Golden tunings with tempered octaves

Hi,

golden tunings like Kornerup's meantone are usually defined with pure
octaves and this constraint along with the number of large and small
steps completely determines the tuning. This means that harmonic
properties beyond the pure octave such as vanishing commas play no
part in determining the tuning.

This is in contrast with the regular mapping paradigm where the
resulting melodic properties are sort of accidental.

But these approaches can be combined if we allow tempered octaves in
golden tunings. We could have GTOP tunings for example which is the
golden tuning that has the lowest max Tenney-weighted error. 5-limit
Meantone would have an octave of 1201.813671648584 cents and a fifth
of 697.2667276651778 cents, I think. Interestingly, it seems that we
must specify the MOS we want to use because for example the above
numbers give L:s of Phi+1 for Meantone[3] while all the rest of the
MOSes have L:s or Phi. This is a bit mysterious to me, why does it
end up being Phi+1?

I'm not trying to be rigorous here so I leave the precise defining
of GTOP tuning to others if needed, you get the gist.

Kalle

Hey, there's a file called meangolden_top.scl in the Scala archive
and it's a preset in the linear temperament dialog too. I didn't know
Manuel had this figured out already! :)

Kalle

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:
>
> Hi,
>
> golden tunings like Kornerup's meantone are usually defined with pure
> octaves and this constraint along with the number of large and small
> steps completely determines the tuning. This means that harmonic
> properties beyond the pure octave such as vanishing commas play no
> part in determining the tuning.
>
> This is in contrast with the regular mapping paradigm where the
> resulting melodic properties are sort of accidental.
>
> But these approaches can be combined if we allow tempered octaves in
> golden tunings. We could have GTOP tunings for example which is the
> golden tuning that has the lowest max Tenney-weighted error. 5-limit
> Meantone would have an octave of 1201.813671648584 cents and a fifth
> of 697.2667276651778 cents, I think. Interestingly, it seems that we
> must specify the MOS we want to use because for example the above
> numbers give L:s of Phi+1 for Meantone[3] while all the rest of the
> MOSes have L:s or Phi. This is a bit mysterious to me, why does it
> end up being Phi+1?
>
> I'm not trying to be rigorous here so I leave the precise defining
> of GTOP tuning to others if needed, you get the gist.
>
> Kalle
>

Kalle wrote :

> the above numbers give L:s of Phi+1 for Meantone[3] while all the
> rest of the MOSes have L:s or Phi. This is a bit mysterious to me,
> why does it end up being Phi+1?

Hi Kalle,
Simply because the sequence of intervals in Phi proportion in the Golden scale is :
diesis - semitone - tone - minor third - fourth - minor 6th - minor 9th - (etc.),
therefore the size proportion between the fourth and the tone is Phi^2, which equals Phi+1, as you know.
- - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:
>
> Kalle wrote :
>
> > the above numbers give L:s of Phi+1 for Meantone[3] while all the
> > rest of the MOSes have L:s or Phi. This is a bit mysterious to me,
> > why does it end up being Phi+1?
>
>
> Hi Kalle,
> Simply because the sequence of intervals in Phi proportion in the
> Golden scale is :
> diesis - semitone - tone - minor third - fourth - minor 6th - minor
> 9th - (etc.),
> therefore the size proportion between the fourth and the tone is
> Phi^2, which equals Phi+1, as you know.
> - - - - - - -
> Jacques

Of course, thanks Jacques!

Now, Paul Erlich gave me this link:

http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm

I don't completely understand why there are first some "alloyed"
MOSes and after them all are golden.

Kalle

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

> Of course, thanks Jacques!
>
> Now, Paul Erlich gave me this link:
>
> http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm
>
> I don't completely understand why there are first some "alloyed"
> MOSes and after them all are golden.
>
> Kalle

Sorry, I haven't read it and I don't see to what your question refers.
What I know is that the "Golden horagrams" and my "fractal Phi waveforms" or my "Golden meta-temperaments" are the same thing and may be you can check if the picture Golden_1267.pdf in my TL files can help to something.
- - - -
Jacques

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

> Now, Paul Erlich gave me this link:
>
> http://www.elvenminstrel.com/music/tuning/horagrams/horagram_intro.htm

Can anyone tell me (offlist) where I can contact David J. Finnamore, the author of this article ? I just tried to write to him but the mailbox is not valid anymore.
- - - -
Jacques