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Message: 3075 - Contents - Hide Contents Date: Tue, 08 Jan 2002 19:15:52 Subject: Re: Dictionary query From: genewardsmith --- In tuning-math@y..., graham@m... wrote: That's where the> "positive"/"negative" terminology is defined, and it is relative to > 12-equal, not Pythagorean. If you'd been paying attention, you could > have checked it. One of Erv Wilson's early Xenharmonikon articles > reiterates this, and another (I think "On Linear Notations ...") extends > it for ETs other than 12.Now all that's left is to get it to make a particle of sense.
Message: 3076 - Contents - Hide Contents Date: Tue, 8 Jan 2002 11:33:42 Subject: Re: please simplify equation From: monz> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Monday, January 07, 2002 4:31 PM > Subject: [tuning-math] Re: please simplify equation > > > Ratios of the sort (a+br)/(c+dr) define an algebraic number > field, which can always be put into the form of a sum of > rational numbers times powers of a single algebraic number r. > In this case, that results in > > (a+br)/(c+dr) = (ac+ad-bd + (bc-ad)r)/(c^2+cd-d^2)Thanks, Gene! That was the final piece of the puzzle which I needed to complete my new Dictionary entry for "golden meantone": Definitions of tuning terms: golden meantone, ... * [with cont.] (Wayb.) love / peace / harmony ... -monz Yahoo! GeoCities * [with cont.] (Wayb.) "All roads lead to n^0" _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3077 - Contents - Hide Contents Date: Tue, 08 Jan 2002 19:55:32 Subject: Re: please simplify equation From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> Ratios of the sort (a+br)/(c+dr) define an algebraic number >> field, which can always be put into the form of a sum of >> rational numbers times powers of a single algebraic number r. >> In this case, that results in >> >> (a+br)/(c+dr) = (ac+ad-bd + (bc-ad)r)/(c^2+cd-d^2) > >> Thanks, Gene! That was the final piece of the puzzle > which I needed to complete my new Dictionary entry > for "golden meantone":Hmmm...if it's in your dictionary, it might be well to be more precise and point out that for a quadratic number field, like the golden ratio field Q(r), that (a+br)/(c+dr) gives all of the elements, but for a field of degree greater than three that would not be so. In general, however, for a field of degee d, elements of the form a_0 + a_1 r + ... + a_{n-1} r^{d-1} with a_i a rational number and r an algebraic number define the number field Q(r) of degree d, where d is the degree of the irrreducible polynomial satisfied by r. Maybe you should just skip the generalities and say that every number of the form (a+br)/(c+dr) with a,b,c,d rational can be reduced by the formula I gave to a unique form A + Br, with A and B rational, and that Q(r) defined by this is an algebraic number field, analogous to the ordinary rational numbers.
Message: 3078 - Contents - Hide Contents Date: Tue, 8 Jan 2002 12:09:47 Subject: Re: please simplify equation From: monz Hi Gene,> From: genewardsmith <genewardsmith@xxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Tuesday, January 08, 2002 11:55 AM > Subject: [tuning-math] Re: please simplify equation > > > Hmmm...if it's in your dictionary, it might be well to > be more precise and point out that for a quadratic number > field, like the golden ratio field Q(r), that (a+br)/(c+dr) > gives all of the elements, but for a field of degree greater > than three that would not be so. In general, however, for > a field of degee d, elements of the form > a_0 + a_1 r + ... + a_{n-1} r^{d-1} with a_i a rational number > and r an algebraic number define the number field Q(r) of > degree d, where d is the degree of the irrreducible > polynomial satisfied by r. > > Maybe you should just skip the generalities and say that > every number of the form (a+br)/(c+dr) with a,b,c,d rational > can be reduced by the formula I gave to a unique form A + Br, > with A and B rational, and that Q(r) defined by this is an > algebraic number field, analogous to the ordinary rational > numbers.Thanks for the further elaboration! But I hesitate to put all of this into the "golden meantone" definition. Wouldn't it be better as part of the "algebraic number" definition? If the latter, then please suggest how the information specific to each Dictionary entry should be placed and how they should be linked. I'm understanding you as saying that golden meantone is an example of a specific type of algebriac number field, and so the Dictionary entries should be written to reflect that. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3079 - Contents - Hide Contents Date: Tue, 08 Jan 2002 20:37:58 Subject: More 72-et 7-tone scales From: genewardsmith More from a 2401/2400 class [0, 3, 12, 26, 35, 49, 63] [3, 9, 14, 9, 14, 14, 9] edges 4 10 18 connectivity 0 1 4 [0, 3, 12, 26, 35, 49, 58] [3, 9, 14, 9, 14, 9, 14] edges 5 10 17 connectivity 0 1 3 [0, 3, 12, 21, 35, 49, 58] [3, 9, 9, 14, 14, 9, 14] edges 3 9 16 connectivity 0 0 2 [0, 3, 12, 21, 35, 49, 63] [3, 9, 9, 14, 14, 14, 9] edges 3 8 16 connectivity 0 0 3 [0, 3, 12, 26, 40, 49, 58] [3, 9, 14, 14, 9, 9, 14] edges 3 8 15 connectivity 0 1 3 [0, 3, 12, 21, 35, 44, 58] [3, 9, 9, 14, 9, 14, 14] edges 3 8 14 connectivity 0 0 1 [0, 3, 17, 26, 35, 49, 58] [3, 14, 9, 9, 14, 9, 14] edges 4 8 13 connectivity 0 1 2 [0, 3, 12, 26, 35, 44, 58] [3, 9, 14, 9, 9, 14, 14] edges 3 7 13 connectivity 0 1 2 [0, 3, 12, 21, 30, 44, 58] [3, 9, 9, 9, 14, 14, 14] edges 2 6 12 connectivity 0 0 1 [0, 3, 17, 26, 35, 44, 58] [3, 14, 9, 9, 9, 14, 14] edges 2 6 10 connectivity 0 0 2
