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Message: 7576 Date: Sat, 11 Oct 2003 17:15:33 Subject: Re: Math Resources From: monz hi Rohan, you left a letter out of the URL. here's the correct one: Application of Vedic Mathematics, Vedic Maths,... * [with cont.] (Wayb.) -monz --- In tuning-math@xxxxxxxxxxx.xxxx "sikkaharyana" <sikkaharyana@y...> wrote:
> Dear Friends, > > There is an website which can help students of any > age to understand math very easily. they r providing > Vedic Mathematics tools using which students can enhance > their calculation speed 10 to 15 times faster than > using conventional methods of calculation.
Message: 7579 Date: Mon, 13 Oct 2003 01:41:58 Subject: T[n] revisited From: Carl Lumma
>Fourththirds[5]
Name[size]
>[16/15, 28/27, 77/75]
The commas.
>[1, -1, 3, -4, -4, 2, -10, 10, -6, -22]
Map?
>[[1, 2, 2, 4, 2], [0, -1, 1, -3, 4]]
Za?
>bad 6476.838089 comp 20.25383770 rms 43.03787612 >graham 7 scale size 5
"graham" is Graham complexity, I assume? What's "comp"?
>Heptadec[9] [36/35, 56/55, 77/75] >[5, 3, 7, 4, -7, -3, -11, 8, -1, -13] >[[1, 1, 2, 2, 3], [0, 5, 3, 7, 4]] >bad 6400.766110 comp 32.19555159 rms 19.64440328 >graham 7 scale size 9 > > >Blackwood[10] >[0, 5, 0, 8, 0, -14] >[[5, 8, 12, 14], [0, 0, -1, 0]] >bad 1662.988586 comp 10.25428060 rms 15.81535241 >graham 5 scale size 10 ratio 2.000000 > > >Pajara[10] >[2, -4, -4, -11, -12, 2] >[[2, 3, 5, 6], [0, 1, -2, -2]] >bad 1550.521632 comp 11.92510946 rms 10.90317748 >graham 6 scale size 10 ratio 1.666667 > > >Pajarous[10] [50/49, 55/54, 64/63] >[2, -4, -4, 10, -11, -12, 9, 2, 37, 42] >[[2, 3, 5, 6, 6], [0, 1, -2, -2, 5]] >bad 6667.906202 comp 43.76707564 rms 12.26714784 >graham 14 scale size 10 > > >Check this out Carl[10] >[2, 1, -4, -3, -12, -12] >[[1, 1, 2, 4], [0, 2, 1, -4]] >bad 2431.810368 comp 9.849243627 rms 25.06824586 >graham 6 scale size 10 ratio 1.666667 > > >Here's another Carl[10] >[2, 6, 6, 5, 4, -3] >[[2, 3, 4, 5], [0, 1, 3, 3]] >bad 2682.600306 comp 11.92510946 rms 18.86388854 >graham 6 scale size 10 ratio 1.666667 > > >NB Carl >[2, 6, 6, 5, 4, -3] >[[2, 3, 4, 5], [0, 1, 3, 3]] >bad 2682.600306 comp 11.92510946 rms 18.86388854 >graham 6 scale size 10 ratio 1.666667
I assume your maple can turn these into generator/ie pairs?
>Dominant sevenths[7] >rms error 20.163 cents > > >Hemifourths[9] >rms error 12.690 > > >Tertiathirds[9] >rms error 12.189 cents > > >Hexadecimal[9] >rms error 18.585
Where, pray tell, is the Official Source of data on these? -Carl
Message: 7580 Date: Tue, 14 Oct 2003 21:01:49 Subject: Chains of fifths and notation From: Gene Ward Smith Basing notation schemes which encompass successivly higher prime limits on a Pythagorean chain-of-fifth (or fourths, etc) method is a necessity, since we start from the 3-limit. To go to the 5-limit, we need to add a comma of the form 2^a 3^b 5^c where c = +-1. If a monzo [a,b,c] gives rise to a temperament with octave period, then the generator mapping will be +-[0, c, -b], and since c=+-1 once again we are back with 3/2, 4/3 etc as generator--a Pythagorean system. Looking at the 5-limit commas, we find really only five reasonable choices, and so five types of notation systems possible for the higher prime limits as well. 16/15 fourth-third systems 135/128 pelogic systems 81/80 meantone systems 32805/32768 schismic systems 2954312706550833698643/2951479051793528258560 [-69, 45, -1] counterschismic systems To go to 7-limit systems, we want 7-limit commas for which the exponent of 7 is +-1, and so forth. Some 7-limit possibilities are: 36/35, 525/512, 64/63, 875/864, 126/125, 225/224, 5120/5103, 65625/65536, 4375/4374 Aside from 33/32, some 11-limit possibilities include: 77/75, 45/44, 55/54, 56/55, 99/98, 100/99, 176/175, 896/891, 385/384, 441/440, 1375/1372, 6250/6237, 540/539, 5632/5625, 131072/130977, 40283203125/40282095616, 6576668672/6576582375, 781258401/781250000, 13841287201/13841203200 Note that while insuring that the count of p-limit commas depends only on the exponent for p is a nice property, it is hardly essential. I think there are advantages, for instance, in using in place of 81/80, 64/63, and 33/32, the commas 5120/5103 and 385/384; they are related by 64/63 = 81/80 5120/5103 33/32 = 81/80 64/63 385/384 In these terms, my example of 77/75 becomes D double flat, up three 81/80 and a 385/384.
