>>> >etter yet would be to simply use the harmonic entropies of the
>>> intervals, which i think i did in a few posts as well.
>>
>> Harmonic entropy sort of makes the idea of rationalizing a scale
>> irrelevant. On the other hand, starting with a *temperament*,
>> it's useful to have a method for snapping it to the lattice in a
>> simple way, as in TM reduction.
>
>yeah but then you break a lot of the consonant connections. which
>makes this whole "rationalization" business look pretty unhealthy to
>me.
Sure.
-C.
Message: 6811 - Contents - Hide Contents
Date: Mon, 12 May 2003 23:15:18
Subject: Re: how come i never saw this before?
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:
>>>> better yet would be to simply use the harmonic entropies of the
>>>> intervals, which i think i did in a few posts as well.
>>>
>>> You're referring to the minimum pairwise entropy posts?
>>
>> no, just multidimensional scaling solutions.
>
> How should I go about finding those posts?
search for "multidimensional scaling" on the tuning list here and in
robert's mills tuning list archive.
Message: 6812 - Contents - Hide Contents
Date: Tue, 13 May 2003 21:34:28
Subject: Efficicency and ambiguity
From: Gene Ward Smith
If I'm understanding Eytan Agmon's paper "Numbers and the Western
Tone-System", he is interested in MOS in an equal temperament which
are efficent and have at most one ambiguous interval. It seems to me that
a 7 or 8 note MOS in Blackwood/15 (with 2/15 as a generator) or a 9 or
10 note MOS of Negri/19 (with a 2/19 generator) would qualify. Eytan
claims to have a proof this is not so, but I don't see how he can in
the face of such examples. Is there anyone following this business who
thinks they could explain this?
I suppose I should look the proof up before the library here closes
for moving.
Message: 6813 - Contents - Hide Contents
Date: Tue, 13 May 2003 23:09:53
Subject: Re: Efficicency and ambiguity
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> If I'm understanding Eytan Agmon's paper "Numbers and the Western
> Tone-System", he is interested in MOS in an equal temperament which
> are efficent and have at most one ambiguous interval. It seems to
me that
> a 7 or 8 note MOS in Blackwood/15 (with 2/15 as a generator) or a 9
or
> 10 note MOS of Negri/19 (with a 2/19 generator) would qualify.
Here are more examples:
15 or 16 notes with a 2/31 generator (Quartaminorthirds/31)
26 or 27 notes with a 2/53 generator
49 or 50 notes with a 2/99 generator
But 2/odd is not the only possibility:
20 or 21 notes of Miracle/41 (4/41 generator.)
51 or 52 notes of Miracle/103
Then there are some nice ones fitting Eytan's conditions:
35 or 37 notes with 35/72 generator
41 or 43 notes with 41/84 generator
Message: 6814 - Contents - Hide Contents
Date: Wed, 14 May 2003 19:39:19
Subject: Re: Efficicency and ambiguity
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
>> If I'm understanding Eytan Agmon's paper "Numbers and the Western
>> Tone-System", he is interested in MOS in an equal temperament which
>> are efficent and have at most one ambiguous interval.
>
> Not at most, exactly one. So your examples don't qualify.
>
> Manuel
why on earth should there be exactly one ambiguous interval? it seems
to me that musical academia has been staring for too long at its 7-
out-of-12-equal navel.
Message: 6815 - Contents - Hide Contents
Date: Wed, 14 May 2003 22:10:26
Subject: Re: Efficicency and ambiguity
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul"
> <manuel.op.de.coul@e...> wrote:
> why on earth should there be exactly one ambiguous interval? it seems
> to me that musical academia has been staring for too long at its 7-
> out-of-12-equal navel.
Good question. However, as we've seen, Eytan actually exterminates
these with an extra assumption that the generator is to be an
approximate fifth. If we solve 1/2+1/(4*n) = log2(3/2) for n,
we get n = 2.94; the nearest interger solution and clearly by far the
best is n = 3, leading to the 12-et. Needless to say, I am not
convinced by all these assumptions, and wish Eytan had presented this
as "here are some interesting conditions on scales, which lead to 12
as their solution".
Message: 6816 - Contents - Hide Contents
Date: Wed, 14 May 2003 12:06:24
Subject: Re: Efficicency and ambiguity
From: Manuel Op de Coul
>If I'm understanding Eytan Agmon's paper "Numbers and the Western
>Tone-System", he is interested in MOS in an equal temperament which
>are efficent and have at most one ambiguous interval.
Not at most, exactly one. So your examples don't qualify.
Manuel
Message: 6817 - Contents - Hide Contents
Date: Wed, 14 May 2003 22:28:01
Subject: Re: Efficicency and ambiguity
From: wallyesterpaulrus
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>> --- In tuning-math@xxxxxxxxxxx.xxxx "Manuel Op de Coul"
>> > why on earth should there be exactly one ambiguous interval? it
seems
>> to me that musical academia has been staring for too long at its
7-
>> out-of-12-equal navel.
>
> Good question. However, as we've seen, Eytan actually exterminates
> these
these? what are these?
Message: 6818 - Contents - Hide Contents
Date: Wed, 14 May 2003 23:13:03
Subject: Some Eytan candidates
From: Gene Ward Smith
By this I mean numbers 4n such that the standard vals for 2n-1 and 2n+1
add up to the standard val for 4n, and all three have badness scores
below 2. All conditions except the requirement that generators be a
fifth are satisfied, and if Paul can call something which isn't an
approximate 5/4 a major third, it seems to me I can call something
which isn't an approximate 3/2 a fifth. Fair is fair. Paul? Is this
legitimate in your view, and if not, why not?
