is an irrational number), the musicians learnt to tune an instrument by
training their ear rather than by applying mathematical principles. Music from
this point of view released itself from mathematical domination.Research Paper
This paper provides an example of how science and beauty are interrelated through music.
Leyan Lo
Mrs. Marionni
Honors World Literature, period 4
13 June 2003
The
Physics of Intonation
Nearly all of Western music, up until the more recent experimenting with microtones, is based on the diatonic scale, having seven notes in an octave. The octave is a very fundamental interval for notes, and relates the frequencies of the notes by the ratio of 2:1. This simple ratio means the notes sound to the ear as if they are the same note. Virtually all of Western music divides the octave into 12 equal semi-tones on which music is based. However, the system is a compromise; the octave has not always been divided this way. Over the ages, there have been four principle tuning systems in wide usage for Western music.
For pure monophonic (one note at a time) musical melody, it does not
matter a great deal what scale or tuning method is used.
However, almost all modern music is polyphonic (many notes sounding
simultaneously). For the music to
sound consonant (in tune), the different notes being sounded simultaneously need
to be related in a way, which is pleasing to the ear. The relationship between the frequencies of notes of the
important musical intervals is what makes them sound pleasing.
The important ratios are as follows:
6:5
minor third
5:4
major third
4:3
perfect fourth
3:2
perfect fifth
8:5
minor sixth
5:3 major sixth
All four of the main tuning
systems considered here were devised around some or most of these intervals.
Western music is generally considered to have started from Pythagoras, the ancient Greek. Pythagoras devised a system based on mathematical principles. He defined the scale around the ratios of the fifth, being in the ratio 3:2 exactly, and the fourth being 4:3 exactly. The difference between these two was then 9:8 (because you need to multiply the fourth ratio by the whole step ratio to end up with the fifth ratio), which he defined as the tone, or whole step. He then divided the octave so that there were the seven notes, as the diatonic scale has today, but to get the mathematics to add up he was left with two semi-tones, which he defined as 256:243.
Thus
the Pythagorean scale has the following intervals:
Cumulative
Intervals: 1
9:8 81:64
4:3 3:2
27:16 243:128
2
Note:
C D
E F
G A
B C
Incremental
Intervals: 9:8
9:8 256:243
9:8 9:8
9:8 256:243
Cents:
204 204
90 204
204 204
90
(The values given as “cents” represent the intervals where the octave is divided into 1200 cents. This is the modern way of representing pitch in such a way that each semitone comprises 100 cents.)
It is interesting to note that
Pythagoras did not recognize the major third, which is distinctly sharp at 81:64
compared with the ideal of 5:4.
The chromatic Pythagorean scale is formed by inserting semitones equal to 114 cents in such a way as to keep all perfect fifths true, except for the interval G#-Eb, which needs to be adjusted so the intervals add up mathematically. This difference is known as the “comma of Didymus.”
Ptolemaic
Tuning is generally referred to as “just intonation.” To create perfect major third intervals, this system alters
one of the fifths, D-A. (And thus
the major sixths are also perfect, except for F-D.)
This makes the triad D-F-A quite unusable, although the others are
perfectly in tune. The Just
Intonation scale employs two different sized tones in the ratios 9:8 and 10:9,
and thus it can hardly be considered satisfactory even for purely melodic music.
Cumulative
Intervals: 1
9:8 5:4
4:3 3:2
5:3 15:8
2
Note:
C D
E F
G A
B C
Incremental
Intervals: 9:8
10:9 16:15
9:8 10:9
9:8 16:15
Cents:
204 182
112 204
182 204
112
The above comments hold true for
scales and intervals based solely on the “white notes.”
Adding the “black notes” to give a full chromatic Just Intonation
scale creates more perfect fifths, but major thirds and sixths which are not
true. In fact the result is three
different sizes of semi-tones. (16:15, 135:128 and 256:243.)
It is thought that this system, although considerably debated, was not
used much.
In Mean-tone Temperament, the major thirds are made exact.
This results in the fifths becoming slightly flattened but in such a way
that the error of the Ptolemaic system is spread out over four fifths.
This reduces the dissonance and makes the fifths more acceptable. The whole tones are also all equal in size, being half the
major third. Melodically the
Mean-tone scale is more acceptable than the Ptolemaic scale.
Cumulative
Intervals: 1
5:4
2
Note:
C D
E F
G A
B C
Cents:
193 193
117 193
193 193
117
The chromatic Mean-tone scale has
semi-tones of two very different sizes: wide 117 cent semi-tones in the diatonic
scale, with 76 cent semi-tones balancing the whole-tone interval of 193 cents.
Mean-tone temperament was designed for keyboard instruments and it was an
acceptable compromise as long as the “black notes” beyond Eb or G# were not
used. The G#:Eb fifth was so bad as to be unusable-it was often
given the name “the wolf.”
In
equal temperament, the octave is divided into 12 equal semi-tones each of 100
cents. The intervals are then built
from the semi-tones. For example a
fifth is 7 semi-tones and a third 5. The
Equal temperament scale is universally used today in Western music.
Cumulative
Intervals: 1
2
Note:
C D
E F
G A
B C
Cents:
200 200
100 200
200 200
100
The chromatic Equal temperament
scale with all semi-tones = 100 cents means that perfectly tuned intervals have
been totally eliminated. However,
the mistuning on fifths is only 2 cents and on thirds 14 cents which the ear
does not appear to mind. The big
advantage is that all keys are equally usable.
With the evolution of this more complicated mathematical model for tuning
an instrument, and with the increased importance of musicality and performance,
music and mathematics in this aspect have lost the close relationship known in
ancient Greek times. As an
even-tempered interval could no longer be expressed as a ratio (
is an irrational number), the musicians learnt to tune an instrument by
training their ear rather than by applying mathematical principles. Music from
this point of view released itself from mathematical domination.
Works
Cited