On Musiquantics
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© bei den Autoren Herausgeber: Bereich Musikinformatik, Musikwissenschaftliches Institut Johannes Gutenberg-Universität Mainz Postfach 3980 55099 Mainz ISSN 0941-0309 Clarence Barlow On Musiquantics Part I: Texts Translators 2002 from the original German Von der Musiquantenlehre: Deborah Richards, Jay Schwartz with the kind support of the Royal Conservatoire The Hague Thereafter translator and editor Clarence Barlow 2 Preface to Part I This work grew out of materials that I assembled for my class in computer music at Cologne Music University from the inception of the class in 1984 until its unfortunate termination in 2005. I named the course Zur Musiquantik (an artificial term, implying ‘on music and quantity’), whence the English title of this book stems. This material was also in my syllabus at the Royal Conservatoire in The Hague under the same English name from 1990-2006, as it has been since then in my position at the University of California Santa Barbara. Driven by the fascination of the connections between music and mathematics, acoustics, phonetics and computer science, and further by my own preoccupation with the quantification of harmony and metre, over the years I gradually put together 32 chapters with numerous illustrations. On Musiquantics has been consciously written in a very compact form, each chapter on two pages, more as a concentrated and comprehensive teaching accompaniment than as an autonomous textbook. It also fulfils a role as a reference book, especially if its content has already been assimilated. It is my hope that my former students, who applied so much patience to this material, some of them repeatedly and frequently attending the course, find in it everything they have learned from me, and that for those who have themselves gone into teaching, the book proves to be useful for their own educational work. I also hope that this book, completed after twenty-four years, helps my current and future students to learn the material more easily than it was possible for their predecessors. I wish to thank all those friends and colleagues who came into contact over the years with the slowly growing book for their valuable suggestions and help. My publisher and dear friend, the now late Prof. Johannes Fritsch (1941-2010), waited with exemplary patience for twenty years for the first release in 2008 of the German version of On Musiquantics. Finally I am most grateful to Prof. Frans de Ruiter, former director of the Royal Conservatoire The Hague, for his support in having the German version translated in 2002 into English (here in its British form) for the use of my students in The Hague. The translation formed a solid base for my continued work on the book, with innumerable corrections and additions and the complete rewriting of Chapters 16-18. 3 On Musiquantics Part I Contents Chapter Page 1 Mathematics 1 : Number Systems 2 2 Mathematics 2 : Two-dimensional Geometric Curves 4 3 Harmony 1 : Intervals of Pitch 6 4 Mathematics 3 : The Linear and the Logarithmic 8 5 Mathematics 4 : Converting Linear↔Logarithmic 10 6 Acoustics 1 : Fundamental Units of Pitch and Loudness 12 7 Harmony 2 : The Frequency Ratio of a Pitch Interval 14 8 Acoustics 2 : Cents and Decibels Compared 16 9 Acoustics 3 : Subjective Loudness and Loudness Level 18 10 Harmony 3 : Classical Just Intonation 20 11 Mathematics 5 : Trigonometry and Analytical Geometry 22 12 Acoustics 4 : Sound Waves and Spectrum 24 13 Informatics 1 : Hardware and Software 26 14 Informatics 2 : Digital and Analogue 28 15 Informatics 3 : Computer Programming 30 16 Informatics 4 : The Programming Language C 32 17 Informatics 5 : General Functions in C 34 18 Informatics 6 : MIDI: Musical Instruments Digital Interface 36 19 Harmony 4 : A Quantitative Approach to Harmonicity 38 20 Harmony 5 : Pitch Rationalisation: Theory 40 21 Harmony 6 : Pitch Rationalisation: Practical Examples 42 22 Metre 1 : A Quantitative Approach to Metre 44 23 Metre 2 : The Measurement of Metric Coherence 46 24 Mathematics 6 : Stochastic Analysis and Synthesis 48 25 Acoustics 5 : Spectral Analysis and Synthesis 50 26 Acoustics 6 : Frequency Modulation and Phase Distortion 52 27 Acoustics 7 : Complex Tones and Noise 54 28 Acoustics 8 : The Bark Scale of Subjective Pitch 56 29 Acoustics 9 : Loudness Summation and Masking 58 30 Acoustics 10 : Sensory Consonance and Dissonance 60 31 Phonetics 1 : Physiological Phonetics: Speech Generation 62 32 Phonetics 2 : Acoustic Phonetics: Formants 64 Part II starts after Page 65 of Part I 4 01 Mathematics 1 – Number Systems Looking at a typical digital clock display, one sees that all