Message: 7581 Date: Wed, 15 Oct 2003 12:04:37 Subject: Re: T[n] revisited From: Carl Lumma Gene? You said NB, and NB I did. -C. At 01:41 AM 10/13/2003, I wrote:
>>Fourththirds[5]
> >Name[size] >
>>[16/15, 28/27, 77/75]
> >The commas. >
>>[1, -1, 3, -4, -4, 2, -10, 10, -6, -22]
> >Map? >
>>[[1, 2, 2, 4, 2], [0, -1, 1, -3, 4]]
> >Za? >
>>bad 6476.838089 comp 20.25383770 rms 43.03787612 >>graham 7 scale size 5
> >"graham" is Graham complexity, I assume? What's "comp"? >
>>Heptadec[9] [36/35, 56/55, 77/75] >>[5, 3, 7, 4, -7, -3, -11, 8, -1, -13] >>[[1, 1, 2, 2, 3], [0, 5, 3, 7, 4]] >>bad 6400.766110 comp 32.19555159 rms 19.64440328 >>graham 7 scale size 9 >> >> >>Blackwood[10] >>[0, 5, 0, 8, 0, -14] >>[[5, 8, 12, 14], [0, 0, -1, 0]] >>bad 1662.988586 comp 10.25428060 rms 15.81535241 >>graham 5 scale size 10 ratio 2.000000 >> >> >>Pajara[10] >>[2, -4, -4, -11, -12, 2] >>[[2, 3, 5, 6], [0, 1, -2, -2]] >>bad 1550.521632 comp 11.92510946 rms 10.90317748 >>graham 6 scale size 10 ratio 1.666667 >> >> >>Pajarous[10] [50/49, 55/54, 64/63] >>[2, -4, -4, 10, -11, -12, 9, 2, 37, 42] >>[[2, 3, 5, 6, 6], [0, 1, -2, -2, 5]] >>bad 6667.906202 comp 43.76707564 rms 12.26714784 >>graham 14 scale size 10 >> >> >>Check this out Carl[10] >>[2, 1, -4, -3, -12, -12] >>[[1, 1, 2, 4], [0, 2, 1, -4]] >>bad 2431.810368 comp 9.849243627 rms 25.06824586 >>graham 6 scale size 10 ratio 1.666667 >> >> >>Here's another Carl[10] >>[2, 6, 6, 5, 4, -3] >>[[2, 3, 4, 5], [0, 1, 3, 3]] >>bad 2682.600306 comp 11.92510946 rms 18.86388854 >>graham 6 scale size 10 ratio 1.666667 >> >> >>NB Carl >>[2, 6, 6, 5, 4, -3] >>[[2, 3, 4, 5], [0, 1, 3, 3]] >>bad 2682.600306 comp 11.92510946 rms 18.86388854 >>graham 6 scale size 10 ratio 1.666667
> >I assume your maple can turn these into generator/ie pairs? >
>>Dominant sevenths[7] >>rms error 20.163 cents >> >> >>Hemifourths[9] >>rms error 12.690 >> >> >>Tertiathirds[9] >>rms error 12.189 cents >> >> >>Hexadecimal[9] >>rms error 18.585
> >Where, pray tell, is the Official Source of data on these? > >-Carl
Message: 7582 Date: Wed, 15 Oct 2003 21:29:54 Subject: Re: Chains of fifths and notation From: George D. Secor --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> Basing notation schemes which encompass successivly higher prime > limits on a Pythagorean chain-of-fifth (or fourths, etc) method is a > necessity, since we start from the 3-limit. To go to the 5-limit, we > need to add a comma of the form 2^a 3^b 5^c where c = +-1. If a
monzo
> [a,b,c] gives rise to a temperament with octave period, then the > generator mapping will be +-[0, c, -b], and since c=+-1 once again
we
> are back with 3/2, 4/3 etc as generator--a Pythagorean system.
