In Chapter II Hindemith discusses 'The Medium'--the sound. It is fascinating to see how Hindemith spends such a great length on the topics regarding the overtone series and its extending branches, unlike most of the other theory books. It is obvious to see this form the foundation for the rest of the book.
This can be seen as a great example of how a composer learns music theory quite differently from the music theorists or performers. Being able to work out the chords, intervals, scales; or analysis pitch class or rows; or see what's been done on the paper is not the essence of music theory for a composer. It is to understand the materials that one can use to compose music and the consequences of the actions; to distinguishes acoustic results with reasons.
pp. 14-15 General Considerations, Overtones.....certain intervals make similar impressions upon all men. When even the man of the lowest level of civilization hears the interval of an octave, he will feel that the upper note is the higher image of the lower. .....the interval of the pure fifth is so unambiguous and independent.....The slightest alteration in the size of an octave or a fifth (perfect, of course).....changes these intervals completely..... .....(poetic comparison) The eye perceives in light which has been split up by a prism a natural series of vibration frequencies. The light of the sun always produces the same immutable series of colors, familiar familiar to us in the rainbow. Now, just as light consists of graduated colors of the spectrum, so a tone consists of many partial tones. The spectrum of the world of sound is the harmonic or overtone series.
pp. 22-23 The Triad
.....In the world of tones, the triad corresponds to the force of gravity. It serves a sour constant guiding point, our unit of measure, and our goal, even in those sections of compositions which avoid it. Must not then a music which consists exclusively of triads provide the highest delight? Pieces of this description, such as those of early Italian choral music, do not, as a matter of fact, belong among the greatest revelations; their uninterrupted sweetness is apt to bore even the gentlest listener. .....our aural nerves, as the result of the intensity of modern life and the surfeit of sounds, are very taut. .....How great the span can be between triads is a question of the hearing habits of the listener and of the ingenuity of the composer. ('as a matter of fact'-->he can say that because he IS Hindemith. Can't help smiling to see he also used the world 'gravity' together with a lot of other poetic metaphors. 'Too many metaphors' the comment I received for the commentary, however, metaphors are everywhere when a composer writes--Hindemith, Stravinsky, Schoenberg etc. How could one, then, express the mysterious beauty when music works? When it can not be seen, or touched, or tasted, or smelled? Understanding the theory might help us to see what's been used to build this piece of work. However knowing which scale or chord been used where won't help us to appreciated the art of music better if one do now know how it sounds like in real time and why it works or does not work. Simply, another metaphor, everyone can cook rice, but not everyone can cook rice right. 'Knowing how' is what distinguishes professionals from the rest--what a composer should know.)
....The feeling for the purity, the harmonic completeness, and the satisfying effect of the triad, which is the same as the unerring judgement in the aural measuring of the octave and the fifth, is accordingly just as natural to us as the body's sense of space. .....The senses of sight and feeling need to make use of memory and comparison in order to arrive at even approximate judgements of size and proportion, and our sense of the passage of time similarly does not permit us to make exact estimates. .....The basic fact of our hearing process reveals to us how closely relate are number and beauty, mathematics and art.
pp. 25-27 Paths to Scale-Formation
.....In contrast to the scales of oriental peoples[sic], as well as to those of mediaeval Europe, our series will not serve exclusively melodic purposes. Not every scale that was originally evolved for melodic purposes is well adapted to the needs of harmonic organization. If a scale is to perform both functions, the intervals must be such that the combinations of tones are as pure as possible (that is, consist of intervals such as are contained in the lower reaches of the overtone series). On the other hand, the grouping of the intervals music not be so rigid that it does not permit of all those little divergences that form on of the greatest charms of melodic expression: the age-old use of impure intonation as an artistic means, the most extreme instance being the purposeful mistuning of subordinate tones in the melody, and the most minute divergence from the pitch being the vibrato, with countless melodic subtleties between these two extremes.
...The next most important interval after the octave, the fifth, opposes considerable resistance to continuous parallel motion. .....the uninterrupted connection of perfect fifths destroys the purity of the octave. But we cannot give up so important an interval altogether, and therefore we adopt it as the contral pillar of our scale. (can't help feeling quite shocked when I realised that I lend on the same conclusion as Hindemith did long before I read his writing. As I often explain the scale to students myself, it is not as a straight line starts from tonic and ends in tonic. It is more like a little universe and each note forms a special interval relationships with the tonic, above and below. Hence the dominant and sub-dominant represent the sound of fifth, the mediant and sub-mediant represent the sound of thirds and so on.)
.....Another method is more productive.....he will divide this space either according to the convenient placing of this fingers, or according to mathematical considerations (in short, the tetrachords form by exactly the same interval relationships). If he then transfers the divisions thus found to the next higher string (and to other still higher) he has achieved a usable scale. The Arabian tonal system (Hindemith talks quite often about Arabian music theory, can't help wonder how much study he had done to it), with its highly developed theory of music, rests on such calculations. Scales so arrived at are admirable material for monophonic music, purely melodic in conception, while for polyphonic music they are practical only to a limited extent, since for the sake of identical fingering on all strings--carrying with it the possibility of parallel motion throughout--purity is neglected. The intervals formed by the tones of the scale do not all have the same proportions as their prototypes in the overtone series. But in polyphonic music, the measuring ear continually seeks the pure intervals of the overtone series, and is dissatisfied not to find them. And to the extent that scales of this sort do contain pure intervals beside the fourth and fifths, their immovable rigidity prevents any free harmonic development. Polyphonic music demands that the tones may at any time be able to change their tonal significance, by relation themselves to changing roots (either of chords or of overtone series). .....It is, however, impossible.....for one tone to perform all these functions without change of pitch.
p. 28 Tempered Tuning
There is no solution of the scale riddle that can reconcile these opposite necessities. Purity must be neglected or the possibility of unhindered polyphony sacrificed (is it still today with the development of electronic musical devices?).
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