Message: 3080 - Contents - Hide Contents Date: Tue, 8 Jan 2002 13:13:32 Subject: observation on golden meantone formula From: monz Hmmm ... take a look at this: Yahoo groups: /tuning-math/files/monz/goldenMT... * [with cont.] Here I plotted the values for c = (8a + 11b) from the golden meantone formula (2^a)*(v^b) = 2^[ (c - b*PHI) / 11]. I found two things interesting about this graph. First, it exactly mirrors the placement of the PHI-related intervals in an interval matrix I made, ordered according to generator number. Secondly ... the plot of the "c" values creates a familiar-looking lattice-diagram-type pattern, where the southwest-to-northeast axis is the inverse of the generator (so it's the "4th") and the southeast-to-northwest axis is the "whole tone". What's going on here? Curious... -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3081 - Contents - Hide Contents Date: Tue, 08 Jan 2002 23:15:56 Subject: 72-et classes of scales based on 4375/4374 From: genewardsmith I show the three vals I used to contruct these classes as well this time. 5 tones [12, 7, 23] [1, 2, 2] [[1, 2, 2, 7], [2, 3, 4, 7], [2, 3, 5, 3]] 7 tones [7, 11, 12] [2, 2, 3] [[2, 3, 4, 7], [2, 3, 5, 3], [3, 5, 7, 10]] 8 tones [7, 5, 16] [5, 1, 2] [[5, 8, 11, 17], [1, 2, 2, 7], [2, 3, 5, 3]] 9 tones [7, 12, 9] [6, 1, 2] [[6, 9, 14, 13], [1, 2, 2, 7], [2, 3, 5, 3]] 10 tones [7, 3, 9] [6, 1, 3] [[6, 9, 14, 13], [1, 2, 2, 7], [3, 5, 7, 10]] [7, 5, 9] [7, 1, 2] [[7, 11, 16, 20], [1, 2, 2, 7], [2, 3, 5, 3]] [7, 11, 5] [5, 2, 3] [[5, 8, 11, 17], [2, 3, 5, 3], [3, 5, 7, 10]] [11, 7, 1] [5, 2, 3] [[5, 8, 12, 13], [2, 3, 4, 7], [3, 5, 7, 10]] 11 tones [12, 2, 7] [1, 2, 8] [[1, 2, 2, 7], [2, 3, 5, 3], [8, 12, 19, 16]] 12 tones [7, 5, 2] [9, 1, 2] [[9, 14, 21, 23], [1, 2, 2, 7], [2, 3, 5, 3]] [11, 1, 6] [5, 5, 2] [[5, 8, 12, 13], [5, 8, 11, 17], [2, 3, 4, 7]] [5, 7, 6] [5, 5, 2] [[5, 8, 12, 13], [5, 8, 11, 17], [2, 3, 5, 3]] [7, 4, 5] [7, 2, 3] [[7, 11, 16, 20], [2, 3, 5, 3], [3, 5, 7, 10]] [10, 2, 7] [1, 3, 8] [[1, 2, 2, 7], [3, 5, 7, 10], [8, 12, 19, 16]] 13 tones [8, 2, 7] [1, 4, 8] [[1, 2, 2, 7], [4, 7, 9, 17], [8, 12, 19, 16]] [7, 6, 3] [6, 3, 4] [[6, 9, 14, 13], [3, 5, 7, 10], [4, 7, 9, 17]] [7, 3, 2] [9, 1, 3] [[9, 14, 21, 23], [1, 2, 2, 7], [3, 5, 7, 10]] 14 tones [7, 1, 2] [9, 1, 4] [[9, 14, 21, 23], [1, 2, 2, 7], [4, 7, 9, 17]] [7, 1, 5] [5, 2, 7] [[5, 8, 11, 17], [2, 3, 5, 3], [7, 11, 17, 16]] 15 tones [2, 11, 5] [5, 2, 8] [[5, 8, 11, 17], [2, 3, 5, 3], [8, 13, 18, 27]] [10, 7, 1] [5, 2, 8] [[5, 8, 12, 13], [2, 3, 4, 7], [8, 13, 19, 23]] [4, 7, 1] [5, 7, 3] [[5, 8, 12, 13], [7, 11, 16, 20], [3, 5, 7, 10]] 16 tones [3, 4, 9] [7, 6, 3] [[7, 11, 16, 20], [6, 9, 14, 13], [3, 5, 7, 10]] 17 tones [2, 9, 5] [7, 2, 8] [[7, 11, 16, 20], [2, 3, 5, 3], [8, 13, 18, 27]] [5, 6, 1] [5, 7, 5] [[5, 8, 12, 13], [7, 11, 16, 20], [5, 8, 11, 17]] 19 tones [1, 6, 5] [7, 5, 7] [[7, 11, 16, 20], [5, 8, 11, 17], [7, 11, 17, 16]] [3, 4, 5] [7, 9, 3] [[7, 11, 16, 20], [9, 14, 21, 23], [3, 5, 7, 10]] [1, 6, 3] [6, 9, 4] [[6, 9, 14, 13], [9, 14, 21, 23], [4, 7, 9, 17]] [2, 7, 5] [9, 2, 8] [[9, 14, 21, 23], [2, 3, 5, 3], [8, 13, 18, 27]] 20 tones [3, 7, 1] [5, 7, 8] [[5, 8, 12, 13], [7, 11, 16, 20], [8, 13, 19, 23]]
Message: 3082 - Contents - Hide Contents Date: Tue, 08 Jan 2002 00:31:51 Subject: Re: please simplify equation From: genewardsmith --- In tuning-math@y..., "monz" <joemonz@y...> wrote:>> Not really. My (3r+1)/(5r+1) is (r+9)/19, >> your (8r+1)/(13r+3) is (r+18)/31, and >> Paul's (2r-1)/(3r-1) = (3-r)/(4-r) = (8-r)/11, >> so these are not the same. > >> Can you show me how you work this magic?Ratios of the sort (a+br)/(c+dr) define an algebraic number field, which can always be put into the form of a sum of rational numbers times powers of a single algebraic number r. In this case, that results in (a+br)/(c+dr) = (ac+ad-bd + (bc-ad)r)/(c^2+cd-d^2) This form of the algebraic numbers in the field Q(r) is unique, since {1, r} are a basis for a vector space over the rationals Q; hence we can determine if two elements of Q(r) are the same by putting them both into this form.> First, your (3r+1)/(5r+1) definitely isn't right anyway. > The exponent of 2 has to be ~0.580178728.By taking the continued fraction for .580178728 I get something very close to 1/1+1/1+1/(1+r), which simplifies to (r+7)/11, which is presumably the meantone fifth you are looking for--Kornerup's, I imagine. (8-r)/11 is the conjugate of (7+r)/11, and will give the right answer only if you replace r with r` = -1-r.> So how do you get (8-r)/11 from (2r-1)/(3r-1) and > (3-r)/(4-r), and why are the other solutions incorrect?The division formula I gave should work--one way to derive it is to multiply numerator and denomiantor by the conjugate of the denominator. I think it should do for the rest of your quest also.
Message: 3083 - Contents - Hide Contents Date: Tue, 08 Jan 2002 01:04:20 Subject: Re: please simplify equation From: genewardsmith --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:> By taking the continued fraction for .580178728 I get something > very close to 1/1+1/1+1/(1+r), which simplifies to (r+7)/11, which is presumably the meantone fifth you are looking for--Kornerup's, I imagine. (8-r)/11 is the conjugate of (7+r)/11, and will give the right answer only if you replace r with r` = -1-r.Whups, I goofed--I was trying to use r, but ended up using r`; so it seems (8-r)/11 is correct.
