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Holder's Comma THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 1/104 Summary 1 Introduction 3 2 Some physics: harmonics, partials and inharmonicity 4 3 Pythagoreas and the classical musical notation 10 4 Zarlino: coming closer to Physics 21 5 The genesis of Harmony: Musica Recta, Musica Ficta 31 6 Time-frequency diagrams and the analysis of music 36 7 The organ and the synthesis of sound colour 43 8 Stringed keyboards: from the harpsichord to the pianoforte 52 THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 2/104 1 Introduction Is there any serious tonal theory behind music? or: why are there 12 notes on each octave on a piano? Many interval systems in the world 7 note gamut (Occidental system) in reality 12 notes are used in modern music but many more have a name! European Medieval and Renaissance music use more refined scales. Middle-Eastern musical systems: 24 notes (“quarter tones”) and many others... THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 3/104 2 Some physics: harmonics, partials and inharmonicity THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 4/104 2.1 What is an octave? Sound ‘modulo...’ a frequency factor 2. Our auditory cortex has got a built-in “logarithmic modulo 2” is it reasonably accurate? General philosophy: just like with vision, the audio information that comes to the conscious level: • is not the sound, • is not the frequencies, • it is actually an attempt at identifying the physical source of the sound. Similarity with vision: Human vision is the process of reconstructing a physical object from its image(s). Human audition is the process of reconstructing an object from the sound(s) it produces. Biological point of view... THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 5/104 2.2 Musical objects and the lattice of harmonics Here, we assume a string is perfectly elastic. Metal strings are relatively recent... • Gut • Copper • Brass • Bronze • Iron • Steel • Nylon • PVF (PolyVinylFluor) • Self overspinning (gut, PVF) • Classical overspinning: • metal (aluminium, brass, silver) over gut or PVF (plucked and bowed instruments) • metal over metal: annealed copper over iron or steel (pianos) • close or open THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 6/104 Fundamental frequency f , harmonic # n has frequency nf . Decomposition into a product of prime numbers: n = 2u × 3x × 5y × 7z × 11t... Striking point + internal damping → reduce higher harmonics Hearing modulo 2 → eliminate 2u The meaningful harmonics are given by n = 3x × 5y × 7z × 11t... Internal damping, choice of striking points and initial conditions (e.g. piano hammer curvature) reduce higher harmonics. Algebraists call this a lattice structure. [Electronicians also work on lattice structures: Boolean algebras are defined as ’distributive complemented lattices’. Karnaugh diagrams are a consequence of this lattice structure] The useful harmonics are given by n = 3x × 5y × 7z and even, in most stringed instruments, by n = 3x THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 7/104 2.3 String vibrating modes tension T = π ρ d2 F2 L2 1 T f requency F = d L π ρ For a perfectly flexible string, partial #n has frequency: n T F = n F = n d L π ρ thus each string produces a series of partials with relative frequencies 1, 2, 3, 4, 5, 6, 7, 8... ‘harmonics’ but in reality, due to string stiffness, the actual frequency of partial #n is higher than the harmonic: 1/2 ( 2) f n = n F 1 + B n using the basic inharmonicity coefficient π2 E S K2 B = T L2 E is the Young’s modulus, only depending on the string’s material. = d For a round section string, K 4 (gyration radius) and π2 E d2 B = 64 ρ For example if B = 0.005 the partials will be 1; 2 (1 + 4B)1/2 ≈ 2 × 1.01; 3 (1 + 9B)1/2 ≈ 3 × 1.022; 4 (1 + 16B)1/2 ≈ 4 × 1.039, etc. THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 8/104 Reference: Fletcher (Neville H.), Thomas D. Rossing, The Physics of Musical Instruments, 2nd edition, Springer1998-99. THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 9/104 3 Pythagoras and the classical musical notation THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 10/104 3.1 Pythagoras and the harmonic 3 We are working in frequencies ‘modulo a factor 2’ Let us choose a frequency FF (say = 1) and work out its