This is an Opt In Archive . We would like to hear from you if you want your posts included. For the contact address see About this archive. All posts are copyright (c).
- Contents - Hide Contents - Home - Section 76000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550 6600 6650 6700 6750 6800 6850 6900 6950
6550 - 6575 -
Message: 6550 - Contents - Hide Contents Date: Thu, 20 Feb 2003 16:45:06 Subject: Re: lattice diagram "levels" of complexity From: manuel.op.de.coul@xxxxxxxxxxx.xxx Carl and Paul wrote:>how about a much simpler example, with much fewer factors? i'm >looking to understand what you mean by "extending the tones out to a >EF Genus in all directions"The only way I know to extend a CPS to a EF genus is to add the remaining combination counts to it. So if you have 2)5 you need to add 1)5, 3)5, 4)5 and 5)5.>> What you never showed is how the definition of stellation on >> George Hart's site requires the 92-tone, and not the 80-tone >> structure. >> >> -Carl>i think manuel has got that down.So if I'm not mistaken the 80-tone structure is the result of the first-order stellation and the 92-tone one of the complete stellation. The former is called "stellated" in Scala and Wilson's stellation is called "superstellated". It's a repeated stellation so that there are no more unstellated chords anymore. Manuel
Message: 6551 - Contents - Hide Contents Date: Thu, 20 Feb 2003 02:39:50 Subject: Re: Reduced generators and special commas From: Carl Lumma>> >The* special comma -- couldn't there be more than one? In >> meantone, i=4 gives 81:80, i=12 the Pythagorean comma, etc. Right? >>I defined it so that there is only one.What's Tenney height? just i? Did Tenney propose this?>12 isn't actually mapped to a 7-limit consonance anyway.Oh, bad example. :( -Carl
Message: 6552 - Contents - Hide Contents Date: Thu, 20 Feb 2003 16:50:59 Subject: Re: scala show data From: manuel.op.de.coul@xxxxxxxxxxx.xxx Carl wrote:>By the way, the installer puts a desktop shortcut >even if I tell it not to. ;)Oh, I'll look after that.>I think it's cool that you are still maintaining it.Thanks, there actually is one blind user (not on the lists) who can't use the GUI-version.>I'm really impressed with the speed of 2.05. Though it >is 9 megs of widgets, and one can't copy and paste to >the Windows clipboard as was possible with the console >version. . .Not completely true because, weird enough, you can do it once. Right click in the main window, select "Enable text editing", then select some text, press ctrl-C and that can be pasted then. Any subsequent copy doesn't work unless you restart the program. Manuel
Message: 6553 - Contents - Hide Contents Date: Thu, 20 Feb 2003 10:56:59 Subject: Re: Reduced generators and special commas From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:>>> *The* special comma -- couldn't there be more than one? In >>> meantone, i=4 gives 81:80, i=12 the Pythagorean comma, etc. Right? >>>> I defined it so that there is only one. >> What's Tenney height? just i? Did Tenney propose this?For p/q in reduced form, p*q; either that or log(p*q).
Message: 6554 - Contents - Hide Contents
Date: Thu, 20 Feb 2003 05:37:41
Subject: Reduced generators and special commas
From: Gene Ward Smith
For a p-limit linear temperament with octave period, we may call the
p-limit rational number q, 1 < q < sqrt(2), the reduced generator if
it defines a generator when approximated by the temperament, and if
it has minimal Tenney height given this condition. We may call the
comma c, c>1 the "special comma" for this temperament if it is the
comma defined by taking the p-limit consonance q^i for i>1 of the
temperament, and equating it to the corresponding JI version of this
consonace. In other words, it is C/q^i or q^i/C, i>1, where C is a
ratio of odd numbers less than or equal to p.
Below are some 7-limit reduced generators and special commas, along
with poptimal generators and TM reduced bases. Note particularly the
pairs of temperaments (Meantone and Flattone, Hemiwuerschmidt and
Hemithird) with the same reduced generator and special comma; these
are very closely related.
