Transcript Scales, Voice

PH 105
Dr. Cecilia Vogel
Lecture 14
OUTLINE
units of pitch intervals
cents, semitones, whole tones, octaves
staves
scales
chromatic, diatonic, pentatonic
consonant intervals
octave, fifth, fourth, major third, minor
third
temperament
equal, just, Pythagorean
Logarithmic Frequency
Measures
Unit
Factor
Equivalent
(equal temp)
cents
1.000578
semitones
1.0595
100 cents
whole tones
1.1225
2 semitones
200 cents
octaves
2
12 semitones
1200 cents
Cents
One cent interval has a ratio of 1.0006
1 cent above 440Hz is
Can you tell the difference between 440 Hz
and 440.25 Hz?
a jnd is a ratio of 1.005
about 8-9 cents
10 cent above 440Hz is
Can you tell the difference between 440 Hz
and 442.55 Hz? (10 cents)
Cents Calculation
Interval, I, in cents is related to the
1200
I log 2 

I
log R R  inverse log 

log 2
 1200 
Example, an octave has a ratio of
1200
I
log?
log 2
Semitone
An octave is often
each semitone is a factor of
multiply 440 Hz (an A) by
you’ll get about 880 Hz
Keys on a piano are separated by
12 semitones in order is a
Musical Staff
Musical notes are
the x-axis is
the y-axis is
Fig 8.9
Only the notes in spaces are written in.
Notes on lines are letters between.
Short lines indicate where sharp/flat would
be , graphically.
Major Diatonic Scale
Western music uses a ____________ instead.
A major diatonic scale has
(the 8th would be an
The intervals are not all semitones
some are
The intervals in major diatonic scale are
Start with any key on the keyboard.
You’ve played a major diatonic scale.
Example
Key of C (major diatonic scale)
play
 CDEFGAB
C to D is a
C#/Db is between
similarly with
E to F is a

Scale on Piano
 one octave on keyboard
ignore the gray for now
Pitch Standard
Current scales based on standard
A4 =
historically lower
Handel’s 422.5 is closer to Ab
Can base your scale on any frequency,
but current instruments are built to
perform well for the standard.
Temperament
Temperament means
how you tune intervals within your scale.
Equal temperament means
all intervals are
each semitone is the
a factor of about 1.06
Keys on a piano are usually tuned to equal
temperament, AKA the tempered scale
Consonance
An octave ratio is a particularly close
relationship in our hearing.
Other simple ratios also tend to be
consonance=
Consonant notes have similar
Example 440 Hz and 660 Hz
both have 1320, 2640, etc as harmonics
Consonant Intervals
See also Table 9.1
Octave interval is simple ratio
Fifth is a simple ratio
Fourth is a simple ratio
Major third is a simple ratio
Minor third is a simple ratio
Temperaments
Tempered Scale or equal temperament
all intervals are
consonant intervals are
Just Scale
consonant intervals are perfect in
other keys are
Pythagorean Scale
fourths and fifths are perfect in
major and minor thirds are
Tempered Scale
The frequencies of 9 octaves of tempered
*not very good
scale are in table 9.2
note
C4
C#/Db
D
D#/Eb
E
F
G
C5
freq(Hz)
261.63
277.18
293.66
311.13
329.63
349.23
392.00
523.25
interval
—
semitone
whole
minor 3rd
major 3rd
fourth
fifth
octave
ratio
1
1.06
1.12
1.19*
1.26*
1.335
1.498
2
simple ratio
6/5 = 1.2
5/4 = 1.25
4/3 = 1.333
3/2 = 1.5
2/1 = 2
Just Diatonic Scale
Just temperament
based on perfect triads
In triad
major 3rd is exactly 5/4
minor 3rd is exactly 6/5
fifth is exactly 3/2
Just Diatonic Scale
To get perfect triads, must sacrifice:
There are two different size whole tones
9/8 (1.125) and 10/9 (1.111).
All semitones are 16/15 (1.067)
but two semitones don’t make whole tone.
so, for example, C# and Db are not the same
Can only tune triads in a particular key
such as C-major
triads will be mistuned in other scales
Just Scale
ratios are perfect in key of C:
note
C4
C#
Db
D
Eb
E
F
G
C5
freq(Hz)
261.63
272.53
279.07
294.33
313.96
327.04
348.84
392.44
523.25
interval
—
whole-semi
semitone
whole
minor 3rd
major 3rd
fourth
fifth
octave
ratio simple ratio
1
 9   15 
  
 8   16 
 16 
 
 15 
9/8
6/5
5/4
4/3
3/2
2
6/5 = 1.2
5/4 = 1.25
4/3 = 1.333
3/2 = 1.5
2/1 = 2
Pythagorean Scale
Pythagorean scale based on
A fifth and a fourth make an octave,
(3/2)(4/3) = __,
so if you tune a fifth, you’ve tuned a
fourth.
To get perfect fifths and fourths in
all scales, must sacrifice:
major and minor thirds are not good
again, C# and Db are not the same
Pythagorean Scale
fourths and fifths perfect
note
C4
C#
Db
D
Eb
E
F
G
C5
freq(Hz)
261.63
279.39
279.07
294.33
310.03
331.22
348.84
392.44
523.25
interval
—
7 5ths- 4 oct
3 oct – 5 5ths
whole
minor 3rd
major 3rd
fourth
fifth
octave
*even worse
ratio
1
7
3 1
   
2 2
simple ratio
4
5
3
2
2


 
3
9/8
1.185*
1.27*
4/3
3/2
2
6/5=1.2
5/4 = 1.25
4/3 = 1.333
3/2 = 1.5
2/1 = 2
Notes on Pythagorean and Just
In C-major scale, both have perfect 4th, 5th
Just has good major thirds in C-major
but bad in other scales.
for example D:A is 1.69, instead of 1.667
Pythagorean has bad major thirds in Cmajor
to have a perfect fifth in another scale.
for example E:C is 1.27 not 1.25, but E:A is
exactly 1.5
Table 9.3 (jnd about 8.6 cents)
Summary
equal pitch intervals are equal frequency factors
jnd, cents, semitone, whole tone, octaves
Scales
chromatic, 12 notes, 1 semitone apart
major diatonic, 7 notes, whole & semitone intervals
pentatonic, 5 notes, whole and 1½ tone intervals
Staff
Temperaments of diatonic scale
equal temperament: equal semitones
just temperament: perfect intervals in one key
Pythagorean temperament: perfect 5ths in any key