Best Partch keyboard mapping

The question was raised last week as to what MOS is "best" for the Partch
43 note scale. From what we already know there are three candidates:

Wilson's 11-limit schismic mapping gets all 43 notes into a 41 note MOS.
Of course this can't be unique, but all of the 41 notes have at least one
counterpart in the Partch scale.

Secor's miracle mapping gets the 11-limit diamond into a 45 note generator
chain. Each interval is represented uniquely. Unfortunately, the Genesis
scale requires more than 45 notes, and no scale with a complete diamond
can fit into the 41 note MOS. The next biggest MOS is 72, which is quite
complex.

Any 72 note keyboard can get the scale uniquely, and be tempered to a high
degree of accuracy.

My criteria for "best" mapping are:

All ratios expressed uniquely (no split keys)

All 43 notes fit in a small MOS. This also means the scale is a subset of
a simple periodicity block.

Looking through the scales in <http://x31eq.com/limit11.key> it
looks like the simplest, unique temperament there is h41&h58. This gets
the whole diamond into the 58 note MOS.

I've written a script to check that both versions of the 43 note scale lie
within the 58 note MOS and it runs okay. I've included it below for those
receiving e-mails who don't lose the indentation. The most complex
intervals are 16:15 and 15:8.

So this 58 note mapping looks like the best one by these criteria. You
may be able to do better by not starting with consistent ETs.

import re, temper

h58 = temper.PrimeET(58,temper.primes[:4])
h41 = temper.PrimeET(41,temper.primes[:4])
scale = h41&h58

for scl in ("partch_43.scl", "partch_43a.scl"):
lastStepSize = 0
for n, d in re.findall(r'(?m)^ (\d+)/(\d+)', open(scl).read()):
vector = temper.factorizeRatio(int(n), int(d))[:5]
octs, gens = scale.byMapping(vector)
if abs(gens)>27:
print "%s/%s too big" %(n, d)
stepSize = temper.dotprod(vector, h58.basis)
if stepSize <= lastStepSize:
print "%s/%s not in sequence" %(n,d)
lastStepSize = stepSize

Graham

--- In tuning-math@y..., graham@m... wrote:
> The question was raised last week as to what MOS is "best" for the
Partch
> 43 note scale. From what we already know there are three
candidates:
>
> Wilson's 11-limit schismic mapping gets all 43 notes into a 41 note
MOS.
> Of course this can't be unique, but all of the 41 notes have at
least one
> counterpart in the Partch scale.
>
> Secor's miracle mapping gets the 11-limit diamond into a 45 note
generator
> chain. Each interval is represented uniquely. Unfortunately, the
Genesis
> scale requires more than 45 notes, and no scale with a complete
diamond
> can fit into the 41 note MOS. The next biggest MOS is 72, which is
quite
> complex.
>
> Any 72 note keyboard can get the scale uniquely, and be tempered to
a high
> degree of accuracy.
>
>
> My criteria for "best" mapping are:
>
> All ratios expressed uniquely (no split keys)
>
> All 43 notes fit in a small MOS. This also means the scale is a
subset of
> a simple periodicity block.
>
> Looking through the scales in
<http://x31eq.com/limit11.key> it
> looks like the simplest, unique temperament there is h41&h58. This
gets
> the whole diamond into the 58 note MOS.

perhaps you or gene could tell us the errors and unison vectors here.

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

> perhaps you or gene could tell us the errors and unison vectors here.

I found all of my so-called "standard" ets up to 100 which represent each note of Genesis, and obtained the following:

58 .3529661000 7.302746897
65 .6150147495 11.35411845
72 .2346001000 3.910001666
73 .5954749001 9.788628497
80 .4116062001 6.174093001
84 .6814441795 9.734916846
87 .4564030700 6.295214756
89 .4717246800 6.360332766
91 .6610206501 8.716755827
94 .3698782499 4.721849998
95 .8571248997 10.82684084

The second column is maximum relative error, and the third maximum
11-limit consonance error in cents. I'm not sure what you want unisons for--do you want to find an 11-limit block containing Genesis?

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > perhaps you or gene could tell us the errors and unison vectors
here.
>
> I found all of my so-called "standard" ets up to 100 which
represent each note of Genesis, and obtained the following:
>
> 58 .3529661000 7.302746897
> 65 .6150147495 11.35411845
> 72 .2346001000 3.910001666
> 73 .5954749001 9.788628497
> 80 .4116062001 6.174093001
> 84 .6814441795 9.734916846
> 87 .4564030700 6.295214756
> 89 .4717246800 6.360332766
> 91 .6610206501 8.716755827
> 94 .3698782499 4.721849998
> 95 .8571248997 10.82684084
>
> The second column is maximum relative error, and the third maximum
> 11-limit consonance error in cents.

gene -- graham did not specify an ET for the scale -- he specified a
linear temperament. so this doesn't answer my question. i'd also like
to know the generator, etc.

