Another talent of the 72-et

If we call the temperament which approximates 3 by (6/5)^6, 5 by
2*(6/5)^5, and 7 by
4*(6/5)^3 "septimal kleismic", then septimal kleismic has an RMS
optimized generator which differs from 19/72 by 0.0026 cents. Of course,
the "7" in question is a twelfth of a tone flat, and we could use the
much better, "catakleismic" value instead, involving
22 generator steps rather than 3, which 72 also does nicely.

--- In tuning@y..., Gene W Smith <genewardsmith@j...> wrote:

/tuning/topicId_36979.html#36979

> If we call the temperament which approximates 3 by (6/5)^6, 5 by
> 2*(6/5)^5, and 7 by
> 4*(6/5)^3 "septimal kleismic", then septimal kleismic has an RMS
> optimized generator which differs from 19/72 by 0.0026 cents. Of
course,
> the "7" in question is a twelfth of a tone flat, and we could use
the
> much better, "catakleismic" value instead, involving
> 22 generator steps rather than 3, which 72 also does nicely.

***What are the notes, again, of this scale, Gene??

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., Gene W Smith <genewardsmith@j...> wrote:

> /tuning/topicId_36979.html#36979

> ***What are the notes, again, of this scale, Gene??

Like miracle, it's not a scale but a temperament. However, scales you might want to consider are:

7 notes

[0, 4, 19, 23, 38, 42, 57, 72]

11 notes

[0, 4, 8, 19, 23, 27, 38, 42, 46, 57, 61, 72]

15 notes

[0, 4, 8, 12, 19, 23, 27, 31, 38, 42, 46, 50, 57, 61, 65, 72]

19 notes

[0, 4, 8, 12, 16, 19, 23, 27, 31, 35, 38, 42, 46,
50, 54, 57, 61, 65\, 69, 72]

Plus, of course, their modal variants.

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

/tuning/topicId_36979.html#36985

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., Gene W Smith <genewardsmith@j...> wrote:
>
> > /tuning/topicId_36979.html#36979
>
> > ***What are the notes, again, of this scale, Gene??
>
> Like miracle, it's not a scale but a temperament. However, scales
you might want to consider are:
>
> 7 notes
>
> [0, 4, 19, 23, 38, 42, 57, 72]
>
> 11 notes
>
> [0, 4, 8, 19, 23, 27, 38, 42, 46, 57, 61, 72]
>
> 15 notes
>
> [0, 4, 8, 12, 19, 23, 27, 31, 38, 42, 46, 50, 57, 61, 65, 72]
>
> 19 notes
>
> [0, 4, 8, 12, 16, 19, 23, 27, 31, 35, 38, 42, 46,
> 50, 54, 57, 61, 65\, 69, 72]
>
> Plus, of course, their modal variants.

***Oh! Thanks, Gene. This kind of thing is extremely useful...
well, to *me* anyway.

What are the special properties of these scales again? I fear I'll
need just "English" since my math skills are well, er...

J. Pehrson

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> ***Oh! Thanks, Gene. This kind of thing is extremely useful...
> well, to *me* anyway.
>
> What are the special properties of these scales again? I fear I'll
> need just "English" since my math skills are well, er...

Joseph,

Kleismic is another name for the chain of minor thirds temperament
described in
http://dkeenan.com/Music/ChainOfMinor3rds.htm
which includes a 9-limit lattice for the 11 note MOS, with chord
diagrams.

To reinforce what Gene said: While kleismic is a microtemperament at
the 5-limit and still pretty good at the 9-limit-sans-7s, it is far
from just (but still usable) for ratios of 7. Although 72-ET
represents 7-limit kleismic quite well, the kleismic approximation to
4:7 is only the _third_best_ available in 72-ET. It has an error of
18.8 cents, unlike the one you are used to in Blackjack.

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

/tuning/topicId_36979.html#37000

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > ***Oh! Thanks, Gene. This kind of thing is extremely useful...
> > well, to *me* anyway.
> >
> > What are the special properties of these scales again? I fear
I'll
> > need just "English" since my math skills are well, er...
>
> Joseph,
>
> Kleismic is another name for the chain of minor thirds temperament
> described in
> http://dkeenan.com/Music/ChainOfMinor3rds.htm
> which includes a 9-limit lattice for the 11 note MOS, with chord
> diagrams.
>
> To reinforce what Gene said: While kleismic is a microtemperament
at
> the 5-limit and still pretty good at the 9-limit-sans-7s, it is far
> from just (but still usable) for ratios of 7. Although 72-ET
> represents 7-limit kleismic quite well, the kleismic approximation
to
> 4:7 is only the _third_best_ available in 72-ET. It has an error of
> 18.8 cents, unlike the one you are used to in Blackjack.

