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Message: 5525 Date: Fri, 08 Nov 2002 20:19:07 Subject: Re: from the realms of private correspondence From: Carl Lumma > Tenney Harmonic Distance. Note that it is only defined for > dyads. I attempted to extend it to triads. Paul claims that > in so doing, I removed its metric status. I also don't see how Tenney HD meets property 4... take dyads 33:26 and 39:22 (Paul's example). -Carl
Message: 5536 Date: Fri, 08 Nov 2002 04:36:31 Subject: Fwd: Re: from the realms of private correspondence From: Carl Lumma >>Your hypothesis is that because psychoacoustic dissonance >>varies from listener to listener, music had to develop a >>grammar to make the perception of dissonance more... >>uniform? > >I don't recall saying that it "has to", or even that >to whatever extent it "ought to", that variation in >perception of dissonance would be the only reason. Not sure how this answers my question... -Carl
Message: 5538 Date: Fri, 08 Nov 2002 04:37:50 Subject: Re: from the realms of private correspondence From: Carl Lumma >>I must not understand property 1. If I have colinear >>points a b c, how is the sum of AB and BC >= AC? > > AB + BC = AC > >The two parts of the line segment AC add up to the whole in >length. Yeah, I must be reading the notation here... Metric -- from MathWorld * ...wrong then? -Carl
Message: 5541 Date: Fri, 08 Nov 2002 22:06:29 Subject: Re: from the realms of private correspondence From: Carl Lumma >>>>>actually, i figured this out once. it's proportional to the >>>>>total edge length of the hyper-rectangle defined by the points. >>>>>but that doesn't make it a metric! >>>> >>>>1. Can you give an example for 4:5:25? >>> >>>ok, in that case the hyper-rectangle collapses down to two >>>dimensions, but the total edge length is unaffected. so no real >>>problem there. the vertices of the hyper-rectangle are the >>>pitches comprising the (non-octave reduced) euler genus whose >>>factors are the notes in your chord. >> >>Come again? What points in the lattice does the rectangle >>intersect? > >its vertices are > >1 4 > >5 20 > >25 100 > >125 500 > >and it also intersects 5 (again), 20 (again), 25 (again), 100 >(again), 2, and 10. My chord is: 25 | 5 | 1 Here are your verticies on the lattice: 125 - x - 500 | | | 25 - x - 100 | | | 5 - x - 20 | | | 1 - x - 4 Obviously, you intend some of these to be on extra dimensions. Why? How did you figure out that the perimeter of these structures would be a consistent taxicab distance for three points? >>>once again, carl -- tenney HD is a metric for *pitches*, not >>>dyads (notwithstanding your "cop-out"). >> >>When did you ever say that? What does it mean? What meaning >>do pitches have in terms of concordance? > >pitches are what you start with when you want to calculate >concordance. the fact that you can transpose them (together) >without affecting the concordance is one of the essential >features in making this a metric at all. I don't understand how a pitch can have concordance. -Carl
Message: 5542 Date: Fri, 08 Nov 2002 05:38:28 Subject: Re: from the realms of private correspondence From: Carl Lumma >>Yeah, I must be reading the notation here... >> >>Metric -- from MathWorld * > >It says the distance from A to B, plus the distance from B to C >(two legs of a triangle) is at least as big as the distance from >A to C (the third leg.) Mint! Of course. Sorry. In which case, I'd think property 4 is the only one my cop-out doesn't meet, and Tenney HD also doesn't meet it. If this really isn't true, I'm hoping someone will refute it. -Carl
Message: 5548 Date: Fri, 08 Nov 2002 23:08:30 Subject: Re: from the realms of private correspondence From: Carl Lumma >>I don't understand how a pitch can have concordance. > >They have it with respect to "1". Dyad. So what you have is a function that assigns a value to pitches, and then you subtract them. The function assigns zero to 1/1 but you're still subtracting. I have a function that assigns a value to triads, and I can subtract them. What's the difference? >Because of the group structure, this automatically gives you >concordance for dyads, which is why thinking aobut it in terms >of norms rather than metrics makes the most sense. If you know what norms are and how to work with them. I'm still struggling with metrics. But do tell. Maybe Paul will follow. -Carl
Message: 5549 Date: Fri, 08 Nov 2002 23:09:53 Subject: Fwd: Re: from the realms of private correspondence From: Carl Lumma --- In tuning-math@y..., <Josh@o...> wrote: > Try it with clarinet timbre, piano timbre and oboe timbre, > and I can practically guarantee that you'll each get an > inconsistent result across timbres. I have, and I don't. Have you? -Carl
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