"father" variant?

So, I was messing around in 21-tone equal, and, well, one thing led to
another, and then I found a 3L+2s scale (are all of these father?
probably not) that tempers out 2200/2197. Which involves 5, 11, and
13. (Though 21 does do 7 well).

"otonal version" (LsLLs): 1/1 13/11 13/10 20/13 20/11 2/1

"utonal version" (sLLsL): 1/1 11/10 13/10 20/13 22/13 2/1

I don't have any idea how much anyone has done on
certain-primes-excluded temperaments, or even temperaments that
approximate, say, 13/10 without approximating 5/4 or 13/8...

I've gotten into these higher harmonics playing on a guitar patch
well-endowed with difference tones.

Up up and away...

--- In tuning-math@yahoogroups.com, "Jacob" <jbarton@...> wrote:
>
> So, I was messing around in 21-tone equal, and, well, one thing led to
> another, and then I found a 3L+2s scale (are all of these father?
> probably not) that tempers out 2200/2197. Which involves 5, 11, and
> 13. (Though 21 does do 7 well).

You can get a {2,5,11,13}-temperament out of this one, of course. I've
been working with 224-et lately, in octoid temperament, and 224
tempers out this "barton" comma. Another equal temperament which does
is 311. If we consider 224&311, we find it has low complexity,
comparitively speaking, for 5, 11, and 13; in fact they have a Graham
complexity of 9. Considering that 224 and 311 means microtempering, we
have here a low-complexity microtemperament for {2,5,11,13}, which is
very interesting. No one has systematically surveyed for such things,
though I've taken a look at a few cases, and Graham may have done
something also.

The generator for this temperament is an 11/8; that's 103 steps of 224
and 143 of 311, either of which is fine for tuning it up. It has MOS
of size 7, 9, 11, 13, 24, 37, 50, 87... Because the {2,5,11,13}
complexity is so low, you can get complete chords for this already
with the 9-note MOS.

This {2,5,11,13} temperament tempers out both 2200/2197 and 6656/6655.
If you also temper out 625/624 and 1575/1573, you get the
corresponding complete 13-limit temperament for 224&311, with wedgie

<<27 8 -67 -1 5 -50 -182 -95 -92 -178 -30 -18 229 262 21||

and prime mapping

[<1 14 6 -28 3 6|, <0 -27 -8 67 1 -5|]

I propose "barton" as the name for 13-limit 224&311.

>> So, I was messing around in 21-tone equal, and, well, one thing led to
>> another, and then I found a 3L+2s scale (are all of these father?
>> probably not) that tempers out 2200/2197. Which involves 5, 11, and
>> 13. (Though 21 does do 7 well).
>
>You can get a {2,5,11,13}-temperament out of this one, of course.

So is that the notation for harmonic basis now? And are we
calling them harmonic bases?

>I've been working with 224-et lately, in octoid temperament,

What's the TM basis for octoid?

>The generator for 224&311 is an 11/8; that's 103 steps of 224
>and 143 of 311, either of which is fine for tuning it up. It has MOS
>of size 7, 9, 11, 13, 24, 37, 50, 87... Because the {2,5,11,13}
>complexity is so low, you can get complete chords for this already
>with the 9-note MOS.

Kyool.

>This {2,5,11,13} temperament tempers out both 2200/2197 and 6656/6655.
>If you also temper out 625/624 and 1575/1573, you get the
>corresponding complete 13-limit temperament for 224&311, with wedgie
>
><<27 8 -67 -1 5 -50 -182 -95 -92 -178 -30 -18 229 262 21||
>
>and prime mapping
>
>[<1 14 6 -28 3 6|, <0 -27 -8 67 1 -5|]
>
>I propose "barton" as the name for 13-limit 224&311.

The introduction of 7 and 3 seem to really hurt it. Maybe the
{2,5,11,13} version should be called barton?

