Tuning-Math Archive Section 11: 10325

Message: 10325 - Contents - Hide Contents Date: Tue, 17 Feb 2004 19:30:17 Subject: Re: top23 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:

> This looks reasonable. Let's go back to the top 23 from Gene's >114...

Gene was using the L_infinity norm of the wedgie there, but never explained why. I used the L_1 norm because that gives you the (hyper) taxicab cross-sectional area of the periodicity unit of the temperament in the Tenney lattice. I'll assume that Gene had some reason for using L_infinity . . . It seems that in all the cases we've looked at, only 7-limit linear's wedgie is "rich" enough so that L_infinity and L_1 don't give virtually identical results . . .

Message: 10326 - Contents - Hide Contents Date: Wed, 18 Feb 2004 18:04:34 Subject: Re: top23 From: Paul Erlich --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> > wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: >>

>>> This looks reasonable. Let's go back to the top 23 from Gene's >>> 114... >>

>> Gene was using the L_infinity norm of the wedgie there, but never >> explained why. >

> It was one of the two obvious choices, and since a linear temperament > is always two vals wedged together, I picked a val-based definition.

I don't get it. Why would a val-based definition lead you to use the L_infinity norm?

Message: 10329 - Contents - Hide Contents Date: Wed, 18 Feb 2004 02:14:55 Subject: Canonical generators for 7-limit planar temperaments From: Gene Ward Smith My ISP conked out, so I installed Juno's free connection, which I recommend for temporary uses like this. It has an incredibly obnoxious ad which can be placed over the top of Yahoo's obnoxious ad. If we have a comma which does not (as 50/49, for instance, will) lead to a part-octave period, we can find generators which are analogous to the period-and-generator generator of a linear temperament, by projecting orthogonally onto a plane in a way which makes the comma dissapper (people may recall my lattice diagrams of this.) We can now do a Minkowski reduction--pick the class representative in the range 1 < q < sqrt(2) which is closest to the origin, as projected, and which has the smallest Tenney norm. Now do the same for second closest. We end up with two rational numbers 1 < q, r < sqrt(2) which serve as generators for the temperament. If we put these together with 2 and the comma, we get a unimodular matrix; inverting this gives a matrix whose columns are the four vals I list. This shows how to map a 7-limit note to octave, the two generators, and the comma; we get the temperament by dropping the comma and retuning. Below you find the comma, then the three generators, followed by the square of the distance from the origin of the two non-octave generators when projected, followed by the ratio of second-nearest to nearest square size. This can be all the way from the same (225/224 and 2401/2400) to quite large (schisma.) On the line below I give the map. 64/63 {2, 3/2, 5/4} [1/7, 5/7] 5 [<1 1 2 4|, <0 -1 0 2|, <0 0 1 0|, <0 0 0 -1|] 81/80 {2, 4/3, 9/7} [1/13, 9/13] 9 [<1 1 0 2|, <0 -1 -4 -2|, <0 0 0 -1|, <0 0 -1 0|] 245/243 {2, 9/7, 7/6} [3/17, 5/17] 5/3 [<1 1 1 2|, <0 1 3 1|, <0 1 1 2|, <0 0 1 0|] 126/125 {2, 6/5, 5/4} [1/7, 5/7] 5 [<1 2 2 1|, <0 1 0 -2|, <0 1 1 1|, <0 0 0 1|] 225/224 {2, 4/3, 15/14} [1/3, 1/3] 1 [<1 1 3 3|, <0 -1 1 0|, <0 0 -1 -2|, <0 0 1 1|] 1728/1715 {2, 7/6, 36/35} [1/5, 1/2] 5/2 [<1 2 3 3|, <0 -2 -3 -1|, <0 1 0 1|, <0 -1 -1 -1|] 1029/1024 {2, 8/7, 35/32} [7/10, 37/10] 37/7 [<1 4 3 2|, <0 3 1 -1|, <0 0 1 0|, <0 1 0 0|] 3136/3125 {2, 28/25, 168/125} [1/19, 18/19] 18 [<1 1 2 2|, <0 1 2 5|, <0 1 0 0|, <0 -1 -1 -2|] 5120/5103 {2, 4/3, 27/20} [1/37, 25/37] 25 [<1 2 4 2|, <0 -1 -3 3|, <0 0 -1 -1|, <0 0 0 -1|] 6144/6125 {2, 35/32, 5/4} [1/5, 1/3] 5/3 [<1 1 2 3| <0 2 0 1|, <0 1 1 -1|, <0 1 0 0|] 32805/32768 {2, 4/3, 448/405} [1/73, 57/73] 57 [<1 2 -1 1|, <0 -1 8 4|, <0 0 0 1|, <0 0 1 1|] 2401/2400 {2, 7/5, 49/40} [3/11, 3/11] 1 [<1 1 3 3|, <0 0 -2 -1|, <0 2 1 1|, <0 -1 0 0|] 4375/4374 {2, 27/25, 10/9} [1/7, 6/35] 6/5 [<1 2 3 3|, <0 -1 -2 1|, <0 -2 -3 -2|, <0 0 0 1|]

