A Music and Mathematics
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A Music and mathematics This appendix covers a topic that is underestimated in its significance by many mathematicians, who do not realize that essential aspects of music are based on applied mathematics. This short treatise is meant to show the fundamentals. It may even pique interest in the laws that govern the sen- suous beauty of music. Additional literature is provided for readers who are particularly interested in this topic. The mathematical proportions of a sounding string were first defined by Py- thagoras, who made his discoveries by subdividing tense strings. This chapter describes changes in scales and tonal systems over the course of musical his- tory. The author of this appendix is Reinhard Amon, a musician and the author of a lexicon on major-minor harmonic systems. The chapter will conclude with exercises in “musical calculation” that show the proximity between music and mathematics. © Springer International Publishing AG 2017 G. Glaeser, Math Tools, https://doi.org/10.1007/978-3-319-66960-1 502 Chapter A: Music and mathematics A.1 Basic approach, fundamentals of natural science The basic building blocks of music are rhythmic structures, certain pitches, and their distances (= intervals). Certain intervals are preferred by average listeners, such as the octave and the fifth. Others tend to be rejected or ex- cluded. If one makes an experimentally-scientific attempt to create a system of such preferred and rejected intervals, one will find a broad correspondence between the preferences of the human ear and the overtone series, which fol- lows from natural laws. Thus, there exists a broad correspondence between quantity (mathematical proportion) and quality (human value judgement). Experiments of this kind have been conducted since the times of Hellenic philosophers – such as Pythagoras of Samos ( 570 480 BC), Archytas of Tarentum ( 380 BC), Aristoxenus ( 330 BC), Eratosthenes (3rd century BC), Didymus ( 30 BC), and Ptolemy ( 150),≈ among− others. Pythagoras≈ founded the religious-political≈ community of the Pythagorean school in Kroton.≈ He was considered to be≈ the incarnation of Apollo and was, even during his lifetime, revered as a god. During such experiments, a string was strung up on a monochord and sub- divided using a measurement scale according to the strict mathematical pro- portions 1 2, 2 3 etc. These parts (their halves, thirds, quarters, etc.) were made to vibrate in relation to the whole. Assuming a base tone C, the following numbererd∶ ∶ tone row is, thus, constructed. The emphasized numbers are “ekmelic tones”. These are prime numbers greater than five. Their exact pitch cannot be represented in the Western classical notation system, whose octaves are only subdivided into twelve separate steps. They are, in fact, a little bit lower than their notation. 1: 2: 3: 4: 5: 6: 7: 8: 9: 10: 11: 12: 13: 14: 15: 16 ↓ ↓ ↓ ↓ C c g c1 e1 g1 b1 c2 d2 e2 f 2 g2 a2 b 2 b2 c3 ♯ ♭ Fig. A.1 tone row by the subdivision of a tense string This row corresponds to the overtone series or harmonic series,1 which is a physical phenomenon that accompanies every sound process. The points that appears next to the numbers transform the distances between the tones into 1Discovered in 1636 by Marin Mersenne. The associated mathematical rules were discovered in 1702 by Joseph Sauveur. A.1 Basic approach, fundamentals of natural science 503 1:2 octave 5:8 minor sixth 15:16 minor second 2:3 fifth 2:3 fifth 5:9 minor seventh 8:9 major second 5:8 minor sixth 3:4 fourth 8:9 major second 5:6 minor third 3:5 major sixth 3:5 major sixth 8:15 major seventh 4:5 major third 5:9 minor seventh 4:5 major third 15:16 minor second 3:4 fourth 8:15 major seventh 5:6 minor third 32:45 tritone 32:45 tritone 1:2 octave TABLE A.1 intervals sorted by sound consolidation and size proportions (C - c octave = 1:2; c - g fifth = 2:3; etc.). The overtones that sound above the base tone (the lowest tone of the row – the one which is notated in the score) are not perceived by us as precisely discernible partial tones, but as “characteristic sounds”. This phenomenon occurs because the strings or columns of air do not only oscillate as a whole, but also in halves, thirds, quarters, . Thus, strings or columns of air do not only produce a single tone, but a sum of tones that produces the characteristic sound. The most important components of Western music – the intervals – can be des- cribed through mathematical proportions of the length of oscillating strings. This is also true for the oscillating columns of air in woodwind instruments. What is more, this row of pitches following from