Tuning-Math Digests messages 6103 - 6127

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Message: 6103

Date: Sun, 19 Jan 2003 12:23:08

Subject: Re: A common notation for JI and ETs

From: David C Keenan

I earlier referred to a "uselessnes" figure-of-demerit for notational 
commas. This was obtained by multiplying the power of 3 in the comma 
(number of fifths) by the size of the comma in cents.

I have since realised that for the purpose of our notation (or any notation 
based on a chain of best fifths) the ideal power of 3 in a comma is not 
zero, but

ideal_power_of_three  =  7 * size_of_comma / size_of_apotome

or

7 * comma_in_cents / 113.685

This ideal value would ensure that the comma always represented the same 
fraction of an apotome irrespective of the size of the fifth.

This can be used to show that a near-ideal half-apotome comma cannot exist. 
It would need to have 3 to the power of 3.5, but the power of 3 must be an 
integer. The best we could do is find a near 57 cent comma with 3^3 or 3^4, 
which is what we have in '|)) the 5:49 comma 392:405 (56.5 c).

We can then define the "slope" of a comma as

slope = actual_power_of_three - ideal_power_of_three

For example (|( as the 5:11' comma 44:45 (38.9 c) has a slope of

2 - 7 * 38.9/113.7 = -0.4

which tells us that it should be extremely good at representing 2/7 of an 
apotome across a wide range of fifth sizes. In fact it works for ETs 
49,56,63,70,77,84 and 91.

I believe we should not use a comma for notating temperaments if it has an 
absolute value of slope greater than 8. The following have slopes between 7 
and 8 and so I think they should also be avoided if possible
'|   as 5' comma
~|(  as 7:13 comma
|~   as 23 comma
)|~  as 19' comma
|))  as 49' diesis

This also says that we should equally allow both
'/|) as 7' diesis  57344:59049
and its apotome complement
.(|\ as 7" diesis  27:28
as you first suggested, since they have slopes of 6.9 and -6.9. In fact 
true apotome complements always have the same absolute value of slope, but 
opposite signs.

This suggests we modify "uselessness" to use slope instead of number of fifths.

uselessness = ABS(slope) * size_of_comma

but in fact I find uselessness fairly useless now, and I'm happy to simply 
limit the absolute slope to 8 and the comma size to 70.17 cents, the 
largest that could conceivably be notated as '((|


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Message: 6105

Date: Sun, 19 Jan 2003 05:50:09

Subject: Re: heuristic and straightness

From: Carl Lumma

>>>shortening the unison vectors makes the temperament worse, but
>>>in a given temperament, this would be counteracted by an
>>>increase in straighness, which makes the temperament better.
>>
>>You lost me.
>
>well, there are probably too many counterfactuals here. why don't
>we start again with any examples from the archives which came up
>in connection with straightness. your choice.

Good idea.  I did find some talk between you and Gene, referring
to results I couldn't find. :(  At one point, you mention low-
badness blocks of one note...  If you pick the example, I'm happy
to attempt to build the block and find the alternate versions...
no promises, though, since I'm new to this.

If we get a good example or two, I'll collect everything and
give you a URL.  Bless you, monz, for doing this sort of thing.

-Carl


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Message: 6108

Date: Sun, 19 Jan 2003 20:16:25

Subject: Re: Calculating geometric complexity and badness

From: Carl Lumma

> 5 limit
> 
> In the 5 limit, if q = 2^u0 3^u1 5^u2 is a comma,
> then [u0, u1, u2] is the wedgie

That's all the wedgie is for a single comma?  I
take it this would be the chromatic uv in the
linear case?

> sqrt(log(3)^2 u1^2 + log(3)^2 u1 u2 + log(5)^2 u2^2)
> 
> is the geometric complexity.

Cool.  I can do that.  ;)

> 11 limit
> 
> Linear
> 
> If 
> 
> p = 2^u0 3^u1 5^u2 7^u3 11^u4
> q = 2^v0 3^v1 5^v2 7^v3 11^v4
> r = 2^w0 3^w1 5^w2 7^w3 11^w4
> 
> are three intervals,

You mean commas, right?

