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114 7-limit temperaments   Message List  
Reply | Forward Message #9052 of 17519 |
Cross-check for TOP 5-limit 12-equal

Wedgie norm for 12-equal:

Take the two unison vectors

|7 0 -3>
|-4 4 -1>

Now find the determinant, and the "area" it represents, in each of
the basis planes:

|7 0| = 28*(e23) -> 28/lg2(5) = 12.059
|-4 4|

|7 -3| = -19*(e25) -> 19/lg2(3) = 11.988
|-4 -1|

|0 -3| = 12*(e35) -> 12 = 12
|4 -1|

sum = 36.047

If I just use the maximum (L_inf = 12.059) as a measure of notes per
acoustical octave, then I "predict" tempered octaves of 1194.1 cents.
If I use the sum (L_1), dividing by the "mystery constant" 3,
I "predict" tempered octaves of 1198.4 cents. Neither one is the TOP
value . . . :( . . . but what sorts of error criteria, if any, *do*
they optimize?

So the cross-checking I found for the 3-limit case in "Attn: Gene 2"
http://groups.yahoo.com/group/tuning-math/message/8799
doesn't seem to work in the 5-limit ET case for either the L_1 or
L_inf norms.

However, if I just add the largest and smallest values above:

28/lg2(5)+19/lg2(3)

I do predict the correct tempered octave (aside from a factor of 2),

1197.67406985219 cents.

So what sort of norm, if any, did I use to calculate complexity this
time? It's related to how we temper for TOP . . .




Sat Jan 31, 2004 8:21 pm

paulerlich
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Message #9052 of 17519 |
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... I'm working to give people what they seem to want. See my next post....
Gene Ward Smith
genewardsmith
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Feb 4, 2004
6:46 am

Wedgie norm for 12-equal: Take the two unison vectors ... Now find the determinant, and the "area" it represents, in each of ... sum = 36.047 If I just use the...
Paul Erlich
paulerlich
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Jan 31, 2004
8:21 pm

... per ... cents. ... TOP ... this ... I used this latter complexity measure to create these graphs: http://groups.yahoo.com/group/tuning_files/files/et3.gif ...
Paul Erlich
paulerlich
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Feb 3, 2004
1:40 am

... Thanks for doing these Paul. ... I'm not familiar enough with 3-limit harmony (or rather ignoring 5-limit harmony) to comment on this, but I think I could...
Dave Keenan
dkeenanuqnetau
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Feb 3, 2004
3:31 am

... with ... which ... interest ... the ... moat ... Same goes for Waage in the 5- (where it's been called Aristoxenean) and 7-limit lineat temperament cases,...
Paul Erlich
paulerlich
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Feb 3, 2004
7:54 pm

... with ... which ... Looks like a 'constellation' -- with 12 stars :( :) ... I also would have liked to see 43 and 50, but I suppose these are just...
Paul Erlich
paulerlich
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Feb 4, 2004
1:28 am

... 12-equal doesn't really make it in as 7-limit for me personally, but I was trying to keep you happy too. ... Yes. And tomorrow someone might come up with...
Dave Keenan
dkeenanuqnetau
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Feb 4, 2004
9:08 am

If someone could help explain this, and/or generalize it to higher dimensions, I'd be thrilled . . . pleeeeeeeeeeeease? Also would like to understand the...
Paul Erlich
paulerlich
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May 25, 2004
6:19 pm

I re-ranked Gene's top 64 using L_1 and got the following top 32. Anything missing? 1. ... 39.8287 -> bad = 57.7058 2. ... 11.7652 -> bad = 235.1092 3. ... ...
Paul Erlich
paulerlich
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Jan 31, 2004
7:23 pm

There's something VERY CREEPY about my complexity values. I'm going to have to accept this as *the* correct scaling for complexity (I'm already convinced this...
Paul Erlich
paulerlich
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Feb 1, 2004
4:41 am

Sorry I skipped this: 4. ... 31.212 -> bad = 300.04 31.212 --> 31 -? 31-note proper MOS ... norm,...
Paul Erlich
paulerlich
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Feb 1, 2004
4:50 am

... Woah! Yes....
Dave Keenan
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Feb 1, 2004
5:32 am

... going ... (I'm ... norm, ... So when I explained this as the number of notes in the bivector, I must have been onto something. It's interesting how the...
Paul Erlich
paulerlich
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Feb 1, 2004
5:58 am

... seem ... Except Blackjack . . ....
Paul Erlich
paulerlich
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Feb 1, 2004
6:19 am

... Wow dude. What sort of DES are these? Not the smallest apparently. -Carl...
Carl Lumma
clumma
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Feb 1, 2004
1:30 pm

... So how are these generators being chosen? Hermite? I confess I don't know how to 'refactor' a generator basis. -Carl...
Carl Lumma
clumma
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Feb 1, 2004
1:43 pm

... No, just assume octave repetition, find the period (easy) and then the unique generator that is between 0 and 1/2 period. ... What do you have in mind?...
Paul Erlich
paulerlich
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Feb 2, 2004
2:33 am

... Isn't it possible to find alternate generator pairs that give the same temperament when carried out to infinity? -Carl...
Carl Lumma
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Feb 2, 2004
3:19 am

... Yup! You can assume tritave-equivalence instead of octave- equivalence, for one thing . . ....
Paul Erlich
paulerlich
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Feb 2, 2004
3:56 am

... And can doing so change the DES series? -Carl...
Carl Lumma
clumma
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Feb 2, 2004
4:29 am

... then ... Well of course . . . can you think of any octave-repeating DESs that are also tritave-repeating?...
Paul Erlich
paulerlich
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Feb 2, 2004
5:14 am

... Right, so when trying to explain a creepy coincidence between complexity and DES cardinalities, might not we take this into account? -Carl...
Carl Lumma
clumma
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Feb 2, 2004
5:26 am

... Sure . . . some of the ones that 'don't work' may be working for tritave-DESs rather than octave-DESs, is that what you were thinking?...
Paul Erlich
paulerlich
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Feb 2, 2004
5:33 am

... Yep! -Carl...
Carl Lumma
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Feb 2, 2004
5:40 am

... norm, ... That's great, Paul. So what's the scaling?...
Gene Ward Smith
genewardsmith
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Feb 1, 2004
6:56 pm

... going ... (I'm ... I'm using your formula from http://groups.yahoo.com/group/tuning-math/message/8806 but instead of "max", I'm using "sum" . . ....
Paul Erlich
paulerlich
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Feb 2, 2004
3:01 am

... So these cosmically great answers are coming from the L1 norm applied to the scaling we got from vals, where we divide by log2(p)'s. What does that mean, I...
Gene Ward Smith
genewardsmith
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Feb 2, 2004
5:18 am

... applied ... The formulas, I think, are basically the same ones being used in the "cross-check" post -- did you have a chance to think about it? ...
Paul Erlich
paulerlich
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Feb 2, 2004
6:04 am

... are ... like ... even ... Gene, how does 4000/3993 (with its factors of 11) get into the description of a 7-limit temperament?...
Paul Erlich
paulerlich
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Mar 2, 2005
6:31 pm

... He did? I can't find this message from Gene Ward Smith, either in my in box or in the Yahoo Groups archive. Is it really out there? Can someone sent me a...
Rich Holmes
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Mar 2, 2005
8:10 pm
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