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Musical Intervals & Scales

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- 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

- Musical intervals are distances of frequency between two notes
- The distance of an octave is a doubling of frequency

pressure

f2 = 2 * f1 Frequency Ratio = 2/1

f1

time

f2

pressure

time

f1

f2

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

- 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

- Are to some degree subjective
- Two notes within each others critical bandwidth sound dissonant
- Other points of dissonance have been noticed

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)

- 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

- 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)

- 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 12/3 2/3 * 2 = 4/3

insert

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

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

do re mi fa so la ti do

exact fourth

exact fifth

slightly off

(should be 5/4)

slightly off

(should be 5/3)

intervals

ratios

- 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

- Has become the standard scale to which all instruments are tuned
- Allows flexibility regarding tonalities that can be used

- 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

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)

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

- 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?

- 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 intervals between each note are what make it a major scale

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- 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

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- 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

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M T

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MT = Minor Third (3 semitones)

ascending intervals

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ascending notes

descending notes

descending intervals

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M T

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ascending intervals

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ascending notes

descending notes

descending intervals

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