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SCIENCE,
MATHEMATICS
AND MUSIC
1
Poetry and Music
Poetry and music are historically related –
chants, songs, incantations
Dance and music similarly related
Early musical accompaniment for song and dance –
Drums
Pipes
Lyre
Lute-like instruments
Harp
2
Poetry / Music relationships
Poetry
Music
Rhyme
Cadence – repetition of
phrase ‘shape’
Rhythm – metre
Beat / accent / time signature
Mood
Modes, major / minor keys
Intonation
Melody
Euphony –
pleasant sound
Harmony
3
Sound Production
Vibrators include –
larynx, reeds, lips, edges, strings, membranes, hollow shapes
Resonators include body cavities (chest, pharynx, sinuses)
boards, pipes, hollow shapes, instrument casing
Methods include –
blowing (physical or mechanical)
plucking, striking, shaking, stroking
4
Musical Instruments – Wind
Blowing
Voice –
larynx and resonance
larynx, little resonance
Lips
trombone
trumpet
french horn
saxophone
Reed
oboe
panpipe
basset horn
reeds
ocarina
5
Wind instruments continued . . .
Pumping
concertina
organ
6
Percussion and striking
triangle
tambourine
castanets
kettle drum
xylophone
7
Strings – plucking . . . . . . . . . and bowing
lyre
harp
(violon) cello
harpsichord
guitar
violin
8
9
Scan original
10
Human audible range from about 16hz to 25/30,000
(upper end lost with age).
i.e. about c-4 to c6 / c7
Concert grand only
C -3
33
C -2
66
C -1
132
C 1 (c)
528
C0
264
C2 (c)
1056
C3 (c)
2112
C4
(c)
4224
- Hertz A
27.5
A
440
11
An Aside on Poetic Rhythm
Like music, traditional poetry is based on a pattern of stressed /
unstressed ‘beats’ – in poetry the beats are syllables and the basic unit is
a poetic ‘foot’. There are four main types and one variant :iamb (‘lame’)
/ ___ / ___ / ___ / ___ / ___
The curfew tolls the knell of parting day
___ / ___ / ___ / ___ /
trochee (‘running’)
Tyger, tyger, burning bright
___ / / ___ / / ___ / / ___ / /
dactyl (‘finger’)
I galloped, Dirk galloped, we galloped all three
anapest (‘reversed’)
/ / ___ / / ___ / / ___ / / ___
The Assyrian came down like the wolf on the fold
spondee, inserted into iamb ___ ___ to indicate heaviness or weariness
The long day wanes, the slow moon climbs
12
These feet are assembled into lines, generally of two to seven feet,
known as dimeter,
trimeter,
tetrameter,
pentameter,
hexameter,
heptameter
Shakespeare’s plays and Gray’s famous elegy owe their stately rhythm
to their iambic pentameters
It’s not always easy to distinguish between iambs and trochees or
dactyls and anapests when there are extra or missing syllables at the
beginning or end of the line – part of ‘poetic licence’.
(Tyger, tyger’ is not a classic trochee as it’s lacking the last syllable)
13
Hymn metres
The metres for hymns, to be sung, are identical to those for other poetry,
but in order to enable a suitable choice of tunes they are classified by
the number of lines per verse (occasionally augmented by the type of
foot).
Short metre (S.M.) 6.6.8.6.
(Blest are the pure in heart)
Common metre (C.M.) 8.6.8.6. (Through all the changing scenes of life)
Long metre (L.M.) 8.8.8.8. (Forth in thy name O Lord I go)
8.7.8.7.D (the D indicates the metre is doubled, i.e. 8 lines)
(Glorious things of thee are spoken)
Many others, including ‘irregular’ – usually with a specifically-written tune
(O come all ye faithful)
14
Note Lengths
The system of note lengths in music is binary, each successive denomination being twice or
half the preceding one. Two obsolete long notes are included for illustration – they also
explain why ‘breve’ (= short) was used for what is now a rarely used long note.
Most modern usage takes the crotchet as the unit of note length and the illustration shows
the number of crotchets equal to the long notes or the number of short notes equal to a
crotchet.
Maxima
(25)
32 x
Semibreve
4x
(22)
Quaver
=
(2-1)
2x
Demisemiquaver 8 x
=
Longa
16 x
(24)
Breve
8x
Minim
2x
(21)
Crotchet
(20)
Semiquaver 4 x
(2-3)
=
Hemidemisemiquaver
(23)
(2-2)
16 x
=
(2-4)
Maxima is 512 (29) times h.d.s.quaver and semibreve is 64 (26) times.
