Gustav Holst
(1874-1934)

The Planets - Saturn, the Bringer of Old Age

Musical Intervals

Refer to Table 10.3 in the text.

Intervals are ratios of frequencies or lengths.

Musical Intervals

Octave (natural interval)

String Length Ratio = 2/1 = 2.00
or ½ = 0.50

Frequency Ratio also = 2.00
or ½ = 0.50

Musical Intervals

Perfect Fifth

Seven semitones

String Length Ratio = 3/2 =1.50
or 2/3 = 0.6667

Frequency Ratio also = 1.50
or 2/3 = 0.6667

Musical Intervals

Musical Intervals

Perfect Fourth

Five semitones

String Length Ratio = 4/3 = 1.33
or ¾ = 0.750

String Length Ratio = 1.333
or ¾ = 0.750

Musical Intervals

Pythagorean Scale

Pythagoras (582-507 BC)

Ratios for intervals: 1.000, 1.333, 1.500, 2.000 (unison, fourth, fifth, octave)

Used these ratios to construct a mathematical scale.

Pythagorean Scale

Used string lengths since frequencies were not known.

How do you divide an octave (1.00 to 2.00) into 8 equal parts?

Or in terms of frequencies an interval such as 220 –440 Hz?

Pythagoras’s Rule

Multiply or divide an existing length (ratio) by 3/2 (=1.500), factor of fifths.

If the result lies between 1 and 2, leave it as it is.

Pythagoras’s Rule

If the answer is less than 1, double it (up an octave)

If the answer is greater than 1, halve it (down an octave)

Pythagoras’s Rule
Step #1

Start with D₄ = 1.000 (293.7 Hz)

Multiply D₄= 1.00 by 1.50 to get 1.5 (the fifth) which is A₄ (440 Hz).

Pythagoras’s Rule
Step #2

Start with D₄ = 1.000 (293.7 Hz)

Divide D₄= 1.00 by 1.50 to get 0.666 and double to get 1.333 (the fourth) which is G₄ (392 Hz).

Pythagoras’s Rule
Step #3

Start with A₄ = 1.500 (440 Hz)

Multiply A₄= 1.50 by 1.50 to get 2.250 and halve to get 1.125 (the major second) which is E₄ (229.6 Hz).

Pythagoras’s Rule
Step #4

Start with G₄ = 1.333 (292 Hz)

Divide G₄= 1.333 by 1.50 to get 0.88888 and double to get 1.777 (the minor seventh) which is C₅ (523.3 Hz).

Pentatonic Scale

These first 5 notes D, E, G, A, C, and D again constitute the 5-note Chinese scale called pentatonic (5 tones)

Greek scales had 7 notes called septatonic

Pythagoras’s Rule
Step #5

Start with E₄ = 1.125 (229.6 Hz)

Multiply E₄= 1.125 by 1.50 to get 1.6875 (the major sixth) which is B₄ (493.9 Hz).

Pythagoras’s Rule
Step #6

Start with C₅ = 1.777 (523.3 Hz)

Divide C₅= 1.777 by 1.50 to get 1.1851 (the minor third) which is F₄ (349.2 Hz).

Pythagoras’s Rule
Step #7

Start with G₄ = 1.333 (292 Hz)

Multiply G₄= 1.333 by 1.50 to get 2.00 (the octave) which is D₅ (587.3 Hz).

Musical Intervals

Dorian Mode

Tone = T and Semitone = s

T s T T T s T

D E F G A B C D

Modes

Scales based on the white keys of the piano

Since there are seven different named keys A, B, C, D, E, F, G, there are seven modes.

Ionian Mode or
Major Scale

T T s T T T s

C D E F G A B C

Aoelian Mode or
Minor Scale

T s T T s T T

A B C D E F G A

Major and
Minor Scales

Major: J.S. Bach “Well Tempered Clavier Book II” Prelude I in C major. (Track #1)

Minor: Prelude IV in C# minor. (Track #7)

Seven Modes

C = Ionian (major scale)

D = Dorian

E = Phrygian (Spanish or Oriental)

F = Lydian (funny, comical)

G = Mixolydian

A = Aeolian (minor scale)

B = Locrian (not used)

Phrygian Mode

(starts on E)

Vaughn Williams

Fantasia on a Theme by Thomas Tallis

Mathematical Basis
for Scales

Multiples of 1.500 generate the same 8 note scale that was found by musicians to be the “right” ones for a musical scale.

Mathematical Basis
for Scales

The fifth is a multiple of 1.500.

The fifth is the 3^rd harmonic. (3h₁/2h₁ = 1.50)

Mathematical Basis
for Scales

Third harmonic of A is E.

Third harmonic of E is B.

Third harmonic of B is F#.

of F# is C#, of C# is G#, of G# is D#, of D# is A#, of A# is F, of F is C, of C is G, of G is D, and of D is back to A.

This is the entire chromatic scale!

Aesthetic Basis
for Scales

The 3^rd harmonic is the lowest and strongest harmonic that is not an octave.