Message: 10330 - Contents - Hide Contents Date: Wed, 18 Feb 2004 22:19:26 Subject: Re: Canonical generators for 7-limit planar temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> Just one question: What is "a 7-limit note" that is mapped by the > four vals listed? Everything else makes sense. Also, how do you > get the Temperament - by dropping the comma and retuning. Say for > 81/80

The "7-limit note" is just a positive rational number of the 7-limit. It can be written as a product of prime numbers, but it can also be written in terms of the octave, generator or generators, and comma or commas. So, for instance, for 81/80-planar, we could use any product 2^a (4/3)^b (9/7)^c (81/80)^d to represent a 7-limit note. We temper by dropping the 81/80 out of the product, and retuning 2, 4/3 and 9/7. The same comment applies to linear temperaments, as for instance 2^a (4/3)^b (81/80)^c (126/125)^d, where we drop both commas.

Message: 10331 - Contents - Hide Contents Date: Wed, 18 Feb 2004 02:17:36 Subject: Re: top23 From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul Erlich" <perlich@a...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote: >

>> This looks reasonable. Let's go back to the top 23 from Gene's >> 114... >

> Gene was using the L_infinity norm of the wedgie there, but never > explained why.

It was one of the two obvious choices, and since a linear temperament is always two vals wedged together, I picked a val-based definition.

