

MusicGoethe is reputed to have said [1], “Geometry is frozen music.” When time is included in the picture, as in Cycles, then we have music which becomes a geometrical collection of frequencies or Vibrations  all of which can be shown to be a series of trigonometric “sine” waves. Harmonies become mathematical ratios which can occur in everything from the Harmony of the Spheres to separate instruments or voices combining the soprano, alto, tenor, baritone and base notes; all meshing coherently and feeling groovy. Every musical pulse consists of numerous sinewave tones. Even a “square wave” is made up of a large number of odd harmonics, and thus by extrapolation, a truly infinite pulse (e.g. the Big Bang or first Ommmmm... of creation) would consist of all possible pure tones. The manner in which musicians examine a spectrum of musical harmonies is, in fact, exactly the same procedure mathematicians call a Fourier Transform. The sum of all musical frequencies thus constitutes the whole of the universe. Pythagoras and the Pythagorean Brotherhood, the members of the philosophical school which followed his teachings, recognized circa 600 B.C.E. what modern physicists and most anyone who has studied the subject of music soon learn, that: 1. Music is geometry, and geometry is music. 2. Music Theory is fundamentally about ratios of numbers. 3. The harmonic nature of music demonstrates the great harmony of creation. 4. Every musical tone or pulse is made up of the sum of many pure sinewaves. 5. Those who learn music do better at mathematics (and vice versa). Possible definitions of “music” and “theory” include: “Music: 1. The art of organizing tones to produce a coherent sequence of sounds intended to elicit an aesthetic response in a listener. 2. Vocal or instrumental sounds having some degree of rhythm, melody, and harmony. “Theory: 1. A system of accepted principles and rules of procedure devised to analyze, predict, or otherwise explain the nature or behavior of a specified set of phenomena.” Scientists, meanwhile, are now beginning to suspect that the human brain is quite literally prewired for music. Sharon Begley [2] has reported that “people can typically remember scores of tunes, and recognize hundreds more” [but not necessarily want to sing them aloud  fortunately]. And yet, these same people can recall only snatches of a few prose passages. Music, on the other hand, can incite passion, belligerence, serenity, fear, or sadness. It is able to do so even with people who have no experience in such things as a particular crescendo in a movie implying the sudden appearance of the bad guy. A bit more controversial is the idea that music can help bridge socioeconomic gaps. It accomplishes this feat by linking music and mathematics. In one study, “After a year of piano, the second graders who received twiceaweek piano training in school scored as well as fourth graders, who did not; half of the second graders scored as well as fifth graders.” The idea music training might help all subjects does not appear to be true. There is something about math and music (proportions, ratios, sequences) all of which underlie both musical and mathematical reasoning. According the Ms. Begley [2], “The temporal lobes of the brain, just behind the ears, act as the music center.” For musicians, who had begun their training before the age of 7, they actually increased the size of their brains  specifically the corpus callosum (the trunk line which connects the brain’s right and left hemispheres). This neural path increase may also explain why the better musicians are not only technically adept (the left brain’s partiality to cognition), but can play with emotion (the right brain’s forte). Even more striking is the fact that mental imagery or mental rehearsals can activate the same regions of the brain as actual practice, and similarly affect the synapses! This aspect of Creating Reality by mental imagery can also be applied outside of musical performances. It also works with sports, professional performances of most any kind, and any anticipated (without fear) and rehearsedinthemind activity. Constantly reviewing emergency procedures in the mind can result in extraordinary skill in an actual emergency. The numerous references to proportion and ratio in music and mathematics deserve a bit more consideration. In the Newsweek article, infants were found to smile when music was played, which consisted of perfect fourths and perfect fifths  i.e. chords or sequences which are separated by either five half steps (as between C and F) or seven half steps (as between C and G), respectively. Babies, however, did not like tritones, where two notes were separated by six half steps (e.g. C and F sharp). In effect, the very young music critics were already well aware of the critical importance of Intervals. Interval An interval is the ratio of frequencies between a base note and another note. A collection of notes is a scale, with tempered scales consisting of notes which have specific sets of intervals which are aesthetically pleasing, and equal temperament scales consisting of notes whose frequencies are multiples of a single ratio. The musical intervals, themselves, are named according to the note in which the interval corresponds. A “fifth”, for example, corresponds to the ratio of frequencies between the fifth and first notes in the major scale  the first note also being called the “tonic”. The intervals in the modern major and minor scales, together with frequency ratios, are: Tonic or first or unison (1:1), Second (9:8), Minor third (6:5), Major third (5:4), Fourth (4:3), Fifth (3:2), Minor sixth (8:5), Major sixth (5:3), Minor seventh (9:5), Major seventh (15:8), and Octave (2:1). There is also a subminor seventh (7:4) and a fifteenth (4:1). From a pleasing to the ear point of view, the fifth (where the fifth note’s frequency is 50% greater