Some composers have used even shorter notes, the semi, demi, hemi cycle is then repeated!
15
Modifications to Note Lengths
Addition of a dot after a note extends its duration by a half. A second
dot adds another quarter, thus:-
=
.
.
=
Staccato shortens a note by about a half and staccatissimo by about
three-quarters. These are indicated by a dot or hyphen respectively
below the note, e.g.
=
=
.
16
Tempo
The note values do not indicate the actual speed of performance. This
is shown either by the appropriate Italian term for the desired speed, or
by metronome marks, or sometimes both.
Term
Prestissimo
Presto
Allegro
Moderato
Andante
Adagio
Larghetto
Largo
Beats per minute
200-208
168-200 eg
120-168
108-120 eg
76-108
66-76
60-66 eg
40-60
= 176
= 110
= 60
17
Rhythm in Music
Like poetry, musical rhythm is based on pattern of stressed/unstressed
‘beats’, arranged mainly in 2’s, 3’s or 4’s. A few composers have
experimented with 5’s, 7’s, etc but they are difficult to sing and even
more so to dance (the two originators of music).
Feet in poetry correspond to bars in music – thus
iamb / trochee = duple or quadruple time
dactyl / anapest = triple time
‘Time signatures’ combine the number of beats in a bar (numerator)
with a measure of beat length (denominator). Crotchets and quavers
are the most used denominators but do not indicate actual speed of
performance this is governed by TEMPO.
18
March
.
Minuet
Waltz
Polka
Tango
19
Pitch
To the human ear, pitch is observed as ‘highness’ or ‘lowness’
Scientifically, frequency of vibration measured in hertz – vibrations per second
Pitch standards are usually related to ‘middle’ C or the A above
Three medieval pitches –
Domestic (virginals), Church (higher than domestic), Military (higher still).
Baroque pitch
about A = 415
Handel’s tuning fork
A = 422.5
Mozart’s piano
about A = 421-422
Military pitch
A = 452.5 – means military band could not play with
orchestra, piano or choir
Philosophical/Scientific pitch C = 256 because repeatedly divisible by 2
A = 426.66
Modern International Standard C = 264, A = 440
Often pressure to raise pitch further, resisted by singers and players of string
instruments – more stress
‘Concert pitch’ has no specific meaning – just ‘higher than normal’, whatever
normal may be
All pitches are approximate and vary with temperature
20
The Diatonic Scale
(Greek = through the tones)
Discovery usually attributed to Pythagoras – experimenting with strings
Earlier civilisations may have been aware – tuning harps, arranging
holes in pipes etc
Pleasant ‘intervals’ between notes appear to have simple ratios of
string length under given tension
Later realisation that pitch is a function of tension and length, and of
‘effective’ length in pipes
Early trumpet-like instruments sometimes up to three metres long for
lower pitch – until discovery of method for bending metal tubes without
crushing – Renaissance in 14th century
The Diatonic scale provides the basis for Just or Pythagorean
Temperament tuning
21
Pythagoras’s Experiment
22
Harmonics
‘Echo’ notes produced by partial vibrations of strings or pipes at higher frequencies
Higher or lower notes that give pleasing effects in combination with fundamental
(Increasing tension also produces change in pitch but is more relevant to tuning)
Simple harmonics in combination produce chords, satisfying ‘filled in’ sound.
Agreeable harmonics have simple ratios to fundamental frequency –
2’s, 3’s and 5’s and combinations (6, 8, 15 etc)
7’s, 11’s, 13’s not agreeable!
Doubling the frequency gives effect to the human ear of ‘same note but higher’ –
doh to doh, soh to soh etc – the octave
Taking diatonic sequence of 24, 27 ------------ 48,
International standard pitch uses multiple of 11,
i.e. 24 x 11 = 264 (middle C)
40 x 11 = 440 (A used for tuning orchestras)
Examples of harmonics follow . . . .
23
Natural and ‘Stopped’ Harmonics
Base note
C = 24
4th natural harmonic
5 times frequency of base note
1/5
E” = 120
1st natural harmonic,
twice frequency of the base note
½
C’ = 48
Stopped harmonic – stopped at ¼
string/pipe length
4/3 times base note, with harmonic
4 times base note
2nd natural harmonic
3 times frequency of base note
1/3
G’ = 72
C” = 96
F = 32
24
More about Harmonics
Higher harmonics (integer multiples of base frequency) give most notes of Diatonic scale
(oddly, except A).