Message: 10332 - Contents - Hide Contents Date: Wed, 18 Feb 2004 22:23:07 Subject: More on chord square planar scales From: Gene Ward Smith Below I give these for 3x3 and 5x5 squares of 225/224 and 2401/2400 planar. I'm using odd numbers since this removes any dependence on the actual generators chosen, though looking at the whole list wouldn't be that bad for even sized squares. Now, however, I only have a prime and inverted form to worry about. What's below is a Maple file which can be read directly into Maple if you have it. # 3x3 and 5x5 chord squares for 225/224, [0,0,1] and [0,1,2] chord generators s3_225 := [[0, 0, 0], [0, 0, -1], [0, 0, 1], [0, 1, 2], [0, -1, -2], [0, -1, -3], [0, 1, 1], [0, -1, -1], [0, 1, 3]]: s5_225 := [[0, 0, 0], [0, -2, -2], [0, 2, 2], [0, 0, -1], [0, 0, 1], [0, 0, -2], [0, 0, 2], [0, 1, 2], [0, -2, -4], [0, -2, -6], [0, -1, - 2], [0, -1, -4], [0, 1, 0], [0, 2, 4], [0, -2, -5], [0, -1, -3], [0, 1, 1], [0, 2, 3], [0, -2, -3], [0, -1, -1], [0, 1, 3], [0, 2, 5], [0, -1, 0], [0, 1, 4], [0, 2, 6]]: # 3x3 and 5x5 chord squares for 2401/2400, [1,1,-1] and [1,1,-2] chord generators s3_2401 := [[0, 0, 0], [1, 1, -1], [1, 1, -2], [-1, -1, 1], [-1, -1, 2], [-2,-2, 3], [0, 0, -1], [0, 0, 1], [2, 2, -3]]: s5_2401 := [[0, 0, 0], [-2, -2, 2], [2, 2, -2], [1, 1, -1], [1, 1, - 2], [-1, -1, 1], [-1, -1, 2], [-2, -2, 3], [0, 0, -1], [0, 0, 1], [2, 2, - 3], [-2, -2, 4], [-4, -4, 6], [-3, -3, 4], [-1, -1, 0], [2, 2, -4], [0, 0, -2], [-3, -3, 5],[1, 1, -3], [-1, -1, 3], [3, 3, -5], [0, 0, 2], [1, 1, 0], [3, 3, -4], [4, 4,-6]]: # corresponding 7-limit scales of notes for above chord lists # 225/224 JI # 20 notes c3_225 := [1, 225/224, 15/14, 49/45, 9/8, 7/6, 135/112, 56/45, 5/4, 21/16, 4/3, 7/5, 45/32, 3/2, 14/9, 45/28, 5/3, 7/4, 28/15, 15/8]: # 44 notes c5_225 := [1, 225/224, 28/27, 21/20, 135/128, 15/14, 49/45, 10/9, 9/8, 2025/1792, 784/675, 7/6, 135/112, 56/45, 5/4, 2025/1568, 21/16, 4/3, 75/56, 2744/2025, 7/5, 45/32, 10/7, 196/135, 3/2, 675/448, 3136/2025, 14/9, 45/28, 10125/6272, 49/30, 224/135, 5/3, 27/16, 675/392, 392/225, 7/4, 16/9, 405/224, 28/15,15/8, 784/405, 6075/3136, 63/32]: # 225/224-reduced using <2, 4/3, 15/14> as generators # 16 notes k3_225 := [[0, 0, 0], [0, 0, 1], [0, 1, -3], [1, -2, 0], [0, 1, -2], [1, -2, 1], [0, 1, -1], [1, -1, -2], [0, 1, 0], [1, -1, -1], [1, -1, 0], [0, 2, -2], [1, -1, 1], [0, 2, -1], [1, 0, -2], [1, 0, -1]]: # 31 notes k5_225 := [[0, 0, 0], [-1, 3, -2], [1, -2, -1], [0, 0, 1], [0, 1, - 3], [-1, 3,-1], [1, -2, 0], [0, 1, -2], [1, -2, 1], [0, 1, -1], [1, -2, 2], [1, -1, -2], [0, 1, 0], [0, 2, -4], [1, -1, -1], [0, 1, 1], [0, 2, -3], [1, -1, 0], [0, 2,-2], [1, -1, 1], [1, 0, -3], [0, 2, -1], [2, -3, 0], [1, -1, 2], [1, 0, - 2], [0, 2, 0], [2, -3, 1], [1, 0, -1], [0, 3, -3], [2, -3, 2], [2, -2, - 2]]: # 2401/2400 JI # 25 notes c3_2401 := [1, 2401/2400, 49/48, 360/343, 15/14, 7/6, 60/49, 49/40, 5/4, 2401/1920, 450/343, 7/5, 10/7, 343/240, 3600/2401, 3/2, 75/49, 49/32, 49/30, 2401/1440, 12/7, 600/343, 7/4, 90/49, 15/8]: # 58 notes c5_2401 := [1, 2401/2400, 49/48, 117649/115200, 864000/823543, 360/343, 18000/16807, 15/14, 343/320, 35/32, 2700/2401, 8/7, 343/300, 225/196, 7/6, 16807/14400, 823543/691200, 60/49, 49/40, 5/4, 2401/1920, 21600/16807, 1080000/823543, 450/343, 21/16, 2401/1800, 75/56, 117649/86400, 480/343, 7/5, 10/7, 343/240, 823543/576000, 16807/11520, 3600/2401, 3/2, 75/49, 49/32, 1296000/823543, 27000/16807, 45/28, 49/30, 117649/72000, 2401/1440, 5764801/3456000, 28800/16807, 12/7, 600/343, 7/4, 16807/9600, 343/192, 216000/117649, 90/49, 28/15, 4500/2401, 15/8, 343/180, 823543/432000]: # 2401/2400-reduced using <2, 7/5, 49/40> as generators # 18 notes k3_2401 := [[0, 0, 0], [1, -2, 0], [-1, 1, 2], [0, -1, 2], [1, -1, - 1], [0, 0, 1], [1, -2, 1], [0, -1, 3], [0, 1, 0], [1, -1, 0], [0, 0, 2], [1, -2, 2], [1, 0, -1], [2, -2, -1], [0, 1, 1], [1, -1, 1], [0, 0, 3], [1, -2, 3]]: # 33 notes k5_2401 := [[0, 0, 0], [1, -2, 0], [-1, 1, 2], [0, -1, 2], [1, -3, 2], [-1, 0,4], [0, 1, -1], [0, -2, 4], [1, -1, -1], [2, -3, -1], [0, 0, 1], [1, -2, 1], [-1, 1, 3], [0, -1, 3], [1, 0, -2], [1, -3, 3], [2, -2, - 2], [0, 1, 0], [1, -1,0], [2, -3, 0], [0, 0, 2], [1, -2, 2], [-1, 1, 4], [0, -1, 4], [1, 0, -1], [2,-2, -1], [0, 1, 1], [1, -1, 1], [2, -3, 1], [0, 0, 3], [1, 1, -2], [1, -2, 3], [2, -1, -2]]:

