Topic: Least squares optimized Phillips scales

2 scales

Thread (4 messages)

From: Gene Ward Smith (2002-10-26)
Subject: Least squares optimized Phillips scales

I did a least-squares fit to 7-limit intervals within 5 cents and 2.5 cents, and the results were revelatory.

! phillips19_1.scl
!
Pauline Phillips 5 cent margin optimized
19
!
84.3139369
151.8583699
199.4177206
268.7134087
383.8258631
400.9748406
467.6441078
500.2450361
584.1379555
668.1811654
699.7491415
768.3582599
852.4717660
899.7040367
967.8513128
1084.072482
1102.929842
1167.567841
2/1

6 major and 5 minor tetrads, 9 major and 7 minor triads, all within an error of 3 cents.

! phillips19_3.scl
!
Pauline Phillips 2.5 cent margin optimized
19
!
84.3723294
153.1326370
199.4516985
268.4554748
384.0592956
400.9792756
467.9551626
500.1311909
583.9525259
668.1674854
699.8776326
768.0935174
854.1073992
899.6726293
968.0465972
1084.139777
1102.934277
1167.481701
2/1

One less minor tetrad, and the error is not much smaller; I think the first version may be what Pauline was shooting for.

From: [email protected] (2002-10-26)
Subject: Re: [tuning] Least squares optimized Phillips scales

In a message dated 10/26/02 8:05:33 AM Central Daylight Time, 
[email protected] writes:


> I think the first version may be what Pauline was shooting for. 
> 
When I invented my scales, I was seeking to play in JI music already composed 
for tempered scales with 12 keyboard notes per octave. And do so with the 
least extra notes anad keys above 12. I found 19 noes to the octave would do 
this when slightly tempered. 

Some of the results were free extras. Such as being to play in C major, 7c 
minor and tertian e minor with the same tuning. And having several secondary 
scales related to these with the same tuning.

Pauline

From: Carl Lumma (2002-10-27)
Subject: Re: Least squares optimized Phillips scales

[Gene wrote...]
>I did a least-squares fit to 7-limit intervals within 5 cents
>and 2.5 cents, and the results were revelatory.

Least-squares fit from what starting scale?  How were they
relevatory?

-Carl

From: Gene Ward Smith (2002-10-27)
Subject: Re: Least squares optimized Phillips scales

--- In tuning@y..., "Carl Lumma" <clumma@y...> wrote:
> [Gene wrote...]
> >I did a least-squares fit to 7-limit intervals within 5 cents
> >and 2.5 cents, and the results were revelatory.
> 
> Least-squares fit from what starting scale?  How were they
> relevatory?

I adjusted the Phillips scale to the 7-limit consonant intervals in included with a margin of 5 cents, and then of 2.5 cents, and it showed the errors needn't be as great, and suggested how the 5-limit substructure looked.