Download L 8-‐9 Musical Scales, Chords , and Intervals, The Pythagorean and

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 equaltempered
scale)
Octave
C↔C
2
2
12
Fifth
C↔G
3/2
6/4 = 3/2
7
Fourth
C↔F
4/3
Major Third
C↔E
81/64
Minor Third
E↔G
Major Sixth
C↔A
Minor Sixth
E↔C
Tonic
C
5
5/4
4
3
27/16
9
8
1
4/4 = 1
none
* a semitone interval corresponds to a frequency difference of about 6%
C
D
E
F
G
A
B
C
* The white notes of the piano give the seven notes of the C-major diatonic scale.
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. Intervals 12-tone scale (chromatic)
8-tone scale (diatonic)
Note span
C -­‐ C
C -­‐ C# C -­‐ D
C -­‐ D# C -­‐ E
C -­‐ F
C -­‐ F#
C -­‐ G
C -­‐ G# C -­‐ A
C -­‐ A# C -­‐ B
C3 -­‐ C4 C3 -­‐ E4 Interval Frequency raIo unison 1/1 semitone 16/15 whole tone (major second) 9/8 minor third
6/5 major third
5/4 perfect fourth
4/3 augmented fourth 45/32 perfect fi`h
3/2 minor sixth
8/5 major sixth
5/3 minor seventh
16/9 (or 7/4) major seventh
15/8 octave 2/1 octave+major third 5/2 Consonant intervals Overlapping harmonics
tonic
120
240
360
480
600
720
840
960
1080 octave
240
480
720
960
fi`h
180 360
540 720
900 fourth
160
320 480
640
800
M third
150
300
450 600
750
900 1050 m third
144
288
432 576 720
864
1008 Dissonant intervals Perceived when harmonics are close enough for beaIng 960 1080 harmonic series Intervals between consecutive harmonics
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 €
€
€
€
f2 2
=
f1 1
f3 3
=
f2 2
f4 4
=
f3 3
f5 5
=
f4 4
f6 6
=
f5 5
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. rd C-­‐Eb, a minor 3
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. th
Similarly a 5 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 Adding intervals means mulIplying frequency raIos major third + minor third perfect fourth + perfect fi`h €
perfect fourth + major third €
perfect fourth + whole tone €
€
5 6 30 3
× =
=
4 5 20 2
4 3 12 2
× = =
3 2 6 1
4 5 20 5
× =
=
3 4 12 3
5 16 80 4
× =
=
4 15 60 3
perfect fi`h octave major sixth perfect fourth more compound intervals raIos larger than 2 can be split up into an octave + something perfect fi`h + perfect fi`h major seventh + minor sixth €
€
3 3 9 2 9
× = = ×
2 2 4 1 8
15 8 15 3 2 3
× = = = ×
8 5 5 1 1 2
Octave + whole tone Octave + perfect fi`h