Pythagoras ahead of his time almost got it right because he was comparing string lengths. A half string vibrates more quickly than twice the frequency due to material stiffness.
Should he have an ipod touch with a signal generator he would be able to measure the frequency of the 2nd harmonic of pleasingly stretched sounded string to be a bit greater than x2!
Therefore the Pythagorean comma is a pseudo problem.
Tuning upwards from a given note by twelve intervals of 3/2 the final note will form an interval with the starting note which is sharper (the correct word here is higher) than the "octave" (mathematical octave, not what the human brain finds a satisfactory octave) by 23.5 cents (the Pythagorean comma).
This is a good thing since the ear likes to hear stretched octaves. (Even +23.5cents at such high frequency may sound flat! Easy to verify by any app like garagband to hear that a very high mathematical C sounds almost like B!)
Good sounding partials are not exact multiples anyway due to the same psycho acoustic reasons so it may well be that right sounding partials will automatically satisfy beat less stretched octaves. So no comma and no beats and at the same time great poetic sound. "Exact" x2 "octaves" sound appallingly flat. Corrected stretched by ear brain octaves and designed partial sound so beautiful they make us feel proud to be part of humanity and nature. Examples are a nicely tuned Steinway piano or a good orchestra with fine instruments and ear brain intonating players and conductors.
Attention to those who change from A440 Hz going below. The way pianos are scaled reducing tension to the strings may make the opposite and make 2nd partials flatter. trying at your own risk you can increase from 440Hz. If you do this on a guitar yes you can go below 440Hz but use thicker strings to keep string tension high if your guitar can stand it.
Should he have an ipod touch with a signal generator he would be able to measure the frequency of the 2nd harmonic of pleasingly stretched sounded string to be a bit greater than x2!
Therefore the Pythagorean comma is a pseudo problem.
Tuning upwards from a given note by twelve intervals of 3/2 the final note will form an interval with the starting note which is sharper (the correct word here is higher) than the "octave" (mathematical octave, not what the human brain finds a satisfactory octave) by 23.5 cents (the Pythagorean comma).
This is a good thing since the ear likes to hear stretched octaves. (Even +23.5cents at such high frequency may sound flat! Easy to verify by any app like garagband to hear that a very high mathematical C sounds almost like B!)
Good sounding partials are not exact multiples anyway due to the same psycho acoustic reasons so it may well be that right sounding partials will automatically satisfy beat less stretched octaves. So no comma and no beats and at the same time great poetic sound. "Exact" x2 "octaves" sound appallingly flat. Corrected stretched by ear brain octaves and designed partial sound so beautiful they make us feel proud to be part of humanity and nature. Examples are a nicely tuned Steinway piano or a good orchestra with fine instruments and ear brain intonating players and conductors.
Attention to those who change from A440 Hz going below. The way pianos are scaled reducing tension to the strings may make the opposite and make 2nd partials flatter. trying at your own risk you can increase from 440Hz. If you do this on a guitar yes you can go below 440Hz but use thicker strings to keep string tension high if your guitar can stand it.
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