-Carl

--- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
Considering that 224 and 311 means microtempering, we
> have here a low-complexity microtemperament for {2,5,11,13}, which
is very interesting. No one has systematically surveyed for such
things, though I've taken a look at a few cases, and Graham may have
done something also.
>
> The generator for this temperament is an 11/8; that's 103 steps of 224
> and 143 of 311, either of which is fine for tuning it up. It has MOS
> of size 7, 9, 11, 13, 24, 37, 50, 87... Because the {2,5,11,13}
> complexity is so low, you can get complete chords for this already
> with the 9-note MOS.

That is a nice 9-note MOS. The large step is awfully close to 7/6 to
be a non-7 temperament, though.

But this is not the father-like temperament I described. That one has
a generator of 13/10 and MOSes of 5, 9, 13, 17, 21, ...(and then I
don't really know where it's converging).

> But this is not the father-like temperament I described. That one has
> a generator of 13/10 and MOSes of 5, 9, 13, 17, 21, ...(and then I
> don't really know where it's converging).

Sorry, that's 5,8,13,21...

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> So, I was messing around in 21-tone equal, and, well, one thing
led to
> >> another, and then I found a 3L+2s scale (are all of these father?
> >> probably not) that tempers out 2200/2197. Which involves 5, 11, and
> >> 13. (Though 21 does do 7 well).
> >
> >You can get a {2,5,11,13}-temperament out of this one, of course.
>
> So is that the notation for harmonic basis now? And are we
> calling them harmonic bases?

I was thinking of them as generators.

> >I've been working with 224-et lately, in octoid temperament,
>
> What's the TM basis for octoid?

{540/539, 1375/1372, 4000/3993}

540/539 says that a 60/49 neutral third is the same as an 11/9 neutral
third, so that the interval from 7/6 to 10/7 is an 11/9. This makes me
think 1-7/6-10/7-5/3 chords are native to the system; in any case the
tuning is adjusted.

1375/1372 says that three 7/5s gives an 11/4.

4000/3993 says that three 11/10s gives a 4/3.

> The introduction of 7 and 3 seem to really hurt it. Maybe the
> {2,5,11,13} version should be called barton?

I was sort of hoping to simplfy matters by calling it the no threes,
no sevens barton or something like that, but this may be a bad idea
since less complex versions of 3 and 7 might be preferred.

>> >You can get a {2,5,11,13}-temperament out of this one, of course.
>>
>> So is that the notation for harmonic basis now? And are we
>> calling them harmonic bases?
>
>I was thinking of them as generators.

We'll need some way to distinguish them from the other
generators of the temperament.

>> >I've been working with 224-et lately, in octoid temperament,
>>
>> What's the TM basis for octoid?
>
>{540/539, 1375/1372, 4000/3993}

It wasn't in Herman's Word file. Speaking of which, there
has got to be an Excel file made. I'll do it unless Herman
wants to. I'm officially requesting contributions.

Paul: do you have this stuff written down somewhere?
Gene?
Herman: do you have a newer version of the Word file?
Graham?

Every temperament that has a name ought to be on it.

>540/539 says that a 60/49 neutral third is the same as an 11/9 neutral
>third, so that the interval from 7/6 to 10/7 is an 11/9. This makes me
>think 1-7/6-10/7-5/3 chords are native to the system; in any case the
>tuning is adjusted.

How would such an adjustment improve the consonance of that chord?

-Carl

Gene Ward Smith wrote:

> You can get a {2,5,11,13}-temperament out of this one, of course. I've
> been working with 224-et lately, in octoid temperament, and 224
> tempers out this "barton" comma. Another equal temperament which does
> is 311. If we consider 224&311, we find it has low complexity,
> comparitively speaking, for 5, 11, and 13; in fact they have a Graham
> complexity of 9. Considering that 224 and 311 means microtempering, we
> have here a low-complexity microtemperament for {2,5,11,13}, which is
> very interesting. No one has systematically surveyed for such things,
> though I've taken a look at a few cases, and Graham may have done
> something also.

It's easy enough to run a survey if you want a survey. Plugging 2.5.11.13 into the script at http://x31eq.com/temper/linear.html gives lots of different versions of this temperament.

