Post on 15-Apr-2018
L 8-‐9 Musical Scales, Chords , and
Intervals, The Pythagorean and Just Scales
History of Western Scales A Physics 1240 Project
by Lee Christy 2010
References to the History
Musical Intervals (roughly in order of decreasing consonance)
Name of Interval
Notes (in key of C major)
Pythagorean Frequency Ratios
Just Frequency Ratios
# Semitones (on equal-tempered scale)
Octave C ↔ C 2 2 12
Fifth C ↔ G 3/2 6/4 = 3/2 7
Fourth C ↔ F 4/3 5
Major Third C ↔ E 81/64 5/4 4
Minor Third E ↔ G 3
Major Sixth C ↔ A 27/16 9
Minor Sixth E ↔ C 8 Tonic C 1 4/4 = 1 none * a semitone interval corresponds to a frequency difference of about 6%
* The white notes of the piano give the seven notes of the C-major diatonic scale.
C D E F G A B C
The raIo of the frequency of C4 to that of C2 is:
a) 2 b) 3 c) 4 d) 8
One octave of the diatonic scale including the tonic and the octave
note contains: a) 5 notes b) 6 notes c) 7 notes d) 8 notes
One octave of the chromaIc scale (including the octave note)
contains: a) 8 notes b) 10 notes c) 11 notes d) 12 notes e) 13notes
A musical scale is a systemaIc arrangement of pitches
Each musical note has a perceived pitch with a parIcular frequency
(the frequency of the fundamental)
Going up or down in frequency, the perceived pitch follows a paXern
One cycle of pitch repeIIon is called an octave.
The interval between successive pitches determines the type of scale.
Note span Interval Frequency raIo C -‐ C unison 1/1 C -‐ C# semitone 16/15 C -‐ D whole tone (major second) 9/8 C -‐ D# minor third 6/5 C -‐ E major third 5/4 C -‐ F perfect fourth 4/3 C -‐ F# augmented fourth 45/32 C -‐ G perfect fi`h 3/2 C -‐ G# minor sixth 8/5 C -‐ A major sixth 5/3 C -‐ A# minor seventh 16/9 (or 7/4) C -‐ B major seventh 15/8 C3 -‐ C4 octave 2/1 C3 -‐ E4 octave+major third 5/2
Intervals 12-tone scale (chromatic) 8-tone scale (diatonic)
Consonant intervals Overlapping harmonics
tonic 120 240 360 480 600 720 840 960 1080
fi`h 180 360 540 720 900 1080
fourth 160 320 480 640 800 960
M third 150 300 450 600 750 900 1050
m third 144 288 432 576 720 864 1008
octave 240 480 720 960
Dissonant intervals
Perceived when harmonics are close enough for beaIng
harmonic series
Fundamental f1
2nd harmonic f2 = 2f1 octave
3rd harmonic f3 = 3f1 perfect fi`h
4th harmonic f4 = 4f1 perfect fourth
5th harmonic f5 = 5f1 major third
6th harmonic f6 = 6f1 minor third
€
f2f1
=21
€
f3f2
=32
€
f4f3
=43
€
f5f4
=54
€
f6f5
=65
Intervals between consecutive harmonics
CT 2.4.5 What is the name of the note that is a major 3rd above E4=330 Hz? A: G B: G# C: A D: A# E: B
Intervals C-‐ D, a second C-‐E, a third C-‐F, a 4th C-‐G, a 5th, C-‐A, a 6th
C-‐B, a (major) 7th, C-‐2C, an octave
C-‐2D, a 9th C-‐2E, a 10th, C-‐2F, an 11th, C-‐2G, a 12th,
C-‐2A, a 13th, etc.
C-‐Eb, a minor 3rd
C-‐Bb, a dominant 7th,
C-‐2Db, a flaXed 9th, etc.
Pythagorean Scale Built on 5ths
A pleasant consonance was observed playing strings whose lengths were
related by the raIo of 3/2 to 1 (demo). Let’s call the longer string C, and the
shorter G, and the interval between G and C a
5th Denote the frequency of C simply by
the name C, etc.
Since f1= V/2L, and LC= 3/2 LG, G =3/2C.
Similarly a 5th above G is 2D, and D= 1/2 (3/2G)= 9/8 C.
Then A is 3/2 D= 27/16 C. Then 2E= 3/2 A or E= 81/64 C, and
B=3/2 E = 243/128 C.
We now have the frequencies for CDE… GAB(2C)
To fill out the Pythagorean scale,
we need F. If we take 2C to be the 5th above F,
then 2C= 3/2F, or F = 4/3 C
Just Scale, Built on Major Triads
We take 3 sonometers to play 3 notes to make a major triad, e.g. CEG. This sounds consonant (and has been the foundaIon of western music for several hundred years), and we
measure the string lengths required for this triad.
We find (demo) that the string lengths have raIos 6:5:4 for the
sequence CEG.
The major triad is the basis for the just scale, which we now develop in a way similar to that of the
Pythagorean scale.
F A C C E G G B D 4 5 6 4 5 6 4 5 6 Now take C to be 1
CT 2.4.5 Suppose we start a scale at E4=330 Hz. What frequency is a (just) perfect 5th above this? A 1650 Hz B: 220 Hz C: 495 Hz D: 660 Hz E: None of these
CT 2.4.5 What is the frequency of the note that is a (just) major 3rd above E4=330 Hz? A: 660 Hz B: 633 Hz C: 512 Hz D: 440 Hz E: 412 Hz
CT 2.4.5 Suppose we start a scale at E4=330 Hz. What frequency is a (just) perfect 5th below this? A 165 Hz B: 220 Hz C: 110 Hz D: 66 Hz E: None of these
compound intervals
major third + minor third
€
54×65
=3020
=32 perfect fi`h
perfect fourth + perfect fi`h
€
43×32
=126
=21 octave
perfect fourth + major third
€
43×54
=2012
=53 major sixth
€
54×1615
=8060
=43 perfect fourth perfect fourth + whole tone
Adding intervals means mulIplying frequency raIos
more compound intervals
perfect fi`h + perfect fi`h
€
32×32
=94
=21×98 Octave + whole tone
major seventh + minor sixth
€
158×85
=155
=31
=21×32 Octave + perfect fi`h
raIos larger than 2 can be split up into an octave + something