The Physics of Music - Physics 15 University of California, Irvine ...

The Physics of Music - Physics 15 

University of California, Irvine 

Instructor: David Kirkby 

dkirkby@uci.edu 

Lecture 10 

• Musical Notes and Scales 

• Scales and Timbre 

• Pythagorean Scale 

• Equal Temperament Scale 

• Unorthodox Scales 

Midterm 

The average score on the midterm was 64%. The average 

on the multiple choice section (73%) was higher than on 

the written sections (59%). 

This average corresponds to C+/B-, which is most likely 

where the final course average will end up and is normal 

for an intro physics course. 

I will be checking that the grades for the two versions 

are consistent, and make adjustments if necessary when 

calculating your final grade. 

Remember that the midterm contributes 25% to your 

final grade for the course (homework is 40%, the final is 

35%). 

Instructor: David Kirkby (dkirkby @uci.edu) 

Physics of Music, Lecture 10, D. Kirkby 2 

Students HectorAleman and Claire Dreyer should see me 

after class today. 

Here are some 

distributions from 

the midterm grades: 

Drop Deadline 

The deadline to drop this course is Friday. 

For Drop Card signatures, see the Physics Undergrad 

Affairs Coordinator: 

Kirsten Lodgard 

klodgard@uci.edu 

137 MSTB 

Homework and midterm scores are posted on the web for 

your reference: 

http://www.physics.uci.edu/undgrad/coursescores.html 



Review of Lecture 9 

In the last lecture, we covered: 

• The perception of combination tones (difference 

tones) 

• Different modes of hearing (analytic/synthetic, 

harmonic/inharmonic) 

• The physical basis for dissonance 

• Theories of pitch perception (the relative importance 

of wavelength and frequency cues) 

Why does a piano 

have 7 white notes 

and 5 black notes 

per octave 



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Musical Scales 

There is an infinite continuum of possible frequencies to 

use in music. 

But, in practice, most music uses only a small (finite) 

number of specific frequencies. 

We call each of these special frequencies a musical note, 

and call a set of notes a musical scale . 

Different cultures have adopted different scales. The 

choice of scale is primarily aesthetic, but some aesthetic 

judgments are heavily influenced by physical 

considerations (e.g., dissonance). 

What can physics tell us about musical scales 


Harmonic Timbres 

Most musical sounds have overtones that are 

approximately harmonic (ie, equally spaced on a linear 

frequency axis). 

This is most likely due to a combination of two related 

factors: 

• The resonant frequencies of many naturally occurring 

resonant systems are approximately harmonic. 

• Your brain is optimized for listening to timbres that 

are approximately harmonic. 

Note that there are examples of naturally occurring 

inharmonic sounds (eg, a hand clap) but we do not perceive 

these as being musical. 


Octaves Rule 

Two notes played together on instruments with harmonic 

timbres sound most consonant (least dissonant) when their 

fundamental frequencies are an exact number of octaves 

apart: 

frequency 

The correct answer to the octave test was #4, although 

most people prefer a slightly bigger octave with a 

frequency ratio of about 2.02:1 that corresponds to #6. 

This preference for slightly stretched octaves may be due 

to our familiarity with listening to pianos which are usually 

deliberately tuned to have stretched octaves (more about 

this in Lecture 14). 

In this sense, an octave is a special interval that we can 

expect will play a special role in any “natural” scale 

(although it is certainly possible to invent un-natural 

scales). 

Try this demonstration to see if you can pick out octaves. 



Subdividing the Octave 

In practice, this means that if a particular frequency is 

included in a scale, then all other frequencies that are an 

exact number of octaves above or below are also included. 

Therefore, choosing the set of notes to use in a scale 

boils down to the problem of how to subdivide an octave. 

Is the choice of how to subdivide an octave purely 

aesthetic, or are there physical considerations that 

prefer certain musical intervals 

Scales and Timbre 

The choice of a scale (subdivisions of an octave) is 

intimately related to the timbre of the instrument that 

will be playing the scale. 

The scale and timbre are related by dissonance: the notes 

of a scale should not sound unpleasant when played 

together. 

For example, most people listening to an “instrument” with 

no overtones (ie, a pure SHM sine wave) will have no 

preference for how to subdivide an octave (and the octave 

is no longer a special interval). 

Try these demonstrations to learn more about the 

connection between scales and timbre. 



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However, most people listening to an instrument with 

harmonic timbre (ie, most “musical” instruments) will have 

a definite preference for certain intervals where 

overtones coincide exactly. 

