RE: [tuning] The Golden Mediant: Complex ratios and metastable in tervals

Dan wrote,

>The Golden Mediant taken (i/j, m/n), or (1/2, 3/5), gives the 2s5L
>Golden Mediant generator of ~696.21¢, and the following temperament:

That's precisely Theovald Kornerup's Phi-based meantone tuning:
http://www.rev.net/~aloe/music/golden.html. Joseph, did you catch that name?
Or shall I give you a further cornerup on what your unintentional pun was?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13033

> Dan wrote,
>
> >The Golden Mediant taken (i/j, m/n), or (1/2, 3/5), gives the 2s5L
> >Golden Mediant generator of ~696.21¢, and the following
temperament:
>
> That's precisely Theovald Kornerup's Phi-based meantone tuning:
> http://www.rev.net/~aloe/music/golden.html. Joseph, did you catch
that name?
> Or shall I give you a further cornerup on what your unintentional
pun was?

Paul... I am indeed dense... but the pun was intentional...
_________ ___ __ _
Joseph Pehrson

Paul H. Erlich wrote,

> That's precisely Theovald Kornerup's Phi-based meantone tuning

Yes, I know (and I should've mentioned it), but has this ever been
generalized in the sense that I'm asking? For instance, here are a
some of the 9-tone Golden Mediant generators I was just looking at for
the "bimodal scale".

The 1s8L/8L1s:

1/1 7/8
8/9
9/10 15/17

Golden Mediant = ~1060.76
LLLLsLLLL

0 139 278 418 557 643 782 922 1061 1200
0 139 278 418 504 643 782 922 1061 1200
0 139 278 365 504 643 782 922 1061 1200
0 139 225 365 504 643 782 922 1061 1200
0 86 225 365 504 643 782 922 1061 1200
0 139 278 418 557 696 835 975 1114 1200
0 139 278 418 557 696 835 975 1061 1200
0 139 278 418 557 696 835 922 1061 1200
0 139 278 418 557 696 782 922 1061 1200
0 139 278 418 557 643 782 922 1061 1200

Golden Mediant = ~1075.23
ssssLssss

0 125 250 374 499 701 826 950 1075 1200
0 125 250 374 576 701 826 950 1075 1200
0 125 250 451 576 701 826 950 1075 1200
0 125 327 451 576 701 826 950 1075 1200
0 202 327 451 576 701 826 950 1075 1200
0 125 250 374 499 624 749 873 998 1200
0 125 250 374 499 624 749 873 1075 1200
0 125 250 374 499 624 749 950 1075 1200
0 125 250 374 499 624 826 950 1075 1200
0 125 250 374 499 701 826 950 1075 1200

The 2s7L/7L2s:

1/2 4/7
5/9
6/11 9/16

Golden Mediant = ~672.85
LLsLLLsLL

0 146 291 381 527 673 819 909 1054 1200
0 146 236 381 527 673 763 909 1054 1200
0 90 236 381 527 617 763 909 1054 1200
0 146 291 437 527 673 819 964 1110 1200
0 146 291 381 527 673 819 964 1054 1200
0 146 236 381 527 673 819 909 1054 1200
0 90 236 381 527 673 763 909 1054 1200
0 146 291 437 583 673 819 964 1110 1200
0 146 291 437 527 673 819 964 1054 1200
0 146 291 381 527 673 819 909 1054 1200

Golden Mediant = ~658.62
ssLsssLss

0 117 234 424 541 659 776 966 1083 1200
0 117 307 424 541 659 848 966 1083 1200
0 190 307 424 541 731 848 966 1083 1200
0 117 234 352 541 659 776 893 1010 1200
0 117 234 424 541 659 776 893 1083 1200
0 117 307 424 541 659 776 966 1083 1200
0 190 307 424 541 659 848 966 1083 1200
0 117 234 352 469 659 776 893 1010 1200
0 117 234 352 541 659 776 893 1083 1200
0 117 234 424 541 659 776 966 1083 1200

The 4L5s/5L4s:

3/4 4/5
7/9
10/13 11/14

Golden Mediant = 940.15
LsLsLsLsL

0 161 260 420 520 680 780 940 1039 1200
0 99 260 359 520 619 780 879 1039 1200
0 161 260 420 520 680 780 940 1101 1200
0 99 260 359 520 619 780 940 1039 1200
0 161 260 420 520 680 841 940 1101 1200
0 99 260 359 520 680 780 940 1039 1200
0 161 260 420 581 680 841 940 1101 1200
0 99 260 420 520 680 780 940 1039 1200
0 161 321 420 581 680 841 940 1101 1200
0 161 260 420 520 680 780 940 1039 1200

Golden Mediant = ~926.15
sLsLsLsLs

0 105 274 378 548 652 822 926 1095 1200
0 169 274 443 548 717 822 991 1095 1200
0 105 274 378 548 652 822 926 1031 1200
0 169 274 443 548 717 822 926 1095 1200
0 105 274 378 548 652 757 926 1031 1200
0 169 274 443 548 652 822 926 1095 1200
0 105 274 378 483 652 757 926 1031 1200
0 169 274 378 548 652 822 926 1095 1200
0 105 209 378 483 652 757 926 1031 1200
0 105 274 378 548 652 822 926 1095 1200

(etc.)

