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!