Different instruments with harmonic timbres have 

different strengths for the various harmonics. These 

differences affect how consonant the preferred intervals 

are but do not change their frequencies. 

Therefore, there is a universal set of preferred 

subdivisions of the octave for instruments with harmonic 

timbres (based on a physical model of dissonance). 

How Finely to Chop the Octave 

Minimizing the dissonance of notes played together on 

instruments with harmonic timbres gives us some guidance 

on how to create a scale with a given number of notes, but 

not on how many notes to use. 

Some of the conventional choices are: 

• Pentatonic: octave is divided into 5 notes (eg, Ancient 

Greek, Chinese, Celtic, Native American music) 

• Diatonic, Modal: octave is divided into 7 notes (eg, 

Indian, traditional Western music) 

• Chromatic: octave is divided into 12 notes (modern 

Western music) 



A Primer on Musical Notation 

The white notes on the piano are named A,B,C,D,E,F,G. 

After G, we start again at A. This reflects the special role 

of the octave: we give two frequencies an octave apart the 

same note name. 

Going up in frequency (towards the right on the keyboard) 

from a white note to its adjacent black note gives a sharp: 

C goes to C # , D goes to D # , etc. 

Similarly, going down in frequency gives a flat: D goes to 

D b , E goes to E b , etc. 

D b 

E b 

C # D # F # G # A # C # D # F # G # A # 

G b A b B b D b E b G b A b B b 

C D E F G A B C D E F G A B C 

C D E F G A B C D E F G A B C 

C # and D b are necessarily the same note on the piano, but 

this is not generally true for all possible scales! 

Physics of Music, Lecture 10, D. Kirkby 15 


Pentatonic Scales 

The usual choice of 5 notes in a pentatonic scale 

corresponds to the black notes on the piano: 

This scale includes the dissonant whole tone (9:8) 

interval, but leaves out the less dissonant major (5:4) and 

minor (6:5) third intervals. Why 

Other choices of 5 notes are also possible. 

Examples: 

• Indian music 

• Chinese music 

• Celtic music: Auld Lange Syne, My Bonnie Lies Over 

the Ocean 

Presumably because music limited to just 5 notes would 

be boring without some dissonance to create tension. 



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Diatonic Scales 

The major and minor scales of Western music are diatonic 

scales, in which the octave is divided into 7 steps. 

The notes of the major scale correspond to the white 

notes on a piano, starting on C. The (natural) minor scale 

corresponds to the white notes starting on A. 

Diatonic scales can also start on any other white note of 

the piano. The results are the modes with names like 

Dorian, Phrygian, Lydian, … 

Most Western music since the 17th century is based on 

major and minor scales. 

Earlier music was primarily modal. 

Example: Gregorian chants 

A 

C 



Chromatic Scales 

Although most Western music is based on diatonic scales, 

it frequently uses scales starting on several different 

notes in the same piece of music (as a device for adding 

interest and overall shape). 

A major scale starting on C uses only white notes on the 

piano, but a major scale starting on B uses all five black 

notes. 

The main reason for adopting a chromatic scale is to be 

able to play pieces based on different scales with the 

same instrument. 

An octave divided into twelve notes includes all possible 

seven-note diatonic scales. 

Not all instruments adopt this strategy. For example, 

harmonicas are each tuned to specific diatonic scales. To 

play in a different key, you need a different instrument 

(or else to master “bending” techniques). 

What exactly should be the frequencies of the 12 notes 

that make up a chromatic scale 



Is there an obvious way to subdivide an octave into twelve 

notes 

Yes: the notes should be equally spaced and include all of 

the special consonant frequency ratios (3:2, 4:3, …). 

To what extent is this actually possible 

After the octave, the fifth (3:2) is the most consonant 

interval for harmonic timbres. The fourth (4:3) is really 

just a combination of the octave and fifth: 

4/3 = (3/2) x (1/2) 


The Circle of Fifths 

We can reach all 12 notes of the chromatic scale by 

walking up or down the piano in steps of a fifth (3:2): 

Going up, we reach all 

white notes of the piano 

except F, and then go 

through the sharps. 

Going down, we hit F first 

and then go through the 

flats. 

Either way, we eventually 

get back to a C (7 octaves 

away) if we start on a C. 