> The generator for this temperament is an 11/8; that's 103 steps of 224
> and 143 of 311, either of which is fine for tuning it up. It has MOS
> of size 7, 9, 11, 13, 24, 37, 50, 87... Because the {2,5,11,13}
> complexity is so low, you can get complete chords for this already
> with the 9-note MOS.

I get a complexity of 9, so there aren't any complete chords in the 9-note MOS. There are two of each in the 11-note MOS.

> This {2,5,11,13} temperament tempers out both 2200/2197 and 6656/6655.
> If you also temper out 625/624 and 1575/1573, you get the
> corresponding complete 13-limit temperament for 224&311, with wedgie
> > <<27 8 -67 -1 5 -50 -182 -95 -92 -178 -30 -18 229 262 21||
> > and prime mapping
> > [<1 14 6 -28 3 6|, <0 -27 -8 67 1 -5|]

That's the one. 1--8 = 9, hence the complexity of 9.

Graham

Jacob wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
> <genewardsmith@...> wrote:

>>The generator for this temperament is an 11/8; that's 103 steps of 224
>>and 143 of 311, either of which is fine for tuning it up. It has MOS
>>of size 7, 9, 11, 13, 24, 37, 50, 87... Because the {2,5,11,13}
>>complexity is so low, you can get complete chords for this already
>>with the 9-note MOS.
> > That is a nice 9-note MOS. The large step is awfully close to 7/6 to
> be a non-7 temperament, though.

I've got two different mappings with a 7-generator 7:6, and neither of them work well as full 13-limit temperaments:

2/25, 48.2 cent generator

basis:
(0.5, 0.040168016583965738)

mapping by period and generator:
[(2, 0), (3, 2), (4, 8), (6, -5), (7, -1), (7, 5)]

mapping by steps:
[(26, 24), (41, 38), (60, 56), (73, 67), (90, 83), (96, 89)]

highest interval width: 13
complexity measure: 26 (50 for smallest MOS)
highest error: 0.009653 (11.584 cents)

and

28/61, 550.6 cent generator

basis:
(1.0, 0.45885703237696263)

mapping by period and generator:
[(1, 0), (-3, 10), (6, -8), (-5, 17), (3, 1), (6, -5)]

mapping by steps:
[(37, 24), (59, 38), (86, 56), (104, 67), (128, 83), (137, 89)]

highest interval width: 28
complexity measure: 28 (37 for smallest MOS)
highest error: 0.014001 (16.801 cents)

Perhaps you can do something with 7:6 as an independent prime interval. I don't think the online script can handle that. Maybe if you multiply through by 3...

> But this is not the father-like temperament I described. That one has
> a generator of 13/10 and MOSes of 5, 8, 13, 21, ...(and then I
> don't really know where it's converging).

Looks like this one, from my (redundant) top 20:

14/37, 454.2 cent generator

basis:
(1.0, 0.3785400201327791)

mapping by period and generator:
[(1, 0), (5, -9), (8, -15), (-1, 10), (8, -12), (9, -14)]

mapping by steps:
[(29, 8), (46, 13), (67, 19), (81, 22), (100, 28), (107, 30)]

highest interval width: 15
complexity measure: 15 (21 for smallest MOS)
highest error: 0.001912 (2.294 cents)
unique

That shows arbitrary mappings for 3 and 7.

Graham

>It's easy enough to run a survey if you want a survey. Plugging
>2.5.11.13 into the script at
>http://x31eq.com/temper/linear.html gives lots of different
>versions of this temperament.

> 17/41, 498.3 cent generator

What's 17/41?

>basis:
>(1.0, 0.41521839935180083)

What's this? It'd be nice to have the TM-reduced commas.

>mapping by period and generator:
>[(1, 0), (2, -1), (-1, 8)]

I thought you agreed to right this on two lines.

>highest interval width: 9

What's this?

>highest error: 0.000181 (0.217 cents)

What's the first value?

What is ET goodness cutoff for?