Dominant seventh poptimal 17/41
[36/35, 64/63] reduced 4/3 special 64/63
Pelogic poptimal 18/41
[36/35, 135/128] reduced 4/3 special 135/128
Negri poptimal 5/48
[49/48, 225/224] reduced 15/14 special 225/224
Kleismic poptimal 14/53
[49/48, 126/125] reduced 6/5 special 126/125
Hemifourth poptimal 4/19
[49/48, 81/80] reduced 8/7 special 49/48
Meantone poptimal 86/205
[81/80, 126/125] reduced 4/3 special 81/80
Porcupine poptimal 8/59
[64/63, 250/243] reduced 10/9 special 250/243
Superpythagorean poptimal 31/76
[64/63, 245/243] reduced 4/3 special 64/63
Beatles poptimal 19/64
[64/63, 686/675] reduced 49/40 special 2401/2400
Flattone poptimal 46/109
[81/80, 525/512] reduced 4/3 special 81/80
Magic poptimal 13/41
[225/224, 245/243] reduced 5/4 special 3125/3072
Nonkleismic poptimal 23/89
[126/125, 1728/1715] reduced 6/5 special 126/125
Semisixths poptimal 44/119
[126/125, 245/243] reduced 9/7 special 245/243
Orwell poptimal 26/115
[225/224, 1728/1715] reduced 7/6 special 1728/1715
Miracle poptimal 17/175
[225/224, 1029/1024] reduced 15/14 special 225/224
Quartaminorthirds poptimal 9/139
[126/125, 1029/1024] reduced 21/20 special 64827/64000
Supermajor seconds poptimal 53/274
[81/80, 1029/1024] reduced 8/7 special 1029/1024
Schismic poptimal 39/94
[225/224, 3125/3087] reduced 4/3 special 5120/5103
Squares poptimal 93/262
[81/80, 2401/2400] reduced 9/7 special 19683/19208
Octafifths poptimal 13/177
[245/243, 2401/2400] reduced 21/20 special 4000/3969
Catakleismic poptimal 52/197
[225/224, 4375/4374] reduced 6/5 special 15625/15552
Hemiwuerschmidt poptimal 37/229
[2401/2400, 3136/3125] reduced 28/25 special 3136/3125
Hemithird poptimal 24/149
[1029/1024, 3136/3125] reduced 28/25 special 3136/3125
Amity poptimal 155/548
[4375/4374, 5120/5103] reduced 128/105 special
4294967296/4254271875
Hemififth poptimal 70/239
[2401/2400, 5120/5103] reduced 49/40 special 2401/2400
Message: 6555 - Contents - Hide Contents Date: Thu, 20 Feb 2003 05:43:52 Subject: Re: lattice diagram "levels" of complexity From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:> so anyway, is "diameter" more-or-less accepted as > the standard terminology for this kind of thing?It's standard math terminology: Graph Diameter -- from MathWorld * [with cont.] Graph Distance -- from MathWorld * [with cont.] Graph Geodesic -- from MathWorld * [with cont.]
Message: 6556 - Contents - Hide Contents Date: Thu, 20 Feb 2003 05:49:57 Subject: Re: Reduced generators and special commas From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <gwsmith@s...>" <gwsmith@s...> wrote: In other words, it is C/q^i or q^i/C, i>1, where C is a> ratio of odd numbers less than or equal to p.This should be, it is the octave reduction of C/q^i or q^i/C, whichever one produces something between 1 and sqrt(2).
Message: 6557 - Contents - Hide Contents Date: Thu, 20 Feb 2003 06:15:10 Subject: Re: Reduced generators and special commas From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <gwsmith@s...>" <gwsmith@s...> wrote: Note particularly the> pairs of temperaments (Meantone and Flattone, Hemiwuerschmidt and > Hemithird) with the same reduced generator and special comma; these > are very closely related.To these we may add Beatles and Hemififth; Negri and Miracle.