Stringed instruments have the 3^rd harmonic.

A scale based on 3^rd harmonics should be the most “natural” or pleasing.

Equal-Tempered Scale

7 Greek modes or Church modes use all of the white keys

Comprise 7 combinations of T=tone and s=semitone sequences

Using a particular mode requires the scale to start on one and only one note.

Need to place semitones anywhere.

Equal-Tempered Scale

Equal-Tempered Chromatic Scale

Major Scales
C Major

Major Scales

Major Scales Quality

Minor Scales
A Minor

Minor Scales

Minor Scales Quality

Diatonic vs. Chromatic

Equal Tempered
vs. Pythagorean

Equal Tempered vs. Pythagorean


	Refer to Table 10.3 in the text.
	Intervals are ratios of frequencies or lengths.


	Octave (natural interval)
		String Length Ratio = 2/1 = 2.00 or ½ = 0.50
		Frequency Ratio also = 2.00 or ½ = 0.50


	Perfect Fifth
		Seven semitones
		String Length Ratio = 3/2 =1.50 or 2/3 = 0.6667
		Frequency Ratio also = 1.50 or 2/3 = 0.6667


	Perfect Fourth
		Five semitones
		String Length Ratio = 4/3 = 1.33 or ¾ = 0.750
		String Length Ratio = 1.333 or ¾ = 0.750


	Pythagoras (582-507 BC)
	Ratios for intervals: 1.000, 1.333, 1.500, 2.000 (unison, fourth, fifth, octave)
	Used these ratios to construct a mathematical scale.


	Used string lengths since frequencies were not known.
	How do you divide an octave (1.00 to 2.00) into 8 equal parts?
	Or in terms of frequencies an interval such as 220 –440 Hz?


	Multiply or divide an existing length (ratio) by 3/2 (=1.500), factor of fifths.
	If the result lies between 1 and 2, leave it as it is.


	If the answer is less than 1, double it (up an octave)
	If the answer is greater than 1, halve it (down an octave)


	Start with D₄ = 1.000 (293.7 Hz)
	Multiply D₄= 1.00 by 1.50 to get 1.5 (the fifth) which is A₄ (440 Hz).


	Start with D₄ = 1.000 (293.7 Hz)
	Divide D₄= 1.00 by 1.50 to get 0.666 and double to get 1.333 (the fourth) which is G₄ (392 Hz).


	Start with A₄ = 1.500 (440 Hz)
	Multiply A₄= 1.50 by 1.50 to get 2.250 and halve to get 1.125 (the major second) which is E₄ (229.6 Hz).


	Start with G₄ = 1.333 (292 Hz)
	Divide G₄= 1.333 by 1.50 to get 0.88888 and double to get 1.777 (the minor seventh) which is C₅ (523.3 Hz).


	These first 5 notes D, E, G, A, C, and D again constitute the 5-note Chinese scale called pentatonic (5 tones)
	Greek scales had 7 notes called septatonic


	Start with E₄ = 1.125 (229.6 Hz)
	Multiply E₄= 1.125 by 1.50 to get 1.6875 (the major sixth) which is B₄ (493.9 Hz).


	Start with C₅ = 1.777 (523.3 Hz)
	Divide C₅= 1.777 by 1.50 to get 1.1851 (the minor third) which is F₄ (349.2 Hz).


	Start with G₄ = 1.333 (292 Hz)
	Multiply G₄= 1.333 by 1.50 to get 2.00 (the octave) which is D₅ (587.3 Hz).


	Scales based on the white keys of the piano
	Since there are seven different named keys A, B, C, D, E, F, G, there are seven modes.


	Major: J.S. Bach “Well Tempered Clavier Book II” Prelude I in C major. (Track #1)
	Minor: Prelude IV in C# minor. (Track #7)


	C = Ionian (major scale)
	D = Dorian
	E = Phrygian (Spanish or Oriental)
	F = Lydian (funny, comical)
	G = Mixolydian
	A = Aeolian (minor scale)
	B = Locrian (not used)


	(starts on E)
	Vaughn Williams
	Fantasia on a Theme by Thomas Tallis


	Multiples of 1.500 generate the same 8 note scale that was found by musicians to be the “right” ones for a musical scale.


	The fifth is a multiple of 1.500.
	The fifth is the 3^rd harmonic. (3h₁/2h₁ = 1.50)


	Third harmonic of A is E.
	Third harmonic of E is B.
	Third harmonic of B is F#.
	of F# is C#, of C# is G#, of G# is D#, of D# is A#, of A# is F, of F is C, of C is G, of G is D, and of D is back to A.
	This is the entire chromatic scale!


	The 3^rd harmonic is the lowest and strongest harmonic that is not an octave.
	Stringed instruments have the 3^rd harmonic.
	A scale based on 3^rd harmonics should be the most “natural” or pleasing.


	7 Greek modes or Church modes use all of the white keys
	Comprise 7 combinations of T=tone and s=semitone sequences
	Using a particular mode requires the scale to start on one and only one note.
	Need to place semitones anywhere.