Message: 10333 - Contents - Hide Contents Date: Wed, 18 Feb 2004 06:39:23 Subject: Re: Canonical generators for 7-limit planar temperaments From: Gene Ward Smith Here is the same list, sorted by Graham-style complexity. By this I mean we take the Graham complexities individually of the two generators, and multiply them, giving the area of the minimum rectangle which contains a tetrad, if the generators are arranged in a square grid. It is striking that 81/80 and 2401/2400 have the same complexity! Complexity 3 64/63 {2, 3/2, 5/4} [1/7, 5/7] 5 [<1 1 2 4|, <0 -1 0 2|, <0 0 1 0|, <0 0 0 -1|] 126/125 {2, 6/5, 5/4} [1/7, 5/7] 5 [<1 2 2 1|, <0 1 0 -2|, <0 1 1 1|, <0 0 0 1|] 1728/1715 {2, 7/6, 36/35} [1/5, 1/2] 5/2 [<1 2 3 3|, <0 -2 -3 -1|, <0 1 0 1|, <0 -1 -1 -1|] Complexity 4 81/80 {2, 4/3, 9/7} [1/13, 9/13] 9 [<1 1 0 2|, <0 -1 -4 -2|, <0 0 0 -1|, <0 0 -1 0|] 225/224 {2, 4/3, 15/14} [1/3, 1/3] 1 [<1 1 3 3|, <0 -1 1 0|, <0 0 -1 -2|, <0 0 1 1|] 1029/1024 {2, 8/7, 35/32} [7/10, 37/10] 37/7 [<1 4 3 2|, <0 3 1 -1|, <0 0 1 0|, <0 1 0 0|] 6144/6125 {2, 35/32, 5/4} [1/5, 1/3] 5/3 [<1 1 2 3| <0 2 0 1|, <0 1 1 -1|, <0 1 0 0|] 2401/2400 {2, 7/5, 49/40} [3/11, 3/11] 1 [<1 1 3 3|, <0 0 -2 -1|, <0 2 1 1|, <0 -1 0 0|] Complexity 5 3136/3125 {2, 28/25, 168/125} [1/19, 18/19] 18 [<1 1 2 2|, <0 1 2 5|, <0 1 0 0|, <0 -1 -1 -2|] Complexity 6 245/243 {2, 9/7, 7/6} [3/17, 5/17] 5/3 [<1 1 1 2|, <0 1 3 1|, <0 1 1 2|, <0 0 1 0|] 5120/5103 {2, 4/3, 27/20} [1/37, 25/37] 25 [<1 2 4 2|, <0 -1 -3 3|, <0 0 -1 -1|, <0 0 0 -1|] Complexity 9 4375/4374 {2, 27/25, 10/9} [1/7, 6/35] 6/5 [<1 2 3 3|, <0 -1 -2 1|, <0 -2 -3 -2|, <0 0 0 1|] 32805/32768 {2, 4/3, 448/405} [1/73, 57/73] 57 [<1 2 -1 1|, <0 -1 8 4|, <0 0 0 1|, <0 0 1 1|]

