PHY2464 - The Physical Basis of Music
PHY
-2464
PHY-2464
Physical Basis of Music
Presentation
Presentation 88
Musical
Musical Elements
Elements B:
B: More
More on
on
Scales
Scales and
and Temperaments
Temperaments
Adapted
Adapted from
from Sam
Sam Matteson’s
Matteson’s
Unit
Unit 22 Sessions
Sessions 20
20 &
& 21
21
Sam
Sam Trickey
Trickey
Feb.
Feb. 7,9
7,9 2005
2005
PHYPHY-2464
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Pres. 8 Musical Elements -B
The tonic of a scale is the pitch that is the basis of
the scale.
The chromatic scale is a series of tones, each
separated from the next by a pitch interval of a
semitone.
The diatonic scale contains a series of (whole)
tone and semitone intervals.
Temperament (and tempering) describes the plan
of intonation compromise that permits multiple
scales.
Tuning describes the matching of pitches
according to some prescription.
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 8 Musical Elements -B
Tempering problem
Equal tempering - the ratio of adjacent
semitone frequencies is 1.059463….
Pythagorean temperament, the ratios of
adjacent semitone frequencies always are
rational fractions (ratios of integers)
The two won’t sound the same!
A brief digression for “keyboard picture” of
scales -
PHYPHY-2464
Pres. 8 Musical Elements -B
Musical Notation
♩♩ ♩
♩
♩
♩♩
440 Hz
♩♩
♩♩ ♩
♩
♩♩♩ C
♩
♩
♩ E F2 B
D E2 G A2
C2 2
2
2
2
3
4
F4
D E4
F3G3 B3C4
A
E3
D3
♩♩
♩
♩♩♩
♩ ♩ ♩G
4
♩
D
A C6
C5 5 F G5 5
B
B5
5
4
A4
E5
4
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 8 Musical Elements -B
Musical Notation
♩♩ ♩
♩
♩
♩♯ ♩♩
♩
♯
♩
♩♯
♩♯ ♩ ♩♯G4♯ B C ♯
♩♯ E F4♯ A ♯
♩♭ ♩♭ D ♯
♭
♩
♩
♩♩♭♩♭
♩
♩
♩
A♭ C♯
E♭
♩ ♩♩
4
♩
PHYPHY-2464
3
3
D3♭ F G ♭
3 3
C3
4
4
5
4
4
B3 ♭
Pres. 8 Musical Elements -B
Musical Intervals
f2 /f1
Name
Unison
1/1
Octave
2/1
Fifth
3/2
Fourth
4/3
Major third 5/4
Minor third 6/5
Semitones
0
12
7
5
4
3
Example
A4 → A4
A4 → A5
A4 → E5
A4 →D5
A4 →C5#
A4 →C5
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 8 Musical Elements -B
Ratios for building the circle of fifths.
A. “An octave is a fifth plus a fourth”
Start at the tonic, f1 and go up a fifth:
f ‘ = 3/2 f1
Go up a fourth: f2 = 4/3 f ‘ = 4/3 × 3/2 f1 = 2 f1
B. “Going up a fifth is same as going down a fourth”
Up a fifth f ‘ = 3/2 f1 vs. down a fourth f” = ¾ f1
Then double to get back into same octave:
f” = 2 × ¾ f1 = 3/2 f1
C. Hidden Gotcha – “An octave is three major thirds”
PHYPHY-2464
Pres. 8 Musical Elements -B
Ratios for building the circle of fifths (continued)
D. “All notes can be visited by going up 12 fifths or
down 12 fourths (and shifting the appropriate
number of octaves)”
Start at tonic, f1 & go up n - fifths: f ‘ = (3/2)n f1
Then shift the multiplier down by enough powers of
(1/2) so that the resulting multiplier is less than
or equal to 2 (to stay in the same octave).
Another Hidden Gotcha:
(3/2)12 = 129.74633… vs. (2)7 = 128
Hmmmm, 12 fifths is a little more than 7 octaves.
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 8 Musical Elements -B
Ratios for building the circle of fifths (continued)
D. (Cont) f ‘ = (3/2)n f1 ; shift back to same octave
(3/2)n shifted (between 1 and 2)
3/2
9/8
27/16
81/64
243/128
729/512
2187/2048 = 1.067871
4/3
16/9
32/27
64/54 etc.