Message: 3084 - Contents - Hide Contents Date: Tue, 08 Jan 2002 03:32:23 Subject: Some 72-et nonatonics From: genewardsmith [0, 9, 18, 30, 39, 44, 53, 58, 67] [9, 9, 12, 9, 5, 9, 5, 9, 5] edges 8 17 27 connectivity 0 2 5 [0, 9, 18, 23, 32, 44, 53, 58, 67] [9, 9, 5, 9, 12, 9, 5, 9, 5] edges 7 17 27 connectivity 0 3 4 [0, 9, 18, 30, 39, 48, 53, 62, 67] [9, 9, 12, 9, 9, 5, 9, 5, 5] edges 9 15 26 connectivity 0 2 5 [0, 9, 18, 30, 39, 44, 53, 62, 67] [9, 9, 12, 9, 5, 9, 9, 5, 5] edges 8 15 26 connectivity 0 2 4 [0, 9, 18, 30, 35, 44, 53, 58, 67] [9, 9, 12, 5, 9, 9, 5, 9, 5] edges 7 16 25 connectivity 0 2 4 [0, 9, 18, 30, 39, 44, 49, 58, 67] [9, 9, 12, 9, 5, 5, 9, 9, 5] edges 8 14 25 connectivity 0 1 4 [0, 9, 18, 27, 39, 48, 53, 62, 67] [9, 9, 9, 12, 9, 5, 9, 5, 5] edges 7 13 25 connectivity 0 1 4 [0, 9, 18, 30, 35, 44, 49, 58, 67] [9, 9, 12, 5, 9, 5, 9, 9, 5] edges 7 15 24 connectivity 0 1 4 [0, 9, 18, 23, 32, 41, 46, 55, 67] [9, 9, 5, 9, 9, 5, 9, 12, 5] edges 6 14 24 connectivity 0 1 3 [0, 9, 18, 27, 32, 41, 53, 62, 67] [9, 9, 9, 5, 9, 12, 9, 5, 5] edges 6 13 24 connectivity 0 1 3 [0, 9, 18, 27, 32, 44, 53, 62, 67] [9, 9, 9, 5, 12, 9, 9, 5, 5] edges 6 13 23 connectivity 0 1 3 [0, 9, 18, 27, 39, 44, 53, 62, 67] [9, 9, 9, 12, 5, 9, 9, 5, 5] edges 6 12 23 connectivity 0 1 4 [0, 9, 18, 27, 32, 41, 53, 58, 67] [9, 9, 9, 5, 9, 12, 5, 9, 5] edges 5 12 22 connectivity 0 1 3
Message: 3085 - Contents - Hide Contents Date: Tue, 08 Jan 2002 09:30:33 Subject: Another 72-et decatonic From: genewardsmith Again, one of these is a clear best-of-class. [0, 5, 14, 23, 26, 35, 44, 49, 58, 63] [5, 9, 9, 3, 9, 9, 5, 9, 5, 9] edges 9 19 33 connectivity 0 2 6
Message: 3087 - Contents - Hide Contents Date: Wed, 09 Jan 2002 21:14:01 Subject: Re: background reading From: genewardsmith --- In tuning-math@y..., "mfeustl" <mfeustl@y...> wrote: A fair amount of the discussion here is still a> little over my head, though-- is there any background reading someone > could suggest to get me up to speed?Some course work would be good. How are you on (1) Linear and multilinear algebra (2) Elementary number theory (3) Abstract algebra Any of these would be good to bone up on. Are you at the University of Toledo? I taught there a few years.
Message: 3088 - Contents - Hide Contents Date: Wed, 9 Jan 2002 14:30:44 Subject: Re: background reading From: monz> From: mfeustl <mfeustl@xxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Wednesday, January 09, 2002 9:55 AM > Subject: [tuning-math] background reading > > > Holy smoke! I did a search on Dowland tuning and I'm delighted to > have found this group. I'm double-majoring in applied math and > instrumental jazz, and I'm always looking for ways to put math and > music together. A fair amount of the discussion here is still a > little over my head, though-- is there any background reading someone > could suggest to get me up to speed?My online Dictionary of Tuning Terms should give you a lot to chew on. Definitions of tuning terms: index, (c) 1998 b... * [with cont.] (Wayb.) And if you haven't already found it, I have a webpage devoted entirely to an exploration of Dowland's tuning. Internet Express - Quality, Affordable Dial Up... * [with cont.] (Wayb.) I just gave a presentation on this in Italy four months ago. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3089 - Contents - Hide Contents
Date: Wed, 09 Jan 2002 06:07:32
Subject: 72-et diatonics vs 46-et diatonics
From: genewardsmith
The first class on my 4375/4374 list was a pentatonic one which we
already did, and the second is the 72-et version of diatonic. I
compare the 72-et version with 46-et in curly braces below it, and
once again a scale with repeated semitones does quite well. This kind
of diatonic seems to deserve some notice. It is also notworthy that it
and some of the other scales actually work better in 72-et than they
do in 46-et.