Do you have a script that starts from commas?

-Carl

Carl Lumma wrote:
>>It's easy enough to run a survey if you want a survey. Plugging >>2.5.11.13 into the script at >>http://x31eq.com/temper/linear.html gives lots of different >>versions of this temperament.
> >>17/41, 498.3 cent generator
> > > What's 17/41?

The generator/period ratio in terms of steps from an MOS.

>>basis:
>>(1.0, 0.41521839935180083)
> > What's this? It'd be nice to have the TM-reduced commas.

It's an octave period and a generator of a fourth. Hence not the father variant of the title.

I don't have a reliable algorithm for getting the correct commas. The unreliable one's there in the library. In this case, it gives 32805:32768 which should be the schisma. That's from

>>> temper.Temperament(29,12,temper.limit5)

17/41, 498.3 cent generator

basis:
(1.0, 0.41521839935180083)

mapping by period and generator:
[(1, 0), (2, -1), (-1, 8)]

mapping by steps:
[(29, 12), (46, 19), (67, 28)]

highest interval width: 9
complexity measure: 9 (12 for smallest MOS)
highest error: 0.000181 (0.217 cents)
unique
>>> ' '.join(['%i:%i'%temper.getRatio(v) for v in _.getUnisonVectors()])
'32805:32768'

I thought the online script was showing equivalences between intervals in the second-order odd-limit, but perhaps I lost that when I changed web hosts.

>>mapping by period and generator:
>>[(1, 0), (2, -1), (-1, 8)]
> > I thought you agreed to right this on two lines.

Only for the prime-limit files. I haven't touched the old odd-limit code.

>>highest interval width: 9
> > What's this?

The largest number of generators to an interval within the odd-limit.

>>highest error: 0.000181 (0.217 cents)
> > What's the first value?

In terms of octaves.

> What is ET goodness cutoff for?

To decide which ETs are used for seeding.

> Do you have a script that starts from commas?

Yes.

http://x31eq.com/temper/vectors.html

Graham

>>> http://x31eq.com/temper/linear.html
>>>
>>> 17/41, 498.3 cent generator
>>
>>
>> What's 17/41?
>
>The generator/period ratio in terms of steps from an MOS.

Considered adding that phrase to the docs?

It'd be nice to get the period in cents.

>>>basis:
>>>(1.0, 0.41521839935180083)
>>
>> What's this? It'd be nice to have the TM-reduced commas.
>
>It's an octave period and a generator of a fourth.

Considered appending "base-2 log" to them?

>I don't have a reliable algorithm for getting the correct commas.
>The unreliable one's there in the library.

Maybe you can translate Gene's maple. Chris can apparently
do it too.

>>>mapping by period and generator:
>>>[(1, 0), (2, -1), (-1, 8)]
>>
>> I thought you agreed to right this on two lines.
>
>Only for the prime-limit files. I haven't touched the old odd-limit code.

Whoops, I meant write. Why not touch?

>>>highest interval width: 9
>>
>> What's this?
>
>The largest number of generators to an interval within the odd-limit.

I thought that was what "complexity measure" was.

>>>highest error: 0.000181 (0.217 cents)
>>
>> What's the first value?
>
>In terms of octaves.

Maybe it should say so.

>> What is ET goodness cutoff for?
>
>To decide which ETs are used for seeding.

I tought so. Are inconsistent ETs bad for seeding?

>> Do you have a script that starts from commas?
>
>Yes.
>
> http://x31eq.com/temper/vectors.html

Aha, sorry, I see that's under "do something else".

-C.

>>> Do you have a script that starts from commas?
>>
>>Yes.
>>
>> http://x31eq.com/temper/vectors.html
>
>Aha, sorry, I see that's under "do something else".

Maybe that link should also appear on the input pages.

>>>>mapping by period and generator:
>>>>[(1, 0), (2, -1), (-1, 8)]
>>>
>>> I thought you agreed to right this on two lines.
>>
>>Only for the prime-limit files. I haven't touched the old odd-limit code.
>
>Whoops, I meant write. Why not touch?