Looking
> at the 5-limit commas, we find really only five reasonable choices, > and so five types of notation systems possible for the higher prime > limits as well. > > 16/15 fourth-third systems > > 135/128 pelogic systems > > 81/80 meantone systems > > 32805/32768 schismic systems > > 2954312706550833698643/2951479051793528258560 > [-69, 45, -1] counterschismic systems > > To go to 7-limit systems, we want 7-limit commas for which the > exponent of 7 is +-1, and so forth. Some 7-limit possibilities are: > > 36/35, 525/512, 64/63, 875/864, 126/125, 225/224, 5120/5103, > 65625/65536, 4375/4374 > > Aside from 33/32, some 11-limit possibilities include: > > 77/75, 45/44, 55/54, 56/55, 99/98, 100/99, 176/175, 896/891, > 385/384, 441/440, 1375/1372, 6250/6237, 540/539, 5632/5625, > 131072/130977, 40283203125/40282095616, 6576668672/6576582375, > 781258401/781250000, 13841287201/13841203200 > > Note that while insuring that the count of p-limit commas depends
only
> on the exponent for p is a nice property, it is hardly essential. I > think there are advantages, for instance, in using in place of > 81/80, 64/63, and 33/32, the commas 5120/5103 and 385/384; they are > related by > > 64/63 = 81/80 5120/5103 > > 33/32 = 81/80 64/63 385/384 > > In these terms, my example of 77/75 becomes D double flat, up three > 81/80 and a 385/384.^
Gene, FYI Dave and I have the following 11-limit commas notated *exactly* with single flags in sagittal: '| for 32768:32805 (5-schisma) |( for 5103:5120 /| for 80:81 |) for 63:64 |\ for 54:55 (| for 45056:45927 as well as the following notated *exactly* with two flags: )|( for 16384:16473 ~|) for 48:49 (|( for 44:45 //| for 6400:6561 /|) for 35:36 /|\ for 32:33 (|) for 704:729 (|\ for 8192:8505^ and a bunch of others notated *exactly* in high-precision sagittal (using the 5-schisma) such as: ./| for 2025:2048 .//| for 125:128 '(|\ for 27:28 --George
Message: 7583 Date: Wed, 15 Oct 2003 18:13:42 Subject: Re: T[n] revisited From: Carl Lumma
>> >[1, -1, 3, -4, -4, 2, -10, 10, -6, -22]
>> >> Map?
> >The wedgie
Ah.
>> >[[1, 2, 2, 4, 2], [0, -1, 1, -3, 4]]
>> >> Za?
> >This is the prime mapping.
Ok, I should have known this. But it never hurts to label these things in your posts. //
>> >Check this out Carl[10] >> >[2, 1, -4, -3, -12, -12] >> >[[1, 1, 2, 4], [0, 2, 1, -4]] >> >bad 2431.810368 comp 9.849243627 rms 25.06824586 >> >graham 6 scale size 10 ratio 1.666667
> >Generator: 358.03 >
>> > >> >Here's another Carl[10] >> >[2, 6, 6, 5, 4, -3] >> >[[2, 3, 4, 5], [0, 1, 3, 3]] >> >bad 2682.600306 comp 11.92510946 rms 18.86388854 >> >graham 6 scale size 10 ratio 1.666667
> >Generators: [600, 128.51]
Thanks, dude! Did you use one of your maple routines to do this? -Carl
Message: 7584 Date: Thu, 16 Oct 2003 17:12:02 Subject: Re: T[n] revisited From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> Did you use one of your maple routines to do this?
Of course--gf7 or gf11 of transpose(mapping to primes).
Message: 7585
Date: Thu, 16 Oct 2003 21:23:11
Subject: Re: Chains of fifths and notation
From: Gene Ward Smith
{{16/15 fourth-third systems
135/128 pelogic systems
81/80 meantone systems
32805/32768 schismic systems
2954312706550833698643/2951479051793528258560
[-69, 45, -1] counterschismic systems}}
We can extend this classification scheme by using planar temperaments
for higher prime limits as well. The wedgie for <81/80, 64/63> is [1,
4, -2, 4, -6, -16], so this goes under the "Dominant Seventh" rubric.
