Musical Techniques
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Dominque Paret is Engineer ESIEE / DEA of physics and was responsible for the technical support of an international leader of electronic components for many years. Professor in numerous engineering schools in France and abroad, he runs a company of consultants/technical experts. He is also a seasoned jazz organist.
Serge Sibony is Engineer ESTP, head of products systems construction, research and development director and Professor in several schools of engineers. A jazz pianist and the author of a series of works on musical composition, he is also a creator and independent developer of sharewares related to music.
Content
Preface xiii
Introduction xv
Part 1. Laying the Foundations 1
Introduction to Part 1 3
Chapter 1. Sounds, Creation and Generation of Notes 5
1.1. Physical and physiological notions of a sound 5
1.1.1. Auditory apparatus 5
1.1.2. Physical concepts of a sound 7
1.1.3. Further information on acoustics and acoustic physiology 8
1.1.4. Idea of minimum audible gap/interval between two frequencies 16
1.1.5. Why have we told this whole story, then? 22
Chapter 2. Generation of Notes 23
2.1. Concept of octave 23
2.1.1. Choice of inner division of an octave 24
2.2. Modes of generation/creation/construction of notes 25
2.3. Physical/natural generation of notes 26
2.3.1. Harmonics 26
2.3.2. Fractional harmonics 26
2.3.3. Initial conclusions 29
2.3.4. Order of appearance and initial naming of the notes 29
2.3.5. A few important additional remarks 32
2.4. Generation of perfect fifth notes 33
2.4.1. Generation with ascending fifths 33
2.4.2. Generation with descending fifths 37
2.4.3. Conclusions on fifth-based constructions of notes 39
2.5. Important remarks on "physical"/"fifths" generation 40
2.6. Generation of tempered notes 40
2.6.1. Notion of the ear's logarithmic sensitivity 41
2.6.2. Examples of electronic generation of tempered notes 43
2.6.3. Relative gaps between tempered and electronic notes 43
2.7. In summary and in conclusion on generation of notes 46
2.8. Comparison of gaps between all the notes thus created 49
2.8.1. Note on pitch-perfect hearing... or is it? 53
Chapter 3. Recreation: Frequencies, Sounds and Timbres 55
3.1. Differences between a pure frequency and the timbre of an instrument 55
3.2. Timbre of an instrument, harmonics and harmony 58
3.2.1. Relations between timbres and spectra 60
3.3. Recomposition of a signal from sine waves 63
3.3.1. Subtractive synthesis 63
3.3.2. Additive synthesis 63
3.3.3. Recreation: harmonic drawbars 64
Chapter 4. Intervals 69
4.1. Gap/space/distance/interval between two notes 69
4.2. Measuring the intervals 70
4.2.1. The savart 70
4.2.2. The cent 71
4.3. Intervals between notes 73
4.3.1. Second interval: major tone and minor tone 74
4.3.2. Major third and minor third interval 75
4.4. Overview of the main intervals encountered 75
4.5. Quality of an interval 76
4.5.1. Instrumentation 76
4.5.2. Tempo 76
4.5.3. Dynamics of amplitudes 76
4.5.4. Register 76
4.6. Reversal of an interval 77
4.7. Commas...ss 77
4.7.1. Pythagorean comma 78
4.7.2. Syntonic comma 79
4.7.3. A few remarks about commas 80
4.7.4. Enharmonic comma 80
4.7.5. Other theoretical commas and a few additional elements 80
4.7.6. Final remarks 82
4.7.7. In summary, commas and C° 83
Chapter 5. Harshness, Consonance and Dissonance 85
5.1. Consonance and dissonance 85
5.1.1. Consonant interval 85
5.1.2. Dissonant interval 86
5.2. Harshness of intervals 86
5.3. Consonance and dissonance, tension and resolution of an interval 87
5.3.1. Consonance of an interval 87
5.3.2. Dissonance of an interval 89
5.3.3. Savarts, ¿F, consonance, pleasing values or beating of frequencies 90
Part 2. Scales and Modes 93
Introduction to Part 2 95
Chapter 6. Scales 97
6.1. Introduction to the construction of scales 97
6.2. Natural or physical scale 98
6.2.1. Harmonics 98
6.3. Pythagorean or physiological diatonic. scale 100
6.3.1. Principle 100
6.3.2. The why and wherefore of the 7-note scale 101
6.3.3. Names of the notes in the Pythagorean scale 104
6.3.4. The series "tone-tone-semi/ tone-tone-tone-tone-semi/tone"? 105
6.3.5. A few comments 106
6.3.6. Uses of the Pythagorean scale, and cases where it cannot be used 107
6.4. Major diatonic scale 108
6.4.1. Intervals present in a major scale 108
6.5. The other major scales 109
