Groven's tuning - another reason why 36 tones

Hi.

I've just read Monz's article on Groven's organ tuning. I'm glad to see this
on your website, Monz. When you are discussing why Groven chose 36 tones for
his tuning, I'd like to add one more. Somewhere else on the web, I found
another paper emphasizing Groven's fondness of Norwegian folk music which
sometimes uses intervals as 11/8 or 13/8. He also wanted to be able to
"imitate" this in some way. He was aware that stacking 30
1/8-schisma-tempered fifths downwards makes an interval which is about 3
cents away from 11/8 (not counting the octave inversions, of course) and
that stacking 33 fifths downwards makes an interval which is about 3 cents
away from 13/8. So it was a good decision for him to choose 36 tones as the
interval of 35 fifths made it possible to approximate 13/9 very well. I'm
just wondering why he also had a 43-tone version of this. So far I haven't
found any new advantages if this other than the possibility of
transposition. What I'd prefer is a 48-tone version (what a coincidence,
another multiple of 12?) which makes it possible to "imitate" one 7-limit
chord very closely as the interval of 7-5 can be approximated well by
stacking 47 fifths. In this case, I prefer to use 1/9-schisma tempering
instead of 1/8-schisma.

The complete tuning follows. Sorry for not substituting the cent sizes of
the JI intervals by their respective ratios. I've just made the scale and
this is what has come out.

! 48temp.scl
!
48-tone chain of 1/9-schisma tempered fifths
48
!
20.85505
41.71010
70.45535
91.31040
112.16545
133.02049
161.76574
182.62079
203.47584
224.33089
273.93119
294.78624
315.64129
336.49634
365.24158
386.09663
406.95168
427.80673
477.40703
498.26208
519.11713
539.97218
568.71743
589.57248
610.42752
631.28257
660.02782
680.88287
701.73792
722.59297
772.19327
793.04832
813.90337
834.75842
863.50366
884.35871
905.21376
926.06881
975.66911
996.52416
1017.37921
1038.23426
1066.97951
1087.83455
1108.68960
1129.54465
1179.14495
2/1

Petr

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@c...> wrote:

He was aware that stacking 30
> 1/8-schisma-tempered fifths downwards makes an interval which is about 3
> cents away from 11/8 (not counting the octave inversions, of course) and
> that stacking 33 fifths downwards makes an interval which is about 3
cents
> away from 13/8.

Doing this adds the commas 352/351 and 625/624 to the schisma,
producing a no-sevens temperament. If we want sevens, one method is to
add 4375/4374 to the mix; this produces a 13-limit linear extension to
the 7-limit "pontiac" temperament with the schisma and ragisma
(4375/4374) as commas. Another approach is to extend garibaldi, the
schisma and 225/224 temperament, which has the property that 1/8
schisma is a poptimal tuning, amd which gives a better badness score
on the measures I've tried. The first approach gives a comma basis of
352/351, 385/384, 625/624 and 729/728; the second a comma basis of
99/98, 176/175, 275/273, 847/845. From this one can conclude it is
comparitively inaccurate in representing intervals involving seven. It
can be treated as 118-edo with a nonstandard mapping; that is, as
<118 187 274 332 408 437|, and in any case in practice both sevens can
be used inconsistently.

So it was a good decision for him to choose 36 tones as the
> interval of 35 fifths made it possible to approximate 13/9 very
well. I'm
> just wondering why he also had a 43-tone version of this.

Why not 41 or 53 tones, which give MOS?

So far I haven't
> found any new advantages if this other than the possibility of
> transposition. What I'd prefer is a 48-tone version (what a coincidence,
> another multiple of 12?) which makes it possible to "imitate" one
7-limit
> chord very closely as the interval of 7-5 can be approximated well by
> stacking 47 fifths. In this case, I prefer to use 1/9-schisma tempering
> instead of 1/8-schisma.

This is the first temperament I discussed, where we add 4375/4374 to
the mix, getting an extended pointiac.

Add five more notes to the chain, and you have Eduardo Sabat-
Garibaldi's Dinarra fretting.

--- In tuning@yahoogroups.com, Petr Paøízek <p.parizek@c...> wrote:

> ! 48temp.scl
> !
> 48-tone chain of 1/9-schisma tempered fifths
> 48
> !
> 20.85505
> 41.71010
> 70.45535
> 91.31040
> 112.16545
> 133.02049
> 161.76574
> 182.62079
> 203.47584
> 224.33089
> 273.93119
> 294.78624
> 315.64129
> 336.49634
> 365.24158
> 386.09663
> 406.95168
> 427.80673
> 477.40703
> 498.26208
> 519.11713
> 539.97218
> 568.71743
> 589.57248
> 610.42752
> 631.28257
> 660.02782
> 680.88287
> 701.73792
> 722.59297
> 772.19327
> 793.04832
> 813.90337
> 834.75842
> 863.50366
> 884.35871
> 905.21376
> 926.06881
> 975.66911
> 996.52416
> 1017.37921
> 1038.23426
> 1066.97951
> 1087.83455
> 1108.68960
> 1129.54465
> 1179.14495
> 2/1
>
>
> Petr
>