Message: 10335 - Contents - Hide Contents Date: Wed, 18 Feb 2004 10:02:11 Subject: Chord lattice generators for 7-limit planars From: Gene Ward Smith In just the same way as for note-classes, we can project the 3D lattice of 7-limit chords to a 2D lattice, and find a Minkowski reduced basis for it, with a shortest and second-shortest size for the generators. I don't see a convincing way to decide between the + and - version of a generator and so normalize as I did before, but I suppose some definition could be given. For examples, consider 225/224 and 2401/2400; the first line is the reduced generator pair, and the second a unimodular lattice which transforms a chord in the usual basis to the two generators, plus a major tetrad on 225/224 or 2401/2400 respectively. This kind of basis can be used to define scales, by taking square arrays of chords in terms of the generators. Below I give the 3x3 square for 2401/2400, which boils 25 JI notes down to 18. The 18 notes to the octave give five major and six minor tetrads, and three supermajor and three subminor tetrads. Inverting, of course, gives six major and five minor tetrads. All of this, of course, is with 2401/2400-planar accuracy, which means effectively JI. 225/224 [[0,0,1], [0,1,2]] [[-2,-2,1], [-1,1,0], [1,0,0]] 2401/2400 [[1,1,-1],[1,1,-2]] [[3,-1,1], [0,-1,-1], [-1,1,0]] ! sqoo.scl 3x3 chord square, 2401/2400 projection of tetrad lattice (612-et tuning) 18 ! 35.294118 84.313725 119.607843 266.666667 350.980392 386.274510 470.588235 582.352941 617.647059 701.960784 737.254902 849.019608 884.313726 933.333333 968.627451 1052.941176 1088.235294 1200.000000 ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 10336 - Contents - Hide Contents Date: Thu, 19 Feb 2004 19:39:56 Subject: Re: Canonical generators for 7-limit planar temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> So, is a,b,c,d a column or a row? (1,1,2,4) or (1,0,0,0) to use the > first one...

I'm not sure which mapping you are looking at--64/63 planar? For 81/80 planar, the val <1 1 0 2| gives you the exponent "a" in the product 2^a (3/2)^b (9/7)^c (81/80)^d, <0 -1 -4 -2| gives "b", <0 0 0 -1| gives you "c", and <0 0 -1 0| gives "d". These vals can be found by inverting the matrix whose rows are the four monzos for the generators.

Message: 10338 - Contents - Hide Contents Date: Thu, 19 Feb 2004 22:30:24 Subject: Re: Canonical generators for 7-limit planar temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> How does a whole val like <1 1 0 2| become an exponent. (Sorry I > really thought I understood this)

That should have been <1 2 4 4|. Suppose we want the product for 5/4; as a monzo, it is |-2 0 1 0>, and we find the exponent for 2 by <1 2 4 4|-2 0 1 0> = 2 Similarly, we get <0 -1 -4 -2|-2 0 1 0> = -4 <0 0 0 -1|-2 0 1 0> = 0 <0 0 -1 0|-2 0 1 0> = -1 and the product is 2^2 (4/3)^(-4) (81/80)^(-1) = 5/4