natural laws contains all important intervals for major-minor tonality. Since modern times, numbers have been used differently in relation to tones – the basis for the description of a pitch now being the frequency, and no longer the length of a string. Here, it is important to understand that frequency is inversely proportional to the length of a sounding string. A ratio of 1:2 with respect to string lengths corresponds to a ratio of 2:1 with respect to frequency, as a string or column of air that is shortened by half oscillates at twice its original frequency. Twelve essential intervals should be given, from consonant intervals (those with a high degree of sound consolidation among both frequencies – from the octave to the minor sixth) to dissonant intervals (those with little or no sound consolidation among both frequencies). Remark: The discovery of the connection between the sensual impression of music and its mathematical fundamentals is described in a story by Pythagoras (Fig. A.2): ⊕ octave 1:2 fifth fifth 2:3 2:3 fourth fourth 3:4 3:4 tempered whole tone 8:9 6 8 9 12 Fig. A.2 Pythagoras’s discovery Fig. A.3 ordering of tonal relationships “One day, during a long walk, Pythagoras pondered the problem of consonance. He wondered whether he would be able to find an aid for his sense of hearing, analogous 504 Chapter A: Music and mathematics to the yardstick or the compass as an aid for the visual sense and to the scale as an aid for the sense of touch. Through a fortunate twist of fate, he passed by a smithy and could clearly hear that several iron hammers striking against anvil, with the exception of one pair of hammers, produced consonant sounds. Full of joy, as if a god supported his plan, he entered the smithy and realized through various experiments that it was the varying weight of the hammers, rather than the blacksmith’s output of energy or the strength of the hammers, that caused the tonal difference. He carefully established the weight of the hammers and their impact, and then returned home. There, he fixed a single nail at a corner of the wall, so that no two nails, each having its specific substance, would falsify the experiment. On the nail, he hung four strings of the same substance, with the same number of wires, the same thickness, and the same rotation, and he attached a weight at the utmost end of each string. All strings had the same length, and by plucking two strings simultaneously, he managed to create the consonances he was looking for, which varied with each pair of strings. With the two weights 12 and 6, he produced an octave and thus determined that the octave is in a ratio of 2 : 1 – something which he had already realized with the weight of the hammers hitting the anvil. With both weights 12 and 8, he received the fifth from which he derived the ratio 3 2. With the weights 12 and 9, he got the fourth and thus the ratio 4 3. When comparing the middle weights 9 and 8, he recognized the interval of the whole tone, which∶ he, then, defined as 9 8. Accordingly, he could define the octave as∶ the union of fifth and fourth, that is, 2 1 3 2 4 3, and the whole tone as the difference of the two intervals, that is 9 ∶8 ~ 3 2 ~ 3 4.”~2 = × The graphical representation (Fig. A.3) of this Pythagorean~ ~ basic~ order of = × ⊕ the tonal ratios shows that, simultaneously with the two fifths (2:3), two quarter intervals (3:4) and a whole tone interval (8:9) are produced. These musical and mathematical rudiments discovered by Pythagoras and his pupils were regarded not as a “primary substance” but as a “primary law”, as it was believed that the the creation of the world and the cosmos was based on invariable numerical proportions. The prevalence of this belief is reflected by the elevated status of mathematics, which is still widely considered as the first of the sciences. This is also due to the influence on a person’s mental balance that is often ascribed to mathematics, as, for instance, in the following quote by the philosopher and mathematician Wilhelm Leibniz (1646–1716): “Music is an unconscious exercise in mathematics in which the mind is unaware that it is dealing with numbers”. 2J. Chailley: Harmonie und Kontrapunkt, p. 7ff. A.2 System formation 505 A.2 System formation Since ancient times, there have been attempts to derive scales and sound systems from the existing tones. To this end, people tried to fill the octave- space with intervals that were preferably of equal size and of harmonious proportions. Remark: The octave – the complete fusion of the two frequencies involved (octave identity) – has a special position among the intervals, as it represents the classifica- tion⊕ framework for all music systems.