Transcript Slides

A little music theory
(mostly notation, names, …and
temperament)
Nature or nurture
Physical: It has nothing to do with human beings. Ex:
beating
Psychophysical, psychological: human anatomy. Ex:
fundamental tracking
Cultural: society dependent. Ex: appreciation of
Beattles songs
Doubling the frequency feels like the same pitch
(pitch periodicity)
f and its harmonics: f, 2f, 3f, 4f, …
2f and its harmonics: 2f, 4f, 6f, …
This is not a cultural phenomena, it seems to be
present in any musical culture.
In Western music the pitch range from f to 2f is
split in 12 steps
(entirely cultural)
f
f0
2 f0
C, C#/Db, D, D#/Eb, E, E#, Fb, F, F#/Gb, G, G#/Ab, A,
A#/Bb, B
or
do, do#/re b, re, re#/mi b, mi, mi#/fa b, fa, fa#, sol,
sol#/la b, la, la#/sib, si
C# D#
...
F# G# A#
C D EF
C2
C3
G
A
B C
C4
...
This has changed historically but now
it’s standard to take:
A4 = 440 Hz
So A5 = 880 Hz, A3 = 220 Hz, …
For the intermediate notes the whole thing is more
contentious (we’ll discuss temperament later)
higher
What about the #’s and b’s ?
C#
Ab
What about the duration of notes ?
half
half
Measure time in beats
four beats in a
measure
this will count
as one beat
slightly
more
complex
several
instruments
Consonance and dissonance
[Let us play some intervals and find what makes
them consonant or dissonant]
C
C# D
D#
E
F
F#
minor major minor major 4th tritone
2nd
2nd
3rd
3rd
G
G# A A#
B C
5th minor major minor major
6th
6th
7th
7th
ratio of frequencies = ratio of small
integers
consonance
Examples:
1/1
unison
2/1
octave = 7 tones
3/2
fifth = 3 ½ tones (actually 1.4983)
4/3
fourth = 2 ½ tones (actually 1.22482)
5/4
major third = 2 tones (actually 1.25991)
Consonance/dissonance and the overtone series
unison = 0 tones
octave = 7 tones
fifth = 3 ½ tones
fourth = 2 ½ tones
major third = 2 tones
consonance beating roughness consonance roughness …
Temperament
Problem: choose the frequencies of the notes
(C, C#, D, …) in order to make the
consonances very good consonances
Remember: the best consonances are
Octaves: 2/1
6 tones = 12 semitones
Fifths: 3/2
3 ½ tones = 7 semitones
Fourths: 4/3
2 ½ tones = 5 semitones
Major thirds: 5/4 2 tones = 4 semitones
…
It is impossible to assign frequencies to the notes
C
C#
D
D#
E
F
F#
G
G# A A#
In such a way as to keep all fifths = 3/2,
fourths = 4/3, … exact
B C
7 octaves  27
C G
3
2
D A E
3
2
3
2
3
2
B
3
2
F#
3
2
C#
3
2
G#
3
2
D# A# F C
3
2
3
2
3
2
12
3
  129.746
2
27 128
not the same
3
2
Pythagorean solution
Make the octaves and fifths
perfect
C
D
E
F
1
9/8
81/64 4/3
33
22
2
27 3
16 2
2
G
A
B
C
3/2 27/16 243/128 2
one tone = 9/8
C
D
1
9/8
½ tone = 256/243
E
F
81/64 4/3
G
B
C
3/2 27/16 243/128 2
1 tone = (256/243)2 = 1.1098…
1 tone = 9/8 = 1.125
A
Pythagorean
comma
1.58
1.60
close, but
not the
same !
Can you hear the bad Pythagorean thirds ?
Perfect third : f2/f1 = 5/4=1.25
Perfect third : f2/f1 = 81/64 = 1.265…
In the Pythagorean temperament some keys are better
than others
Samuel Barber's Adagio for Strings
courtesy of
G. Moore
C
Ab
Other temperaments
Pythagorean: good fifth (except one), bad thirds
Just: some thirds and fifths are good (tonic, dominant
and subdominant of some keys)
Meantone: better thirds than fifths
...
Equal temperament: split the difference equally among
notes. Nothing is perfect, nothing is too bad
Recap of Music Theory
half tone tone
C3
C4
same interval = same ratio of frequencies
Consonances: sensation of calm and repose
Frequency ratios
2/1
3/2
4/3
5/4
name
octave
fifth
forth
major third
Dissonances: sensation of tension
Frequency ratios
name
729/512
tritone
243/128
minor second
Temperament: an assignment of
frequencies to all twelve notes from C to B
It is impossible to find a temperament where all the
octaves and fifths are perfect
Pythagorean: all octaves and all but one fifth are perfect.
One fifth is very off (pythagorean comma).
Well or equal : split the differences equally. Every
semitone = 1.059…
Equal temperament
C C#
D
D#
E
F
F#
G
G# A A#
r
r2
r12=2
r12  2  r  12 2 1.05946...
B C
Nothing too good, nothing too bad …
Fifths: r7 = 1.498 instead of 3/2=1.5
Fourths: r5 = 1.3348 instead of 4/3=1.3333
Thirds: r3=1.25992 instead of 5/4=1.25
…