Taking base as 24 and reducing all harmonics to 1st octave:-
Harmonic
Frequency
Reduce to 1st Octave Note
1st
24
24
C (doh)
nd
2
48
÷ 2 =
24
C )1st octave)
3rd
72
÷ 2 =
36
G (soh)
th
4
96
÷ 2 =
24 [sic] 48?
C (2nd octave)
5th
120
÷ 4 =
30
E (mi)
6th
144
÷ 4 =
36
G (soh again)
th
7
168
÷ 4 =
42
A♯ Bb ??
8th
192
÷ 8 =
24
C (3rd octave)
9th
216
÷ 8 =
27
D (re)
th
10
240
÷ 8 =
30
E again
11th
264
÷ 8 =
33
below F♯ ??
th
12
288
÷ 8 =
36
G again
13th
312
÷ 8 =
39
between Ab & A ??
14th
336
÷ 8 =
42
Like 7th harmonic?
15th
360
÷ 8 =
45
B (te)
16th
384
÷ 16 =
24
C (4th octave)
Most others are unpleasant (?? above) except
18th = D, 20th = E, 24th = G
27th gives 40.5, discordant A in C major, perfect soh in D major (= 1½ x 27)
25
Harmonics of Diatonic Scale
24 27 30
doh
re
C
D
mi
E
32 36 40
fah soh lah
F
G
A
45 48
te
B
60
72
96
frequency
Tonic sol-fa
doh
C’ D’ E’ F’
192 Relative
G’ A’
B’
C’’
C’’’ etc
Principal note
26
Mode, Key, Modulation and Temperament
Modes used for church music in Middle Ages for variation in mood – joy, sadness,
repentance.
Modes use different starting notes but remain on the ‘white notes’ of the keyboard
Four Ambrosian modes start on D, E, F, G
Gregorian additions on C, A, B
Glarus added modes for five intermediate semitones – eventually 12 modes in all.
The Ambrosian and Gregorian are most widely used:C Hypolydian Mode VI (Gregorian)
D
Dorian Mode III (Ambrosian)
E
Phrygian Mode III (Ambrosian – not well liked)
F
Lydian Mode V (Ambrosian)
G
Mixolydian Mode VII (Ambrosian)
A
Hypodorian Mode II ((Gregorian)
B
Hypophrygian Mode IV (Gregorian
(D) An additional Hypomixolydian Mode VIII, also not favoured
Modes are still occasionally used in modern music – Vaughan Williams used them in
some works
Best known example is ‘Greensleeves’ (Hypodorian) though it has a final ‘te-doh’
cadence, so it ends in A minor
27
Keys
Early music based on C major (as we know it), Modes simply used different
start/finish notes, occasionally with final semitone cadence (te-doh)
Late 15th/early 16th century composers (eg Palestrina, Byrd) began to use
various starting notes but with diatonic intervals preserved (they are distorted in
modes except Hypolydian Mode VI)
Thus for starting note G, we need F♯, for start on F we need B , and for E we
need F ♯, G ♯, C ♯, D ♯
Nowadays music that stays strictly in one key is considered boring – key shift =
Modulation – introduces accidentals - ♯ and b
By the Bach era modulation was standard and chromatics (shifting in
semitones) entered the musical armoury.
It became apparent that instruments tuned to diatonic intervals in a particular
key produced unpleasant clashes when modulating to certain other keys.
These dissonances became known as ‘the wolf’ – they bite your ears!
The ‘wolf’ can also occur with inadequately designed instruments when the
natural harmonic of the resonator (violin body, wind instrument tube, keyboard
casing etc) clashes with the harmonics of the note being played – like a glass
jangling when a certain note is played!
28
Comparison of Keys (major, minor) and Modes
29
‘The Wolf’
For most human ears, the lower limit of perception of change in pitch is about
one tenth to one eighth of a tone, corresponding to 1% to 1½% in frequency.
For a dissonance in a chord the threshold may be lower. The discrepancies
from true harmonies in modulation for an instrument tuned diatonically can
reach 1.3% - this causes the wolf.
As an example, a modulation from C major to F major gives the following
relative frequencies:
F
G
A
B
C
D
E
F
32
36 40
*
48
54 60 64
* B not defined
What we require for true diatonics, starting with F, are:
F
G
A
B
C
D
E
F
32
36 40 42.67 48
53.33
60
64
The D will be noticeably out of tune unless the instrument can be retuned to F
as keynote.
The discrepancy in D is about 1¼%, and for other keys similar problems occur.
30
Perfect Fifths – The Tuner’s Dilemma
The octave and the fifth (C – G, E – B etc) are easy to tune by ear due to their
harmonics.