However, <81/80, 36/35> and <81/80, 5120/5103> are equally D7 systems,
giving the same wedgie, and very closely related to using 64/63 to
notate. We pass to the 11-limit by adding 33/32; however the same
wedgie is obtainable by adding 55/54 or 385/384, so any of the nine
systems <81/80, {36/35,64/63,5120/5103}, {33/32,55/54,385/384}> is
equally well an 11-limit D7 notation scheme.
Another very logical system falling under the general Meantone heading
is Septimal Meantone; here to 81/80 we add 126/125, 225/224 or
3136/3125. If now for the 11-limit we add 99/98, 176/175, 441/440,
1375/1372 or 5632/5625, we get 11-limit Meantone; if instead we add
385/384 or 540/549 we get Meanpop systems.
George tells us that each of 36/35, 64/63 and 5120/5103 is sagittally
symbolized, and so is 55/54 (but not 385/384,) so D7 systems are
pretty well covered. So far as Septimal Meantone goes, none of
126/125, 225/224 or 3136/3125 seems to have a symbol, nor do 99/98,
176/174, 441/440, 1375/1372, 5632/5625, 385/384 or 540/539. A symbol
for 385/384 and one for either of 126/125 or 225/224 and either of
99/98 or 176/175 would be nice if other 81/80 systems were to be included.
Message: 7586 Date: Thu, 16 Oct 2003 00:16:01 Subject: Re: T[n] revisited From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
> >Fourththirds[5]
> > Name[size] >
> >[16/15, 28/27, 77/75]
> > The commas. >
> >[1, -1, 3, -4, -4, 2, -10, 10, -6, -22]
> > Map?
The wedgie
> >[[1, 2, 2, 4, 2], [0, -1, 1, -3, 4]]
> > Za?
This is the prime mapping.
> >bad 6476.838089 comp 20.25383770 rms 43.03787612 > >graham 7 scale size 5
> > "graham" is Graham complexity, I assume? What's "comp"?
Geometric complexity--using the old-style natural log defintion. Generator: 455.25
> >Heptadec[9] [36/35, 56/55, 77/75] > >[5, 3, 7, 4, -7, -3, -11, 8, -1, -13] > >[[1, 1, 2, 2, 3], [0, 5, 3, 7, 4]] > >bad 6400.766110 comp 32.19555159 rms 19.64440328 > >graham 7 scale size 9
Generator 141.16
> > > >Blackwood[10] > >[0, 5, 0, 8, 0, -14] > >[[5, 8, 12, 14], [0, 0, -1, 0]] > >bad 1662.988586 comp 10.25428060 rms 15.81535241 > >graham 5 scale size 10 ratio 2.000000
Generators: [240, 90.61]
> > > >Pajara[10] > >[2, -4, -4, -11, -12, 2] > >[[2, 3, 5, 6], [0, 1, -2, -2]] > >bad 1550.521632 comp 11.92510946 rms 10.90317748 > >graham 6 scale size 10 ratio 1.666667
Generators: [600, 108.81]
> >Pajarous[10] [50/49, 55/54, 64/63] > >[2, -4, -4, 10, -11, -12, 9, 2, 37, 42] > >[[2, 3, 5, 6, 6], [0, 1, -2, -2, 5]] > >bad 6667.906202 comp 43.76707564 rms 12.26714784 > >graham 14 scale size 10
Generators: [600, 109.88]
> > > >Check this out Carl[10] > >[2, 1, -4, -3, -12, -12] > >[[1, 1, 2, 4], [0, 2, 1, -4]] > >bad 2431.810368 comp 9.849243627 rms 25.06824586 > >graham 6 scale size 10 ratio 1.666667
Generator: 358.03
> > > >Here's another Carl[10] > >[2, 6, 6, 5, 4, -3] > >[[2, 3, 4, 5], [0, 1, 3, 3]] > >bad 2682.600306 comp 11.92510946 rms 18.86388854 > >graham 6 scale size 10 ratio 1.666667
Generators: [600, 128.51]
> Where, pray tell, is the Official Source of data on these?
Didn't know there was such a thing.
Message: 7587 Date: Thu, 16 Oct 2003 21:40:39 Subject: can someone check this data? From: Paul Erlich Yahoo groups: /tuning-math/files/Paul/ * [with cont.]
Message: 7588 Date: Thu, 16 Oct 2003 21:41:01 Subject: can someone check this data? From: Paul Erlich Yahoo groups: /tuning-math/files/Paul/test.html * [with cont.]
