For medieval church music, the answer was that only harmonies of octaves, fifths and fourths were permissible. For Harry Partch, ratios built out of prime numbers less than 11 are acceptable (he also includes 9, "because it is an odd number". He mentions 15 in this same context and it is clear to me that he is saying that numbers with two prime factors have a value greater than other numbers but less than primes). He halted at 11 and did not proceed to 13 simply because he thought that gave him enough to be getting on with.
A simple mathematical explanation of consonance is that it is a measure of how often frequency intervals line up. An octave is extremely consonant because it has the maximum possible agreement between two different frequencies (the wavelength is half or double that of the other). A fifth is very consonant because three wavelengths of the fifth frequency line up with two wavelengths of its tonic.
So, we can measure consonance by multiplying together a ratio's numerator and denominator (Partch calls these the odentity and udentity, I don't know if this is common practice). It's important to have the simplest equivalent fraction, for example, 2/1 is correct but 4/2 is not. This is because what we're actually after is the product of the identities (prime factors) of the note. Here are some examples:
2/1 (octave) - dissonance 2
3/2 (fifth) - dissonance 6
It makes no sense to try and use this method to calculate the consonance of equally tempered intervals because the ratios involved are irrational. The values that result only reflect your level of accuracy in calculating them.
I'm no closer to defining the point at which consonance becomes dissonance, but I now have a method for objectively comparing intervals and stating whether one is more or less consonant than another.
I wonder if modern music theory understands this as plainly as I have set it out? Partch understands the general principle but survives only by rules of thumb. I need a more advanced textbook.
A simple mathematical explanation of consonance is that it is a measure of how often frequency intervals line up. An octave is extremely consonant because it has the maximum possible agreement between two different frequencies (the wavelength is half or double that of the other). A fifth is very consonant because three wavelengths of the fifth frequency line up with two wavelengths of its tonic.
So, we can measure consonance by multiplying together a ratio's numerator and denominator (Partch calls these the odentity and udentity, I don't know if this is common practice). It's important to have the simplest equivalent fraction, for example, 2/1 is correct but 4/2 is not. This is because what we're actually after is the product of the identities (prime factors) of the note. Here are some examples:
2/1 (octave) - dissonance 2
3/2 (fifth) - dissonance 6
5/4 (just major third) - dissonance 20
45/32 (a small just tritone) - dissonance 1440 (this is universally agreed to be dissonant)It makes no sense to try and use this method to calculate the consonance of equally tempered intervals because the ratios involved are irrational. The values that result only reflect your level of accuracy in calculating them.
I'm no closer to defining the point at which consonance becomes dissonance, but I now have a method for objectively comparing intervals and stating whether one is more or less consonant than another.
I wonder if modern music theory understands this as plainly as I have set it out? Partch understands the general principle but survives only by rules of thumb. I need a more advanced textbook.
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