A piano or harpsichord tuner would start from one note, say middle C, and tune the
upper and lower octaves from it. An obvious next step is to tune the Gs, i.e. the
fifths from each C, with a check on the G octaves.
From G, the fifths progress via D, A, E, B, then into sharps.
Downwards from C, he would tune F, then B, E and on through the flats and even
double flats to D
When he reaches B♯ upwards & D downwards, he expects to arrive back at C.
Tuning in perfect fifths, ratio 3/2 or their inverse, the perfect fourths (4/3), he
arrives at B♯ about a quarter-tone sharp of C, and at D about a quarter-tone flat
of C.
In fact, it happens that only F, G and D have their pure diatonic frequencies, while
E, A and B are distorted. Also none of the ‘enharmonics’, like C♯/D , F♯/G ,
concur.
31
Tuning by Octaves, fifths and fourths
(starting Co = middle C)
32
The Problem of Perfect Fifths
Mathematically, the problem is that 12 rising fifths correspond closely,
but not exactly, to 7 rising octaves.
With ratios of 3/2 for the fifth
and 2 for the octave,
(3/2)12 = 129.746…
27 = 1.28
a discrepancy of 129.746 / 128/000 = 1.01364 or about 1.4%.
Thus the relative frequency for B♯ becomes 24 x 1.01364 = 24.33
and for D
becomes 24 x 1.01364 = 23.67
33
The Modulation Problem (Diatonic Scale)
The Diatonic scale gives specific ratios between the frequencies of successive
notes – five whole tones and two semi-tones. However the whole tones vary
between 9/8 and 10/9, as shown below for the key of C major.
Unfortunately, even a change to a closely related key such as from C to G or C to F
distorts these ratios.
In the diagram _ indicates that the ratio is not determined until :* a new semitone is introduced
If we set F♯ and Bb to the frequencies required for the G major and F major keys
respectively, it turns out that they are not suitable for many of the more remote keys
C major.
C _____ D ______ E ______ F ______ G ______ A ______ B ______C
9/8
10/9
16/15
9/8
10/9
9/8
16/15
G major
G ______ A ______ B ______ C ______ D ______ E ______ F ______ G
10/9
9/8
16/15
9/8
10/9
_
_
*
F major
F ______ G ______ A ______ B ______ C ______ D ______ E ______ F
9/8
10/9
_
*
_
9/8
10/9
16/15
34
More Problems with Modulation
The modulation problem is exacerbated when two or more keys require the same sharps
or flats – they may require different tunings for the same apparent note! Although we now
think of C♯ being the same as Db, F♯ the same as Gb, etc, this is not strictly true.
A comparison of the keys F, D, A, E and B (all major) illustrates the more general problem.
NOTE
C
C♯ / D
D
D♯ / E
E
F
F♯ / G
G
G♯ / A
A
A♯ / B
B
C
C
24
-27
-30
32
-36
-40
-45
48
F
24
-26.6
-30
32
-36
-40
42.6
-48
KEY
D
-25.31
27
-30.38
-33.75
36
-40.5
-45
--
A
-25
26.6
-30
-33.3
-37.5
40
-45
--
E
-25
-28.12
30
-33.75
-37.5
40
-45
--
B
-25.31
-28.31
30
-33.75
-37.5
-42
45
-35
Killing the Wolf
During Bach’s time, musicians experimented with changes in tuning, especially of
harpsichords, claviers etc, adjusting the frequency of certain notes to avoid the
wolf. This worked well in some closely related keys but made matters worse in
others more remote.
Bach was known to tweak the tuning of a note on his harpsichord, even during play!
Such devices were not appropriate for ‘fixed pitch’ instruments like oboes, flutes,
trumpets, organs.
Another experiment involved introducing extra strings, eg for pairs of notes like A
and B. This obviously became cumbersome – one keyboard experiment had 53
keys within the octave to cope with the 3, 4 or even 5 variants of each note.
The most widely adopted approach was Meantone. It retained the perfect fifth on
the keynote and also major thirds (C to E, F to A etc) but adjusted the notes in
between. There would be a different Meantone for each base key.
Meantone tuning allowed acceptable modulation to three sharp or flat major keys
and one minor each way. Other keys would still suffer the wolf – even worse than in
‘Just’ Temperament.
36
More about Meantone
For meantone centred on C (and again setting C = 24) the following
frequencies result.
C
D
E
F
G
A
B
C
24
26.83
30
32.2
36
40.25
45
48
The sharps / flats would be about halfway between the adjacent whole tones,
eg about
25.4 for C♯ / D
38.1 for A♯ / B etc.
