A pretty weird but definitely interesting scale

Hi everyone.

I would be very curious if someone had mentioned this before -- at least I haven't found nor a single word about it. Similarly to the way BP has a period of 3/1 and approximates 1:3:5:7:9, this is a 4/1-periodic linear temperament and approximates 1:4:7:10:13:16 so there's some kind of "triharmony" in it -- or whatever I could call it. It has two interval sizes, one being about 158 cents and another one being quite close to the usual 12-EDO semitone. The generator is ~357.076254 cents, which is the 9th root of 32/5.
I think this could be a similarly interesting possibility as BP is. Unfortunately, so far I've not found a way to easily set up the tuning in the XG format but I hope I'll think of a solution soon because, personally, I can't wait to make a new piece in it. The scale has 20 tones and looks like this:

! triharmon.scl
!
The triharmonic scale
20
!
99.53378
257.54248
357.07625
456.61003
614.61873
714.15251
8/5
971.69498
1071.22876
1170.76254
1328.77124
1428.30502
5/2
1685.84749
1785.38127
1943.38997
2042.92375
2142.45752
2300.46622
4/1

Petr

I was reading about the "47-step, non-octave scale within the framework of the double
octave" by Heinz Bohlen http://members.aol.com/bpsite/pythagorean.html
and I see some similarities with yours.

--- In tuning@yahoogroups.com, Petr Pařízek <p.parizek@...> wrote:
>
> Hi everyone.
>
> I would be very curious if someone had mentioned this before --

if you really look at the 12th as an interval of equivalence BP can look like this type of diamonds
http://anaphoria.com/images/BPdiamond.gif
http://anaphoria.com/images/BPdia2.gif

iranief wrote:
>
> I was reading about the "47-step, non-octave scale within the > framework of the double
> octave" by Heinz Bohlen http://members.aol.com/bpsite/pythagorean.html > <http://members.aol.com/bpsite/pythagorean.html>
> and I see some similarities with yours.
>
> --- In tuning@yahoogroups.com <mailto:tuning%40yahoogroups.com>, Petr > Pařízek <p.parizek@...> wrote:
> >
> > Hi everyone.
> >
> > I would be very curious if someone had mentioned this before --
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/index.html>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main/index.asp> 88.9 FM Wed 8-9 pm Los Angeles

iranief wrote:
> I was reading about the "47-step, non-octave scale within the framework of the double > octave" by Heinz Bohlen http://members.aol.com/bpsite/pythagorean.html
> and I see some similarities with yours.
> > --- In tuning@yahoogroups.com, Petr Pařízek <p.parizek@...> wrote:
>> Hi everyone.
>>
>> I would be very curious if someone had mentioned this before -- They do seem to be related. The harmonic resources are similar. The triharmonic scale can be extended to 47 notes with 27 large steps of 58.47 cents and 20 small steps of 41.06 cents, compared with the 51.06 cent steps of the equally tempered non-octave scale and the varying sizes of the intervals in the just scale on the Bohlen-Pierce site.

> They do seem to be related. The harmonic resources are similar. The
> triharmonic scale can be extended to 47 notes with 27 large steps of
> 58.47 cents and 20 small steps of 41.06 cents, compared with the 51.06
> cent steps of the equally tempered non-octave scale and the varying
> sizes of the intervals in the just scale on the Bohlen-Pierce site.

And there's yet another possibility which lies somewhere in-between: If you take the 20 tone set and fill the larger steps with the smaller ones, you get a 27 tone scale, which is something neither the webpage nor I have mentioned. Still, 47 seems to me to be quite a lot of notes.

Petr

Petr Par�zek wrote:
>> They do seem to be related. The harmonic resources are similar. The
>> triharmonic scale can be extended to 47 notes with 27 large steps of
>> 58.47 cents and 20 small steps of 41.06 cents, compared with the 51.06
>> cent steps of the equally tempered non-octave scale and the varying
>> sizes of the intervals in the just scale on the Bohlen-Pierce site.
> > And there's yet another possibility which lies somewhere in-between: If you > take the 20 tone set and fill the larger steps with the smaller ones, you > get a 27 tone scale, which is something neither the webpage nor I have > mentioned. Still, 47 seems to me to be quite a lot of notes.

For a 2-octave scale it's not bad (less than 24 notes per octave), but still, a generalized keyboard with at least a 7x7 array of keys would be useful to have (a 14x8 array for 4 octaves).

The 4:7:10:13 chord would fit nicely under the fingers with a generalized keyboard mapping of 357.076254 cents across and 99.533778 cents down to the right. Put your fingers on X-G-U-K or C-H-I-L to get an idea. That won't be much help until generalized keyboards are more common, but maybe someday...