19:16:15 temps again

OK, more mathematics has been done... the results are, I hope,
practically Scala-ready.

#1 (nicknamed 19berger)
C 1
C# 256/243
D 151/135
Eb 19/16
E 304/243
F 4864/3645
F# 2215/1576
G 2423/1620
A 271/162
Bb 8417/4728
B 15/8

... numbers for G,D,A,F#,Bb are rational approximations to regular
tempering of fifths between C-...-E, B-.-C#, Eb-.-F resp.

(Is there a formula or website which spits out 'nice' rational
approximations to any given precision of decimal representation?)

#2 (nicknamed 19otti)
C 1
C# 135/128
D 573/512
Eb 19/16
E 2565/2048
F 171/128
F# 45/32
G 383/256
G# 2431/1536
A 3429/2048 (or 226/135)
Bb 57/32
B 15/8

... here only G,D,A and G# are 'tempered'. Moreover C-G-D-(A)-E uses a
division of the 96/95 comma as 384:383:382:381:380. This one seemed
to work out a bit neater.

~~~T~~~

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> (Is there a formula or website which spits out 'nice' rational
> approximations to any given precision of decimal representation?)

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/cfCALC.html

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> OK, more mathematics has been done... the results are, I hope,
> practically Scala-ready.
>
> #1 (nicknamed 19berger)
> C 1
> C# 256/243
> D 151/135
> Eb 19/16
> E 304/243
> F 4864/3645
> F# 2215/1576
> G 2423/1620
> A 271/162
> Bb 8417/4728
> B 15/8

I'd like it a whole lot better if it had twelve notes. Why is G#/Ab
missing? Anyway, here it is as a Scala scale:

! 19berger.scl
Tom dent's 19berger scale
11
!
256/243
151/135
19/16
304/243
4864/3645
2215/1576
2423/1620
271/162
8417/4728
15/8
2

> ... numbers for G,D,A,F#,Bb are rational approximations to regular
> tempering of fifths between C-...-E, B-.-C#, Eb-.-F resp.
>
> (Is there a formula or website which spits out 'nice' rational
> approximations to any given precision of decimal representation?)

Why not let Scala do it for you? It may use continued fractions, I
don't know.

> #2 (nicknamed 19otti)

This is two different scales. Which is it reaally?

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
> >
> >
> > OK, more mathematics has been done... the results are, I hope,
> > practically Scala-ready.
> >
>
> I'd like it a whole lot better if it had twelve notes. Why is G#/Ab
> missing? Anyway, here it is as a Scala scale:

Grr, something flew off while cutting and pasting... E-G# should be
512/405, therefore G# turns out to be the rather ghastly 155648/98415,
as shown below in the corrected version.

> ! 19berger.scl
> Tom Dent's 19berger scale
> 12
> !
> 256/243
> 151/135
> 19/16
> 304/243
> 4864/3645
> 2215/1576
> 2423/1620
> 155648/98415
> 271/162
> 8417/4728
> 15/8
> 2

> > (Is there a formula or website which spits out 'nice' rational
> > approximations to any given precision of decimal representation?)
>
> Why not let Scala do it for you? It may use continued fractions, I
> don't know.

I'm only getting things into Scala format so that people can do
whatever they usually do with it (someone said one could check which
previously invented scale they are closest to). I've never seen the
program myself.

Here's the second one formatted.

! 19otti.scl
Tom Dent's 19otti scale
12
!
135/128
573/512
19/16
2565/2048
171/128
45/32
383/256
2431/1536
3429/2048
57/32
15/8
2

>
> This is two different scales. Which is it really?

They are both them really... yes, there are two different scales,
tending slightly more towards Kirnbergery or Vallotti-y respectively.

~~~T~~~

> > This is two different scales. Which is it really?
>
> They are both them really... yes, there are two different
> scales, tending slightly more towards Kirnbergery or
> Vallotti-y respectively.

I think Gene meant, the -otti has an option note. Which
note should we assume you wanted?

-Carl

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:
>
>
> OK, more mathematics has been done... the results are, I hope,
> practically Scala-ready.
>
> ...
> #2 (nicknamed 19otti)
> C 1
> C# 135/128
> D 573/512
> Eb 19/16
> E 2565/2048
> F 171/128
> F# 45/32
> G 383/256
> G# 2431/1536
> A 3429/2048 (or 226/135)
> Bb 57/32
> B 15/8

Hey, Tom, I'm impressed! There's lots of proportional beating in
both the major & minor triads if A is 3429/2048. (Forget about
226/135.)

> ... here only G,D,A and G# are 'tempered'.

It appears that the C is tempered, too, because F:C isn't 2:3.

--George

Carlos, et al,

Well, I just posted about a lack of mention of the
dual scales generated by 34, sent it off, and then
found mention of said double scales in the very next
e-mail!

Aarrrrrrr! It's a conspiracy, I'm tellin' ya!

M

--- Carl Lumma <clumma@yahoo.com> wrote:

> > > This is two different scales. Which is it
> really?
> >
> > They are both them really... yes, there are two
> different
> > scales, tending slightly more towards Kirnbergery
> or
> > Vallotti-y respectively.
>
> I think Gene meant, the -otti has an option note.
> Which
> note should we assume you wanted?
>
> -Carl
>
>

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Once more, the figures below are the correct values:

> ! 19berger.scl
> Tom Dent's 19berger scale
> 12
> !
> 256/243
> 151/135
> 19/16
> 304/243
> 4864/3645
> 2215/1576
> 2423/1620
> 155648/98415
> 271/162
> 8417/4728
> 15/8
> 2

! 19otti.scl
Tom Dent's 19otti scale
12
!
135/128
573/512
19/16
2565/2048
171/128
45/32
383/256
2431/1536
3429/2048
57/32
15/8
2

On the second one, George said:
> Hey, Tom, I'm impressed! There's lots of proportional beating in
> both the major & minor triads if A is 3429/2048. (Forget about
> 226/135.)

Hmmm... after I calculated there was such an insignificant difference
between the two A's, I figured why not go with the 308-381-381-383-384
division of the 19-comma. I had no thoughts at all of proportionality.

> > ... here only G,D,A and G# are 'tempered'.
>
> It appears that the C is tempered, too, because F:C isn't 2:3.
>

OK, it's in a sort of grey area between just and tempered - because
the 19th harmonic was the point of the construction, I used the fact
that you can construct a 19-limit 'schisma' or 'kleisma' 513/512 and
effectively temper fifths by it.

However, the other 'tempered' intervals are ratios that have nothing
to do with 19-limit, just convenient approximations.

Anyone found out what the scales most closely resemble?

~~~T~~~