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desire for meantone with an 11-limit interval on piano   Message List  
Reply | Forward Message #66543 of 85208 |
Re: desire for meantone with an 11-limit interval on piano

--- In tuning@yahoogroups.com, Kurt Bigler <kkb@...> wrote:
multiply all ratios by factor 264=11*3*2*2 in order to
obtain integral absolute pitch-frequencies
> > ! bihexany.scl
> > Hole around [0, 1/2, 1/2, 1/2]
> > 12
> > !
C4 264 from 1/1 middle C=264Hz
C# 280 > 35/33
(D4 297 > 9/8) inserted instead Cb=B# for smoothening the 5ths
Eb 308 > 7/6
E4 330 > 5/4
F4 336 > 14/11
F# 360 > 15/11
G4 396 > 3/2
G# 420 > 35/22
A4 440 > 5/3 !normal pitch 440Hz
Bb 462 > 7/4
B4 480 > 20/11
(Cb=B# 504 > 21/11)
C5 528 > 2

or arranged in a cycle of a dozen partial tempered 5ths

C 33,..,264 begin, starting @ middle-C
G 99:=33*3,198,396
D (37,74,148,296)297:=99*3
A 55,110(111:=37*3),220,440 ! (111:110)*(297:296)=81:80 the SC
E (41,82,164)165:=55*3,330
B 15,30,60,,(123:=41*3)120,240,480 ! (165:164)*(41:40)=33:32
F# 45:=15*3,..,360
C# 35,70,(135:=45*3)140,280 ! 140:135=28:27
G# (13,26,52,104)105:=35*3,210,420
Eb (39:=13*3,78)77,154,308
!(105:104)*(77:78)=2695:2704=299.4444.../300.4444...
Bb (7,...,228)231:=77*3,462 ! 231:228=77:76
F (11,22)21:=7*3,42,84,168,336
C 33:=11*3 cycle closed, ready done.

i.m.o: heavy harsh tempering, but it works.
A.S.






Fri May 26, 2006 8:53 am

a_sparschuh
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Message #66543 of 85208 |
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Gene and others, ... I seem to recall that Gene or someone had made some reasonable lattice representation of this (in 2-d projection), but I can't find it,...
Kurt Bigler
voxdig
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May 26, 2006
6:34 am

... multiply all ratios by factor 264=11*3*2*2 in order to obtain integral absolute pitch-frequencies ... C4 264 from 1/1 middle C=264Hz C# 280 > 35/33 (D4 297...
a_sparschuh
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May 26, 2006
8:54 am

... I'm not ... The first neat fact to bear in mind is that everything in the 7-limit can be expressed in terms of 2, 49/40, 10/7, and 2401/2400, and that last...
Gene Ward Smith
genewardsmith
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May 26, 2006
9:13 am

Gene, ... Thanks, I'll take a look at that. I've got a bit of reviewing to do to understand all this, but it is a good exercise. But maybe you can give me a...
Kurt Bigler
voxdig
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May 27, 2006
1:33 am

... Yeah, if somebody could plot that, I'd love to see it. ... Sorry to butt in, but I think the 2401/2400 thing was just an example -- a 7-limit one. The...
Carl Lumma
clumma
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May 27, 2006
2:01 am

... Since (441/440)^2/(243/242) = 2401/2400, it also tempers that out. Plus, (243/242)/(441/440) = 540/539....
Gene Ward Smith
genewardsmith
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May 27, 2006
3:09 am

... I'm working on getting Mathematica to do this. I can probably provide a png file when I have something that looks decent. ... Heck, no problem. Fancy...
Kurt Bigler
voxdig
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May 27, 2006
3:33 am

... Great! ... You can think in terms of 81/80 reducing the 5-limit to a linear thing -- and back it up with examples at your piano! ... I think this is...
Carl Lumma
clumma
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May 27, 2006
5:15 am

... Yes. But it doesn't give you a 5-limit structure. One consequence of tempering out 2401:2400 is that you have true neutral thirds: the perfect fifth can...
Graham Breed
x31eq
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May 27, 2006
6:45 am

... Well here's a first cut. It's not very "informational": http://breathsense.com/published/bihexany1.png I can add labelling etc. once i understand it...
Kurt Bigler
voxdig
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May 27, 2006
7:06 am