[0, 12, 23, 35, 46, 58, 65]
[12, 11, 12, 11, 12, 7, 7]
edges 8 11 18 connectivity 1 2 4
{[0, 8, 15, 23, 30, 38, 42]
[8, 7, 8, 7, 8, 4, 4]
edges 8 9 17 connectivity 1 1 4}
[0, 12, 23, 35, 42, 54, 65]
[12, 11, 12, 7, 12, 11, 7]
edges 11 12 17 connectivity 2 2 4
{[0, 8, 15, 23, 27, 35, 42]
[8, 7, 8, 4, 8, 7, 4]
edges 11 11 16 connectivity 2 2 3}
[0, 12, 24, 35, 47, 54, 61]
[12, 12, 11, 12, 7, 7, 11]
edges 6 10 17 connectivity 0 1 3
{[0, 8, 16, 23, 31, 35, 39]
[8, 8, 7, 8, 4, 4, 7]
edges 7 7 15 connectivity 1 1 3}
[0, 12, 23, 35, 46, 53, 65]
[12, 11, 12, 11, 7, 12, 7]
edges 9 10 16 connectivity 2 2 3
{[0, 8, 15, 23, 30, 34, 42]
[8, 7, 8, 7, 4, 8, 4]
edges 9 10 16 connectivity 2 2 3}
[0, 12, 24, 35, 42, 54, 61]
[12, 12, 11, 7, 12, 7, 11]
edges 8 10 16 connectivity 1 2 4
{[0, 8, 16, 23, 27, 35, 39]
[8, 8, 7, 4, 8, 4, 7]
edges 8 8 15 connectivity 1 1 4}
[0, 12, 24, 35, 42, 54, 65]
[12, 12, 11, 7, 12, 11, 7]
edges 9 10 15 connectivity 1 1 3
{[0, 8, 16, 23, 27, 35, 42]
[8, 8, 7, 4, 8, 7, 4]
edges 9 9 15 connectivity 1 1 3}
[0, 12, 24, 35, 47, 54, 65]
[12, 12, 11, 12, 7, 11, 7]
edges 7 9 15 connectivity 0 1 3
{[0, 8, 16, 23, 31, 35, 42]
[8, 8, 7, 8, 4, 7, 4]
edges 8 8 14 connectivity 1 1 3}
[0, 12, 24, 35, 47, 58, 65]
[12, 12, 11, 12, 11, 7, 7]
edges 5 8 15 connectivity 0 1 2
{[0, 8, 16, 23, 31, 38, 42]
[8, 8, 7, 8, 7, 4, 4]
edges 6 7 15 connectivity 0 1 2}
[0, 12, 23, 34, 46, 53, 65]
[12, 11, 11, 12, 7, 12, 7]
edges 8 8 14 connectivity 1 1 3
{[0, 8, 15, 22, 30, 34, 42]
[8, 7, 7, 8, 4, 8, 4]
edges 8 10 16 connectivity 1 2 4}
[0, 12, 24, 35, 46, 58, 65]
[12, 12, 11, 11, 12, 7, 7]
edges 5 7 14 connectivity 0 0 2
{[0, 8, 16, 23, 30, 38, 42]
[8, 8, 7, 7, 8, 4, 4]
edges 5 7 16 connectivity 0 0 3}
[0, 12, 24, 36, 47, 54, 61]
[12, 12, 12, 11, 7, 7, 11]
edges 4 7 14 connectivity 0 0 2
{[0, 8, 16, 24, 31, 35, 39]
[8, 8, 8, 7, 4, 4, 7]
edges 6 7 15 connectivity 1 2 2}
[0, 12, 24, 31, 42, 54, 65]
[12, 12, 7, 11, 12, 11, 7]
edges 8 8 13 connectivity 1 1 2
{[0, 8, 16, 20, 27, 35, 42]
[8, 8, 4, 7, 8, 7, 4]
edges 8 9 15 connectivity 1 1 3}
[0, 12, 24, 35, 42, 53, 65]
[12, 12, 11, 7, 11, 12, 7]
edges 7 8 13 connectivity 0 0 2
{[0, 8, 16, 23, 27, 34, 42]
[8, 8, 7, 4, 7, 8, 4]
edges 7 7 13 connectivity 0 0 3}
[0, 12, 24, 35, 46, 53, 65]
[12, 12, 11, 11, 7, 12, 7]
edges 5 6 12 connectivity 0 0 2
{[0, 8, 16, 23, 30, 34, 42]
[8, 8, 7, 7, 4, 8, 4]
edges 5 6 13 connectivity 0 0 2}
[0, 12, 24, 31, 43, 54, 65]
[12, 12, 7, 12, 11, 11, 7]
edges 6 6 11 connectivity 0 0 1
{[0, 8, 16, 20, 28, 35, 42]
[8, 8, 4, 8, 7, 7, 4]
edges 6 7 13 connectivity 0 0 2}
[0, 12, 24, 36, 47, 54, 65]
[12, 12, 12, 11, 7, 11, 7]
edges 4 5 11 connectivity 0 0 2
{[0, 8, 16, 24, 31, 35, 42]
[8, 8, 8, 7, 4, 7, 4]
edges 5 6 13 connectivity 0 1 2}
[0, 12, 24, 36, 47, 58, 65]
[12, 12, 12, 11, 11, 7, 7]
edges 2 4 11 connectivity 0 0 1
{[0, 8, 16, 24, 31, 38, 42]
[8, 8, 8, 7, 7, 4, 4]
edges 3 5 14 connectivity 0 0 1}
[0, 12, 24, 36, 43, 54, 65]
[12, 12, 12, 7, 11, 11, 7]
edges 4 4 9 connectivity 0 0 1
{[0, 8, 16, 24, 28, 35, 42]
[8, 8, 8, 4, 7, 7, 4]
edges 4 5 12 connectivity 0 0 2}
Message: 3090 - Contents - Hide Contents Date: Thu, 10 Jan 2002 05:25:50 Subject: Re: background reading From: genewardsmith --- In tuning-math@y..., "mfeustl" <mfeustl@y...> wrote:> Hi--Looks like I'm pretty much in the right place at time with > coursework-- I'm taking abstract algebra this semester, and I'm in my > second semester of numerical methods/analysis. And yeah, I'm at UT. > Love it there. :)Go Rockets. :) Who's your prof for abstract algebra?
Message: 3091 - Contents - Hide Contents Date: Thu, 10 Jan 2002 23:34:57 Subject: Re: All in the spirit of friendship, Gene From: paulerlich --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:>but I'll try to work more like this.I think this will be helpful for everyone, and again I apologize for my crankiness -- I got no sleep last night aside from a short piece of music that came to me in a dream -- and departed my mind almost as quickly.>In that spirit, can you explain >if the 27-et hyperpythagorean system is positive, or something else?27-tET is considered a _triply positive_ system. See page 8 of http://www.anaphoria.com/xen2.PDF - Type Ok * [with cont.] (Wayb.) . . .