In-Reply-To: <a3mvf9+db4f@eGroups.com>
Gene:

> > I found all of my so-called "standard" ets up to 100 which
> represent each note of Genesis, and obtained the following:
> >
> > 58 .3529661000 7.302746897
> > 65 .6150147495 11.35411845
> > 72 .2346001000 3.910001666
> > 73 .5954749001 9.788628497
> > 80 .4116062001 6.174093001
> > 84 .6814441795 9.734916846
> > 87 .4564030700 6.295214756
> > 89 .4717246800 6.360332766
> > 91 .6610206501 8.716755827
> > 94 .3698782499 4.721849998
> > 95 .8571248997 10.82684084
> >
> > The second column is maximum relative error, and the third maximum
> > 11-limit consonance error in cents.

Thanks for that. I had checked the prime-mappings below 58 to verify that
there isn't a simpler constant structure lurking. Checking they don't
uniquely represent each note in the right order. I also checked that 53
couldn't work with any 11-limit mapping so long as you preserved its good
5-limit mapping. There may be other possible inconsistent mappings. I
don't know if you'd want to use them.

Paul:
> gene -- graham did not specify an ET for the scale -- he specified a
> linear temperament. so this doesn't answer my question. i'd also like
> to know the generator, etc.

The first step is to find an ET or constant structure that uniquely
represents each note in sequence. Only then can you start looking for a
linear temperament with an MOS that doesn't take you beyond that number of
notes. Showing there aren't any valid constant structures below 58 is
sufficient proof that there are no better mappings by my criteria,
although some other mapping of 58 may be equally good.

Here's the temperament listing. You could have got it from the .key list
I linked to, or the CGI for ETs to linear temperaments.

29/99, 351.5 cent generator

basis:
(1.0, 0.29293920388170502)

mapping by period and generator:
[(1, 0), (1, 2), (-5, 25), (-1, 13), (2, 5)]

mapping by steps:
[(58, 41), (92, 65), (135, 95), (163, 115), (201, 142)]

highest interval width: 25
complexity measure: 25 (41 for smallest MOS)
highest error: 0.005264 (6.317 cents)
unique

128:99 =~ 9:7
14:11 =~ 81:64

consistent with: 41, 58

For commatic unison vectors, I get

243:242, 540:539 and 896:891

Graham

--- In tuning-math@y..., graham@m... wrote:
> In-Reply-To: <a3mvf9+db4f@e...>
> Gene:
>
> > > I found all of my so-called "standard" ets up to 100 which
> > represent each note of Genesis, and obtained the following:
> > >
> > > 58 .3529661000 7.302746897
> > > 65 .6150147495 11.35411845
> > > 72 .2346001000 3.910001666
> > > 73 .5954749001 9.788628497
> > > 80 .4116062001 6.174093001
> > > 84 .6814441795 9.734916846
> > > 87 .4564030700 6.295214756
> > > 89 .4717246800 6.360332766
> > > 91 .6610206501 8.716755827
> > > 94 .3698782499 4.721849998
> > > 95 .8571248997 10.82684084
> > >
> > > The second column is maximum relative error, and the third
maximum
> > > 11-limit consonance error in cents.
>
> Thanks for that. I had checked the prime-mappings below 58 to
verify that
> there isn't a simpler constant structure lurking. Checking they
don't
> uniquely represent each note in the right order. I also checked
that 53
> couldn't work with any 11-limit mapping so long as you preserved
its good
> 5-limit mapping. There may be other possible inconsistent
mappings. I
> don't know if you'd want to use them.
>
> Paul:
> > gene -- graham did not specify an ET for the scale -- he
specified a
> > linear temperament. so this doesn't answer my question. i'd also
like
> > to know the generator, etc.
>
> The first step is to find an ET or constant structure that uniquely
> represents each note in sequence. Only then can you start looking
for a
> linear temperament with an MOS that doesn't take you beyond that
number of
> notes. Showing there aren't any valid constant structures below 58
is
> sufficient proof that there are no better mappings by my criteria,
> although some other mapping of 58 may be equally good.
>
> Here's the temperament listing. You could have got it from
the .key list
> I linked to, or the CGI for ETs to linear temperaments.
>
>
> 29/99, 351.5 cent generator
>
> basis:
> (1.0, 0.29293920388170502)
>
> mapping by period and generator:
> [(1, 0), (1, 2), (-5, 25), (-1, 13), (2, 5)]
>
> mapping by steps:
> [(58, 41), (92, 65), (135, 95), (163, 115), (201, 142)]
>
> highest interval width: 25
> complexity measure: 25 (41 for smallest MOS)
> highest error: 0.005264 (6.317 cents)
> unique
>
> 128:99 =~ 9:7
> 14:11 =~ 81:64
>
> consistent with: 41, 58
>
>
> For commatic unison vectors, I get
>
> 243:242, 540:539 and 896:891
>
>
> Graham

thanks graham!

you and gene between you have made enough valuable discoveries to
fill several issues of xenharmonikon. i encourage both of you to
begin working on publishable papers as soon as possible!