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

The ratio 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 chromatic 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 systematic arrangement of pitches Each musical note has a perceived pitch with a particular frequency (the frequency of the fundamental) Going up or down in frequency, the perceived pitch follows a pattern One cycle of pitch repetition is called an octave. The interval between successive pitches determines the type of scale.

Note spanInterval Frequency ratio C - Cunison 1/1 C - C#semitone 16/15 C - Dwhole tone (major second) 9/8 C - D#minor third 6/5 C - Emajor third 5/4 C - Fperfect fourth 4/3 C - F#augmented fourth 45/32 C - Gperfect fifth 3/2 C - G#minor sixth 8/5 C - Amajor sixth 5/3 C - A#minor seventh 16/9 (or 7/4) C - Bmajor seventh 15/8 C 3 - C 4 octave 2/1 C 3 - E 4 octave+major third 5/2 Intervals 12-tone scale (chromatic) 8-tone scale (diatonic)

Consonant intervals Overlapping harmonics tonic fifth fourth M third m third octave Dissonant intervals Perceived when harmonics are close enough for beating

harmonic series Fundamental f 1 2 nd harmonic f 2 = 2f 1 octave 3 rd harmonic f 3 = 3f 1 perfect fifth 4 th harmonic f 4 = 4f 1 perfect fourth 5 th harmonic f 5 = 5f 1 major third 6 th harmonic f 6 = 6f 1 minor third Intervals between consecutive harmonics

CT 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 5 th, C-A, a 6 th C-B, a (major) 7 th, C-2C, an octave C-2D, a 9 th C-2E, a 10 th, C-2F, an 11 th, C-2G, a 12 th, C-2A, a 13 th, etc.

C-E b, a minor 3 rd C-B b, a dominant 7th, C-2D b, a flatted 9th, etc.

Pythagorean Scale Built on 5ths

A pleasant consonance was observed playing strings whose lengths were related by the ratio 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 5 th Denote the frequency of C simply by the name C, etc.

Since f 1 = V/2L, and L C = 3/2 L G, G =3/2C. Similarly a 5 th 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 5 th 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 foundation 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 ratios 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 Now take C to be 1

CT 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 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 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 thirdperfect fifth perfect fourth + perfect fifth octave perfect fourth + major thirdmajor sixth perfect fourth perfect fourth + whole tone Adding intervals means multiplying frequency ratios

more compound intervals perfect fifth + perfect fifthOctave + whole tone major seventh + minor sixth Octave + perfect fifth ratios larger than 2 can be split up into an octave + something