Transcript Musical Intervals and Scales

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
pressure
2
f1
1.5
1
0.5
0
time
-0.5
-1
-1.5
f2
-2
f 2 = 2 * f1
Frequency Ratio = 2/1
Intervals: The Fifth
2.5
pressure
2
f1
1.5
1
0.5
0
time
-0.5
-1
-1.5
-2
f2
-2.5
f2 = 3/2 * f1
Frequency Ratio = 3/2
Musical Intervals
Frequency ratio
1/1
2/1
3/2
4/3
5/3
5/4
6/5
8/5
Interval
Unison
Octave
Fifth
Fourth
Major Sixth
Major Third
Minor Third
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
(100Hz)
OK
3/2
(150Hz)
OK
1
3/2
(100Hz) (150Hz)
9/4
(225Hz)
Descend one
octave
( / 2)
27/8
81/16
(337.5Hz) (506.25Hz)
Descend one
octave
( / 2)
Descend two
octaves
( / 4)
243/32
(759.38Hz)
Descend two
octaves
( / 4)
9/4
27/8
81/16
243/32
(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)
The Pythagorean Scale
• 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
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
Problems with Pythag
exact fourth
exact fifth
slightly off
(should be 5/4)
1
9/8
9/8
81/64
9/8
slightly off
(should be 5/3)
4/3
256/243 9/8
3/2
27/16
9/8
243/128
9/8
2
256/243
intervals
ratios
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
S
T
T
S
MT
S
MT = Minor Third (3 semitones)
Melodic C Minor
ascending
intervals
T
S
T
T
T
T
S
ascending notes
descending notes
T
S
T
T
S
T
T
descending
intervals
Harmonic C Sharp Minor
T
S
T
T
S
MT
S
Melodic C Sharp Minor
ascending
intervals
T
S
T
T
T
T
S
ascending notes
descending notes
T
S
T
T
S
T
T
descending
intervals