Message: 10341 - Contents - Hide Contents Date: Fri, 20 Feb 2004 03:12:36 Subject: Semisixths From: Gene Ward Smith As a 5-limit temperament, semisixths is defined by the semisixths comma, or medium semicomma, of 78732/78125. This gives a decent 5-limit temperament, but the generator makes a sharp 9/7 and so we are immediately led onwards to the 7-limit. Here we have two commas, 126/125 and 245/243. Two generators now get us to a 5/3, and so we have a 9/7-9/7-6/5 magic triad as a ubiquituous semisixths chord, based on (5/3)/(9/7)^2 = 245/243. Of course as usual 126/125 gives us 6/5-6/5-6/5-7/6 diminished seventh chords as a natural part of the system. We get a decent 11-limit system by adding 176/175 as a comma, but this gets us up to a Graham complexity of 31; in the 7-limit the Graham complexity is 13, and in the 9-limit 14, so Semisixths[19] is plentifully supplied with tetrads. The 7-limit tetrads, of course, project down to a one-dimensional system; in the case of semisixths the generator is [1, -1, -1]; this corresponds to the 7/6-35/24-5/3-35/18 chord. Organzing the tetrads by the generator is a useful start towards organizing your understanding of the harmony of a 7-limit linear temperaments in my experience.

Message: 10342 - Contents - Hide Contents Date: Fri, 20 Feb 2004 07:14:46 Subject: Hemiwuerschmidt From: Gene Ward Smith While I'm pondering what tuning system to try next, I've got an opportunity to present the sort of stuff I like to work out in advance. Here is hemiwuerschmidt, giving the tetrad generator of [2,1,0] and the tetrads within various DE scales (19, 25, 31, 37 notes) in terms of that generator. Then something which is especially nice when it is symmetric, but useful even when it isn't, as here--the number of common notes of two tetrads, separated by n generators. To get the same thing for a minor tetrad (which will represented by an odd number, if 0 is a major tetrad) we read the table in the minus direction for plus generator steps. For example, I want to know the number of notes in common for tetrad 5 and tetrad 9, where 5 is odd and hence is a minor tetrad; 9-5=4, so I look up -4, and see it has one common note with 0; hence 5 will have a common note with 9, +4 steps up. Generator: 28/25 TM basis: <2401/2400, 3136/3125> Tetrad generator: [2, 1, 0] Tetrads in DE scales s19: [0, 2, 4, 9, 11, 13] s25: [0, 2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 21, 23, 25] s31: [0, 2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 35, 37] s37: [0, 2, 4, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 43, 45, 47, 49] Common notes of tetrads -32: 1 -31: 0 -30: 0 -29: 0 -28: 1 -27: 0 -26: 0 -25: 0 -24: 0 -23: 1 -22: 1 -21: 0 -20: 0 -19: 2 -18: 0 -17: 0 -16: 0 -15: 1 -14: 0 -13: 2 -12: 0 -11: 0 -10: 1 -9: 2 -8: 0 -7: 0 -6: 1 -5: 0 -4: 1 -3: 1 -2: 0 -1: 0 0: 4 1: 0 2: 0 3: 0 4: 1 5: 0 6: 1 7: 0 8: 0 9: 2 10: 1 11: 0 12: 0 13: 2 14: 0 15: 0 16: 0 17: 0 18: 0 19: 2 20: 0 21: 0 22: 1 23: 0 24: 0 25: 0 26: 0 27: 0 28: 1 29: 0 30: 0 31: 0 32: 1 ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 10344 - Contents - Hide Contents Date: Sat, 21 Feb 2004 15:38:42 Subject: Re: poincare duality From: Herman Miller Gene Ward Smith wrote:

> By the way, there's a discussion of this page > > Exterior power - Wikipedia, the free encyclopedia * [with cont.] (Wayb.) = > > Exterior power - Wikipedia, the free encyclopedia * [with cont.] (Wayb.) > > on the corresponding talk page ("discuss this page.") I was thinking > of writing something on the wedge product which could actually be > understandable by non-mathematicians, and Charles was not too keen on > the idea. However, I see that immediately afterward he moved material > from the Grassmann algebra page to this one, giving it an entire new > introductory section, and thereby making it more concrete, just as we > had discussed. I'd be interested if people here could tell me if it > makes any sense to them, and how much.