Is your prime-limit code online?

-C.

Carl Lumma wrote:
>>>>http://x31eq.com/temper/linear.html
>>>>
>>>>17/41, 498.3 cent generator
>>>
>>>
>>>What's 17/41?
>>
>>The generator/period ratio in terms of steps from an MOS.
> > Considered adding that phrase to the docs?

That ratio is documented at

http://x31eq.com/temper.html

which is still the best documentation anywhere.

> It'd be nice to get the period in cents.

It's usually trivial to calculate once you know the number of periods to the octave. You get that from the mapping.

>>>>basis:
>>>>(1.0, 0.41521839935180083)
>>>
>>>What's this? It'd be nice to have the TM-reduced commas.
>>
>>It's an octave period and a generator of a fourth.
> > Considered appending "base-2 log" to them?

In general, it's in terms of the equivalence interval, not a base-2 log of everything. It always gives the generator in cents when it knows the prime intervals are prime numbers. If you don't know what the other numbers are you shouldn't be missing any information.

>>I don't have a reliable algorithm for getting the correct commas.
>>The unreliable one's there in the library.
> > Maybe you can translate Gene's maple. Chris can apparently
> do it too.

It depends on a function that removes the torsion in a way I don't understand. I haven't studied the documentation to see what it's doing.

>>>>mapping by period and generator:
>>>>[(1, 0), (2, -1), (-1, 8)]
>>>
>>>I thought you agreed to right this on two lines.
>>
>>Only for the prime-limit files. I haven't touched the old odd-limit code.
> > Whoops, I meant write. Why not touch?

The odd-limit code is more complex, so I preferred to work on the prime-limit code last time I worked on this. The stringifications were never intended to be user-friendly temperament definitions. Over the years I haven't written the code to display them better in HTML. If I ever do, it will work fine with the odd-limit library as it is.

>>>>highest interval width: 9
>>>
>>>What's this?
>>
>>The largest number of generators to an interval within the odd-limit.
> > I thought that was what "complexity measure" was.

No, you multiply by the number of periods to either the equivalence interval or the octave.

>>>>highest error: 0.000181 (0.217 cents)
>>>
>>>What's the first value?
>>
>>In terms of octaves.
> > Maybe it should say so.

It would if the libraries were properly documented.

>>>What is ET goodness cutoff for?
>>
>>To decide which ETs are used for seeding.
> > I tought so. Are inconsistent ETs bad for seeding?

Good ETs tend to lead to good rank2 temperaments. In some cases a rank 2 temperament class isn't good enough to have 2 consistent ETs. If you want those anyway (and they're important for high prime limits with moderate complexity and error) you need to seed with inconsistent ETs.

Consistent ETs, where they exist, are best for defining a temperament. It means there's no ambiguity about the mapping and so you only need to give the number of notes to the octave.

Graham

At 12:54 AM 3/13/2006, you wrote:
>Carl Lumma wrote:
>>>>>http://x31eq.com/temper/linear.html
>>>>>
>>>>>17/41, 498.3 cent generator
>>>>
>>>>
>>>>What's 17/41?
>>>
>>>The generator/period ratio in terms of steps from an MOS.
>>
>> Considered adding that phrase to the docs?
>
>That ratio is documented at
>
>http://x31eq.com/temper.html
>
>which is still the best documentation anywhere.

Not just anywhere; it's located under "source code".

>> It'd be nice to get the period in cents.
>
>It's usually trivial to calculate once you know the number of periods
>to the octave. You get that from the mapping.

Sometimes it's just easiest to glance at the period and generator
and start chaining away.

>>>>>basis:
>>>>>(1.0, 0.41521839935180083)
>>>>
>>>>What's this? It'd be nice to have the TM-reduced commas.
>>>
>>>It's an octave period and a generator of a fourth.
>>
>> Considered appending "base-2 log" to them?
>
>In general, it's in terms of the equivalence interval, not a base-2 log
>of everything.