6.6. Scales and chromatic scales 109
6.6.1. Chromatic scale 110
6.6.2. Chromatic scales 110
6.7. Tempered scale 114
6.7.1. Principle of the tempered scale 114
6.7.2. Comparisons between physical, Pythagorean and tempered scales 115
6.8. Other scales 117
6.9. Pentatonic scale 117
6.9.1. A little history, which will prove important later on 117
6.9.2. Theory 118
6.9.3. Reality 120
6.9.4. Relations between major and minor pentatonic scales 123
6.9.5. Pentatonic scale and system 124
6.10. "Blues" scale 125
6.11. Altered scale and jazz scale 126
6.12 "Tone-tone" (whole-tone) scale 127
6.13. Diminished scale or "semitone/tone" scale 128
6.14. In summary 128
6.15. Technical problems of scales 129
6.15.1. Scale and transposition 130
6.15.2. Alterations 132
Chapter 7. Scales, Degrees and Modes 135
7.1. Scales and degrees 135
7.2. Degree of a note in the scale 136
7.3. Interesting functions/roles of a few degrees of the scale 136
7.4. Modes 137
7.4.1. The numerous modes of a major scale 138
7.4.2. The original minor modes and their derivatives 142
7.4.3. A few normal modes 143
Part 3. Introduction to the Concept of Harmony: Chords 145
Introduction to Part 3 147
Chapter 8. Harmony 149
8.1. Relations between frequencies 149
8.2. How are we to define the concept of harmony? 150
Chapter 9. Chords 151
9.1. The different notations 151
9.1.1. Convention of notations for notes 151
9.2. Chords 152
9.3. Diatonic chords 153
9.3.1. Diatonic chords with 3 notes: "triads" 154
9.3.2. 4-note diatonic chords known as "seventh" chords" 155
9.4. "Fourth-based" chords 157
9.4.1. Convention of notations of the chords 157
9.5. Chord notations 158
9.5.1. In the major scale 159
9.5.2. In minor scales 161
9.5.3. Scales and chords 166
9.5.4. List of common chords 169
9.5.5. Table of frequently used chords 171
9.6. What do these chords sound like? 173
9.6.1. In statics 173
9.6.2. In dynamics 173
9.7. Temporal relations between chords 174
9.8. Melody line 175
9.9. Peculiarities and characteristics of the content of the chord 175
9.10. Relations between melodies and chords 175
9.11. The product of the extremes is equal to the product of the means 176
Part 4. Harmonic Progressions 179
Introduction to Part 4 181
Chapter 10. Some Harmonic Rules 183
10.1. Definition of a chord and the idea of the color of a chord 183
10.1.1. Notations used 183
10.1.2. Equivalent or harmonious chords 184
10.2. A few harmonic rules 184
10.2.1. The eight fundamental syntactic rules 185
10.2.2. Rules of assembly 186
10.2.3. Next steps 187
10.2.4. Descending chromatism rule 188
10.2.5. Justifications of the eight harmonic rules by descending chromatism 190
10.3. Conclusions on harmonic rules 193
Chapter 11. Examples of Harmonic Progressions 195
11.1. Harmonic progressions by descending chromatism 195
11.1.1. Example 1 195
11.1.2. Example 2 196
11.1.3. Example 3 197
11.2. Codes employed for writing progressions 198
11.2.1. Key changes in a progression 199
11.2.2. Detailed example of decoding of progressions 202
11.3. Hundreds, thousands of substitution progressions... 204
11.3.1. Major scale, the best of 204
11.3.2. List of harmonious progressions 206
11.4. Chromatism in "standards" 213
11.5. Families of descending chromatisms 214
11.5.1. Family: "1 chromatism at a time" 215
11.5.2. Family: "up to two descending chromatisms at once" 217
11.5.3. Family: "up to 3 descending chromatisms at once" 220
11.5.4. Family: "up to 4 ascending and descending chromatisms at once" 220
11.5.5. Conclusions 225
Chapter 12. Examples of Harmonizations and Compositions 227
12.1. General points 227
12.2. Questions of keys 228
12.3. Example of reharmonization 228
12.3.1. Blue Moon (by Lorenz Hart and Richard Rodgers) 229
12.3.2. Summertime (by G. Gershwin) 239
12.3.3. Sweet Georgia Brown (by Bernie, Pinkard and Casey) 243
12.4. Example of harmonization 247
12.4.1. Madagascar (by Serge Sibony) 247
12.5. Conclusion 252
Conclusion 253
Appendix 255
Glossary 273
Bibliography 279
Index 281
1
Sounds, Creation and Generation of Notes
To begin this book, which has the ambitious aim of serving as a passport to harmony in the musical domain, it is perfectly normal to offer a recap of a few elementary aspects, which are absolutely necessary for the workings of our auditory apparatus (ear + brain + education + civilization + etc.) which will, ultimately, be the adjudicator of all this work. Thus. 1; 2; 1, 2, 3, 4!
