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SCIENCE, MATHEMATICS

AND MUSIC

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Poetry and MusicPoetry and music are historically related – chants, songs, incantationsDance and music similarly relatedEarly musical accompaniment for song and dance –

DrumsPipesLyreLute-like instrumentsHarp

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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 – Harmonypleasant sound

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Sound ProductionVibrators 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

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Musical Instruments – WindBlowingVoice – larynx and resonance

larynx, little resonance

Lips

trombone trumpet french horn saxophone

Reed

oboe panpipe basset horn reeds ocarina

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Wind instruments continued . . .

Pumping

concertina

organ

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Percussion and striking

triangle

tambourine

castanets

kettle drum

xylophone

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Strings – plucking . . . . . . . . . and bowing

lyre harp

(violon) cello

harpsichord guitarviolin

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Scan original

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Human audible range from about 16hz to 25/30,000 (upper end lost with age). i.e. about c-4 to c6 / c7

A27.5

A 440

C -3

33C -2

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- Hertz -

Concert grand only

C -1

132C 0

264C 1 (c)528

C2 (c)1056

C3 (c) 2112

C4 (c) 4224

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An Aside on Poetic RhythmLike 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 wearinessThe long day wanes, the slow moon climbs

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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)

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Hymn metresThe 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)

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Note LengthsThe 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 32 x (25) Longa 16 x (24) Breve 8 x (23)

Semibreve 4 x (22) Minim 2 x (21) Crotchet (20)

Quaver 2 x = (2-1) Semiquaver 4 x = (2-2)

Demisemiquaver 8 x = (2-3) Hemidemisemiquaver 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!

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Modifications to Note LengthsAddition 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.

= = .

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TempoThe 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 Beats per minute

Prestissimo 200-208Presto 168-200 eg = 176Allegro 120-168Moderato 108-120 eg = 110Andante 76-108Adagio 66-76Larghetto 60-66 eg = 60Largo 40-60

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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 – thusiamb / trochee = duple or quadruple timedactyl / 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.

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March

MinuetWaltz

Polka

Tango

.

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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 = 415Handel’s tuning fork A = 422.5Mozart’s piano about A = 421-422Military pitch A = 452.5 – means military band could not play with orchestra, piano or choirPhilosophical/Scientific pitch C = 256 because repeatedly divisible by 2

A = 426.66Modern 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

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The Diatonic Scale (Greek = through the tones)

Discovery usually attributed to Pythagoras – experimenting with stringsEarlier 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

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Pythagoras’s Experiment

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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 . . . .

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Natural and ‘Stopped’ HarmonicsBase 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 length4/3 times base note, with harmonic 4 times base note

2nd natural harmonic3 times frequency of base note 1/3 G’ = 72

C” = 96 F = 32

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More about HarmonicsHigher 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 Note1st 24 24 C (doh)2nd 48 ÷ 2 = 24 C )1st octave)3rd 72 ÷ 2 = 36 G (soh)4th 96 ÷ 2 = 24 [sic] 48? C (2nd octave)5th 120 ÷ 4 = 30 E (mi)6th 144 ÷ 4 = 36 G (soh again)7th 168 ÷ 4 = 42 A♯ Bb ??8th 192 ÷ 8 = 24 C (3rd octave)9th 216 ÷ 8 = 27 D (re)10th 240 ÷ 8 = 30 E again11th 264 ÷ 8 = 33 below F♯ ??12th 288 ÷ 8 = 36 G again13th 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 = G27th gives 40.5, discordant A in C major, perfect soh in D major (= 1½ x 27)

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Harmonics of Diatonic Scale24 27 30 32 36 40 45 48 60 72 96 192 Relative

frequencydoh re mi fah soh lah te doh Tonic sol-fa

C D E F G A B C’ D’ E’ F’ G’ A’ B’ C’’ C’’’ etc Principal note

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Mode, Key, Modulation and TemperamentModes 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 keyboardFour Ambrosian modes start on D, E, F, GGregorian additions on C, A, BGlarus 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 worksBest known example is ‘Greensleeves’ (Hypodorian) though it has a final ‘te-doh’ cadence, so it ends in A minor

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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!

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Comparison of Keys (major, minor) and Modes

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‘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 64The 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.

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Perfect Fifths – The Tuner’s DilemmaThe 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.

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Tuning by Octaves, fifths and fourths(starting Co = middle C)

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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

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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 majorG ______ A ______ B ______ C ______ D ______ E ______ F ______ G 10/9 9/8 16/15 9/8 10/9 _ * _

F majorF ______ G ______ A ______ B ______ C ______ D ______ E ______ F

9/8 10/9 _ * _ 9/8 10/9 16/15

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More Problems with ModulationThe 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 KEY

C F D A E BC 24 24 -- -- -- --C♯ / D -- -- 25.31 25 25 25.31D 27 26.6 27 26.6 -- --D♯ / E -- -- -- -- 28.12 28.31E 30 30 30.38 30 30 30F 32 32 -- -- -- --F♯ / G -- -- 33.75 33.3 33.75 33.75G 36 36 36 -- -- --G♯ / A -- -- -- 37.5 37.5 37.5A 40 40 40.5 40 40 --A♯ / B -- 42.6 -- -- -- 42B 45 -- 45 45 45 45C 48 48 -- -- -- --

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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.

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More about MeantoneFor meantone centred on C (and again setting C = 24) the following frequencies result.

C D E F G A B C24 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 A B CMeantone C

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Equal TemperamentThe 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.

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The Equal Temperament ScaleTwelve 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 G G♯/A A A♯/B B C24.00 25.43 26.94 28.54 30.24 32.04 33.94 35.96 38.10 40.36 42.76 45.31 48

compared with the diatonic 24 27 30 32 36 40 45 48

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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)

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Other Musical ScalesNumerous 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 F Ab 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♯

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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.

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‘Twinkle, Twinkle, Little Star’ in other musical scales

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Thank you –

any questions?