Where do I input the equivalence interval?

>>>>>highest interval width: 9
>>>>
>>>>What's this?
>>>
>>>The largest number of generators to an interval within the odd-limit.
>>
>> I thought that was what "complexity measure" was.
>
>No, you multiply by the number of periods to either the equivalence
>interval or the octave.

Ah, right.

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> Every temperament that has a name ought to be on it.

Certainly one that has actual music written in it, and I'm polishing
my Octoid[72] piece now. Anyway, it's one of the better microtemperaments.

> >540/539 says that a 60/49 neutral third is the same as an 11/9 neutral
> >third, so that the interval from 7/6 to 10/7 is an 11/9. This makes me
> >think 1-7/6-10/7-5/3 chords are native to the system; in any case the
> >tuning is adjusted.
>
> How would such an adjustment improve the consonance of that chord?

By moving the 60/49 closer to 11/9. I'm using the 224-et tuning, and in
224-et, 60/49 and 11/9 are the same, and the tuning is less than a
cent away from JI 11/9.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> I get a complexity of 9, so there aren't any complete chords in the
> 9-note MOS. There are two of each in the 11-note MOS.

Right, 9-9=0, sorry.

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> > Maybe you can translate Gene's maple. Chris can apparently
> > do it too.
>
> It depends on a function that removes the torsion in a way I don't
> understand. I haven't studied the documentation to see what it's doing.

Eh? I start from wedgies, which have their torsion pre-removed.

Me:
>>It depends on a function that removes the torsion in a way I don't >>understand. I haven't studied the documentation to see what it's doing.

Gene:
> Eh? I start from wedgies, which have their torsion pre-removed.

Yes, but your first-stage algorithm for getting unison vectors from the wedgie (a wedge product of something with each prime, I think) can give a set of intervals with torsion. It's supposed to be the Hermite reduction that removes the torsion, but I never worked out how it did that.

Graham

Carl Lumma wrote:

> It wasn't in Herman's Word file. Speaking of which, there
> has got to be an Excel file made. I'll do it unless Herman
> wants to. I'm officially requesting contributions.
> > Paul: do you have this stuff written down somewhere?
> Gene?
> Herman: do you have a newer version of the Word file?
> Graham?
> > Every temperament that has a name ought to be on it.

I haven't updated the Word file since I put it up. And I don't own Excel, so if I did anything it'd be a Works or OpenOffice spreadsheet.

I haven't been very systematic about making note of every named temperament and every variant temperament name, I'm afraid. There've been lots of threads that I've glanced over or ignored, depending on how busy or interested I was at the time, and a couple of times I set the list to nomail. What would be nice is to have an article on one of the wikis (Keenan Pepper has a tuning wiki page that might be an appropriate place for a file like that) so that anyone can add any temperament that's missing, fill in any alternative names or other data, and include links to relevant pages. I could sign up for the tuning wiki and start an article if that sounds like something useful.

>I haven't been very systematic about making note of every named
>temperament and every variant temperament name, I'm afraid. There've
>been lots of threads that I've glanced over or ignored, depending on how
>busy or interested I was at the time, and a couple of times I set the
>list to nomail. What would be nice is to have an article on one of the
>wikis (Keenan Pepper has a tuning wiki page that might be an appropriate
>place for a file like that) so that anyone can add any temperament
>that's missing, fill in any alternative names or other data, and include
>links to relevant pages. I could sign up for the tuning wiki and start
>an article if that sounds like something useful.

Great idea!

Too bad MediaWiki doesn't support spreadsheets.

I'll try to set up a page in the wee hours tonight or tomorrow.
Or maybe somebody could beat me to it. I'd like to roughly copy Paul's
spreadsheet...

/tuning/database?method=reportRows&tbl=10

It should ideally have only R2 temperaments, with name as the
primary key, and then maybe...

old name(s), wedgie, comma(s), map, TOP period/generator

...and perhaps TOP damage and complexity (whatever those are).
Perhaps it should cover the 5-11 limits.

-Carl