Has this type of a generalization has been proposed before?

- dan

Dan wrote,

>Has this type of a generalization has been proposed before?

Essentially, that is what Wilson is doing with his horagrams.

Dan Stearns wrote:

> http://www.egroups.com/message/tuning/13035
>
> ... here are a some of the 9-tone Golden Mediant generators
> I was just looking at for the "bimodal scale".

Tables of numbers never mean as much to me as pictures
representing them, so I used Excel to make graphs of the
pitches in Dan's scales, and put them on a webpage at:

http://www.ixpres.com/interval/td/stearns/gmbimodal.htm

Dave Keenan's excellent observation, that the Golden Mediant
proposed in his recent paper with Margo Schulter is an operation
on ratios whereas recent proposals by Dan and others are
logarithmic in nature, was included.

BTW, a late 'welcome back, Dave' from me too. But I do hope
you reconsider your 'brief return' and stick around for good.
I find your work to be among the most valuable on the list,
and I deeply regret the decisions of both Daniel Wolf and
Kraig Grady to leave.

(Dave, you've probably gathered by now that Kraig, who
considers himself decidedly a JI composer, needed a break
from the heated arguments about what exactly a JI is; but
you probably missed Daniel Wolf's departure and might want
to know the reason... it's much more prosaic: as the web-based
version of the Tuning List gets more and more popular, people
are tending to post more frequent short posts rather than
infrequent longer ones, and in Hungary Mr. Wolf is paying
for every minute [or second?] of download time; it simply
got too expensive and time-consuming for him.)

-monz
http://www.ixpres.com/interval/monzo/homepage.html

Paul H. Erlich wrote,

> Essentially, that is what Wilson is doing with his horagrams.

Alright, I'll give that another, or better look then.

BTW, here's the "golden mediant" 8s2L decatonic scale...

a/b c/d = 3/4 1/1
x = 493.2011
ssLssssLss:

0 107 214 386 493 600 707 814 986 1093 1200
0 107 280 386 493 600 707 880 986 1093 1200
0 173 280 386 493 600 773 880 986 1093 1200
0 107 214 320 427 600 707 814 920 1027 1200
0 107 214 320 493 600 707 814 920 1093 1200
0 107 214 386 493 600 707 814 986 1093 1200
0 107 280 386 493 600 707 880 986 1093 1200
0 173 280 386 493 600 773 880 986 1093 1200
0 107 214 320 427 600 707 814 920 1027 1200
0 107 214 320 493 600 707 814 920 1093 1200
0 107 214 386 493 600 707 814 986 1093 1200

And here's the 11-tET (22-tET) mediants...

a/b c/d = 5/6 4/5
x = 488.5166
ssLssssLss:

0 111 223 377 489 600 711 823 977 1089 1200
0 111 266 377 489 600 711 866 977 1089 1200
0 154 266 377 489 600 754 866 977 1089 1200
0 111 223 334 446 600 711 823 934 1046 1200
0 111 223 334 489 600 711 823 934 1089 1200
0 111 223 377 489 600 711 823 977 1089 1200
0 111 266 377 489 600 711 866 977 1089 1200
0 154 266 377 489 600 754 866 977 1089 1200
0 111 223 334 446 600 711 823 934 1046 1200
0 111 223 334 489 600 711 823 934 1089 1200
0 111 223 377 489 600 711 823 977 1089 1200

dan

Dan Stearns wrote,

>And here's the 11-tET (22-tET) mediants...

What exactly do you mean by that?

Paul H. Erlich wrote,

> What exactly do you mean by that?

Just that the 11-tET 1s4L mapping falls between those mediants, i.e.,
a/b, c/d = 5/6, 4/5:

1/1 3/4
4/5
5/6 7/9
6/7 9/11 11/14 10/13

So for the second "static symmetrical major" decatonic I used X =
P/(n*d) where "P" = a given periodicity; and P = 600 there, "n" =
(b+phi*d) and "d" = (a+phi*c).

dan

Dan, is the reference to 11-tET just superfluous and confusing, or is there
some meaning
to it, beyond it being the first convergent after 5-tET and 6-tET along the
"noble" path to the ultimate interval size you use? If the latter, then you
might want to say something like "11, 17, 28, . . . . tET", right?

Paul H. Erlich,

> Dan, is the reference to 11-tET just superfluous and confusing, or
is there some meaning to it,

No, I can see how it could be confusing.

> then you might want to say something like "11, 17, 28, . . . . tET",
right?

Yes that's right, it's just not exactly the way I had it framed in my
mind.

dan