Three Fundamental Facts Frequency Pitch middle A is often 440 Hz but not necessarily Any pitch class can be duplicated by multiplying the frequency by 21 or 12 Musical intervals are associated with ratios multiply a given frequency by a given ratio traverse a given interval in pit ID: 661862
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Slide1
Tuning Basics
INART 50
Science of MusicSlide2
Three Fundamental Facts
Frequency ≠ Pitch
(middle A is often 440 Hz, but not necessarily)
Any pitch class can be duplicated by multiplying the frequency by 2/1 or 1/2
Musical intervals are associated with ratios: multiply a given frequency by a given ratio, traverse a given interval in “pitch space.”
To span the same distance in the opposite direction, flip the ratio.Slide3
Pythagoras
An interval based on 2/1 (or ½) is profound since pitch class is duplicated.
The next most profound
interval is based on 3/2
It’s consonant
It establishes the primacy
of the numbers 1, 2, and 3
It sets up a variety of symmetries:
2
/1
1/1
diapason
(“through all”)Slide4
Pythagoras
An interval based on 2/1 (or ½) is profound since pitch class is duplicated.
The next most profound
interval is based on 3/2
It’s consonant
It establishes the primacy
of the numbers 1, 2, and 3
It sets up a variety of symmetries:
1/1
2
/1
3/2
2/3
Go up and down by 3/2Slide5
Pythagoras
An interval based on 2/1 (or ½) is profound since pitch class is duplicated.
The next most profound
interval is based on 3/2
It’s consonant
It establishes the primacy
of the numbers 1, 2, and 3
It sets up a variety of symmetries:
1/1
2
/1
3/2
2/3
Transpose by 2/1
4/3Slide6
Pythagoras
An interval based on 2/1 (or ½) is profound since pitch class is duplicated.
The next most profound
interval is based on 3/2
It’s consonant
It establishes the primacy
of the numbers 1, 2, and 3
It sets up a variety of symmetries:
1/1
2
/1
3/2
4/3
4/3
4/3Slide7
Pythagoras
1/1
2
/1
3/24/3Continue to derive new pitch classes by traversing intervals of 3/2.
Transpose all ratios to fall within the range between 1/1 and 2/1.
3/2
x
3/2 = 9/4
transpose by ½:
9/4 x 1/2 = 9/89/8Slide8
Pythagoras
1/1
2
/1
3/24/3Continue to derive new pitch classes by traversing intervals of 3/2.Transpose all ratios to fall within the range between 1/1 and 2/1.
9/8
x
3/2 = 27/16
9/8
27/16Slide9
Pythagoras
1/1
2
/1
3/24/3Continue to derive new pitch classes by traversing intervals of 3/2.Transpose all ratios to fall within the range between 1/1 and 2/1.
27/16
x
3/2 = 81/32
9/8
27/16
transpose by ½:81/32 x 1/2 = 81/6481/64Slide10
Pythagoras
1/1
2
/1
3/24/3Continue to derive new pitch classes by traversing intervals of 3/2.Transpose all ratios to fall within the range between 1/1 and 2/1.
81/64
x
3/2 = 243/128
9/8
27/16
81/64243/128Slide11
Pythagoras
1/1
2
/1
3/24/3Continue to derive new pitch classes by traversing intervals of 3/2.Transpose all ratios to fall within the range between 1/1 and 2/1.
This is the basis of the major scale:
9/8
27/16
81/64
243/128
7 pitch classes2 step sizes, large and smallProblems:The ratios get increasingly awkward, and less consonant.
After 12 such successive pitch classes, the result is extremelyclose to 2/1, but not quite. The symmetry doesn’t hold up.Slide12
Pythagorean Tuning(based on perfect fifths)
1/1
9
/
881/644/33/227/16
243/128
2
/1
9
/
89/8
256/2439/8
9
/
8
9
/8256/243Slide13
Just Intonation
As an alternative, consider the pitch classes of the first 6 harmonics
With a fundamental
f
of 100 Hz, the 6 harmonics are:100 ( f )200 ( 2f )
300 ( 3
f
)
400 ( 4
f )500 ( 5f )600 ( 6
f )Slide14
Just Intonation
As an alternative, consider the pitch classes of the first 6 harmonics
The ratios of these harmonics to the fundamental are:
100 (
f )200 ( 2f )
300 ( 3
f
)
400 ( 4
f )500 ( 5f )600 ( 6f )
1/12/13/14/15/1
6/1Slide15
Just Intonation
As an alternative, consider the pitch classes of the first 6 harmonics
For simplicity, consider just the ratios:
1/1
2/13/1
4/1
5/1
6/1Slide16
Just Intonation
As an alternative, consider the pitch classes of the first 6 harmonics
The pitch class ratio of each harmonic may be found by transposing them down by octaves (multiplications of ½) until the ratio lies between 1/1 and 2/1:
1/1
2/13/1
4/1
5/1
6/1
1/1
2/1
3/1 x ½ = 3/24/1 x ½ = 4/2 = 2/15/1 x ½ = 5/2;
6/1 x ½ = 6/2; 5/2 x ½ = 5/46/2 x
½ = 6/4 = 3/2
5/4 is very close to the Pythagorean third at 81/64.