harmonics 3x As we work modulo a factor 2, let us see how the numbers 3x × 2y are spread on the scale of frequencies. CD E F G A B C The sequence of the first 7 Pythagorean harmonics: FCGDAEB (natural order), or ABCDEFG (frequency order): tonality of C. This is one of the simplest possible musical systems. It is possible to make music using 7 tones. Their order of appearance is F C G D A E B. Actually they are named after the order of their frequencies: A B C D E F G. The tradition in the Western musical systems is to call C (the second tone of the sequence) the Tonic. F the Subdominant and G the Dominant. If we take the tonic C as a point of departure, the interval widths are: large, large, small, large, large, large, small. Now it is of course possible to use another note than F as a point of departure of the sequence FCGDAEB. For example lets take C. The new sequence is: CGDAEBX where X is a new frequency. We have to give a name to X; as the name F is no longer used in this new system, we recycle the name F and write the sign # to avoid any confusion with the old F. This is how key signatures are made – and why there is one # in G major, two ## in D major, etc. X = F# CD E F G A B C The new sequence using C as a point of departure (tonality of G) THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 11/104 Reb Mib Solb Lab Sib Reb Do Do# Re Re# Mi Fa Fa# Sol Sol# La La# Si Do Do# Enriching the scale: more tones upwards and backwards This is the basis of the standard musical notation and the infinite Pythagorean scale: ... Ebb Bbb Fb Cb Gb Db Ab Eb Bb F C G D A E B F# C# G# D# A# E# B# Fx Cx Gx ... tone of C: –––^––––––––––––––– no alteration tone of G: –––^––––––––––––––– one # tone of D: –––^––––––––––––––– ## tone of A: –––^––––––––––––––– ### tone of F: –––^––––––––––––––– one b tone of Bb: –––^––––––––––––––– bb tone of Eb: –––^––––––––––––––– bbb etc. hence the classical ‘key signatures’ which tell the tonality: here are the most frequent ones: THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 12/104 The most frequent key signatures and the corresponding tonalities. THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 13/104 Tuning with pure "fifths" (harmonic no. 3) shows the Pythagorean comma: 312 / 2something THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 14/104 ...and some flats. This may be pushed to infinity, an endless story... THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 15/104 THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 16/104 The infinite system of music as it is written: THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 17/104 The Pythagorean scale is infinite. This is because the equation: − 3x 2 y = 1 has no solution with positive integers. However we should be so glad to be able to eventually fall back into our shoes rather than having an infinity of keys at each octave – in other terms, are there approximate solutions to this equation? The first solution is x = 7, y = 11. 2187 3x 2y = ≈ 1.068 2048 which is a very crude approximation. It would result in dividing the octave into 7 equal intervals. The second solution is x = 12, y = 19 . 531441 3x 2y = = 1.0136 524288 which is the basis for the ‘equal temperament’, a very old system (China, ca. 2000 B.C.?) that has re- appeared in Europe during the late 18th Century, as a simplification of the more refined previous systems. The third solution is x = 43, y = 68. 3.282569674 E20 3x 2y = = 1.1122 2.951479052 E20 and the fourth solution is x = 53, y = 84. 1.938324567 E25 3x 2y = = 1.0021 1.934281311 E25 The latter is remarkably accurate. It results into the octave being divided into 53 equal micro-intervals, called ‘Holder’s Comma’. The difference with the exact Pythagorean frequencies are smaller than anything audible. THE FOUNDATIONS OF MUSICAL SYSTEMS Jean Louchet 18th December 2012 Hogeschool Gent page 18/104 Reb Eb Gb Ab Bb Db C C# D D# E F F# G G# A A# B C C# The 53 interval system: a good approximation - one Pythagorean comma is not far from 1/53 octave (Holder comma). If we express the Pythagorean system into Holder’s commas, we have: C - D 9 commas D - E 9 commas E - F 4 commas F - G 9 commas G - A 9 commas A - B 9 commas B - C 4 commas The interval between a note and the homologous sharp or flat note (e.g.
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