1.1. Physical and physiological notions of a sound
1.1.1. Auditory apparatus
In acoustic science, sound is a vibration propagating through gases, liquids and solids. For humans, generally, it is the vibration of a mass of air, driven by a tiny variation in air pressure, with varying rapidity, which, via the outer ear, vibrates the membrane of the hearer's eardrums and stimulates nerve endings situated in the inner ear (see Figure 1.1).
1.1.1.1. Outer ear
The auditory canal in the "outer" ear is in the shame of an acoustic horn, decreasing in diameter as we approach the bottom - i.e. the eardrum.
1.1.1.2. Middle ear
The middle ear contains the eardrum and three tiny bones, respectively called the hammer, anvil and the stirrup, which, together, make up the "ossicular chain". The hammer and the anvil form a fairly inflexible joint called the "incudomalleolar joint". The vibrations of masses of air in the auditory canal cause the eardrum to vibrate. These mechanical vibrations are then transmitted along the ossicular chain mentioned above, and then into the inner ear through the oval window.
Figure 1.1. Diagram of the human auditory apparatus (source: Wikipedia). For a color version of this figure, please see www.iste.co.uk/paret/musical.zip
The mode of simplified propagation of vibrations in the inner ear is essentially as follows: the lines of the concentric zones of iso-amplitude of certain frequencies are parallel to the "handle" (shaft) of the hammer, with, for the membrane of the eardrum, zones of vibration with greater amplitude than the handle.
As the middle ear forms a cavity, overly high external pressure may perforate the eardrum. In order to ensure and re-establish a pressure balance on both sides of the eardrum (inner/outer), the middle ear is connected to the outside world (the nasal cavities) via the Eustachian tubes.
1.1.1.3. Inner ear
The inner ear contains not only the organ of hearing, but also the vestibule and the semicircular ducts, the organ of balance (not shown in the figure), responsible for perception of the head's angular position and its acceleration. Microscopic motions of the stirrup are transmitted to the "cochlea" via the oval window and the vestibule.
The cochlea is a hollow organ filled with a fluid called endolymph. It is lined with sensory hair cells (having microscopic hairs - cilia - which serve as sensors), which cannot regenerate once lost. They have tuft-like protruding structures: stereocilia. These cells are arranged all along a membrane (the basilar membrane), which divides the cochlea into two chambers. Together, the hair cells and the membranes to which they connect make up the "organ of Corti".
The basilar membrane and the hair cells are set in motion by the vibrations transmitted through the middle ear. Along the cochlea, each cell has a preferential frequency to which it responds, so that on receiving the information, the brain can differentiate the frequencies (the pitches) making up the different sounds. The hair cells nearest the base of the cochlea (oval window, nearest to the middle ear) tend to respond to high-pitched sounds, and those situated at its apex (final coil of the cochlea), on the other hand, respond to low-frequency sounds.
It is the hair cells which carry out mechanical- (i.e. pressure-based) electrical transduction of the original signal: they transform the motions of their cilia into nerve signals, sent via the auditory nerve. It is this signal which is interpreted by the brain as a sound whose tone height (pitch) corresponds to the group of cells stimulated.