Message: 7589 Date: Thu, 16 Oct 2003 21:57:39 Subject: Re: Chains of fifths and notation From: Gene Ward Smith --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> We can extend this classification scheme by using planar temperaments > for higher prime limits as well.
If we insist on having commas of the form 2^a 3^b p^{+-1}, we don't
get anything as nice as we have for D7 systems:
11-limit meantone
[81/80, 59049/57344, 387420489/369098752]
meanpop
[81/80, 59049/57344, 17537553/16777216]
Message: 7590 Date: Thu, 16 Oct 2003 18:51:52 Subject: Re: can someone check this data? From: Carl Lumma Awesome! Sorry I can't help check it. How'd you get this html? 'd be nice to have a database with column-sort. If you're doing this in Excel, you can export as a CSV file, which Yahoo's database thingy can then import (I assume that's how you did the 5-limit database?). Oh, and where are the TM-reduced bases? -Carl
Message: 7591 Date: Fri, 17 Oct 2003 16:24:35 Subject: Re: can someone check this data? From: Manuel Op de Coul So I've picked a few to check, and got the same results. Only one very slight difference in the RMS error of meantone, you have 4.217731 and I got 4.217730, but for the rest they were exactly the same.
>no, they're 5-limit linear temperaments, and 2 minus 1 equals 1.
Yes, I was temporarily confused. Manuel
Message: 7592 Date: Fri, 17 Oct 2003 16:50:37 Subject: Re: can someone check this data? From: Manuel Op de Coul Now I checked more and found a few more differences in RMS values. I'm beginning to worry about my square root routine. Could someone else verify, for example for counterschismic, is the RMS error 0.026391 as Paul gives or 0.026394? Gene, Graham? Thanks, Manuel
Message: 7593 Date: Fri, 17 Oct 2003 12:44:15 Subject: Re: can someone check this data? From: Carl Lumma
>> Oh, and where are the TM-reduced bases?
> >the comma, silly goose!
Weird, I didn't see the right side of the page the first time. -Carl
Message: 7594 Date: Fri, 17 Oct 2003 19:46:14 Subject: Re: [5-11]-prime-space p-block and 13edo From: monz since the ASCII lattice i posted below got messed up by Yahoo (even "Expand Messages" didn't work for me this time), here's a good quality image of the lattice: Yahoo groups: /tuning_files/files/monz/[5-11]- * [with cont.] primespace-13edo.gif or Sign In - * [with cont.] (Wayb.) --- In tuning-math@xxxxxxxxxxx.xxxx "monz" <monz@a...> wrote:
> > so i set up a prime-space consisting of the > euler-genus 5^(-1...+5) * 11^(-2...+3) . > > > and found these two unison-vectors: > > > ~cents [2,5,11]-monzo ratio > > 56.30525686 [ 0 3 -2] 125/121 > > 26.58125482 [-15 2 3] 33275/32768 > > > ... and voila, out popped 13edo. > > > the 13-tone JI periodicity-block given by these > unison-vectors is shown on this lattice: > > 1/1 is the reference note, the unison-vectors > are shown by "X", and notes not included in the > periodicity-block are shown as "()" > > > ()------()--------X-----------()----------() > | | | | | > ()------()----3025/2048---15125/8192------() > | | | | | > ()-----55/32---275/256-----1375/1024---6875/4096 > | | | | | > 1/1-----5/4-----25/16-------125/64------625/512 > | | | | | > ()------()------25/22-------125/88--------() > | | | | | > ()------()-------()------------X----------()
-monz
Message: 7595 Date: Fri, 17 Oct 2003 20:54:03 Subject: Re: can someone check this data? From: Graham Breed Manuel Op de Coul wrote:
> Now I checked more and found a few more differences in RMS values. > I'm beginning to worry about my square root routine. Could someone > else verify, for example for counterschismic, is the RMS error > 0.026391 as Paul gives or 0.026394? Gene, Graham?
Yes, my RMS routine was much wronger, but I've fixed that now. I get
4.217730 cents for meantone and 0.026394 for this 53&306 thing.