Message: 6558 - Contents - Hide Contents Date: Thu, 20 Feb 2003 07:22:24 Subject: Re: Reduced generators and special commas From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <gwsmith@s...>" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <gwsmith@s...>" > <gwsmith@s...> wrote: > > Note particularly the>> pairs of temperaments (Meantone and Flattone, Hemiwuerschmidt and >> Hemithird) with the same reduced generator and special comma; these >> are very closely related. >> To these we may add Beatles and Hemififth; Negri and Miracle.And Kleismic and Nonkleismic, and Dominant Sevenths and Superpythagorean. Do I need to make a computer do this?
Message: 6559 - Contents - Hide Contents Date: Thu, 20 Feb 2003 01:34:23 Subject: Re: Reduced generators and special commas From: Carl Lumma>For a p-limit linear temperament with octave period, we may call the >p-limit rational number q, 1 < q < sqrt(2), the reduced generator if >it defines a generator when approximated by the temperament, and if >it has minimal Tenney height given this condition. We may call the >comma c, c>1 the "special comma" for this temperament if it is the >comma defined by taking the p-limit consonance q^i for i>1 of the >temperament, and equating it to the corresponding JI version of this >consonace. In other words, it is C/q^i or q^i/C, i>1, where C is a >ratio of odd numbers less than or equal to p.*The* special comma -- couldn't there be more than one? In meantone, i=4 gives 81:80, i=12 the Pythagorean comma, etc. Right? -Carl
Message: 6560 - Contents - Hide Contents Date: Sat, 22 Feb 2003 04:25:10 Subject: 2401/2400 From: Gene Ward Smith I was looking into why my attempt to find an analog to Miracle with 2401/2400 a comma and with a generator between 50/49 and 49/48 didn't work, and found out something fascinating--such a thing is algebraically impossible. However, we can find an analogue of sorts. For a 7-limit wedgie to have 2401/2400 as a comma, it must be of the form [4b - 2a, a, b, -4c - 5a, -2c - 5b, c] with a, b, c relatively prime. This tells us, for instance, that a fifth always requires an even number of generator steps to be represented, and so a fifth can never be a generator for such a temperament. If we look at the number of generator steps needed for 49/48 (or equivalently, 50/49), which is the first component of the wedge of the above wedgie with the Monzo representation of 49/48, we get that it is 2a-2b, so we can't have what I was looking for, which is that this be 1. However, we *can* get something with two generator steps, with a wedgie [2n, n+2, n+1, -5n - 4c - 10, -5n - 2c - 5, c]. Such a temperament will have a generator of 7/5; of course (7/5)^2 = 49/25 = 2 (49/50) and so this has the required two-step relationship to 50/49 (and so 49/48.) If we look for wedgies of low badness in this form, we find them for n = -48, -15, -13 and most especially, n = 20. This gives us a good microtemperament, which is the closest thing available as an analogue to Miracle unless you are willing to count things like Ennealimmal, with non-octave periods. The wedgie is [40, 22, 21, -58, -79, -13], the mapping [[1, 21, 13, 13], [0, -40, -22, -21]], 83/171 is a poptimal generator, and it can be described as 68&171 in terms of standard vals. The TM basis is [2401/2400, 48828125/48771072].
Message: 6561 - Contents - Hide Contents Date: Sat, 22 Feb 2003 21:07:41 Subject: Re: Paul's poptimal graph From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <gwsmith@s...>" <gwsmith@s...> wrote:> By the way, Paul, I am terrifically impressed that you could > calculate enough points to draw a decent graph and find a minimum. > How in the world did you manage?i have the matlab optimization toolbox. 1000 points took a few seconds, probably less than 1 second had i used the 2.4 gigahertz machine instead. i actually calculated 10000 points, taking a few minutes, before plotting. to find the minimum, i just minimized the function of p which minimizes the p-norm of the errors (in the 5-limit). so i had to write the first (logically, second) minimization as a matlab function (1 line) and the second (logically, first) minimization as another matlab function (4 lines) called by the first.