digits, appearing in the same place, consist of various combinations of the same seven dashes. By individually switching these dashes on or off, the desired form is achieved. Is it possible to achieve other symbols under these conditions? If yes, how many, and what symbols are they? The answer to these questions can be concluded from Γ01: one sees 128 various symbols, some of which are relatively familiar (viz. 73 shaded and wholly connected ones, reduceable to 28 basic mirrorable forms: ÆÇÈÉÊËÌÍÎÏÐÑÒÓÔÕÖ×ØÙÚÛÜÝÞßàá; in addition, all of them are serpentine, with the exception of the last 7). The (im-)possibility of additional symbols using these seven dashes can be tested as follows: The seven dashes are first brought into a fixed order of observation (e.g. |¥|¦|§|¨|©|ª|«|), so that they can be examined in turn and in the same manner with each combination to be tested. The digits 0 and 1 can be assigned to represent the conditions ‘on’ and ‘off’, respectively; the state of all dashes switched off can be written 0000000 – switching on the last dash yields the representation 0000001. If we list out all such combinations of 0 and 1 systematically (e.g. 0000000, 0000001, 0000010, 0000011, 0000100, 0000101, ... 1111001, 1111101, 1111111), we see that there are exactly 128 of them. If instead of the seven dashes only four were to be examined, our enumeration would appear as follows: 0000, 0001, 0010, 0011, 0100, 0101, 0110, 0111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111. The arrangement of the digits 0 and 1 is called a binary number representation (from the Latin bini meaning ‘in pairs’), because it is based solely on two digits, 0 and 1 – the more common number system employing the ten digits 0 to 9 is called a decimal system (from the Latin decem = ‘ten’). The decimal number 5741 equals the summation 5000+700+40+1 or 5ž103 + 7ž102 + 4ž101 + 1ž100 (n0 = 1 for any number n); the binary number ijklmn correspondingly means iž25 + jž24 + kž23 + lž22 + mž21 + nž20 or 32i+16j+8k+4l+2m+n, bearing in mind that the digits i to n have the value 0 or 1 only. 101101 (binary) is therefore equal to 32+0+8+4+0+1 = 45 (decimal). Not only are the biological properties of the human hands with their ten fingers decisive for the global decimal system; in the numerals of various cultures of the world, the origins of these symbols can be recognised as the drawings of fingers and hands, as for example in the so-called ‘Arabic’ numerals common in the Western world. 2 1 shows one finger pointing upwards, 2 and 3 show two and three fingers pointing left. According to Victor Goldschmidt (1932), 5 depicts an open hand held downward and 0 a clenched fist representing ten – it was not until late in history that the role of the zero was adapted here (note also the dial of older telephones, where 0 triggers ten impulses). Furthermore 4 signifies, among other things, five minus one (‘Ы’), 6 depicts a hand with an upright index finger as 5+1 (‘Ь’), and 9 shows ten minus one (‘Э’). Finally we see in 7 and 8 – similar to 2 and 3 – again two and three fingers, no doubt of the other added hand. The decimal and binary number systems differ in their bases, 10 for the former and 2 for the latter. Comparable to the 10 decimal digits (0 to 9) there are only 2 binary digits (0 and 1). In two-digit numbers in a number system, each possible first digit can be succeeded by every possible second digit; hence there are 102=100 two-digit decimal numbers and 22=4 two-digit binary numbers. It can thus be seen that the quantity of all n-digit numbers amounts to Bn, where B is the base of the system. The largest n-digit n B-based number amounts to B _1, e.g. the largest 7-digit binary number 1111111b = 7 (64+32+16+8+4+2+1)d = 127d = 2 _1. (The subscripts b and d mean ‘binary’ and ‘decimal’ respectively.) The smallest non-negative number of every base is 0; the highest 7-digit binary 7 number is 1111111b or 127d. There are 2 =128 distinct 7-digit binary numbers; this indicates that every number between 0b and 1111111b (i.e. between 0d and 127d) corresponds to an individual binary number and to one of the 128 depicted combinations of the seven dashes. Bases other than 2 and 10 are not only possible, they are also commonly used (especially in data processing) – the octal system (Latin octo = ‘eight’) is based on 8, the more widespread hexadecimal system (hex from the Greek for ‘six’) on 16: the values 10d to 15d are hereby represented by the symbols À Á Â Ã Ä Å, a convention even less creative than the proposal of the American Duodecimal Society in the 1940s that 10d and 11d be written as ‘X’ and ‘E’ (spoken dek and el). New digit symbols would have been more exciting, as e.g.