... It helps if you know that the stick figure with two legs is an ontonal tetrad, and standing on his head is a utonal tetrad. You can also see two hexagons...
Gene Ward Smith
genewardsmith
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May 27, 2006
8:40 am

... You have the origin labled, so that's 1/1. Then the stick figure off to the right, with 1/1 the left leg, is the 1-5/4-3/2-7/4 hexad. The 3/2 is the right...
Gene Ward Smith
genewardsmith
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May 27, 2006
9:02 am

... seems to me ... Not really. It isn't like 225/224, where you can look at it as a way of mashing the 7s down into the 5-limit lattice. Here the horizontal ...
Gene Ward Smith
genewardsmith
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May 27, 2006
9:21 am

... Even more, a 24-tone chain allows good combinations of both 7-limit and 11-limit, like "B-D#-F#-Gx-C#-Fb". And if you are willing to accept a 29-tone...
Petr ParĂ­zek
p.parizek@...
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May 16, 2006
8:59 pm

back in the mid 80's when i only had a 12 tone marimba to retune to go along with my set of dallesandro tubes with the full 1-3-5-7-9-11 CPS, i settled on...
Kraig Grady
banaphshu
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May 17, 2006
2:28 pm

... Hiya Kraig- did I get this right? ! Kraig Grady's dual [5 7 9 11] hexany scale. 12 ! 35/33 10/9 7/6 14/11 15/11 81/56 3/2 45/28 135/77 81/44 27/14 2 ! Gene...
Carl Lumma
clumma
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May 18, 2006
7:15 am

... No; mine is constructed in a way analogous to the construction of a hexany as a lattice hole. This is a lattice hole in an 11-limit lattice....
Gene Ward Smith
genewardsmith
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May 18, 2006
10:04 pm

... I'd love to hear more about the difference. -Carl...
Carl Lumma
clumma
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May 19, 2006
9:01 pm

... Here are some relevant articles: http://groups.yahoo.com/group/tuning-math/message/11813 http://groups.yahoo.com/group/tuning-math/message/11833 ...
Gene Ward Smith
genewardsmith
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May 20, 2006
1:10 am

... Yes, well I think I grok all that. The scale I gave was supposed to be two hexanies (as you point out, in the 11-limit, hexanies are consonant chords) a...
Carl Lumma
clumma
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May 20, 2006
2:59 am

Hi all, ... Carl, The ratios are clearly different; those you cite are more complex, but the step sizes are less extreme and overall, their average deviation ...
Yahya Abdal-Aziz
yahya_melb
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May 18, 2006
2:07 pm

... Great, Yahya! -Carl...
Carl Lumma
clumma
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May 19, 2006
9:03 pm

i am going to assume this is right as i don't have time to figure it out right now. Gene's looks like it has a similar construction cause it is outwardly...
Kraig Grady
banaphshu
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May 18, 2006
2:22 pm

... Try out the following circle of a dozen 5ths in absolute frequencies: C 33 cps or Hz, "the fundamental base of the chord G 99 :=33*3 "just pure 5th D...
a_sparschuh
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May 19, 2006
1:41 pm

... I really can't understand what this means. Why did you round everything to whole numbers of hertz? Are the octaves really supposed to be stretched by 10-20...
Keenan Pepper
keenan_pepper
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May 19, 2006
2:42 pm

... Perhaps you will be able to comprehend the numbers a little bit more, if you get the same series represented in ascending order, resorted upwards in seize...
a_sparschuh
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May 19, 2006
6:53 pm

... Dear Keenan, here comes additional the conversion-step into the middle octave too: C4 264:= 33*8 > > C 33 G4 396:= 99*4 > > G 99 D4 297_______ > > D...
a_sparschuh
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May 20, 2006
3:53 pm

... There are still some simple arithmetic errors here. 123*4 = 492, not 462, but I suspect that's just a typo. I can't understand how you get from B to F#. It...
Keenan Pepper
keenan_pepper
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May 20, 2006
4:52 pm

... ___ B4 492:=123*4 > > B (61,122)123 "instead former-wrong "typo": 462 ... The !!! 5th B>F# has to be flattend down by the product of (123/122)*(122/121) =...
a_sparschuh
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May 22, 2006
5:13 pm

... It sounds harsh to my ears too. That's neat about "alphorn fa" though, maybe I'll work that into a Wikipedia article. ... LOL, and what if we allow...
Keenan Pepper
keenan_pepper
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May 24, 2006
2:47 am
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