Message: 6562 - Contents - Hide Contents Date: Sat, 22 Feb 2003 08:31:17 Subject: Paul's poptimal graph From: Gene Ward Smith By the way, Paul, I am terrifically impressed that you could calculate enough points to draw a decent graph and find a minimum. How in the world did you manage?
Message: 6563 - Contents - Hide Contents Date: Sat, 22 Feb 2003 23:31:35 Subject: Re: Paul's poptimal graph From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <gwsmith@s...>" > <gwsmith@s...> wrote:>> By the way, Paul, I am terrifically impressed that you could >> calculate enough points to draw a decent graph and find a minimum. >> How in the world did you manage? >> i have the matlab optimization toolbox. 1000 points took a few > seconds, probably less than 1 second had i used the 2.4 gigahertz > machine instead. i actually calculated 10000 points, taking a few > minutes, before plotting. > > to find the minimum, i just minimized the function of p which > minimizes the p-norm of the errors (in the 5-limit). so i had to > write the first (logically, second) minimization as a matlab function > (1 line) and the second (logically, first) minimization as another > matlab function (4 lines) called by the first.I guess I should consider getting Matlab. Maple would take hours getting even one of the points. Algebraically, using resultants and Sturm sequences one can deal with the problem in another way, but it is a huge mess involving high order polynomials.
Message: 6564 - Contents - Hide Contents Date: Mon, 24 Feb 2003 03:29:18 Subject: Algebraic functions for 5-limit temperaments From: Gene Ward Smith If b is the "brat" as defined on the tuning group, then the fifth and the third, for any linear temperament, will be algebraic functions of b (and vice versa.) If we choose a b value, and specialize one of the polynomials below, then one of its real roots will be the fifth or third we seek; similarly, inserting a value for a fifth or third allows us to compute b. Here's an example of what you might do with this. We know 2^(47/81) is a good value for a meantone fifth, being poptimal. If we set f equal to this value in the meantone fifth polynomial below and solve for b, we get -6.01, which is awfully close to -6. If we set b=-6 in the polynomial, we get 15f^4 - 10f - 60 (3f^4 - 2f - 12 if we reduce to lowest terms), one of whose roots gives us a poptimal meantone fifth with a brat of -6. Comma 81/80 Meantone Polynomial for fifth (-2*b+3)*f^4-10*f+10*b Polynomial for major third 625/16*b^4+75/2*(2*b-3)^2*b^2*t^2-10*(2*b- 3)^3*b*t^3+(2*b-3)^4*t^4+(-125*b^4+375/2*b^3-625/4)*t Comma 2048/2025 Diaschismic Polynomial for fifth 25*f^6-50*f^5*b+25*f^4*b^2-8192*(2*b-3)^2 Polynomial for major third -1280000+16*(2*b-3)^4*t^6-160*b*(2*b-3) ^3*t^5+600*b^2*(2*b-3)^2*t^4-1000*b^3*(2*b-3)*t^3+625*t^2*b^4 Comma 15625/15552 Kleismic Polynomial for fifth 15625*f^6+(-69120*b^4+18432*b^5-438*b+138240*b^3- 2048*b^6-23328-155520*b^2)*f^5+234375*f^4*b^2- 312500*f^3*b^3+234375*b^4*f^2-93750*b^5*f+15625*b^6 Polynomial for major third 6250*t^6+32*(2*b-3)^5*t^5-400*b*(2*b-3) ^4*t^4+2000*b^2*(2*b-3)^3*t^3-5000*b^3*(2*b-3)^2*t^2+6250*b^4*(2*b-3) *t-3125*b^5 Comma 32805/32768 Schismic Polynomial for fifth 5*f^9-5*f^8*b-192+128*b Polynomial for major third -12500000+256*(2*b-3)^8*t^9-5120*b*(2*b-3) ^7*t^8+44800*b^2*(2*b-3)^6*t^7-224000*b^3*(2*b-3)^5*t^6+700000*b^4* (2*b-3)^4*t^5-1400000*b^5*(2*b-3)^3*t^4+1750000*b^6*(2*b-3)^2*t^3- 1250000*b^7*(2*b-3)*t^2+390625*t*b^8 Comma 128/125 Augmented Polynomial for fifth 125*f^3-375*b*f^2+375*b^2*f+3*b^3+864*b-432- 576*b^2 Polynomial for major third t^3-2 Comma 135/128 Pelogic Polynomial for fifth 5*f^4-5*f^3*b-24+16*b Polynomial for major third 500+8*(2*b-3)^3*t^4-60*b*(2*b-3) ^2*t^3+150*b^2*(2*b-3)*t^2-125*t*b^3 Comma 250/243 Porcupine Polynomial for fifth 32*(2*b-3)^3*f^5+125*f^3-375*b*f^2+375*b^2*f- 125*b^3 Polynomial for major third 128*(2*b-3)^5*t^5-1600*b*(2*b-3)^4*t^4-(- 3125+216000*b^2-64000*b^5-432000*b^3+288000*b^4)*t^3-20000*b^3*(2*b-3) ^2*t^2+25000*b^4*(2*b-3)*t-12500*b^5 Comma 2109375/2097152 Orwell Polynomial for fifth 3125*f^8-15625*f^7*b+31250*f^6*b^2- 31250*f^5*b^3+15625*f^4*b^4-3125*f^3*b^5+8192*(2*b-3)^5 Polynomial for major third 32000+8*(2*b-3)^3*t^8-60*b*(2*b-3) ^2*t^7+150*b^2*(2*b-3)*t^6-125*t^5*b^3
Message: 6565 - Contents - Hide Contents Date: Mon, 24 Feb 2003 13:55:46 Subject: Well behaved brats for other temperaments From: Gene Ward Smith We can use the same magic values for the brat (-6, -1, -1/6, 0, 1/4, 2/3, 1, 3/2, 2, 4, 9, infinity) and also get nice beat ratios. I checked Schismic, Kleismic, Diaschismic, and Orwell. For the latter two the only magic value which works is b=3/2, with perfect fifths, but in neither case is this poptimal. Schismic and Kleismic were much more like Meantone; for Schismic both b=-1 and b=-1/6 are poptimal, whereas for Kleismic b=2 is poptimal.
Message: 6566 - Contents - Hide Contents Date: Mon, 24 Feb 2003 07:46:05 Subject: Re: Algebraic functions for 5-limit temperaments From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <gwsmith@s...>" <gwsmith@s...> wrote:> If b is the "brat" as defined on the tuning group, then the fifth and > the third, for any linear temperament, will be algebraic functions of > b (and vice versa.) If we choose a b value, and specialize one of the > polynomials below, then one of its real roots will be the fifth or > third we seek; similarly, inserting a value for a fifth or third > allows us to compute b.I prefer to use numerical methods since a spreadsheet is all you need for them. e.g. for meantone just set up a column with decimal fractions from 1/11 (.091) to 1/3 (0.333) in increments of 0.001, representing the fractions of a comma by which the 5th is tempered. Call this x. Then in the next two columns calculate the corresponding tempering of the major and minor thirds, y=4x-1 and z=1-3x. Then from those calculate the three beat ratios in the major triad 3/2*z/y, 5/3*y/x, 5/2*z/x and the three in the minor triad, z/y, 2*y/x, 2*z/x. Then scan down the columns until you find the beat ratios you are looking for and note the corresponding value in the comma fraction column (x).
Message: 6567 - Contents - Hide Contents Date: Mon, 24 Feb 2003 10:50:13 Subject: Re: Algebraic functions for 5-limit temperaments From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> I prefer to use numerical methods since a spreadsheet is all you need > for them.If you have the tools to use it, my method rules, since, your method takes longer and fails to deliver the precise answer. However, if you want to extend this business to the 7-limit and need therefore to look at approximations, this is something to try.