Pres. 8 Musical Elements -B
The Circle of Fifths:
A
463.53
A♯ , B♭
F
C
742.5 F♯,G♭
,G♭
695.3
660
495
B
782.22
G♯,A♭
,A♭
G
835.32
Ratio = 3:2
440 Hz
E
PHYPHY-2464
(3/2)n
3/2
9/4
27/8
81/16
243/32
729/64
2187/128
2/3
4/9
8/27
16/54
D♯,E♭
,E♭
626.48
D
n
1
2
3
4
5
6
7
-1
-2
-3
-4
521.48
C♯,D♭
,D♭ 556.88
586.67
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 8 Musical Elements -B
Making sense of Cents
Pitch interval for two frequencies f2 , f1
Interval in cents = Ι(¢)
= 3986.31 log (f2 / f1 )
Example: major third (5/4)
Ι(¢) = 3986.31 log (5/4) =386.21 ¢
Intervals in cents , Ι(¢), ADD
PHYPHY-2464
Pres. 8 Musical Elements -B
Musical Intervals in Cents
f2 /f1 Semitones
Name
Unison
1/1
0
Octave
2/1
12
Fifth
3/2
7
Fourth
4/3
5
Major third
5/4
4
Minor third
6/5
3
Size in cents
0
1200
701.954
498.044
386.313
315.640
Something odd: Nominally 1 semitone = 100 ¢
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 8 Musical Elements -B
Musical Intervals in Cents
Octave
= 12 semitones = 1200 ¢
Fifth + Fourth = 7 semitones + 5 semitones
= 701.954 ¢ + 498.044 ¢ = 1199.999 ¢
≈ 1200 ¢
3 Major thirds = 3 × 4 semitones
= 3 × 386.313 ¢ =1158.939 ¢
≈ 1159 ¢
“lesser diesis” = 1200 – 1159 = 41 ¢
This error has to be somewhere in tuning of thirds
PHYPHY-2464
Pres. 8 Musical Elements -B
Diatonic scales
„
Major diatonic: TTSTTTS where
– T=go up a Tone, S= go up a Semitone
„
„
„
If you start at C on the keyboard, this is all white
keys
Doing so corresponds to picking a “key
signature” – defines “tonic” and tells the
intervals to be used.
An example is shown in Hall, Fig. 18.6, as
follows.
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 8 Musical Elements -B
Diatonic scales (continued)
™ Start at C and tune down a fifth (2:3) and up
an octave for “perfect” F and also go up a
fifth (3:2) to get perfect G.
™ Then go up a major third (5:4) from C to get
E and up a major third from F to get A. Also
go up a major third from G to get B.
™ Go down a fifth from A to get D.
™ Tune the C an octave up
Gives C, D, E, F, G, A, B, C
PHYPHY-2464
Pres. 8 Musical Elements -B
Diatonic scales continued
So we have C, D, E, F, G, A, B, C
„ The intervals in cents:
fifths = 701.954 ¢ , fourths = 498.044 ¢, major thirds =
386.313 ¢,
„
(Nominally one semitone is 100 ¢)
Suppose someone needs to start with a tonic of Aflat.
Aflat. It
is supposed to be two major thirds above C or about 772
¢ above the C. But G# is supposed to be one semitone up
from G and the G was constructed as a perfect fifth (702
¢) above the C, so G# = 702 + 96.5 = 798.5 ¢ (I’
(I’ve used
1/3 of a major third for the semitone)
„ Here is an Aflat not equal to a G# Supposedly enharmonic
notes turn out not to be.
be.
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 8 Musical Elements -B
„ Can we escape enharmonic notes that aren’t?
„ Idea – Don’t mix fifths & thirds (dubious esthetics!)
„ Circle of fifths starting at Aflat: Aflat →Eflat →Bflat →F
→C →G →D →A →E →B →F# →C# →G#
„ Is the G# the same pitch 7 octaves up as the
Aflat
where we started?
„NO! We’v already seen it as a power of 2. In cents, go
up twelve fifths:
Ι(¢) = 12 × 701.954 = 8423.4 ¢
whereas 7 octaves = 8400 ¢
“ditonic comma” = 8423.4 – 8400 = 23.4 ¢
PHYPHY-2464
Pres. 8 Musical Elements -B
„ Inescapable compromises
„ Must spread the ditonic comma and the lesser
dieses around somewhere in our tuning.
„ Different compromises are called temperaments.
„ “Equal tempering” spreads all the errors
uniformly. Result is better fifths at the expense of
thirds.
„ “Quarter comma meantone” favors thirds but
yields one awful fifth (the “wolf” at 738 ¢); the
others are “narrow” (697 ¢) but usable.
„ Study figures 18.7 through 18.11 in Hall
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 8 Musical Elements - B
Summary (and reminders):
„ See handout “Rudiments of Scales & Temperaments”
„ While an octave is a fourth plus a fifth, it is not
possible to tune, for example, perfect major thirds
(5:4) and octaves at the same time, nor is it possible
to have 7 octaves in tune by fifths alone, etc.
„ Our perceptions of pitch match mathematical reality.
While rational fractions (ratios of integers) are very
appealing, our pitch apparatus also deals with
irrational numbers (e.g. 21/12) very well.
„ Temperaments are most important for fixed pitch
instruments (pianos, harpsichords, harps, organs)