Musical Intervals & Scales

• Creator of instruments will need to define the tuning of that instrument

• Systems of tuning depend upon the intervals (or distances of frequency) between notes

Intervals

• Musical intervals are distances of frequency between two notes

• The distance of an octave is a doubling of frequency

Intervals: The Octave

time

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

f2

f1pressure

f2 = 2 * f1 Frequency Ratio = 2/1

Intervals: The Fifth

f2 = 3/2 * f1 Frequency Ratio = 3/2

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

time

pressure

f2

f1

Musical Intervals

Frequency ratio Interval1/1 Unison2/1 Octave3/2 Fifth4/3 Fourth5/3 Major Sixth5/4 Major Third6/5 Minor Third8/5 Minor Sixth

Consonance & Dissonance

• Commonly used intervals are commonly used because they sound good

• When two or more tones sound pleasing together this is known as consonance

• When they sound harsh, jarring, or unpleasant this is known as dissonance

Consonance & Dissonance

• Are to some degree subjective

• Two notes within each others critical bandwidth sound dissonant

• Other points of dissonance have been noticed

Scales

Aimed at creating:

‘a discrete set of pitches in such a way as to yield the maximum possible number of consonant combinations (or the minimum possible number of dissonances) when two or more notes of the set are sounded together.’

Roederer (1975: 153)

The Pythagorean Scale

• Step 1 - Ascend in fifths

1 3/2 (3/2)2 (3/2)3 (3/2)4 (3/2)5

(100Hz) (150Hz) (225Hz) (337.5Hz) (506.25Hz) (759.38Hz)

or

1 3/2 9/4 27/8 81/16 243/32

The Pythagorean Scale

• Step 2 - bring into the range of a single octave by descending in whole octave steps

1 3/2 9/4 27/8 81/16 243/32

(100Hz) (150Hz) (225Hz) (337.5Hz) (506.25Hz) (759.38Hz)

Descend one octave

( / 2)

Descend one octave

( / 2)

Descend two octaves

( / 4)

Descend two octaves

( / 4)

OK OK

1 3/2 9/4 27/8 81/16 243/32

(100Hz) (150Hz) (112.5Hz) (168.75Hz) (126.56Hz) (189.84Hz)

The Pythagorean Scale

• Step 3 - arrange the notes obtained in ascending order

1 9/8 81/64 3/2 27/16 243/128 2

(100Hz) (112.5Hz) (126.56Hz) (150Hz) (168.75Hz) (189.84Hz) (200Hz)

• Step 4 - create the fourth by descending a fifth and then moving up an octave

2/3 1 2/3 2/3 * 2 = 4/3

The Pythagorean Scale

insert

(100Hz) (112.5Hz) (126.56Hz) (133.33) (150Hz) (168.75Hz) (189.84Hz) (200Hz)

1 9/8 81/64 4/3 3/2 27/16 243/128 2

do re mi fa so la ti do

Problems with Pythag

1 9/8 81/64 4/3 3/2 27/16 243/128 2

9/8 9/8 256/243 9/8 9/8 9/8 256/243

intervals

ratios

exact fourth exact fifthslightly off (should be 5/4) slightly off

(should be 5/3)

Problems with Pythag

• More problems are created when same method is used to extend to a chromatic scale

• For example, two different semitone intervals are created; this limits the number of keys that music can be played in

The Equal Tempered Scale

• Has become the standard scale to which all instruments are tuned

• Allows flexibility regarding tonalities that can be used

The Equal Tempered Scale

• Achieved by creating 12 equally spaced semi-tonal divisions

i = 21/12 = 1.059463

• Requires all of the intervals within an octave to be slightly mistuned

The Equal Tempered Scale

For example, the ratio of notes:

• a fifth (3/2 = 1.5) apart is tuned to 1.4987 (0.087% flat)

• a sixth apart (5/3 = 1.6667) is tuned to 1.6823 (0.936% sharp)

Intervals

• In equal temperament are measured by the number of letter names between two notes (both of whose letter names are included)

Third

Minor Third

Fourth

Fifth

Sixth

Minor Sixth

Tones & Semitones

• Moving up a semitone is moving up one key on the keyboard

• Moving up a tone is moving up two keys on the keyboard

• A fifth involves moving up how many semitones?

The Major & Minor Scales

• A scale is an alphabetic succession of notes ascending or descending from a starting note

• Beginning with the note C the succeeding white notes of the keyboard form the C major scale

The C Major Scale

• The intervals between each note are what make it a major scale

C Major

T T S T T T S

Major Scales

• Move up one note but keep the same intervals between the notes and the scale C Sharp Major is found

• This is the next Major Scale

• Continue this process to find all twelve Major Scales

C Sharp Major

T T S T T T S

The Minor Scales

• A different pattern of intervals produces all of the Harmonic Minor Scales

• The Melodic Minor Scales are a variation of these, their intervals change depending upon whether the scale is ascended or descended

Harmonic C Minor

T TS T S M T S

MT = Minor Third (3 semitones)

Melodic C Minor

T TS T T T S

T TS T S T T

ascending intervals

descending intervals

ascending notes

descending notes

Harmonic C Sharp Minor

T TS T S M T S

Melodic C Sharp Minor

T TS T T T S

T TS T S T T

ascending intervals

descending intervals

ascending notes

descending notes