Message: 6568 - Contents - Hide Contents Date: Mon, 24 Feb 2003 11:52:06 Subject: Some well-behaved brats for meantone From: Gene Ward Smith I checked integer brat ratios, looking for values where all three brat ratios were integers or their inverses, and did the same for inverse integer brats. The result was this list: [-6, 3, -2], [-1, 1, -1], [1, 1/5, 5], [2, -1/5, -10], [4, -1, -4], [9, -3, -3], [1/4, 1/2, 1/2] Here the first number is the brat, the others the other beat ratios. The b=1 fifth is very flat, but the others are all usable, and b=-6, as we remarked before, is even poptimal. Robert's b=2 doesn't look to be that good compared to some of the rest of these; the b=-1 is particularly striking; it's pretty close to 2/7-comma, and even closer to 5/17-comma. The exact value of the fifth is the positive real root of f^4-2f-2. By the way, Paul, have you any comments on the psychoacoustics of all of this? What do you think of Robert's idea?
Message: 6569 - Contents - Hide Contents Date: Mon, 24 Feb 2003 12:28:36 Subject: Re: Some well-behaved brats for meantone From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <gwsmith@s...>" <gwsmith@s...> wrote:> I checked integer brat ratios, looking for values where all three > brat ratios were integers or their inverses, and did the same for > inverse integer brats. The result was this list: > > [-6, 3, -2], [-1, 1, -1], [1, 1/5, 5], [2, -1/5, -10], [4, -1, -4], > [9, -3, -3], [1/4, 1/2, 1/2]If we look for everything with either infinity or a Tenney height of ten or less for all three ratios, we add to the above [2/3, 1/3, 2], [-1/6, 2/3, -1/4], [0, 3/5, 0], [infinity, infinity, -5/2], [3/2, 0, infinity]. The b=2/3 system is a "flattone" system, in the vicinity of the 26 et; the two "infinity" systems are of course 1/4-comma meantone and JI.
Message: 6570 - Contents - Hide Contents Date: Tue, 25 Feb 2003 00:52:48 Subject: Re: Some well-behaved brats for meantone From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <gwsmith@s...>" <gwsmith@s...> wrote:> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith <gwsmith@s...>" > <gwsmith@s...> wrote: >>> I checked integer brat ratios, looking for values where all three >> brat ratios were integers or their inverses, and did the same for >> inverse integer brats. The result was this list: >> >> [-6, 3, -2], [-1, 1, -1], [1, 1/5, 5], [2, -1/5, -10], [4, -1, -4], >> [9, -3, -3], [1/4, 1/2, 1/2] >> If we look for everything with either infinity or a Tenney height of > ten or less for all three ratios, we add to the above > > [2/3, 1/3, 2], [-1/6, 2/3, -1/4], [0, 3/5, 0], > [infinity, infinity, -5/2], [3/2, 0, infinity]. > > The b=2/3 system is a "flattone" system, in the vicinity of the 26 et; > the two "infinity" systems are of course 1/4-comma meantone and JI.You mean Pythagorean. JI is not a linear temperament. I made this same mistake myself recently. Of course you (and I, when I made the mistake) meant that the fifths are just; so-called "3-limit JI"; but this is an oxymoron to most people, and in any case we are talking 5-limit here. Yes. I can see the same results in my spreadsheet. However, where you find something to be very close to say 5/17-comma, I find it to be exactly so (to the limit of accuracy of IEEE floating-point numbers). This is because (and this answers a question of Paul Erlich's on the tuning list in response to a 4 year old post of mine), like Bob Wendell, I am using the approximation that frequency deviation is proportional to comma fraction deviation (i.e. log frequency deviation). This is an extremely good approximation in the range from 0 to about 20 cents. The few thousandths of a cent error that result in the size of the generator are of no interest to anyone but a mathematician, and it means that one can deal with linear equations rather than polynomials. And of course, just as you do with the polynomials, you could solve these approximate linear equations algebraicly instead of numerically to get the rational comma fraction solutions that are familiar to people.