5-limit: 4, 8, 12
7-limit: 8, 12, 16, 32, 40, 56, 60, 72, 84, 224
11-limit: 16, 128, 176, 224, 320, 364, 460, 764, 1164
13-limit: 12, 16, 84, 176, 224, 320, 364, 624, 764, 1048
17-limit: 24, 224, 248, 296, 316, 320, 344, 364, 412, 460
19-limit: 24, 224, 248, 296, 316, 320, 344, 364, 412, 416, 460, 528,
768, 904, 1116
Message: 6819 - Contents - Hide Contents
Date: Wed, 14 May 2003 23:03:07
Subject: Re: Efficicency and ambiguity
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>> Good question. However, as we've seen, Eytan actually exterminates
>> these
>
> these? what are these?
Systems with an odd et n and 2/n as generator.
Message: 6820 - Contents - Hide Contents
Date: Thu, 15 May 2003 00:30:28
Subject: Scala and Agmon diatonic systems
From: Gene Ward Smith
I loaded the 24-et, 13-note MOS with 13/24 generator into Scala to
check whether it connected to any Arabic scales. I didn't find out,
since when I tried to compare to the scales in my scl directory, which
I got by donwloading and unzipping from the Scala site, it said that
every one of the scales had a read error, then that it had encountered
a bug and that I should send a bug report to Manuel. This is, among
other things, the bug report.
I did note, however, that it gave the following information about it:
Scale is a Winograd/Gammer deep scale
Scale is an Agmon diatonic system
Rothenberg stability 77/78
Lumma stability 1/2
Message: 6821 - Contents - Hide Contents
Date: Thu, 15 May 2003 00:34:15
Subject: Re: Efficicency and ambiguity
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>> Good question. However, as we've seen, Eytan actually exterminates
>> these
>
> these? what are these?
MOS of size (n+1)/2 within an n-et, such that n is odd and 2/n is the
generator.
Message: 6822 - Contents - Hide Contents
Date: Thu, 15 May 2003 03:57:29
Subject: Re: Scala and Agmon diatonic systems
From: Gene Ward Smith
--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:
> I loaded the 24-et, 13-note MOS with 13/24 generator into Scala to
> check whether it connected to any Arabic scales. I didn't find out,
> since when I tried to compare to the scales in my scl directory, which
> I got by donwloading and unzipping from the Scala site, it said that
> every one of the scales had a read error, then that it had encountered
> a bug and that I should send a bug report to Manuel. This is, among
> other things, the bug report.
I tried to do the same under W98, and this time the zip file would not
upzip correctly.
Message: 6823 - Contents - Hide Contents
Date: Thu, 15 May 2003 12:26:27
Subject: Re: Efficicency and ambiguity
From: Manuel Op de Coul
>> >hy on earth should there be exactly one ambiguous interval? it seems
>> to me that musical academia has been staring for too long at its 7-
>> out-of-12-equal navel.
>> these? what are these?
>MOS of size (n+1)/2 within an n-et, such that n is odd and 2/n is the
>generator.
Yeah, those have no ambiguous interval, leaving only the even n with
exactly one ambiguous interval.
Manuel
Message: 6824 - Contents - Hide Contents
Date: Sat, 17 May 2003 04:59:35
Subject: Efficient MOS
From: Gene Ward Smith
Those with good memories may recall the notation n+m;s I introduced to
describe MOS on ets. Here the n+m part defines the generator and et;
if we assume n>m and d = gcd(n,m) then if q/r is the next element of
the nth row of the Farey sequence after m/n, so that if u = m/d and v
= n/d then u*r-v*q=1, the generator Gen(n,m) = (q+r)/(u+v), the
period is 1/d, and the equal temperament is n+m. Then n+m;s means the
s-element scale in the n+m-et with generators Gen(n,m) and 1/d.
In terms of this notation, Eytan is considering 2*n+1+2*n-1;2*n+1.
This suggests how to put things in a wider context, by looking at
n+m;n for
various values of n-m togeter with n mod n-m.
Difference of one
We get scales n+1+n;n+1, with et 2*n+1 and generator 2/(2*n+1)
Differences of two
(1) Scales 2*n+2+2*n;2*n+2 with generators [1/2, 1/(2*n+1)] in the
4*n+2 et
(2) Scales 2*n+1+2*n-1;2*n+1 with generator (2*n+1)/(4*n) in the 4*n
et; if you add to this the very strange requirement that the generator
maps from 3/2, you get Eytan's diatonic systems
Differences of three
(1) Scales 3*n+3+3*n;3*n+3 with generators [1/3, 1/(2*n+1)] in the
6*n+3 et
(2) Scales 3*n+2+3*n-1;3*n+2 with generator (2*n+1)/(6*n+1) in the
6*n+1 et
(3) Scales 3*n+1+3*n-1;3*n+1 with generator (4*n)/(6*n-1) in the
6*n-1 et
And so forth...
Previous Next
6000
6050
6100
6150
6200
6250
6300
6350
6400
6450
6500
6550
6600
6650
6700
6750
6800
6850
6900
6950
6800 -
6825 -