(5/4 = 80/64)This sounds more consonant, as it is a naturally-occurring harmonic.
Thus, the first 6 harmonics are often said to be:
fundamental – octave – fifth – octave – third - fifthSlide17
Just Intonation
If the pitch classes of the first 6 harmonics are also found for the fourth (4/3) and fifth (3/2), two new pitch classes appear that are close to Pythagorean ratios, but are simpler and more consonant.
1/1
2/1
3/14/15/1
6/1
1/1
2/1
3/2
2/1
5/43/2 1/1
???
?
?
?
4/3
??
?
?
?
?
3/2
To find the pitch classes, do the same procedure as the previous slide, but start with a ratio of 4/3 instead of 1/1. Repeat, starting with a ratio of 3/2.
Multiply the starting ratio by values 1-6, then transpose down by octaves (multiply by ½) until the ratio falls between 1/1 and 2/1.Slide18
Pythagorean Tuning(based on perfect fifths)
1/1
9
/
881/644/33/227/16
243/128
2
/1
9
/
89/8
256/2439/8
9
/
8
9
/8256/243
Just Tuning
(based on natural harmonics)
1/1
9
/
8
5
/4
4
/
3
3
/
2
5
/3
15/8
2
/1
9
/
8
10/9
16/15
9
/
8
10/9
9
/
8
16/15
Problem:
It’s hard to transpose (change the fundamental pitch).
e.g., if in the middle of a piece one decides to
modulate
, considering the pitch 3/2 as
do,
then a complete scale is not available.
The distance from 1/1 to 9/8 is 9/8. But the tone that is the same distance from 3/2, 3/2
x
9/8 = 27/16, is not in the scale.Slide19
Pythagorean Tuning(based on perfect fifths)
1/1
9
/
881/644/33/227/16
243/128
2
/1
9
/
89/8
256/2439/8
9
/
8
9
/8256/243
Just Tuning
(based on natural harmonics)
1/1
9
/
8
5
/4
4
/
3
3
/
2
5
/3
15/8
2
/1
9
/
8
10/9
16/15
9
/
8
10/9
9
/
8
16/15
The above just scale was created by the Greek philosopher Ptolemy.
It is sometimes called the Ptolemaic just scale.
There are a variety of just scales that musicians explore.
What all the scales have in common is that they are all based on simple ratios, taken from the pitch classes of natural harmonics.
e.g., a scale that had the pitch class of the 17
th
harmonic would include a ratio derived by transposing the 17
th
harmonic down by octaves until its pitch class is found to be a ratio falling between 1/1 and 2/1:
17/1
x
½ = 17/2;
17/2
x
½ = 17/4;
17/4
x
½ = 17/8;
17/8
x
1/2 =
17/16Slide20
Equal Temperament
As a compromise, since the early 1700s, Western music has used a scale that divides the octave (diapason) into equal perceptual steps.
With 12 equal steps per octave, any note may be used as the first note of a scale, and all notes are available.
The intervals are not as consonant as just intonation, but the compromise has been considered with the sacrifice as a change of scale can take place in the middle of a piece, without having to stop to retune the instrument.
n
= 0, 1, 2, … 12
To derive frequencies in
twelve tone equal temperament
, start with a frequency
f
, and multiply it by 2n/12 for n = 0-12.Slide21
Pythagorean Tuning(based on perfect fifths)
1/1
9
/
881/644/3 3/227/16
243/128
2
/1
9
/
89/8
256/2439/8
9
/
8
9
/8256/243
Just Tuning
(based on natural harmonics)
1/1
9
/
8
5
/4
4
/
3
3
/
2
5
/3
15/8
2
/1
9
/
8
10/9
16/15
9
/
8
10/9
9
/
8
16/15
Twelve Tone Equal Temperament
(based on perceptually equal subdivisions of the octave)
n
= 0, 1, 2, … 12Slide22
The cent measurement
In order to compare tuning systems and intervals, the
cent
increment was created.
A cent is 1/100 of a semitone, or 1/1200 of an octave:
n
= 0, 1, 2, … 1200
Thus, differences among tuning systems can be quantified.
An equal tempered fifth, for example, is 700 cents; a 3/2 just fifth is 701.955 cents, indicating a difference of almost 2 cents.
An interval ratio,
r, may be converted to cents, c
, by the equationSlide23
The cent measurement
In order to compare tuning systems and intervals, the
cent
increment was created.
A cent is 1/100 of a semitone, or 1/1200 of an octave:
n
= 0, 1, 2, … 1200
Thus, differences among tuning systems can be quantified.
An equal tempered fifth, for example, is 700 cents; a 3/2 just fifth is 701.955 cents, indicating a difference of almost 2 cents.
An interval ratio,
r, may be converted to cents, c
, by the equatione.g., the interval in cents for the ratio 3/2 is
1200 x log
10(3/2)/log10(2) = 1200 x 0.1761/0.301 = 701.955