User:FloraC/Critique on Functional Just System: Difference between revisions

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To explain why such phenomena are not desirable, we may say M3 has complexity of 4 since it is four fifths from the tonic, and m3 has 3. A2 on the other hand has 9 and d4 has 8. So it feels literally like a little gap. Their presence renders a minority of harmonics substantially more complex and difficult to grasp than others. The 157th overtone is an instance of such – it falls between M3 and m3 and is a d4.

That leads to the questionable nature of the current choice of radius. 65/63 is, in fact, an artificial number in the context of Pythagorean tuning. The author backed that in order to avoid two cases of absurdity that ~~has~~ evaluated to be of equal extent, 33/32 should be allowed as a comma whereas 32/31 should not, and that 65/63 simply does that work. However, the perception of interval classes is largely a subjective matter, and that 31/16 to be interpreted as a P8 is equally absurd as 11/8 to be interpreted as a d5 is a hasty assertion. The exact reason why 32/31, despite only differing from 33/32 by 1024/1023 (1.69 cents), should take a different function, is absent.

That leads to the questionable nature of the current choice of radius. 65/63 is, in fact, an artificial number in the context of Pythagorean tuning. The author backed that in order to avoid two cases of absurdity that have evaluated to be of equal extent, 33/32 should be allowed as a comma whereas 32/31 should not, and that 65/63 simply does that work. However, the perception of interval classes is largely a subjective matter, and that 31/16 to be interpreted as a P8 is equally absurd as 11/8 to be interpreted as a d5 is a hasty assertion. The exact reason why 32/31, despite only differing from 33/32 by 1024/1023 (1.69 cents), should take on a different function, is absent.

When I say 65/63 is an artificial number, I do mean there are natural options which, although appearing numerically complex to human users, a sophisticated system should not shy away from. One of them is sqrt (256/243), which the author admits to have considered and rejected. This number as a radius perfectly covers the range of M2–m3, therefore disallowing any prime harmonics to be hosted by AA1 or dd4. A similar option is sqrt (2187/2048), which measures exactly half of a sharp/flat accidental and as a radius perfectly covers the range of m3–M3, therefore disallowing any prime harmonics to be hosted by A2 or d4. From my perspective, 11/8 should be a P4, but I would not like to take sides on whether 31/16 should be a M7 or a P8. Instead, I argue for sealing up the gap between m3 and M3 in the choice of radius, and suggest sqrt (2187/2048) to accomplish that.

Using sqrt (2187/2048) certainly has impacts on some intervals and the first "victim" is the 31st harmonic. However, it is not necessarily a bad move, as is explained above. One of the confusing parts is that 33/31 is now a type of P1, but if you see 32/31 as a substitute of 33/32, that is not difficult to understand. To compensate, no prime harmonics can ever be hosted by augmented or diminished intervals, and hence, no Pythagorean intervals differing from another by a Pythagorean comma can host prime harmonics.

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 To explain why such phenomena are not desirable, we may say M3 has complexity of 4 since it is four fifths from the tonic, and m3 has 3. A2 on the other hand has  9 and d4 has 8. So it feels literally like a little gap. Their presence renders a minority of harmonics substantially more complex and difficult to grasp than others. The 157th overtone is an instance of such – it falls between M3 and m3 and is a d4.
-That leads to the questionable nature of the current choice of radius. 65/63 is, in fact, an artificial number in the context of Pythagorean tuning. The author backed that in order to avoid two cases of absurdity that has evaluated to be of equal extent, 33/32 should be allowed as a comma whereas 32/31 should not, and that 65/63 simply does that work. However, the perception of interval classes is largely a subjective matter, and that 31/16 to be interpreted as a P8 is equally absurd as 11/8 to be interpreted as a d5 is a hasty assertion. The exact reason why 32/31, despite only differing from 33/32 by 1024/1023 (1.69 cents), should take a different function, is absent.
+That leads to the questionable nature of the current choice of radius. 65/63 is, in fact, an artificial number in the context of Pythagorean tuning. The author backed that in order to avoid two cases of absurdity that have evaluated to be of equal extent, 33/32 should be allowed as a comma whereas 32/31 should not, and that 65/63 simply does that work. However, the perception of interval classes is largely a subjective matter, and that 31/16 to be interpreted as a P8 is equally absurd as 11/8 to be interpreted as a d5 is a hasty assertion. The exact reason why 32/31, despite only differing from 33/32 by 1024/1023 (1.69 cents), should take on a different function, is absent.
 When I say 65/63 is an artificial number, I do mean there are natural options which, although appearing numerically complex to human users, a sophisticated system should not shy away from. One of them is sqrt (256/243), which the author admits to have considered and rejected. This number as a radius perfectly covers the range of M2–m3, therefore disallowing any prime harmonics to be hosted by AA1 or dd4. A similar option is sqrt (2187/2048), which measures exactly half of a sharp/flat accidental and as a radius perfectly covers the range of m3–M3, therefore disallowing any prime harmonics to be hosted by A2 or d4. From my perspective, 11/8 should be a P4, but I would not like to take sides on whether 31/16 should be a M7 or a P8. Instead, I argue for sealing up the gap between m3 and M3 in the choice of radius, and suggest sqrt (2187/2048) to accomplish that.
 Using sqrt (2187/2048) certainly has impacts on some intervals and the first "victim" is the 31st harmonic. However, it is not necessarily a bad move, as is explained above. One of the confusing parts is that 33/31 is now a type of P1, but if you see 32/31 as a substitute of 33/32, that is not difficult to understand. To compensate, no prime harmonics can ever be hosted by augmented or diminished intervals, and hence, no Pythagorean intervals differing from another by a Pythagorean comma can host prime harmonics.