Message: 6571 - Contents - Hide Contents Date: Tue, 25 Feb 2003 02:05:07 Subject: Re: Some well-behaved brats for meantone From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:> You mean Pythagorean. JI is not a linear temperament.I saw this after posting, but it didn't seem worth another posting to correct it.> Yes. I can see the same results in my spreadsheet. However, where you > find something to be very close to say 5/17-comma, I find it to be > exactly so (to the limit of accuracy of IEEE floating-point numbers).It's the linearization business I just posted about on tuning.> And of course, just as you do with the polynomials, you could solve > these approximate linear equations algebraicly instead of numerically > to get the rational comma fraction solutions that are familiar to people. See above.
Message: 6572 - Contents - Hide Contents Date: Tue, 25 Feb 2003 04:18:29 Subject: Miracle brats From: Gene Ward Smith Miracle is not especially notable as a 5-limit system, but it seems to me having triads that beat in sync could be interesting anyway. Howbeit, here is the lowdown: Secor polynomial: 5*s^13-5*b*s^7+8*b-12 = 0 Secor for b=0, 5-limit poptimal: 1.06966311063807 or 116.587791307739 cents From root of 440: Fifth F = 659.074923710535 Hz Major third T = 549.229103092098 Hz Beat of fifth 3*440 - 2*F = 1.85015257893 Hz Beat of major third 5*440 - 4*T = 3.08358763161 Hz Beat of minor third 6*T - 5*F = 0 Hz Fundamental beat = fifth beat / 3 = third beat / 5 = .6167175263 Hz Secor for b=-1/6, 7-limit poptimal: 1.06965100291870 or 116.568195030691 cents From root of 440: Fifth F = 659.030163813727 Hz Major third T = 549.272622860298 Hz Beat of fifth 3*440 - 2*F = 1.93967237255 Hz Beat of major third 5*440 - 4*T = 2.90950855881 Hz Beat of minor third 6*T - 5*F = .48491809315 Hz Fundamental beat is minor third beat = .48491809315 Hz Secor for b=2/3, 11-limit poptimal: 1.06973058849676 or 116.696999787326 cents From root of 440: Fifth F = 659.324422707746 Hz Major third T = 548.986634061637 Hz Beat of fifth 3*440 - 2*F = 1.35115458451 Hz Beat of major third 5*440 - 4*F = 4.05346375345 Hz Beat of minor third 5*F - 6*T = 2.70230916891 Fundamental beat is fifth beat = 1.35115458451 Hz
Message: 6573 - Contents - Hide Contents Date: Wed, 26 Feb 2003 21:04:34 Subject: Re: A Property of MOS/DE Scales From: Carl Lumma Kalle wrote...>I am aware of these more sophisticated (and also more elegant) >tools and possibilities. So if anyone is interested please read >the thread in tuning list and continue on tuning-math.Anybody else interested in reviving this thread? Relevant messages... Yahoo groups: /tuning/message/41371 * [with cont.] Yahoo groups: /tuning/message/41383 * [with cont.] Yahoo groups: /tuning-math/message/5132 * [with cont.] Yahoo groups: /tuning-math/message/5136 * [with cont.] -Carl
Message: 6574 - Contents - Hide Contents Date: Thu, 27 Feb 2003 18:23:23 Subject: Re: A Property of MOS/DE Scales From: wallyesterpaulrus --- In tuning-math@xxxxxxxxxxx.xxxx "Carl Lumma <ekin@l...>" <ekin@l...> wrote:> Kalle wrote...>> I am aware of these more sophisticated (and also more elegant) >> tools and possibilities. So if anyone is interested please read >> the thread in tuning list and continue on tuning-math. >> Anybody else interested in reviving this thread? > > Relevant messages... > > Yahoo groups: /tuning/message/41371 * [with cont.] > Yahoo groups: /tuning/message/41383 * [with cont.] > > Yahoo groups: /tuning-math/message/5132 * [with cont.] > Yahoo groups: /tuning-math/message/5136 * [with cont.] > > -Carlyes, i am very interested!!
6000 6050 6100 6150 6200 6250 6300 6350 6400 6450 6500 6550 6600 6650 6700 6750 6800 6850 6900 6950
6550 - 6575 -