Message: 3092 - Contents - Hide Contents Date: Thu, 10 Jan 2002 05:38:47 Subject: Re: Some 8-tone 72-et scales From: clumma>I've decided to add the 9-limit numbers to my set of measures; that >gives us a better idea of the nature of these scales, in that we >can see how much of the 11-limit harmony, if any, actually involves >11. Good! > [0, 7, 14, 30, 37, 53, 60, 67] > [7, 7, 16, 7, 16, 7, 7, 5] > edges 11 17 21 22 connectivity 2 3 5 5This scale looks better than it would have. 5 at the 9 limit is better than at the 11 limit. That's why I think we should be normalizing against limit and cardinality of the scale. -Carl
Message: 3093 - Contents - Hide Contents Date: Thu, 10 Jan 2002 18:59:18 Subject: Re: Dictionary query From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 10, 2002 1:56 PM > Subject: [tuning-math] Re: Dictionary query > > > Today, on these lists, we tend to call negative systems "meantone" > and positive systems "schismic". The reason 700 cents was chosen as > the dividing line between "negative" and "positive" is that when the > fifth is below 700 cents, the "meantone" (+4 fifths) approximation to > the 5/4 is better than the "schismic" (-8 fifths) approximation to > the 5/4. When the fifth is above 700 cents, the "schismic" > approximation to the 5/4 is better than the "meantone" approximation > to the 5/4. I might differ, saying that there is a "gray area", and > also factoring the 6/5 into consideration . . . but the definitions > are well-established and there is no reason to favor ones which could > breed potential contradictions. > > As for your definition pages, Monz, they definitely give the wrong > idea. Positive systems should be characterized by the fraction of a > _schisma_ that the fifths differ from just -- this is the relevant > measure of them. Knowing what fraction of a syntonic comma a positive > system's fifth might have been _increased_ by is irrelevant for > understanding the functioning of the system, and is potentially > misleading.Thanks very much for that, Paul. So how does it look now? Definitions of tuning terms: positive system, ... * [with cont.] (Wayb.) -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3094 - Contents - Hide Contents Date: Thu, 10 Jan 2002 06:24:10 Subject: 72-et scale classes based on 385/384 From: genewardsmith Some of the three-step-size scale types have shown up before, but some have not; I notice that Blackjack is also in here. 7 tones [7, 12, 7, 16] [2, 1, 2, 2] [[2, 3, 5, 5, 7], [1, 2, 2, 3, 4], [2, 3, 4, 5, 8], [2, 3, 5, 6, 6]] 8 tones [7, 7, 5, 16] [2, 3, 1, 2] [[2, 3, 5, 5, 7], [3, 5, 6, 8, 12], [1, 2, 2, 3, 4], [2, 3, 5, 6, 6]] 9 tones [7, 12, 7, 4] [2, 3, 2, 2] [[2, 3, 5, 5, 7], [3, 5, 7, 9, 10], [2, 3, 4, 5, 8], [2, 3, 5, 6, 6]] [7, 7, 12, 9] [2, 4, 1, 2] [[2, 3, 5, 5, 7], [4, 6, 9, 11, 14], [1, 2, 2, 3, 4], [2, 3, 5, 6, 6]] 10 tones [7, 5, 7, 9] [5, 1, 2, 2] [[5, 8, 12, 14, 17], [1, 2, 2, 3, 4], [2, 3, 4, 5, 8], [2, 3, 5, 6, 6]] [7, 5, 7, 11] [2, 3, 3, 2] [[2, 3, 5, 5, 7], [3, 5, 7, 9, 10], [3, 5, 6, 8, 12], [2, 3, 5, 6, 6]] [7, 9, 7, 3] [2, 3, 4, 1] [[2, 3, 5, 5, 7], [3, 5, 7, 9, 10], [4, 6, 9, 11, 14], [1, 2, 2, 3, 4]] 11 tones [7, 7, 12, 2] [2, 6, 1, 2] [[2, 3, 5, 5, 7], [6, 9, 14, 17, 20], [1, 2, 2, 3, 4], [2, 3, 5, 6, 6]] [7, 12, 4, 3] [2, 3, 4, 2] [[2, 3, 5, 5, 7], [3, 5, 7, 9, 10], [4, 6, 9, 11, 14], [2, 3, 4, 5, 8]] 12 tones [7, 7, 5, 2] [4, 5, 1, 2] [[4, 6, 9, 11, 14], [5, 8, 12, 14, 17], [1, 2, 2, 3, 4], [2, 3, 5, 6, 6]] [7, 2, 7, 10] [2, 3, 6, 1] [[2, 3, 5, 5, 7], [3, 5, 7, 9, 10], [6, 9, 14, 17, 20], [1, 2, 2, 3, 4]] [5, 7, 7, 4] [3, 5, 2, 2] [[3, 5, 7, 9, 10], [5, 8, 12, 14, 17], [2, 3, 4, 5, 8], [2, 3, 5, 6, 6]] [7, 7, 5, 2] [7, 2, 1, 2] [[7, 11, 16, 20, 24], [2, 3, 5, 5, 7], [1, 2, 2, 3, 4], [2, 3, 5, 6, 6]] [2, 7, 5, 11] [2, 3, 5, 2] [[2, 3, 5, 5, 7], [3, 5, 6, 8, 12], [5, 8, 12, 14, 17], [2, 3, 5, 6, 6]] [5, 7, 2, 9] [3, 5, 2, 2] [[3, 5, 6, 8, 12], [5, 8, 12, 14, 17], [2, 3, 4, 5, 8], [2, 3, 5, 6, 6]] 13 tones [2, 7, 7, 3] [3, 4, 5, 1] [[3, 5, 7, 9, 10], [4, 6, 9, 11, 14], [5, 8, 12, 14, 17], [1, 2, 2, 3, 4]] [7, 7, 2, 3] [7, 2, 3, 1] [[7, 11, 16, 20, 24], [2, 3, 5, 5, 7], [3, 5, 7, 9, 10], [1, 2, 2, 3, 4]] 14 tones [5, 7, 2, 10] [2, 6, 5, 1] [[2, 3, 5, 5, 7], [6, 9, 14, 17, 20], [5, 8, 12, 14, 17], [1, 2, 2, 3, 4]] [5, 2, 7, 7] [3, 4, 5, 2] [[3, 5, 6, 8, 12], [4, 6, 9, 11, 14], [5, 8, 12, 14, 17], [2, 3, 5, 6, 6]] [4, 7, 8, 3] [7, 2, 3, 2] [[7, 11, 16, 20, 24], [2, 3, 5, 5, 7], [3, 5, 7, 9, 10], [2, 3, 4, 5, 8]] [5, 4, 7, 3] [3, 4, 5, 2] [[3, 5, 7, 9, 10], [4, 6, 9, 11, 14], [5, 8, 12, 14, 17], [2, 3, 4, 5, 8]] 15 tones [7, 5, 2, 3] [7, 2, 5, 1] [[7, 11, 16, 20, 24], [2, 3, 5, 5, 7], [5, 8, 12, 14, 17], [1, 2, 2, 3, 4]] 16 