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Using the circle of fifths, we can calculate the frequency 

of any note we reach going up as: 

starting 

note 

f = f 0 x (3/2) x (3/2) x … x (3/2) / 2 / 2 / … / 2 

Steps up in fifths 

Steps down 

in octaves 

A similar method works for each step down by a fifth: 

f = f 0 / (3/2) / (3/2) / … / (3/2) x 2 x 2 x … x 2 

Steps down in fifths 

Steps up 

in octaves 

What happens when we get back to our original note 

For example, after going 12 fifths up, we get back to a C 

that is 7 octaves up which corresponds to a note: 

f = f 0 x (3/2) 12 / (2) 7 = f 0 (531441/524288) = 1.014 f 0 

We end up close but not exactly where we started! 



Pythagorean Scale 

If we ignore this problem of not getting back to where we 

started, we end up with the set of notes corresponding to 

the Pythagorean scale. 

The Pythagorean scale has the feature that all octave and 

fifth intervals are exact (and therefore so are fourths). 

But the Pythagorean scale also has some shortcomings: 

• The frequencies we calculate for the black notes 

depend on whether we are taking steps up or down, so 

C # and D b are different notes! 

• The semitones from E to F and B to C are bigger than 

the semitones from C to C # and D b to D. 

• Frequency ratios for intervals other than 8ve, 4th, 5th 

depend on which note you start from, and can be far 

from the ideal ratios. 



Alternative Scales 

Since the major (5:4) and minor (6:5) 3rd intervals are 

important for diatonic music, several alternative scales 

have been proposed that have these intervals better in 

tune (ie, closer to their ideal frequency ratios) without 

sacrificing the octave, fourth, and fifth too much. 

Some alternatives that I will not discuss are the meantone 

and just intonation scales (see Sections 9.3-9.4 in the 

text for details). 

These scales both improve the tuning of intervals but 

sound differently depending on the choice of starting 

note for a diatonic scale (Beethoven described D b -major 

as “majestic” and C-major as “triumphant”). They also give 

different frequencies for C # and D b , etc. 


Equal Temperament Scale 

The scale that is most widely used today is the equal 

temperamentscale. 

This scale is the ultimate compromise for an instrument 

that is tuned infrequently and for which the performer 

cannot adjust the pitch during performance. 

The equal temperament scale gives up on trying to make 

any intervals (other than the octave) exactly right, but 

instead makes the 12 notes equally spaced on a logarithmic 

scale. 

Listen to the difference between equally-spaced notes on 

linear and logarithmic scales: 


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Mathematically, each semitone corresponds to a frequency 

ratio of 2 1/12 = 1.059, so that 12 semitones exactly equals 

an octave. 

The equal temperament scale has the main advantage that 

all intervals sound the same (equally good or bad) 

whatever note you start from. 

Therefore, diatonic scales played from different notes 

(eg, C-major, D-major, …) are mathematically identical 

except for their absolute frequency scale (which most 

people have no perception of). 

Unorthodox Scales 

Instead of dividing the octave into 12 equally spaced 

notes, we can divide it into any number of equally spaced 

notes. 

Listen to these scales with different numbers of notes: 

• 12 notes (standard equal-tempered chromatic scale) 

• 13 notes 

• 8 notes 

Why aren’t 13 and 8 note scales popular Because they are 

more dissonant than 12 note scales when two or more 

notes are played together with harmonic timbres. 



Unorthodox Instruments 

Some instruments designed to play unorthodox scales have 

actually been built and played: 

Unorthodox Instruments 

Although most real acoustic instruments have 

approximately harmonic timbres, artificial instruments can 

be electronically synthesized to have any timbres. 

Fokker organ designed 

to play a 31-note scale 

http://www.xs4all.nl/~huygensf/english/index.html 

In particular, we can create instruments that are less 

dissonant when played in non-standard scales. 

The results are interesting and easy to listen to 

(compared with the Fokker organ). For example: 

• 11-note scale: 

• 19-note scale: 

http://eceserv0.ece.wisc.edu/~sethares/mp3s/ 



Frequency Standardization 

Most people have no perception of absolute pitch so it is 

not surprising that we managed for a long time without any 

standard definition of the frequency of middle C. 

In 1877, the A 4 pipes on organs reportedly ranged from 

374 - 567 Hz (corresponding to the modern range F-C # ). 

The modern standard is A 4 = 440 Hz and was adopted in 

1939. 

Summary 

We covered the following topics: 

• Musical Notes and Scales 

• Scales and Timbre 

• Pythagorean Scale 

• Equal Temperament Scale 

• Unorthodox Scales 



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