diatonic Scale with Just Intonation
(C = 256 Hz Scientific Scale)

This particular scale is heptatonic, which means it contains seven intervals: three major, two minor, and two semitone. It's also called diatonic, which is an archaic reference that literally means "across the tones". When the ratios come in this order — major, minor, semi, major, minor, major, semi — it's called a major scale. In the diagram below, we have a C major scale.


Ratios of Adjacent Notes in a Major Scale with Just Intonation
(C = 256 Hz Scientific Scale)

Musical intervals are cyclic over an octave, so the sequence repeats as you go up to the next octave or down to the lower octave. There's no reason I have to start playing a scale on C like the diagram above shows. Why not start playing on A as the diagram below shows? This still gets you through an octave with all the nice intervals selected over the course of Western history. When the ratios come in this order — major, semi, major, minor, semi, major, minor — it's called a minor scale. In the diagram below we have an A minor scale,

[DIAGRAM OF A MINOR SCALE]
Ratios of Adjacent Notes in a Minor Scale with Just Intonation
(C = 256 Hz Scientific Scale)

The problem (if that's the right word) is that not all the intervals are of the same size. 9:8 major whole tone, 10:9 minor whole tone, and 16:15 semitone. The 16:15 diatonic or just semitone appears between the third and fourth and the seventh and octave. It's called a semitone since it's approximately half the size of the whole intervals. But two half intervals do not equal a whole. In fact it's a little too big.

two diatonic semitones  16   16  =  256  = 1.13778
15 15 225
one major whole tone          9  = 1.125
8
one minor whole tone          10  = 1.111 …
9

A compromise value of 25:24 called the classic or small semitone (a minor whole tone 10:9 divided by diatonic semitone 16:15) comes closer, but as the name implies, it's a bit small.

two small semitones  25   25  =  625  = 1.08507
24 24 576

Sharps (♯) and flats (♭) are different as a result of …

If you don't like semitones (and why don't you?) you could eliminate them. Take the diatonic scale and drop the third and seventh notes (the ones at the bottom of the semitone intervals). This gives you a scale with five notes — a pentatonic scale.

[DIAGRAM OF PENTATONIC SCALE]
Ratios of in a Pentatonic Scale with Just Intonation (C = 256 Hz Scientific Scale)

Diatonic + Pentatonic = Chromatic

How about an octatonic, "string of pearls" scale?

What about microtonal scales? Nothing in this book.

equal temperament


Ratios of the Chromatic Intervals in a Scale with Equal Temperament
(A = 440 Hz American Standard Scale)

Less consonant (full of beats), but easier in terms of transposition.

noise

Music is sound with a discrete structure. Noise is sound with a continuous structure.

Notes on noise that must be condensed to eliminate copyrighted text and paraphrased.