tones [4, 3, 9, 4] [2, 7, 3, 4] [[2, 3, 5, 5, 7], [7, 11, 16, 19, 25], [3, 5, 7, 9, 10], [4, 6, 9, 11, 14]] [3, 8, 4, 3] [2, 3, 9, 2] [[2, 3, 5, 5, 7], [3, 5, 7, 9, 10], [9, 14, 21, 25, 31], [2, 3, 4, 5, 8]] [2, 5, 7, 5] [6, 3, 5, 2] [[6, 9, 14, 17, 20], [3, 5, 6, 8, 12], [5, 8, 12, 14, 17], [2, 3, 5, 6, 6]] [2, 5, 7, 5] [6, 4, 5, 1] [[6, 9, 14, 17, 20], [4, 6, 9, 11, 14], [5, 8, 12, 14, 17], [1, 2, 2, 3, 4]] 17 tones [2, 5, 5, 9] [7, 3, 5, 2] [[7, 11, 16, 19, 25], [3, 5, 6, 8, 12], [5, 8, 12, 14, 17], [2, 3, 5, 6, 6]] [2, 5, 7, 3] [7, 4, 5, 1] [[7, 11, 16, 20, 24], [4, 6, 9, 11, 14], [5, 8, 12, 14, 17], [1, 2, 2, 3, 4] ] [4, 1, 7, 3] [7, 3, 5, 2] [[7, 11, 16, 20, 24], [3, 5, 7, 9, 10], [5, 8, 12, 14, 17], [2, 3, 4, 5, 8]] [3, 7, 2, 8] [2, 6, 8, 1] [[2, 3, 5, 5, 7], [6, 9, 14, 17, 20], [8, 13, 19, 22, 28], [1, 2, 2, 3, 4]] [1, 6, 3, 7] [2, 3, 8, 4] [[2, 3, 5, 5, 7], [3, 5, 7, 9, 10], [8, 13, 19, 22, 28], [4, 6, 9, 11, 14]] 18 tones [7, 3, 2, 1] [7, 2, 8, 1] [[7, 11, 16, 20, 24], [2, 3, 5, 5, 7], [8, 13, 19, 22, 28], [1, 2, 2, 3, 4]] 19 tones [4, 4, 3, 5] [7, 2, 7, 3] [[7, 11, 16, 20, 24], [2, 3, 5, 5, 7], [7, 11, 16, 19, 25], [3, 5, 7, 9, 10] ] [5, 4, 3, 3] [3, 9, 5, 2] [[3, 5, 7, 9, 10], [9, 14, 21, 25, 31], [5, 8, 12, 14, 17], [2, 3, 4, 5, 8]] [2, 5, 5, 7] [9, 3, 5, 2] [[9, 14, 21, 25, 31], [3, 5, 6, 8, 12], [5, 8, 12, 14, 17], [2, 3, 5, 6, 6]] [3, 5, 4, 4] [7, 3, 4, 5] [[7, 11, 16, 19, 25], [3, 5, 7, 9, 10], [4, 6, 9, 11, 14], [5, 8, 12, 14, 17 ]] 20 tones [5, 3, 7, 1] [3, 8, 4, 5] [[3, 5, 7, 9, 10], [8, 13, 19, 22, 28], [4, 6, 9, 11, 14], [5, 8, 12, 14, 17 ]] 21 tones [1, 5, 3, 8] [6, 8, 6, 1] [[6, 9, 14, 17, 20], [8, 13, 19, 22, 28], [6, 8, 14, 17, 19], [1, 2, 2, 3, 4 ]] [6, 1, 3, 1] [7, 2, 8, 4] [[7, 11, 16, 20, 24], [2, 3, 5, 5, 7], [8, 13, 19, 22, 28], [4, 6, 9, 11, 14 ]] [2, 5, 2, 5] [6, 9, 5, 1] [[6, 9, 14, 17, 20], [9, 14, 21, 25, 31], [5, 8, 12, 14, 17], [1, 2, 2, 3, 4 ]] 22 tones [4, 3, 1, 4] [7, 7, 3, 5] [[7, 11, 16, 20, 24], [7, 11, 16, 19, 25], [3, 5, 7, 9, 10], [5, 8, 12, 14, 17]] [5, 1, 5, 2] [7, 8, 5, 2] [[7, 11, 16, 20, 24], [8, 13, 19, 22, 28], [5, 8, 12, 14, 17], [2, 3, 4, 5, 8]] [1, 5, 3, 7] [7, 8, 6, 1] [[7, 11, 16, 20, 24], [8, 13, 19, 22, 28], [6, 8, 14, 17, 19], [1, 2, 2, 3, 4]] [2, 5, 2, 3] [7, 9, 5, 1] [[7, 11, 16, 20, 24], [9, 14, 21, 25, 31], [5, 8, 12, 14, 17], [1, 2, 2, 3, 4]] 24 tones [5, 3, 2, 1] [7, 8, 4, 5] [[7, 11, 16, 20, 24], [8, 13, 19, 22, 28], [4, 6, 9, 11, 14], [5, 8, 12, 14 , 17]] [4, 2, 3, 5] [6, 8, 9, 1] [[6, 9, 14, 17, 20], [8, 13, 19, 22, 28], [9, 14, 21, 25, 31], [1, 2, 2, 3, 4]] [5, 3, 1, 6] [3, 8, 9, 4] [[3, 5, 7, 9, 10], [8, 13, 19, 22, 28], [9, 14, 21, 25, 31], [4, 6, 9, 11, 14]] 25 tones [4, 2, 3, 1] [7, 8, 9, 1] [[7, 11, 16, 20, 24], [8, 13, 19, 22, 28], [9, 14, 21, 25, 31], [1, 2, 2, 3 , 4]] 27 tones [5, 2, 1, 3] [7, 7, 8, 5] [[7, 11, 16, 20, 24], [7, 11, 16, 19, 25], [8, 13, 19, 22, 28], [5, 8, 12, 14, 17]] 28 tones [5, 3, 1, 1] [7, 8, 9, 4] [[7, 11, 16, 20, 24], [8, 13, 19, 22, 28], [9, 14, 21, 25, 31], [4, 6, 9, 11 , 14]]
Message: 3095 - Contents - Hide Contents Date: Thu, 10 Jan 2002 00:50:40 Subject: Some 8-tone 72-et scales From: genewardsmith I've decided to add the 9-limit numbers to my set of measures; that gives us a better idea of the nature of these scales, in that we can see how much of the 11-limit harmony, if any, actually involves 11. [0, 7, 14, 30, 37, 53, 60, 67] [7, 7, 16, 7, 16, 7, 7, 5] edges 11 17 21 22 connectivity 2 3 5 5 [0, 7, 14, 30, 37, 44, 60, 67] [7, 7, 16, 7, 7, 16, 7, 5] edges 11 18 21 21 connectivity 1 3 4 4 [0, 7, 14, 21, 37, 44, 60, 67] [7, 7, 7, 16, 7, 16, 7, 5] edges 9 15 19 21 connectivity 0 2 3 4 [0, 7, 14, 21, 37, 53, 60, 67] [7, 7, 7, 16, 16, 7, 7, 5] edges 7 13 18 21 connectivity 0 2 3 5 [0, 7, 14, 21, 37, 44, 49, 56] [7, 7, 7, 16, 7, 5, 7, 16] edges 9 17 19 20 connectivity 1 3 4 4 [0, 7, 14, 21, 37, 44, 51, 56] [7, 7, 7, 16, 7, 7, 5, 16] edges 8 17 18 20 connectivity 0 3 3 4 [0, 7, 14, 21, 37, 44, 51, 67] [7, 7, 7, 16, 7, 7, 16, 5] edges 8 16 18 20 connectivity 0 2 2 4 [0, 7, 14, 21, 28, 44, 51, 67] [7, 7, 7, 7, 16, 7, 16, 5] edges 6 13 15 19 connectivity 0 1 1 3 [0, 7, 14, 21, 28, 44, 60, 67] [7, 7, 7, 7, 16, 16, 7, 5] edges 5 11 15 19 connectivity 0 1 2 3 [0, 7, 14, 21, 28, 35, 51, 56] [7, 7, 7, 7, 7, 16, 5, 16] edges 4 13 13 18 connectivity 0 2 2 3 [0, 7, 14, 21, 28, 44, 51, 56] [7, 7, 7, 7, 16, 7, 5, 16] edges 6 14 15 18 connectivity 0 2 2 4 [0, 7, 14, 21, 28, 35, 51, 67] [7, 7, 7, 7, 7, 16, 16, 5] edges 3 11 13 18 connectivity 0 1 2 3