In Meantone C, acceptable modulation to G, D or A (on the sharp side) or F, B
and E on the flat side. A and D minors were also satisfactory. More remote
keys (B, F♯, D etc) would become discordant.
The comparison between Meantone C and Just Temperament can be
illustrated thus, on a logarithmic scale.
Just Temperament (Diatonic)
C
D
E
F
G
A
B
C
|________|________|____|_________|________|__________|_____|
C
D
E
F
G
Meantone C
A
B
C
37
Equal Temperament
The principle of Equal Temperament, with all semitone ratios equal, was
certainly known to the Greeks and possibly earlier. Some writings of Aristoxenos
(c350 BC, a student of Aristotle) are preserved and demonstrate this.
We require twelve equal semitone ratios within the octave. With a 2:1 ratio in
frequency for the octave, this means the semitone must become 2 1/12th .Using
a computer or modern calculator, or by logarithms, this is easily evaluated as
1.059463……
Although the Greeks did not have these methods available, their skill with
fractions would enable them to solve the problem thus:We know that 16/15 is too large (raised to the 12th power it is 2.1694….. (about
2 1/6th). 17/16 is also too large (2.07 approx), but 18/17 is slightly too small
(1.9856 approx). By interpolation, we find 178/168 is close enough (2.001376)
and this reduces to 89/84, or 1.059524 in decimal notation.
Applying this ratio twelve times yields the ‘well tempered’ or Equal Temperament
chromatic scale.
For most people, the discrepancies from ‘Just Temperament’ (the Diatonic
scale) are not noticeable of objectionable. Although some purists prefer Just or
Meantone for early music played on period instruments, for most listeners they
sound a little strange.
38
The Equal Temperament Scale
Twelve equal semitone ratios in the octave.
Frequency ratio 2 1/12th = 1.0595
Close to 89/84 instead of 16/15 for the diatonic semitone.
All notes are slightly distorted except the octave.
None of the discrepancies offensive and most listeners unaware of them.
Permits free modulation among all keys without the wolf.
Sharp and adjacent flat semitones are merged –
C♯/D , D ♯ /E , F♯ /G , G ♯ /A , A♯ /B (enharmonics)
The whole tone becomes 1.1225 (449/400) instead of 1.125 or 1.111 (9/8 or 10/9)
For base note 24, the other notes become:C
C♯/D
D D♯/E E
F
F♯/G
24.00
24
25.43 26.94
27
G
G♯/A
A
A♯/B
B
C
28.54 30.24 32.04 33.94 35.96 38.10 40.36 42.76 45.31 48
compared with the diatonic
30
32
36
40
45
48
39
Comparisons of Just, Meantone and
Equal Temperaments (logarithmic scaling)
For Just and Meantone temperaments, ♯ and semitones vary within shaded bands.
For Equal Temperament, all semitone intervals are in the same ratio (1.0595)
40
Other Musical Scales
Numerous musical scales have been used at various times and by differen cultures.
Chinese and other Oriental music has a tradition of the Pentatonic scale,
represented by C D E G A C in Western notation. (Note that the diatonic semitones
(F, B) are missing).
C D
E G
A
C
Another Pentatonic scale with equal intervals has the three versions, based on
minor thirds:C Eb
F♯
A
C
C♯ E G Bb C♯
D
Ab
F
B
D
The chord based on these five notes is often used in Western music, resolving onto
a major or minor triad. The whole tone is hexatonic, in two versions:C
D
E F♯ G♯
A♯ C
C♯
D♯
F
G
A
B
C♯
41
Other Musical Scales (continued)
The chromatic scale reappears in 20th century music as the Twelve
Note scale. The difference is that instead of the semitone being used
for embellishment or modulation, all notes are treated as equally
important – in other words there are no major or minor keys as such.
Microtone scales use intervals of less than a semitone. Some Asian
music uses quarter-tones and a few composers have experimented
with one-eighth tones. They need to be played on unfretted strings,
wind instruments with a slide (trombones) or produced electronically.
Like many of the other variant scales, they do not appeal to Western
ears versed in the Diatonic or Equal Temperament tradition.
Finally, there is the Bagpipe scale. This scale, unique to traditional
Scottish music, has a ‘drone’ note played continuously, plus just over
an octave (G to A) of almost conventional notes, except that the C and
F are slightly sharp of true diatonic tuning.
42
‘Twinkle, Twinkle, Little Star’ in other musical scales
43
Thank you –
any
questions?
44