Ennealimmal 45 vs Miracle 41

Paul asked how these compared, so I fired up Scala and checked. I got
28 major and 28 minor triads in Miracle41, all of which are
extendable to tetrads, as well as 27 supermajor and 27 subminor
triads. In Ennealimmal45, by way of contrast, I got 28 major and 28
minor triads, all of which are extendable to tetrads, as well as 27
supermajor and 27 subminor triads. Ennealimmal45 cheats by having
four extra notes per octave, but makes up for it by being in much
better tune, for those people who might notice or care.

Paul called it "immensely complex", but I think that overstates it.
Ennealimmal45 can be done by stacking five 9-ets, separated by
something very close to 36/35. Each octave is simply a 5x9 rectangle
of 45 notes.

For those who want to try this comparison out here they are:

! ennea45.scl
!
Ennealimmal-45, in a 7-limit least-squares tuning
45
!
35.3350
48.9992
84.3342
97.9983
133.3333
168.6684
182.3325
217.6675
231.3316
266.6667
302.0017
315.6658
351.0008
364.6650
400.0000
435.3350
448.9992
484.3342
497.9983
533.3333
568.6684
582.3325
617.6675
631.3316
666.6667
702.0017
715.6658
751.0008
764.6650
800.0000
835.3350
848.9992
884.3342
897.9983
933.3333
968.6684
982.3325
1017.6675
1031.3316
1066.6667
1102.0017
1115.6658
1151.0008
1164.6650
2/1

! miracle41.scl
!
Miracle-41, in a 7-limit least-squares tuning
41
!
34.2705
68.5411
82.3024
116.5729
150.8435
185.1140
198.8754
233.1459
267.4164
301.6870
315.4483
349.7188
383.9894
418.2599
432.0213
466.2918
500.5623
534.8328
548.5942
582.8647
617.1353
651.4058
665.1672
699.4377
733.7082
767.9787
781.7401
816.0106
850.2812
884.5517
898.3130
932.5836
966.8541
1001.1246
1014.8860
1049.1565
1083.4271
1117.6976
1131.4589
1165.7295
2/1

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> Paul asked how these compared, so I fired up Scala and checked. I
got
> 28 major and 28 minor triads in Miracle41, all of which are
> extendable to tetrads, as well as 27 supermajor and 27 subminor
> triads. In Ennealimmal45, by way of contrast, I got 28 major and 28
> minor triads, all of which are extendable to tetrads,

Something's not adding up, Gene. How can there be 28 of _anything_,
when the scale exactly repeats itself 9 times per octave? I get 18
major and 18 minor tetrads all extendable to triads.

--- In tuning@y..., "paulerlich" <paul@s...> wrote:

> Something's not adding up, Gene. How can there be 28 of _anything_,
> when the scale exactly repeats itself 9 times per octave? I get 18
> major and 18 minor tetrads all extendable to triads.

Sorry, you are right. I guess I read the data cross-eyed.

Paul:
> > Something's not adding up, Gene. How can there be 28 of _anything_,
> > when the scale exactly repeats itself 9 times per octave? I get 18
> > major and 18 minor tetrads all extendable to triads.

Gene:
> Sorry, you are right. I guess I read the data cross-eyed.

So it's 18 for Ennealimmal45 and 28 for Miracle41?

Graham

--- In tuning@y..., graham@m... wrote:
> Paul:
> > > Something's not adding up, Gene. How can there be 28 of
_anything_,
> > > when the scale exactly repeats itself 9 times per octave? I get
18
> > > major and 18 minor tetrads all extendable to triads.
>
> Gene:
> > Sorry, you are right. I guess I read the data cross-eyed.
>
> So it's 18 for Ennealimmal45 and 28 for Miracle41?

Right--my "18" somehow ended up looking a lot like "28" and I misread
it. Paul spotted it because he actually thought about it.

I get, for intervals:

3/2: 27 35
5/4: 18 34
7/4: 27 39
5/3: 36 28
7/6: 45 33
7/5: 36 36

and for extra,

9/7: 27 27

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_31110.html#31110

Gene... would you mind please running by me how the
word "Ennealimmal" is derived again?? Initially, I thought it was
something vaguely off color...

JP

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:

> Gene... would you mind please running by me how the
> word "Ennealimmal" is derived again?? Initially, I thought it was
> something vaguely off color...

The large limma is 27/25; if you take nine of them you get
(27/25)^9 = 7625597484987 / 3814697265625 = 1.9990046. This tells us
that 2*(25/27)^9 is very close to 1; and it has been given the name
"ennealimma" because nine limmas down and an octave up make an
ennealimma (ennea = nine in ancient Greek.) The 5-limit tuning system
based on the ennealimma, and the 7-limit extension of that, have
thereby been given the name "ennealimmal". The 7-limit ennealimmal
system has generators of a ninth of an octave, which is almost
exactly a limma, and something very close to 36/35.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_31110.html#31168

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
>
> > Gene... would you mind please running by me how the
> > word "Ennealimmal" is derived again?? Initially, I thought it
was something vaguely off color...
>
> The large limma is 27/25; if you take nine of them you get
> (27/25)^9 = 7625597484987 / 3814697265625 = 1.9990046. This tells
us that 2*(25/27)^9 is very close to 1; and it has been given the name
> "ennealimma" because nine limmas down and an octave up make an
> ennealimma (ennea = nine in ancient Greek.) The 5-limit tuning
system based on the ennealimma, and the 7-limit extension of that,
have thereby been given the name "ennealimmal". The 7-limit
ennealimmal
> system has generators of a ninth of an octave, which is almost
> exactly a limma, and something very close to 36/35.

Hi Gene!

Thanks for the update. This "limma" then, at 27/25 must be different
from the "limma" that Joe Monzo refers to in his dictionary of
256/243 or the "diatonic semitone" of Pythagorean tuning at 90 cents.

When I was reading Paul's _Introduction to Periodicity Blocks_ again,
I had a question about that, since I wondered if the transposition of
90 cents he refers to when creating a "Periodicity Block" of a
pentatonic scale could be termed a "semitone."

I soon dismissed that idea, since I thought a "semitone" was only a
legitimate reference to something connected with 12-tET, but now I
see that the 90 cent interval can, indeed, be called a "diatonic
semitone..." That seems a little peculiar to me. If that's
the "semitone" then what is the "whole tone?" I must be missing
something...

In any case, it looks as though your "limma" has nothing to do with
this Pythagorean limma... ??

Thanks!

Joseph