Message: 3096 - Contents - Hide Contents Date: Thu, 10 Jan 2002 10:02:19 Subject: Still more 72-et decatonics From: genewardsmith More from 4375/4374 [0, 9, 16, 23, 32, 39, 46, 53, 62, 69] [9, 7, 7, 9, 7, 7, 7, 9, 7, 3] edges 14 25 29 36 connectivity 1 3 4 7 [0, 7, 16, 23, 30, 39, 46, 53, 60, 69] [7, 9, 7, 7, 9, 7, 7, 7, 9, 3] edges 14 24 28 36 connectivity 1 3 4 7 [0, 9, 16, 23, 30, 39, 46, 53, 60, 69] [9, 7, 7, 7, 9, 7, 7, 7, 9, 3] edges 13 24 28 36 connectivity 1 3 4 6 [0, 7, 14, 23, 30, 37, 46, 53, 60, 69] [7, 7, 9, 7, 7, 9, 7, 7, 9, 3] edges 13 23 28 35 connectivity 1 2 3 6 [0, 7, 14, 23, 30, 37, 44, 53, 60, 69] [7, 7, 9, 7, 7, 7, 9, 7, 9, 3] edges 12 23 27 35 connectivity 0 1 2 6 [0, 7, 16, 23, 30, 39, 46, 53, 62, 69] [7, 9, 7, 7, 9, 7, 7, 9, 7, 3] edges 13 22 26 35 connectivity 1 2 3 6 [0, 7, 16, 23, 30, 37, 46, 53, 62, 69] [7, 9, 7, 7, 7, 9, 7, 9, 7, 3] edges 11 21 25 35 connectivity 0 1 2 6 [0, 9, 16, 23, 30, 39, 46, 55, 62, 69] [9, 7, 7, 7, 9, 7, 9, 7, 7, 3] edges 11 20 25 35 connectivity 0 2 3 6 [0, 9, 16, 23, 32, 39, 46, 53, 60, 69] [9, 7, 7, 9, 7, 7, 7, 7, 9, 3] edges 11 23 27 34 connectivity 0 2 3 6 [0, 9, 16, 25, 32, 39, 46, 53, 62, 69] [9, 7, 9, 7, 7, 7, 7, 9, 7, 3] edges 11 23 26 34 connectivity 0 2 3 6 [0, 7, 16, 23, 32, 39, 46, 53, 60, 69] [7, 9, 7, 9, 7, 7, 7, 7, 9, 3] edges 10 21 25 34 connectivity 0 2 3 6 [0, 7, 14, 21, 30, 37, 44, 53, 60, 69] [7, 7, 7, 9, 7, 7, 9, 7, 9, 3] edges 11 21 24 33 connectivity 0 1 1 5 [0, 7, 14, 21, 30, 37, 46, 53, 60, 69] [7, 7, 7, 9, 7, 9, 7, 7, 9, 3] edges 10 19 23 33 connectivity 0 2 2 5 [0, 7, 14, 23, 30, 37, 44, 51, 60, 69] [7, 7, 9, 7, 7, 7, 7, 9, 9, 3] edges 9 20 23 32 connectivity 0 0 1 5 [0, 9, 16, 25, 32, 39, 46, 53, 60, 69] [9, 7, 9, 7, 7, 7, 7, 7, 9, 3] edges 8 20 23 32 connectivity 0 2 3 5 [0, 9, 18, 25, 32, 39, 48, 55, 62, 69] [9, 9, 7, 7, 7, 9, 7, 7, 7, 3] edges 10 20 22 32 connectivity 0 0 0 4 [0, 7, 14, 23, 30, 37, 46, 53, 62, 69] [7, 7, 9, 7, 7, 9, 7, 9, 7, 3] edges 10 19 22 32 connectivity 0 1 1 5 [0, 7, 14, 23, 30, 39, 46, 53, 62, 69] [7, 7, 9, 7, 9, 7, 7, 9, 7, 3] edges 10 18 21 32 connectivity 0 2 2 5 [0, 7, 16, 23, 30, 37, 44, 53, 62, 69] [7, 9, 7, 7, 7, 7, 9, 9, 7, 3] edges 8 18 21 32 connectivity 0 0 1 5 [0, 7, 14, 21, 30, 39, 46, 53, 60, 69] [7, 7, 7, 9, 9, 7, 7, 7, 9, 3] edges 8 15 19 32 connectivity 0 1 1 5 [0, 7, 14, 23, 32, 39, 46, 53, 60, 69] [7, 7, 9, 9, 7, 7, 7, 7, 9, 3] edges 7 16 21 32 connectivity 0 1 2 5 [0, 7, 16, 23, 30, 37, 44, 51, 60, 69] [7, 9, 7, 7, 7, 7, 7, 9, 9, 3] edges 8 19 21 31 connectivity 0 1 2 5 [0, 7, 14, 23, 30, 37, 44, 53, 62, 69] [7, 7, 9, 7, 7, 7, 9, 9, 7, 3] edges 9 18 20 31 connectivity 0 0 0 4 [0, 7, 16, 25, 32, 39, 46, 53, 60, 69] [7, 9, 9, 7, 7, 7, 7, 7, 9, 3] edges 7 17 20 31 connectivity 0 1 2 5 [0, 9, 18, 25, 32, 39, 46, 53, 60, 69] [9, 9, 7, 7, 7, 7, 7, 7, 9, 3] edges 6 17 20 31 connectivity 0 1 3 5 [0, 7, 14, 23, 32, 39, 46, 53, 62, 69] [7, 7, 9, 9, 7, 7, 7, 9, 7, 3] edges 8 16 19 31 connectivity 0 1 1 4 [0, 9, 16, 25, 32, 41, 48, 55, 62, 69] [9, 7, 9, 7, 9, 7, 7, 7, 7, 3] edges 7 17 20 30 connectivity 0 1 1 4 [0, 9, 18, 25, 32, 41, 48, 55, 62, 69] [9, 9, 7, 7, 9, 7, 7, 7, 7, 3] edges 8 18 20 29 connectivity 0 0 0 3 [0, 7, 14, 21, 30, 37, 46, 53, 62, 69] [7, 7, 7, 9, 7, 9, 7, 9, 7, 3] edges 7 16 18 29 connectivity 0 1 1 4 [0, 9, 16, 23, 32, 41, 48, 55, 62, 69] [9, 7, 7, 9, 9, 7, 7, 7, 7, 3] edges 6 14 18 29 connectivity 0 1 1 4 [0, 7, 14, 21, 30, 37, 44, 53, 62, 69] [7, 7, 7, 9, 7, 7, 9, 9, 7, 3] edges 8 17 18 28 connectivity 0 0 0 3 [0, 7, 14, 23, 30, 39, 46, 55, 62, 69] [7, 7, 9, 7, 9, 7, 9, 7, 7, 3] edges 7 15 17 28 connectivity 0 2 2 4 [0, 9, 18, 25, 34, 41, 48, 55, 62, 69] [9, 9, 7, 9, 7, 7, 7, 7, 7, 3] edges 5 15 17 28 connectivity 0 0 0 3 [0, 7, 14, 21, 30, 39, 46, 53, 62, 69] [7, 7, 7, 9, 9, 7, 7, 9, 7, 3] edges 7 14 16 28 connectivity 0 1 1 4 [0, 9, 16, 25, 34, 41, 48, 55, 62, 69] [9, 7, 9, 9, 7, 7, 7, 7, 7, 3] edges 4 13 16 28 connectivity 0 1 1 4 [0, 7, 14, 23, 30, 37, 46, 55, 62, 69] [7, 7, 9, 7, 7, 9, 9, 7, 7, 3] edges 7 15 17 27 connectivity 0 1 1 3 [0, 7, 16, 25, 32, 41, 48, 55, 62, 69] [7, 9, 9, 7, 9, 7, 7, 7, 7, 3] edges 5 14 15 26 connectivity 0 0 0 3 [0, 9, 18, 27, 34, 41, 48, 55, 62, 69] [9, 9, 9, 7, 7, 7, 7, 7, 7, 3] edges 4 13 15 26 connectivity 0 0 0 2 [0, 7, 16, 23, 32, 41, 48, 55, 62, 69] [7, 9, 7, 9, 9, 7, 7, 7, 7, 3] edges 4 12 14 26 connectivity 0 1 1 3 [0, 7, 14, 21, 30, 37, 46, 55, 62, 69] [7, 7, 7, 9, 7, 9, 9, 7, 7, 3] edges 5 13 14 25 connectivity 0 1 1 3 [0, 7, 14, 21, 30, 39, 46, 55, 62, 69] [7, 7, 7, 9, 9, 7, 9, 7, 7, 3] edges 5 12 13 25 connectivity 0 1 1 3 [0, 7, 16, 25, 34, 41, 48, 55, 62, 69] [7, 9, 9, 9, 7, 7, 7, 7, 7, 3] edges 3 11 12 23 connectivity 0 0 0 2 [0, 7, 14, 23, 32, 41, 48, 55, 62, 69] [7, 7, 9, 9, 9, 7, 7, 7, 7, 3] edges 3 10 11 21 connectivity 0 0 1 2 [0, 7, 14, 21, 30, 39, 48, 55, 62, 69] [7, 7, 7, 9, 9, 9, 7, 7, 7, 3] edges 4 10 10 21 connectivity 0 0 0 3