Graham
Message: 7596 Date: Fri, 17 Oct 2003 00:12:31 Subject: Re: Chains of fifths and notation From: Dave Keenan --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> George tells us that each of 36/35, 64/63 and 5120/5103 is sagittally > symbolized, and so is 55/54 (but not 385/384,) so D7 systems are > pretty well covered. So far as Septimal Meantone goes, none of > 126/125, 225/224 or 3136/3125 seems to have a symbol, nor do 99/98, > 176/174, 441/440, 1375/1372, 5632/5625, 385/384 or 540/539. A symbol > for 385/384 and one for either of 126/125 or 225/224 and either of > 99/98 or 176/175 would be nice if other 81/80 systems were to be
included. Some of these will be exactly symbolised in the final system, using accented symbols, it's just that they have not yet been finalised to the satisfaction of both George and I, and at present we're devoting our time to the article explaining the basic (unaccented) system. It's highly probable that each of the following will be the primary meaning of some accented sagittal symbol. 385/384 126/125 225/224 99/98 But the following are very unlikely to be primary symbol interpretations, because there are other more popular (usually less complex) kommas which are very close to them (typically within 0.4 c). In that case you can simply use the symbol for that nearby komma and explain somewhere that that's what you are doing (assuming you don't also need that symbol for its primary purpose, which is fairly unlikely). We refer to this as using a symbol in a secondary role. 176/175 3136/3125 441/440 1375/1372 5632/5625 540/539 Regards, -- Dave Keenan
Message: 7597 Date: Fri, 17 Oct 2003 03:22:17 Subject: Re: Chains of fifths and notation From: Dave Keenan This is a long overdue reply to Gene in "Re: Polyphonic notation" on the tuning list Yahoo groups: /tuning/message/47497 * [with cont.] --- In tuning@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@xxxxxxxxxxx.xxxx "Dave Keenan" <d.keenan@b...> wrote: >
> > We have also used a temperament to help decide on the actual symbols > > to be used for the comma ratios in the superset. This is an > > 8-dimensional temperament with a maximum error of 0.39 cents. The 8 > > dimensions relate to the 9 flags (including the accent mark) that
> make
> > up the symbols, less one degree of freedom because a certain > > combination is set equal to the apotome.
> > What commas are being tempered out?
I'm afraid I don't know. I just specified certain combinations of generators to approximate certain 23-limit ratios (which were themselves commas, but were not being tempered out) and solved numerically for the generators. If you're still interested, maybe there's some other information I could give you if you wanted to work them out. But it may have to wait some time.
> > Well I think the fact that we have Graham proposing MIRACLE > > temperament with 10 nominals and Gene proposing ennealimmal > > temperament with 9 nominals should make it clear that there is > > unlikely to ever be agreement on which is the ultimate temperament
> for
> > notating everything else including ratios.
> > My proposal is simply intended to notate effective 7-limit JI > (extendible to 11-limit) not everything.
Fine.
Message: 7598 Date: Fri, 17 Oct 2003 08:20:06 Subject: [5-11]-prime-space p-block and 13edo From: monz playing around with my software, i noticed that 13edo, which does not have anything resembling a 3:2 "perfect-5th", gives a not-really-horrible approximation of 5:4 (just a little worse than that of 12edo, but in the opposite direction), and a very good approximation of 11:8 : 2^(4/13) = ~369.2307692 cents, ~17.08294463 less than 5:4 2^(6/13) = ~553.8461538 cents, ~2.528211481 more than 11:8 so i set up a prime-space consisting of the euler-genus 5^(-1...+5) * 11^(-2...+3) . and found these two unison-vectors: ~cents [2,5,11]-monzo ratio 56.30525686 [ 0 3 -2] 125/121 26.58125482 [-15 2 3] 33275/32768 ... and voila, out popped 13edo. the 13-tone JI periodicity-block given by these unison-vectors is shown on this lattice: 1/1 is the reference note, the unison-vectors are shown by "X", and notes not included in the periodicity-block are shown as "()" ()------()--------X-----------()----------() | | | | | ()------()----3025/2048---15125/8192------() | | | | | ()-----55/32---275/256-----1375/1024---6875/4096 | | | | | 1/1-----5/4-----25/16-------125/64------625/512 | | | | | ()------()------25/22-------125/88--------() | | | | | ()------()-------()------------X----------() i'd be interested in any feedback anyone has on this. -monz
Message: 7599 Date: Fri, 17 Oct 2003 11:06:54 Subject: Re: [5-11]-prime-space p-block and 13edo From: Manuel Op de Coul I will add this scale to the archive. Speaking of lattices, I have improved the lattice player in Scala. Now it will also display nonrational intervals, so one can use it for the "bingocard" type. Go to Analyse:Lattice and player, click the 2D play button, tick the "Nearest in emtpy positions" checkbox and set an amount of rows and columns. Manuel
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