I propose we go with either these 19 (in weighted-complexity order)

neutral thirds

meantone

pelogic

augmented

porcupine

diminished

diaschismic

small diesic

quarter fourths

kleismic

twin meantone

half meantone-fourth

half meantone-fifth

minimal diesic

schismic

wuerschmidt

tiny diesic

orwell

amt

or if you want a bit more complexity add

seventh of major third

semisuper

parakleismic

twin schismic

half schismic-fourth

half schismic-fifth

bringing the total up to 26.

To see what they mean and see how they might be arrived at using

Gene's badness, modified to use Paul's weighted complexity, see the

latest incarnation of my spreadsheet.

http://dkeenan.com/Music/5LimitTemp.xls.zip

They require an error cutoff of 35 cents and the short list has a

weighted complexity cutoff of 10 while the long list has 12.

Any objections? Short list or long?

>neutral thirds

>meantone

>pelogic

>augmented

>porcupine

>diminished

>diaschismic

>small diesic

>quarter fourths

>kleismic

>twin meantone

>half meantone-fourth

>half meantone-fifth

>minimal diesic

>schismic

>wuerschmidt

>tiny diesic

>orwell

>amt

What you're calling small diesic is magic. I wonder why 5-limit

miracle didn't make Gene's original list... otherwise the short

list has everything I care about.

-Carl

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

> What you're calling small diesic is magic.

Sorry, my mistake.

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

> I propose we go with either these 19 (in weighted-complexity order)

> neutral thirds

> meantone

> pelogic

> augmented

> porcupine

> diminished

> diaschismic

> small diesic = magic

> quarter fourths = quadrafourths

> kleismic

> twin meantone = garbage

> half meantone-fourth = ditto

> half meantone-fifth = ditto

> minimal diesic

> schismic

> wuerschmidt

> tiny diesic

> orwell

> amt

What happened to chromic?

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

> What happened to chromic?

I don't need it on the list, and by your badness it is _much_ worse

than any of those on the list. You'd have to include lots of others if

you included chromic. I should never have tried to name it. Sorry.

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

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

> What happened to chromic?

Oh I think I realise why you ask, since you wouldn't want it by your

own badness measure. You think I'm violating my own badness measure.

The thing is, I've changed my parameters from 7.4 cents and 0.5 power,

to 5.5 cents and 0.43 power to make it agree better with your list.

See my latest version spreadsheet

http://dkeenan.com/Music/5LimitTemp.xls.zip

I've fixed the names.

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

> See my latest version spreadsheet

> http://dkeenan.com/Music/5LimitTemp.xls.zip

>

> I've fixed the names.

i don't think this is correct, dave:

"Flat_badness = complexity^2*error with cutoffs"

that's not flat at all, right, gene?

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

> Oh I think I realise why you ask, since you wouldn't want it by your

> own badness measure. You think I'm violating my own badness measure.

Nope. My 5-limit badness is 774, not something to set the world on fire, but it did pass your measure, as you say. I want a name for it because my 7-limit badness score is 173, so we have a decent 5-limit

temperament which heats up in the 7-limit.

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

> i don't think this is correct, dave:

>

> "Flat_badness = complexity^2*error with cutoffs"

>

> that's not flat at all, right, gene?

Not for 5-limit.

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

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

>

> > Oh I think I realise why you ask, since you wouldn't want it by

your

> > own badness measure. You think I'm violating my own badness

measure.

>

> Nope. My 5-limit badness is 774, not something to set the world on

fire, but it did pass your measure, as you say.

>

That was before I went back to using Paul's weighted complexity. Now

it comes out badder than pelogic and semisuper (and with my new

parameters 5.5c and 0.43 power, badder than seventh of major third

[-9,7] and parakleismic).

> I want a name for it because my 7-limit badness score is 173, so we

> have a decent 5-limit temperament which heats up in the 7-limit.

Being good at 7-limit is a good reason to have a name for it, but

obviously should have no bearing on whether or not it's included in

the 5-limit list.

If you modify your badness to use log-of-odd-limit weighted complexity

you will see that its requirement of 9 generators to the fifth

combined with its so-so 3 cent rms error, gives it a badness of 1113

that pushes it way down the list. The worst on my proposed list so far

is only 625 by this badness (quadrafourths).

But I don't mind including it so long as it doesn't require you to

include anything else. By your badness with weighted complexity, there

may well be some others in between quadrafourths and "chrome" that I

will find objectionable. There are certainly some that _you_ will find

objectionable: the half and twin neutral thirds and kleismics (that

I'm happy to omit).

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

> > twin meantone = garbage

> > half meantone-fourth = ditto

> > half meantone-fifth = ditto

Paul, you say they are not temperaments because they do not map to a

_the_ JI lattice, and are therefore merely "tuning systems". But of

course they map to two (or more) disconnected JI lattices which is a

hell of a lot better than a mere "tuning system". How about we call

them degenerate temperaments?

Gene, since you feel so strongly about it I'll go with Paul's earlier

suggested compromise. Leave them out of the lists. Mention somewhere

in the paper that these modifications are possible and result in a

degenerate temperament having the same errors as the parent

temperament but twice the complexity, and a different pair of

(generator, period) and therefore different MOS. Use these meantone

ones as the example, then say no more about them.

Agreed?

Anyone else object?

I just added degeneracy detection to my spreadsheet.

http://dkeenan.com/Music/5LimitTemp.xls.zip

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

> Paul, you say they are not temperaments because they do not map to a

> _the_ JI lattice, and are therefore merely "tuning systems". But of

> course they map to two (or more) disconnected JI lattices which is a

> hell of a lot better than a mere "tuning system".

Eh? Why is that? So does the 7-et+5-et system, which is the sort of thing I want to avoid; these degenerate temperaments are headed in that direction.

How about we call

> them degenerate temperaments?

Sounds OK; I don't want to give the impression that they are the same kind of thing as a "regular" regular temperament. Mentioning them makes sense to me, dwelling on them doesn't.

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

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

> > > twin meantone = garbage

> > > half meantone-fourth = ditto

> > > half meantone-fifth = ditto

>

> Paul, you say they are not temperaments because they do not map to

a

> _the_ JI lattice, and are therefore merely "tuning systems". But of

> course they map to two (or more) disconnected JI lattices which is

a

> hell of a lot better than a mere "tuning system". How about we call

> them degenerate temperaments?

decent, though torsion, rather than contorsion, would seem closer to

the usage of 'degeneracy' that i'm familiar with from physics and

math, when applied to temperaments.

>> hell of a lot better than a mere "tuning system". How about we call

>> them degenerate temperaments?

>

>decent, though torsion, rather than contorsion, would seem closer to

>the usage of 'degeneracy' that i'm familiar with from physics and

>math, when applied to temperaments.

Agree. They're not instances of degenerate temperament. They're

instances of superposed, perfectly legitimate temperament_s_.

-Carl