Topic: Groven's tuning - another reason why 36 tones

1 scales

Thread (3 messages)

From: Petr Pařízek (2005-11-21)
Subject: 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

From: Gene Ward Smith (2005-11-21)
Subject: Re: Groven's tuning - another reason why 36 tones

--- In tuning@yahoogroups.com, Petr Paøízek  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.

From: wallyesterpaulrus (2005-11-21)
Subject: Re: Groven's tuning - another reason why 36 tones

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

--- In tuning@yahoogroups.com, Petr Paøízek  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
>