Message: 3098 - Contents - Hide Contents Date: Thu, 10 Jan 2002 19:13:58 Subject: Re: Dictionary query From: monz> From: paulerlich <paul@xxxxxxxxxxxxx.xxx> > To: <tuning-math@xxxxxxxxxxx.xxx> > Sent: Thursday, January 10, 2002 7:05 PM > Subject: [tuning-math] Re: Dictionary query > >>> Thanks very much for that, Paul. So how does it look now? >> Definitions of tuning terms: positive system, ... * [with cont.] (Wayb.) >> >> >> >> -monz >> Unfortunately, 22 is not a schismic temperament . . . this is my > fault, of course . . . I later alluded to the correct definition in > conversation with Gene, as you can see . . . I'm a bit too tired to > correct this now, but I'm sure Graham or John Chalmers can help you > if they're available before I can get back to you.OK, but if they don't post anything here, please do give me more info.> P.S. Monz, why do you like to keep incomplete/incorrect > definitions/descriptions at the top of your dictionary pages, or even > in there at all? Why not attempt for the more precise, univerally > agreed-on definitions/examples first, and then post > alternate/intermediary-stages-in-someone's-thinking stuff later, > preferably on entirely separate webpages?Umm... because I'm a decent author but a lousy editor? The Dictionary is always a work-in-progress, and I prefer to simply amend definitions that are not complete. But if they really are *incorrect*, then please, by all means, not only give me the correct information, but also tell me what to get rid of! I have no problem deleting something that really is wrong. (I may not get around to it as quickly as any of us would like... but that's another story...) If even one other person here agrees with you that the commatic description of positive systems is absolutely useless, then they're history. Let me know. -monz _________________________________________________________ Do You Yahoo!? Get your free @yahoo.com address at Yahoo! Mail - The best web-based email! * [with cont.] (Wayb.)
Message: 3099 - Contents - Hide Contents Date: Thu, 10 Jan 2002 19:44:44 Subject: Re: Optimal 5-Limit Generators For Dave From: paulerlich --- In tuning-math@y..., graham@m... wrote:> Paul:>> It should be quite straightforward to prove. How could you tell >> whether 50:49 produces torsion or not in an octave-invariant >> formulation? >> Do you care about it being [dis]proven, then?Sure, but this works pretty well as a "proof" for me.> I expect your algorithm for > generating periodicity blocks will solve everything.How so? It doesn't detect torsion . . . I needed Gene's fix, which takes the powers of 2 into account, to do that.> But I haven't looked > it up because people keep saying they aren't interested, while >asking more > and more questions.Looked what up?> It won't change anything musically.Not sure what you mean.> > > Paul:>> I thought Gene showed that the common-factor rule only works in the >> octave-specific case. >> I don't remember him considering the adjoint, rather than the wedge > product. But we may not need it anyway.Gene, any enlightenment?> > Me:>>> Pairs of ETs with >>> torsion don't work with wedge products either. It may be that the>> sign of the>>> mapping can be used to disambiguate them. Otherwise, give the>> range of generators>>> as part of the definition. > > Paul:>> You've lost me. Gene, any comments? >> Meaning contorsion here. The octave-specific wedge product can remove it, > but not use it. An octave-equivalent wedge product (the octave- equivalent > mapping) will treat such systems, wrongly, as requiring a division of the > octave. But starting from ETs it does make more sense to use > octave-specific vectors in the first place. Perhaps we should only ask if > unison vectors can work in an octave-equivalent system, in which case this > problem doesn't apply. > > Me:>>> Wouldn't it be nice to say whether or not Fokker's methods would>> have worked if he>>> had run into torsion? > > Paul:>> I'm pretty sure the answer is no. Gene? >> The main thing we've added to Fokker (after Wilson) is the mapping, > instead of merely counting the number of notes in the periodicity block. > The Monz-shruti example gives a periodicity block with more notes than you > need for the temperament, but the mappings still come out.What do you mean, "the mappings still come out"? The 3:2, for example, is not always represented by the same number of steps.> There are more > insidious examples of torsion where the mappings don't work either. The > problem being that octave-equivalent matrices don't differentiate commatic > torsion from systems requiring a period that isn't the octave.Exactly my point.
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