****Got it! Thanks, Dave. That clears it up for me. Why can I read
*your* paragraphs and understand them and not Gene's?? Either Gene's
paragraphs are too dense, or *I* am too dense. [Just joking...]

Thanks!

Joseph

--- In tuning@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:

> To reinforce what Gene said: While kleismic is a microtemperament at
> the 5-limit and still pretty good at the 9-limit-sans-7s, it is far
> from just (but still usable) for ratios of 7. Although 72-ET
> represents 7-limit kleismic quite well, the kleismic approximation to
> 4:7 is only the _third_best_ available in 72-ET. It has an error of
> 18.8 cents, unlike the one you are used to in Blackjack.

Of course, the musical possibilities of having nicely tuned 5-limit chords coupled with harsher 7-chords is a familiar one; think
1-5/4-3/2-16/9 = 1-5/4-3/2-9/5 in meantone. I'm working on a magic piece at the moment, but this is on my list of things to try.

>****Got it! Thanks, Dave. That clears it up for me. Why can I read
>*your* paragraphs and understand them and not Gene's?? Either Gene's
>paragraphs are too dense, or *I* am too dense. [Just joking...]

Fear not Joseph, I'm lucky to understand half of what Gene says. Not
any fault of his -- his style is clean and efficient -- and the part I
do understand is totally cool!

-Carl

--- In tuning@y..., Carl Lumma <carl@l...> wrote:

/tuning/topicId_36979.html#37015

> >****Got it! Thanks, Dave. That clears it up for me. Why can I
read *your* paragraphs and understand them and not Gene's?? Either
Gene's paragraphs are too dense, or *I* am too dense. [Just
joking...]
>
> Fear not Joseph, I'm lucky to understand half of what Gene says.
Not any fault of his -- his style is clean and efficient -- and the
part I do understand is totally cool!
>
> -Carl

***Thanks, Carl! That's encouraging! I was beginning to get
paranoid...

jp

--- In tuning@y..., Carl Lumma <carl@l...> wrote:

Not
> any fault of his -- his style is clean and efficient -- and the
part I
> do understand is totally cool!

Part of a good style is to be aware of one's audience, and since I
had Joseph in mind I perhaps should have explained certain matters.
Could mean I'm dense. :)

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

/tuning/topicId_36979.html#37021

> --- In tuning@y..., Carl Lumma <carl@l...> wrote:
>
> Not
> > any fault of his -- his style is clean and efficient -- and the
> part I
> > do understand is totally cool!
>
> Part of a good style is to be aware of one's audience, and since I
> had Joseph in mind I perhaps should have explained certain matters.
> Could mean I'm dense. :)

***That's great, Gene. "Dumb it down" if you don't mind... :)

You're always coming up with great stuff, so I'd like to understand
it, if I can. I would also take Johnny Reinhard's suggestion very
seriously to get some music to him. You have the math background to
*really* plow into some new terrain!!!!!

Joseph

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., Gene W Smith <genewardsmith@j...> wrote:
>
> > /tuning/topicId_36979.html#36979
>
> > ***What are the notes, again, of this scale, Gene??
>
> Like miracle, it's not a scale but a temperament. However, scales
you might want to consider are:

> 19 notes
>
> [0, 4, 8, 12, 16, 19, 23, 27, 31, 35, 38, 42, 46,
> 50, 54, 57, 61, 65\, 69, 72]

this is one of the 19-toners i gave you quite some time ago, right
before the miracle breakthrough. in fact, i think this was the very
first one -- i called it 5-limit, because the 7-limit approximations
are off by a full twelfth tone (a full degree of 72-equal).

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_36979.html#37024

> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > > --- In tuning@y..., Gene W Smith <genewardsmith@j...> wrote:
> >
> > > /tuning/topicId_36979.html#36979
> >
> > > ***What are the notes, again, of this scale, Gene??
> >
> > Like miracle, it's not a scale but a temperament. However, scales
> you might want to consider are:
>
> > 19 notes
> >
> > [0, 4, 8, 12, 16, 19, 23, 27, 31, 35, 38, 42, 46,
> > 50, 54, 57, 61, 65\, 69, 72]
>
> this is one of the 19-toners i gave you quite some time ago, right
> before the miracle breakthrough. in fact, i think this was the very
> first one -- i called it 5-limit, because the 7-limit
approximations
> are off by a full twelfth tone (a full degree of 72-equal).

***Oh sure! Now I remember when we went through those. We went
through several different subsets of around that number of notes,
which eventually lead to the "Miracle..."

I'll find those in my notes.

Joseph