I've been pondering this, and I think there is a strong argument in favor of using 53. The top of the poptimal range for septimal kleismic and the bottom of the poptimal range for 5-limit kleismic coincide at the minimax generator of 3^(1/6), which is the same for both. This is the only generator which is poptimal in both limits, but of course 53, which has a much better fifth than 72, comes a lot closer. Moreover, kleismic is more important as a 5-limit system (where it is very strong) than as a 7-limit system, and I think we should try to use one et for all of the versions of a temperament when we can. I vote (if that is how this is done) for 53. Anyone else care to chime in?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith

<genewardsmith@j...>" <genewardsmith@j...> wrote:

> I've been pondering this, and I think there is a strong argument in

favor of using 53. The top of the poptimal range for septimal kleismic

and the bottom of the poptimal range for 5-limit kleismic coincide at

the minimax generator of 3^(1/6), which is the same for both. This is

the only generator which is poptimal in both limits, but of course 53,

which has a much better fifth than 72, comes a lot closer. Moreover,

kleismic is more important as a 5-limit system (where it is very

strong) than as a 7-limit system, and I think we should try to use one

et for all of the versions of a temperament when we can. I vote (if

that is how this is done) for 53. Anyone else care to chime in?

<

The way this standardisation stuff has been done in the past, is not

by voting as such, but by consensus. That is, we keep presenting

arguments for and against various options, with as little ego

investment and as much praise of the other people's ideas as possible,

until everyone who has ever expressed an opinion on it, either agrees

or says they no longer care.

So here goes me.

While I liked the pattern that kleismic makes in 72-ET, as pointed out

by George, I'm swayed by Gene's arguments above. By most people's

reckoning kleismic is one of the top four 5-limit temperaments. At the

7-limit there are two extensions of kleismic that might be considered.

Both are down past number ten on anyone's list.

However, if you _were_ using kleismic for 7-limit with the least

complex 7's, you would probably be dissatisfied with a 53-ET based

notation.

Lets try approaching it in an ET-independent manner, considering only

the 5-limit map

2:3 is 6 gens

4:5 is 5 gens

5:6 is 1 gen

BTW the two 7-limit extensions have

4:7 is 3 gens

or

4:7 is 22 gens

I'll use / and \ as 5-comma (80:81) up and down symbols and FCGDAEB

and #b have their usual Pythagorean relationships. Then the following

is clearly a correct notation for a chain of 19 notes of kleismic.

Alternative names are given underneath some note names.

A#\\\ C#\\ E\ G Bb/ Db// E#\\\ G#\\ B\

Bbb/// Fb///

D F/ Ab// Cb/// D#\\ F#\ A C/ Eb// Gb///

B#\\\ Fx\\\

Which ET notation best preserves this?

53: /| /|\ (|)

or

72: /| |) /|\

or should we use something else such as

/| //| .//|

Since we've got symbols for 11 commas in both of those ET notations,

we really should check whether they are valid in any sensible 11-limit

extensions of kleismic.

Gene or Graham, have any 11-limit kleismics turned up in your

searches. If so, what maps?

I wrote:

> Which ET notation best preserves this?

> 53: /| /|\ (|)

> or

> 72: /| |) /|\

> or should we use something else like

> /| //| .//|

>

> Since we've got symbols for 11 commas in both of those ET notations,

> we really should check whether they are valid in any sensible 11-limit

> extensions of kleismic.

>

> Gene or Graham, have any 11-limit kleismics turned up in your

> searches. If so, what maps?

In the case of the 53-ET based notation it could also be a 1,3,5,11

temperament (no 7s).

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:

> > Gene or Graham, have any 11-limit kleismics turned up in your

> > searches. If so, what maps?

The more complex one, "catakleismic", has an 11-limit extension which I've also called "catakleismic", which uses -21 generators for 11.

The 300th row of the Farey sequence goes

19/72 < 71/269 < 52/197 < 33/125

Of these, 52/197 is poptimal for the 7-limit, and 71/269 for the 11-limit; I would suggest 72-et for notating catakleismic, therefore, although 125 would be possible also.

--- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>" <d.keenan@u...> wrote:

> However, if you _were_ using kleismic for 7-limit with the least

> complex 7's, you would probably be dissatisfied with a 53-ET based

> notation.

Why? Neither one is using the best value of the "7" of the et in question, and in both cases the 7-limit intervals are much more out of tune than the 5-limit intervals. It is cheesy no matter how you notate it, and I don't see why it is any *more* dissapointing for 53 than for 72.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith

<genewardsmith@j...>" <genewardsmith@j...> wrote:

> --- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"

<d.keenan@u...> wrote:

> > However, if you _were_ using kleismic for 7-limit with the least

> > complex 7's, you would probably be dissatisfied with a 53-ET based

> > notation.

>

> Why? Neither one is using the best value of the "7" of the et in

question, and in both cases the 7-limit intervals are much more out

of tune than the 5-limit intervals. It is cheesy no matter how you

notate it, and I don't see why it is any *more* dissapointing for 53

than for 72.

Since 7 is +22 and 11 is -21 in the series of ~5:6's in 53, 72, and

125, these are the normal positions for the kleismic temperament. So

we should be using symbols of a different nature to notate a lot of

the tones closer to the origin, which would seem to call for the 5^2

symbol, //|. The notation for 125 uses not only this, but also both

the 7-comma and 11-diesis symbols if the tuning is extended far

enough to take in ratios of 7 and 11, so 125 seems like a logical

choice to me.

--George

--- In tuning-math@yahoogroups.com, "gdsecor <gdsecor@y...>" <gdsecor@y...> wrote:

> > Why? Neither one is using the best value of the "7" of the et in

> question, and in both cases the 7-limit intervals are much more out

> of tune than the 5-limit intervals. It is cheesy no matter how you

> notate it, and I don't see why it is any *more* dissapointing for 53

> than for 72.

>

> Since 7 is +22 and 11 is -21 in the series of ~5:6's in 53, 72, and

> 125, these are the normal positions for the kleismic temperament.

We're not on the same page here, so it's no wonder we haven't come to an agreement on notation. The above system I have been calling "Catakleismic", reserving "Kleismic" for the system which approximates 7/4 by three minor third generators, and hence uses the

875/864 comma. Since this is how "Kleismic" was described when introduced, I think we need to stick to that name.

I agree that Catakleismic, with kernel generated by

[225/224, 4375/4374], is best notated by 72-et. I am proposing that

Kleismic, with kernel generated by [49/48, 126/125] be notated by

53-et.

--- In tuning-math@yahoogroups.com, "gdsecor <gdsecor@y...>"

<gdsecor@y...> wrote:

> --- In tuning-math@yahoogroups.com, "Gene Ward Smith

> <genewardsmith@j...>" <genewardsmith@j...> wrote:

> > --- In tuning-math@yahoogroups.com, "Dave Keenan <d.keenan@u...>"

> <d.keenan@u...> wrote:

> > > However, if you _were_ using kleismic for 7-limit with the least

> > > complex 7's, you would probably be dissatisfied with a 53-ET based

> > > notation.

> >

> > Why? Neither one is using the best value of the "7" of the et in

> question, and in both cases the 7-limit intervals are much more out

> of tune than the 5-limit intervals. It is cheesy no matter how you

> notate it, and I don't see why it is any *more* dissapointing for 53

> than for 72.

You're right. I hadn't realised that 72-ET's 7 wasn't kleismic's least

complex 7.

> Since 7 is +22 and 11 is -21 in the series of ~5:6's in 53, 72, and

> 125, these are the normal positions for the kleismic temperament. So

> we should be using symbols of a different nature to notate a lot of

> the tones closer to the origin, which would seem to call for the 5^2

> symbol, //|. The notation for 125 uses not only this, but also both

> the 7-comma and 11-diesis symbols if the tuning is extended far

> enough to take in ratios of 7 and 11, so 125 seems like a logical

> choice to me.

This agrees better with the ET-independent notation I gave too.

So

/ // ///

would become

/| //| (|)