Yahoo Tuning Groups Ultimate Backup tuning Necessary conditions for real existence

On Tue, Feb 14, 2012 at 5:20 PM, genewardsmith
<genewardsmith@...> wrote:
>
> One such condition I would suggest is that the wedgie W should be recoverable from the octave equivalent portion, meaning that if you take (Wb(2)b'K, where K = <1 log2(3) log2(5) ... log2(p)|, and round it to the nearest integer, it should equal W. This is not a very stringent condition, but depending on your philosophical views it may or may not distress you to learn that JI is recoverable, and hence might possibly really exist.

This logical exposition was destroyed by the Argument That Yahoo
Supports Unicode fallacy.

-Mike

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Feb 14, 2012 at 5:20 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > One such condition I would suggest is that the wedgie W should be recoverable from the octave equivalent portion, meaning that if you take (Wv2)^K, where K = <1 log2(3) log2(5) ... log2(p)|, and round it to the nearest integer, it should equal W. This is not a very stringent condition, but depending on your philosophical views it may or may not distress you to learn that JI is recoverable, and hence might possibly really exist.
>
> This logical exposition was destroyed by the Argument That Yahoo
> Supports Unicode fallacy.
>
> -Mike
>

🔗Mike Battaglia <battaglia01@...>

On Tue, Feb 14, 2012 at 5:25 PM, genewardsmith
<genewardsmith@...> wrote:
>
> One such condition I would suggest is that the wedgie W should be recoverable from the octave equivalent portion, meaning that if you take (Wv2)^K, where K = <1 log2(3) log2(5) ... log2(p)|, and round it to the nearest integer, it should equal W. This is not a very stringent condition, but depending on your philosophical views it may or may not distress you to learn that JI is recoverable, and hence might possibly really exist.

What meaning does it have to take the exterior product of some val and
the JIP? And then what happens if you round it?

-Mike

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> What meaning does it have to take the exterior product of some val and
> the JIP? And then what happens if you round it?

Examples: <<0 0 1 0 2 2|| v 2 = <0 0 0 -1|. <0 0 0 -1|^JIP = <<0 0 1 0 1.585 2.322||; rounded is <<0 0 1 0 2 2||. Status: recoverable. This temperament (which tempers out 5/4 and 4/3, God help us) might exist. It is the highest-error recoverable 7-limit temperament.

<<0 2 0 3 0 -5|| v 2 = <0 0 -2 0|. <0 0 -2 0|^JIP = <<[0 2 0 3.167 0 -5.615||, rounded is <<0 2 0 3 0 -6||. This temperament (which merely tempers out 9/8 and 7/6, but which is more complex) is not recoverable. Both, especially the first, are pretty close to the line.

🔗Herman Miller <hmiller@...>

On 2/15/2012 12:29 AM, genewardsmith wrote:
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia<battaglia01@...> wrote:
>
>> What meaning does it have to take the exterior product of some val and
>> the JIP? And then what happens if you round it?
>
> Examples:<<0 0 1 0 2 2|| v 2 =<0 0 0 -1|.<0 0 0 -1|^JIP =<<0 0 1 0 1.585 2.322||; rounded is<<0 0 1 0 2 2||. Status: recoverable. This temperament (which tempers out 5/4 and 4/3, God help us) might exist. It is the highest-error recoverable 7-limit temperament.
>
> <<0 2 0 3 0 -5|| v 2 =<0 0 -2 0|.<0 0 -2 0|^JIP =<<[0 2 0 3.167 0 -5.615||, rounded is<<0 2 0 3 0 -6||. This temperament (which merely tempers out 9/8 and 7/6, but which is more complex) is not recoverable. Both, especially the first, are pretty close to the line.

Such examples may have interest as mathematical objects, but I think we need at least some minimum conditions when considering potential temperaments. One that occurs to me is that each of the primes must have a distinct mapping. With the <<0 0 1 0 2 2|| temperament both 3/1 and 5/1 have the same mapping. We could call this a "degenerate" temperament, so it technically would still be a temperament, but we're more likely to be interested in non-degenerate temperaments.

Although <<0 2 0 3 0 -5|| isn't degenerate if you only count the primes, 6/1 and 7/1 do have the same mapping. I think it's reasonable to define a kind of temperament which has distinct mappings for the harmonic series up to a certain point (perhaps the first gap, 11/1 in the case of 7-limit temperaments). I think it's also reasonable to insist on tunings in which each successive tempered interval in the harmonic series is greater than the previous interval. In other words, an ascending scale. You could still consider temperaments that violate these conditions as mathematical curiosities, but they're not likely to be of any musical use. (Not as temperaments, that is, although you could use them as warping transformations.)

🔗gbreed@...

We're most likely to be interested in temperaments that are low in error and complexity. It's as simple as that.

Graham

------Original message------
From: Herman Miller <hmiller@...>
To: <tuning@yahoogroups.com>
Date: Wednesday, February 15, 2012 9:13:45 PM GMT-0500
Subject: Re: [tuning] Re: Necessary conditions for real existence

Such examples may have interest as mathematical objects, but I think we
need at least some minimum conditions when considering potential
temperaments. One that occurs to me is that each of the primes must have
a distinct mapping. With the <<0 0 1 0 2 2|| temperament both 3/1 and
5/1 have the same mapping. We could call this a "degenerate"
temperament, so it technically would still be a temperament, but we're
more likely to be interested in non-degenerate temperaments.

Although <<0 2 0 3 0 -5|| isn't degenerate if you only count the primes,
6/1 and 7/1 do have the same mapping. I think it's reasonable to define
a kind of temperament which has distinct mappings for the harmonic
series up to a certain point (perhaps the first gap, 11/1 in the case of
7-limit temperaments). I think it's also reasonable to insist on tunings
in which each successive tempered interval in the harmonic series is
greater than the previous interval. In other words, an ascending scale.
You could still consider temperaments that violate these conditions as
mathematical curiosities, but they're not likely to be of any musical
use. (Not as temperaments, that is, although you could use them as
warping transformations.)

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