than the first note’s frequency) is noteworthy. For a lyre, for example, by taking C as the root note, this ancient instrument could be tuned to the notes C, F, G, C’ (with ratios: 1, 4/3, 3/2, and 2), while the combinations C, F, and C’ (1, 4/3, and 2) and C, G, C’ (1, 3/2, and 2) would result in euphony. According to Nicomaeus, lyres were tuned to these notes at least until the time of Orpheus. Scales Scales are all about establishing certain intervals which give the most pleasing sound, or the best euphony. Life is comparatively simple with single instruments, but when groups of instruments gather together in a jazz trio, a marching band, or a symphony orchestra, it becomes a bit more complicated. The diatonic scale is ancient, but a variation due to Ptolemy (and dating from the second century A.D.) has been incorporated into modern intonation. The scale was rediscovered in the late 15th century, and eventually became the basic scale used in western music. In particular, all of the church modes (dorian, phrygian, lydian, myxolydian, and aeolian) are rotations of the socalled major diatonic scale. Diatonic scales (as well as other scales) work differently for different starting notes. Thus the introduction of the concept of key. Music written in one intonation, for example, would have to be rewritten if the scale (or starting note) were shifted, in order to preserve consonance. The scale used on piano and fretted instruments is actually an approximation to the exact ratios of the diatonic scale, because modern western music uses an equal temperament scale, allowing music to be transposed while slightly sacrificing the euphony of chords. The Pythagorean scale is a particular tuning for the diatonic scale and consists of only two intervals: 9:8 (the second) and 256:243. Pythagoras (570504 B.C.E) is usually given credit for discovering that vibrating strings with lengths the ratios of small whole numbers of each other produced a pleasing harmony. The 9:8 intervals between notes were chosen and then the gaps filled in with hemitones, where one hemitone equals a ratio 256:243. A hemitone (1.0535) is, however, significantly less than half a Pythagorean tone (i.e. Ö(9/8) =1.0606). The good news, however, is that the Pythagorean scale contains four fifths and five fourths, which is better than what can be attained from any other eight notes. And the fourths and fifths, corresponding to the magical numbers of 5 and 7 half steps, are just more indications of the geometrical basis of music  and quite possibly why 5 and 7 sided geometry is aesthetically pleasing, as well as being musically harmonious. Equal Temperament The problem with a major scale, minor scale, or any combination of scales which have unequal intervals is that musical melodies cannot be readily transposed to a different tonic (key). For instance, since a major scale is defined to have exact ratios of frequencies 9:8, 5:4, 4:3, 3:2, 5:3, 2:1, etc., changing the tonic from a C to a D (using a C with a frequency of 264 Hertz, i.e. cycles per second) in a major scale would result in the following: note C D E F G A B C’ D’ Key of C 264 297 330 352 396 440 495 528 594 Key of D &; 297 334 371 396 445 495 557 594 Obviously, while D, G, B, and D’ have the same frequency in both keys (and E and A are close), F and C’ are significantly off. To be able to play a given piece of music in either the key of C or D would thus require separate keyboard keys (or frets on a fretted instrument) in order to obtain the same note. This would cause problems in transposition (playing a tune a fixed number of steps lower, for example), because this would require frequencies which the instrument was incapable of playing. The equal temperament scale (invented by Andreas Werckmeister, in 1691) was introduced to avoid this serious musical problem. Equal temperament makes the ratio between each half step a constant. This allows a song written in one key to be shifted up any number of half steps, and still contain exactly the same harmony (although the frequencies themselves will be altered). Notes can then be defined by octaves  an octave representing a doubling in frequency  and still produce a pleasing sound when played simultaneously with the same note of another octave. The number of subdivisions in each octave is, in principle, arbitrary. For harmony’s sake, however, most notes in the scale must have frequencies which differ from other notes by ratios of small whole numbers. The best scale will thus contain a large number of notes which are resonant, which is the manner in which musical harmony developed historically. The ear has essentially a logarithmic response (as does the eye, explaining astronomers’ use of the magnitude system for apparent brightness), so that the perceived difference between notes is the same if their frequencies are spaced as a power law. For fretted and keyboard instruments (which can access only a discrete number of frequencies), the entire frequency range can now be partitioned into a number of discrete notes spaced at equal logarithmic intervals. Music may indeed hath charms to soothe the savage beast. But it’s all in the mathematics. Thus in the next encounter with a savage beast  say the drooling monsters from Aliens  the heroine might consider either humming a few bars of “Don’t Worry, Be Happy,” sing a happy tune, or begin quoting Prime or Fibonacci Numbers. Or, if you’re really clever, simply say, “The sum of the squares of the sides of a right triangle equals the square of the hypotenuse.” Just be sure and pronounce “hypotenuse” correctly! And do it with a flair for the musical version. Sacred Geometry Cycles Vibrations Coda (Harmony of the Spheres) Scherzo (Indianapolis) _________________________ References: [2] Sharon Begley, “Music on the Mind”, Newsweek, July 24, 2000, pages 5052.


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