Thus, we have briefly recapped the set (mechano-electric + brain) of our likes and dislikes, which it is important to please, and therefore stroke the hairs as much as possible in the direction of the grain!
Following this brief interlude, let us now return to simple physics.
1.1.2. Physical concepts of a sound
A sound is represented by a physical signal - a variation in pressure in the ear - whose characteristics are primarily represented by three parameters:
- - the set of instantaneous frequencies making up the acoustic signal - in physical terms, the spectrum or spectral content (the frequency values are expressed in Hertz, representing a certain number of wave variations per second);
- - the acoustic level, expressed in the form of pressure (Pascal), power (acoustic Watts or else transposed into decibels - dB) or in acoustic intensity (W/m²);
- - the respective evolutions of the amplitudes and spectra of the sound as a function of the time (temporal evolution) between the sound's appearance and its complete cessation (the conventional English terms are attack, decay, sustain and release time) - see Figure 1.2.
Figure 1.2. Attack R1, decay R2 and R3, sustainn L3 and releaase time R4. For a color version of this figure, please see www.iste.co.uk/pparet/musical.zzip
The simplest and purest sound corresponds to a single-frequency sine wave, at constant amplitude, and sustained (see Figure 1.3). This is all very well, but hearers will very soon get bored of it!
Figure 1.3. Single-frequency, constant-amplitude, sustained sine wave
1.1.3. Further information on acoustics and acoustic physiology
In order to help readers better comprehend and prevent unfortunate misunderstandings, here are few useful definitions.
1.1.3.1. Energy (acoustic)
By its very principle, a sound source diffuses acoustic energy E. Like any respectable form of energy, this is measured in Joules (J). The acoustic energy is linked to the movement of the air molecules propagating in the acoustic wave.
1.1.3.2. Power (acoustic)
If an acoustic source emits a sound with energy E (in Joules) over a time period ?t (in seconds), the acoustic power Pow of that source is the amount of energy emitted over that time period, and is defined as:
Power = E/?t
The (acoustic) power is measured in Watts (W).
Table 1.1. Acoustic power of several sources
1.1.3.3. Pressure (acoustic)
The pressure Pres results from a force F (in Newton or kg per m/s²), applied to a surface S (in square meters, m²). Its value, therefore, is defined as:
In acoustics, we distinguish between two particular types of pressure values:
- atmospheric pressure "P0" (also known as static pressure), which is the pressure exerted by the atmosphere (all molecules of air) on the Earth, on all humans and, of course, on their eardrums. Its value varies somewhat with changing weather conditions, but we can state that, on average, it is:
- acoustic pressure "p" (also known as dynamic acoustic pressure). As a sound wave propagates, the air molecules which are set in motion cause slight local variations in the atmospheric pressure. This dynamic variation of pressure is what we called acoustic pressure "p".
This acoustic pressure p exerts a new force on the eardrum. This causes it to vibrate, so it transmits the waves to the brain via the mechanisms of the middle- and inner ear, as described above.
Thus, at a given point in space, the total pressure is:
NOTE.- The atmospheric pressure P0 is an ambient pressure, known as the "absolute" pressure, and therefore always positive, whilst the acoustic pressure p is a fluctuation around atmospheric pressure, and hence can either be positive (overpressure) or negative (pressure deficit).
1.1.3.4. Intensity (acoustic)
Acoustic intensity, or sound intensity, corresponds to a quantity of acoustic energy E (in Joules) which, over a period ?t (in seconds), traverses a surface area (be it real or virtual) S (in m2). Thus, it is defined as:
If we suppose that the sound source radiates uniformly in all directions in space (i.e. it is a source said to be isotropic, or homogeneous), it will emit waves with spherical wavefronts. If the radius of the sphere is r, its surface area will be equal to S = 4pr² and the acoustic intensity received over the spherical whole of a wavefront would be equal to:
Thus, in the case of an isotropic source with power Pow, the acoustic (sound) intensity decreases in inverse proportion to the square of the distance r from the source. In addition, the sound reception of the human ear to the amplitude of a sound is not directly proportional to its amplitude, but is proportional to its logarithm. Thus:
- - doubling the power of the sound source is equivalent to increasing the sound level by 3 dB;
- - quadrupling that...
