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Message: 6836 Date: Fri, 30 May 2003 22:54:14 Subject: ...continued... From: Carl Lumma Gene wrote... >If T is a linear temperament, and T[n] a scale (within an octave) of >n notes, then if the number of generator steps for an interval q times >the number of periods in an octave is +-n, q is a chroma for T[n]. In >terms of the programs I sent you, a7d(T,q)[1] = +-n. > >>Thanks. I believe that is the chromatic uv. There should only be >>one for a given T[n]. > >Any one of these, times a comma of T[n], will be another one; hence >they are infinite in number. Yeah, but this is true for any interval in the temperament. What isn't clear to me is: 1. The choice of a chroma q and pi(p)-1 commas (where p is the harmonic limit) specifies a linear temperament, right? It does not specify an n for T[n] or a tuning for T, but it does specify a family of maps (and if we're lucky, a canonic map), right? 2. Yet above it appears that changing n in T[n] changes the chroma (but obviously not the temperament, T). Therefore, we have a problem, unless changing n can only change the chroma among the family of comma-transposed chroma for that T... ? Finally, note that I'm still confused about prime- vs. odd-limit as regards pi(p)-1. Obviously I'm assuming prime-limit here, but should pi(p) be changed to ceiling(p/2)? That is, how many commas does a 9-limit linear temperament require? Paul? -Carl
Message: 6837 Date: Fri, 30 May 2003 23:02:02 Subject: Re: ...continued... From: Carl Lumma Just a note; for the purposes of the below message, and from now on, I intend "chroma" = "chromatic unison vector" and "comma" = "commatic unison vector". -Carl >Gene wrote... > >>If T is a linear temperament, and T[n] a scale (within an octave) of >>n notes, then if the number of generator steps for an interval q times >>the number of periods in an octave is +-n, q is a chroma for T[n]. In >>terms of the programs I sent you, a7d(T,q)[1] = +-n. >> >>>Thanks. I believe that is the chromatic uv. There should only be >>>one for a given T[n]. >> >>Any one of these, times a comma of T[n], will be another one; hence >>they are infinite in number. > >Yeah, but this is true for any interval in the temperament. > >What isn't clear to me is: > >1. >The choice of a chroma q and pi(p)-1 commas (where p is the harmonic >limit) specifies a linear temperament, right? It does not specify >an n for T[n] or a tuning for T, but it does specify a family of maps >(and if we're lucky, a canonic map), right? > >2. >Yet above it appears that changing n in T[n] changes the chroma (but >obviously not the temperament, T). Therefore, we have a problem, >unless changing n can only change the chroma among the family of >comma-transposed chroma for that T... ? > > >Finally, note that I'm still confused about prime- vs. odd-limit as >regards pi(p)-1. Obviously I'm assuming prime-limit here, but should >pi(p) be changed to ceiling(p/2)? That is, how many commas does a >9-limit linear temperament require? Paul? > >-Carl
Message: 6838 Date: Sat, 31 May 2003 11:40:30 Subject: Re: ...continued... From: Carl Lumma >No, the chroma has nothing to do with defining the temperament; it >defines the scale, given the temperament. >Gene says pi(p)-2 commas, which will be correct if you're counting 2. >It does specify the n, but not the tuning for T. If you don't want n, >you don't need the chroma. Thanks. Got it. >This should be pi(p)-2 commas, and no chroma, or 2 vals. In general >pi(p)-n commas, or n vals, specifies an (n-1)-temperament. Can someone give the vals for 5-limit meantone? >> Finally, note that I'm still confused about prime- vs. odd-limit as >> regards pi(p)-1. Obviously I'm assuming prime-limit here, but should >> pi(p) be changed to ceiling(p/2)? That is, how many commas does a >> 9-limit linear temperament require? Paul? > >Exactly as many as a 7-limit linear temperament. >... linear independence. So the 5-limit (2-3-5) requires one comma. >So does the 2-3-7 limit. And so would a system composed of octaves, >fifths, and 7:5 tritones, although it uses 4 prime numbers. Ok, but what about stuff like (2-3-5-9) where we don't have linear independence but wish to consider 9 as consonant as 3 or 5? How does visualization in terms of blocks work on a lattice with a 9-axis? -Carl
Message: 6842 Date: Sat, 31 May 2003 12:05:27 Subject: Re: ...continued... From: Graham Breed Carl Lumma wrote: > 1. > The choice of a chroma q and pi(p)-1 commas (where p is the harmonic > limit) specifies a linear temperament, right? It does not specify > an n for T[n] or a tuning for T, but it does specify a family of maps > (and if we're lucky, a canonic map), right? Gene says pi(p)-2 commas, which will be correct if you're counting 2. It does specify the n, but not the tuning for T. If you don't want n, you don't need the chroma. > 2. > Yet above it appears that changing n in T[n] changes the chroma (but > obviously not the temperament, T). Therefore, we have a problem, > unless changing n can only change the chroma among the family of > comma-transposed chroma for that T... ? No, there's no problem. > Finally, note that I'm still confused about prime- vs. odd-limit as > regards pi(p)-1. Obviously I'm assuming prime-limit here, but should > pi(p) be changed to ceiling(p/2)? That is, how many commas does a > 9-limit linear temperament require? Paul? It generally goes by prime numbers, or more generally by prime intervals -- that is a set of intervals none of which can be arrived at by adding and subtracting the other ones. This is like linear independence. So the 5-limit (2-3-5) requires one comma. So does the 2-3-7 limit. And so would a system composed of octaves, fifths, and 7:5 tritones, although it uses 4 prime numbers. Graham
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