The definition there seems easier to understand than the one from Mathworld that was posted way back, but that could just be that I've spent enough time looking at actual wedgies used in temperaments and seeing how they relate to commas and maps that the concept in general makes more sense now. The problem is that this page describes in an abstract sense what a wedge product is rather than explaining how to calculate it. What would be more useful to musicians is a page describing wedge products strictly as applied to vals and monzos, rather than the more general definition that mathematicians need, with clear algorithms and examples of how to wedge vals to get linear temperaments, how to find commas and maps from wedgies, and anything else that might be of interest in making music. I'd also start with the 7-limit examples, since it's in the 7-limit that wedgies start being useful in representing temperaments. One problem with the The Wedge Product * [with cont.] (Wayb.) page is that the 7-limit examples are the last thing on the page. I think that any page intended for musicians should start with these 7-limit definitions of monzo product and val product, with examples of familiar temperaments like meantone, and progress from there to explain how to generalize these definitions and use them for different kinds of temperaments. And technical terms such as "abelian group", which aren't immediately relevant to the musical use of wedge products, should not be in an introductory page. The Wiki page starts right out with references to tensor products and concepts from Grassman algebra theory that aren't of much interest to someone wanting to figure out the mathematics of temperaments, so it's even less useful than the Xenharmony page in that respect. ________________________________________________________________________ ________________________________________________________________________ ------------------------------------------------------------------------ Yahoo! Groups Links <*> To visit your group on the web, go to: Yahoo groups: /tuning-math/ * [with cont.] <*> To unsubscribe from this group, send an email to: tuning-math-unsubscribe@xxxxxxxxxxx.xxx <*> Your use of Yahoo! Groups is subject to: Yahoo! Terms of Service * [with cont.] (Wayb.)

Message: 10345 - Contents - Hide Contents Date: Sat, 21 Feb 2004 01:32:31 Subject: Re: What the numbers mean From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> Gene, is there a way to calculate the period values, or is it just > done "ad hoc?" (I had the same question regarding Graham's matrices, > I guess the answer would apply to both situations) Thanx

I don't know what you mean by the period values, but I'm sure there's a way to calculate it. What, exactly, is a period value--the number of periods to an octave in a linear temperament?

Message: 10347 - Contents - Hide Contents Date: Sat, 21 Feb 2004 02:54:35 Subject: JIP From: Paul Erlich /root/tentop.htm * [with cont.] (Wayb.) If the JIP is not a point in the original space it operates on, then it probably shouldn't be referred to as a point. Rather, it seems to measure pitch, so why not refer to it as PITCH or something? (You probably asked me if I preferred that already but I had even less idea what it meant then.)

Message: 10348 - Contents - Hide Contents Date: Sat, 21 Feb 2004 03:06:03 Subject: Who's Val, anyway? From: Paul Erlich I sympathize with the confusion over the term "Val". You have named its dual after our friend Monz. Graham's approach has always been, I think, to work with these directly, so why not refer to them with the term "Breed" (which happens to be descriptive as well!)?

Message: 10349 - Contents - Hide Contents Date: Sat, 21 Feb 2004 03:20:33 Subject: Re: What the numbers mean From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad" <paul.hjelmstad@u...> wrote:

> I mean the like <1 1 3 3| below. So actually the number of periods > in a mapping (what I should have said). I this case of course they > are octaves. > > [<1 1 3 3|, <0 6 -7 -2|],

The first 1 of <1 1 3 3| is the number of periods per octave; in this case the octave is the period. The others depend on precisely which generator you choose--is it to be an approximate 15/14, or maybe a 28/15 or 15/7 instead? If it is a usual secor, s, then we must choose a so that a+6s is an approximate 3, and so forth. Very recently I posted how, if you know the wedgie, solving for x,y,z in <1 x y z| ^ <0 6 -7 -2| = wedgie gives you the various possible values, which is another approach. Finally, you can define a certain set of vals directly from the wedgie, and Hermite-reduce a matrix of these, and get one canonically-defined possibility. So, it's not so much ad hoc as an embarrassent of riches.