![]() ![]() Similarily, we could note that the major chord as a whole occurs early in the harmonic series (overtones 4,5,6) which is related to what Scott has written. (What I have ignored here are little differences in frequencies between the usual equal temperament tuning and the actual frequencies of the overtones.) This makes it plausible that replacing it by a perfect fourth may improve the overall consonance. ![]() ![]() Since the augmented forth occurs very far down the harmonic series of the root, it involves a very complicated frequency relationship. What we get is the lydian scale which differs from the major scale by a single note: it contains the augmented fourth instead of the perfect fourth. This simple rule already gives us almost the major scale. Successively stacking perfect fifths and adjusting the octaves should give a scale with few dissonances between the different notes because each note occurs early in the harmonic series of other notes and therefore many simple frequency relationships occur. The first and the second overtones are the ocatve and the perfect fifth. If you look at the harmonic series, there are a number of ways to make it plausible that the major scale sounds consonant under this premise. Let the basic premise be that simple frequency relationships correspond to consonance while complicated frequency relationships correspond to dissonance. But generally, if you play a note, lower overtones are more present than higher overtones.) (The relative intensities of the overtones vary depending on the instrument which is what gives each instrument its characteristic sound. So beside the root, you get overtones which build up the harmonic series. If you play a single note on a musical instrument, many different oscillation modes are excited. ![]() Actually given the Neanderthal brain size compared to us modern humans, that might not be much of an insult. Criticizing it negatively may get you placed in the Neanderthal Hall of Fame. Somewhat like modern art: folks love it or hate it. IMO, neither terribly productive or interesting. Now we have a sort of 'everyone's scale is right, and nobody's scale is wrong, therefore whatever I do is worthy' point of view. Composers like Ives, Henry Cowell, Arnold Schonberg, and John Cage attempted to undo that, pretty much from the start of the 20th century. G Zarlino (Renaissance) wrote several treatises on harmony and tunings, which started the ball rolling toward the more modern Well Tempered scale. Microtonal music does not do well in the notational context of music the way we learned it in elementary school. Some are pentatonic (5 tones not 8), example Peruvian 'pan flute'. These tones did sound well together in a chord.įirst off, the well tempered tunings we use nowadays were not employed extensively until about the time of J S Bach. My copy of From Polychords to Polya: Adventures In Musical Combinatorics came in handy for the above, but it is dear - Amazon shows used copies at $60 USD on up! I haven't read it, but The Math Behind the Music looks interesting, and is reasonably priced.ĭid you ever hear a 'wolf tone' or a dissonant piece of music? Charles Ives come to mind - people hated some of his compositions for excessive use of dissonance. Adding to the confusion, Greek nomenclature for their three standard tunings for 4 stringed instruments were called diatonic (for instance, A-G-F-E), chromatic (A-Gb-F-E), and enharmonic (A-Gbb-F(q)-E) where what I'm showing as F(q) is F lowered by a quarter-tone, and G-flat-flat would be close to an evenly tempered F, but isn't exactly as note spacing (usually) followed a 3 cycle Pythagorean 'just' intonation scheme. In an evenly tempered scale, the diatonic interval of a perfect forth (E and A, for instance) sounds exactly the same as the augmented third chromatic interval E and G# because, being evenly tempered, the frequencies of G# and A notes are identical in just intonations they aren't the same frequencies. This is one of the many places where things get weird. Western music also has several forms of 'just' intonation which also have different, untempered spacings between semitones, and a whole crop of other intonation systems abound in non-western musical traditions. ), although even this isn't clear cut well temperament, as in Bach's Well-Tempered Clavier, is but one system of tweaking temperament intervals so pleasing-sounding chords can be played over an instrument's entire tonal range. Modern western music (and my guitar necks) are mostly built around a 12 note, evenly tempered chromatic scale based on a logarithmic interval of #\sqrt 2 # between semitones (for instance, between C and C#, C# to D. As unlikely as it sounds, I'd say you are both right. All discussions of music and math fall immediately into the deep end of the pool. ![]()
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