> Spacial

You the man, Gene!

> Badness
> 
> If r is error (usually rms error), c is complexity, n is the
> number of primes in the p-limit (n=pi(p)) and d is the
> codimension, which is to say the number of commas used to
> define the temperament,

Does it matter how many we temper out here?

> then
> 
> badness = r c^(n/d)
> 
> is the log-flat badness measure.

Great!  I always thought the critical exponent had to be
determined empirically.

-Carl


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Message: 6109

Date: Sun, 19 Jan 2003 20:24:13

Subject: Re: heuristic and straightness

From: Carl Lumma

>>>>>shortening the unison vectors makes the temperament worse,
>>>>>but in a given temperament, this would be counteracted by
>>>>>an increase in straighness, which makes the temperament
>>>>>better.
>>>>
>>>>You lost me.
>>>
>>>well, there are probably too many counterfactuals here. why
>>>don't we start again with any examples from the archives
>>>which came up in connection with straightness. your choice.
>>
>>Good idea.  I did find some talk between you and Gene,
>>referring to results I couldn't find. :(  At one point, you
>>mention low-badness blocks of one note...  If you pick the
>>example, I'm happy to attempt to build the block and find the
>>alternate versions... no promises, though, since I'm new to
>>this.
>
>what are we talking about, anyway? (no offence)

I'm trying to figure out what you meant in the top paragraph,
there...

>>>>>shortening the unison vectors makes the temperament worse,

If we heuristically ignore the sizes, then yes.

>>>>>but in a given temperament, this would be counteracted by
>>>>>an increase in straighness, which makes the temperament
>>>>>better.

???  If straightness is maximal when the uvs are maximally
orthogonal, how does this mean the uvs have gotten shorter?
I thought it was a *decrease* in straightness that made the
difference/sum vector shorten, making the temperament *worse*.


Ultimately, I'm trying to figure out how changing the
straigtness can make a temperament "worse" but keep badness
constant.  Either badness is broken or it doesn't!

-Carl


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Message: 6110

Date: Sun, 19 Jan 2003 20:31:10

Subject: Re: New file uploaded to tuning-math

From: Carl Lumma

Paul,

These are tha bomb!

-Carl


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Message: 6112

Date: Mon, 20 Jan 2003 19:18:51

Subject: Re: An 11-limit linear temperament top 100 list

From: Carl Lumma

>If we define "epimericity" for p/q > 1 reduced to its lowest
>form as log(p-q)/log(q),

Now we're talking.  Looks a lot like the "cent" heuristic...

>then as I suggested in Message 4458,

Apparently, *then* we were talking.  I remember reading that,
too, but I just didn't have a clue where it was coming from.
Now, I think I'm catching on.

-Carl


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Message: 6119

Date: Mon, 20 Jan 2003 08:23:14

Subject: Re: Superparticular temperaments

From: Carl Lumma

>In the 5-limit case, it is clearly not so, unless we
>think meantone is the last word. In the 7-limit case,
>we do miss some important systems, but the inclusion
>of some fairly obvious commas would catch them.
//
>There are 150 7-limit superparticular linear temperaments;

I dunno Gene; looks like a bust to me.  I wonder what
other easy stuff can be done to fractions to study commas.
There are jacks, but they're a subset of the above...
While we would expect superparticulars to be the smallest
intervals of a given complexity, there must be a cleaner
way of doing this... has anything been done on 'badness for
commas'?

-C.


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Message: 6120

Date: Mon, 20 Jan 2003 08:26:20

Subject: Re: Calculating geometric complexity and badness

From: Carl Lumma

I wrote..
> > In the 5 limit, if q = 2^u0 3^u1 5^u2 is a comma,
> > then [u0, u1, u2] is the wedgie
> 
> That's all the wedgie is for a single comma?  I
> take it this would be the chromatic uv in the
> linear case?

D'oh, I meant commatic.

-C.


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