Rank 2 parametric scalar badness results

I've coded up a search for rank 2 temperament searches using the parametric badness measure I've been talking about. At this stage there may still be bugs but it seems to be working.

The general behavior is that any given parameter is a reasonably restrictive search. So there's no parameter that'll give the Middle Path temperaments at the top. On the flip side, there are few temperament classes that can make the number 1 spot for any parameter. Pajara and magic don't make it. The best goes straight from dominant to meantone to miracle to ennealimmal.

Some benchmarks then. The form Ek as a parameter multiplying complexity to get simple badness is easiest to understand. Ek is a kind of weighted error. One obvious value for it is 1 cent per octave. The top 10 is then (with octave equivalent wedgies)

1) meantone <1 4 10]
2) magic <5 1 12]
3) miracle <6 -7 -2]
4) orwell <7 -3 8]
5) garibaldi <1 -8 -14]
6) sensisept <7 9 13]
7) myna <10 9 7]
8) kleismatic microtemperament? <6 5 22]
9) pajara <2 -4 -4]
10) valentine? <9 5 -3]

That doesn't include ennealimmal, of course. If you want it then you should try a lower error "cutoff". 0.5 cents per octave gives the top 10 of

1) miracle <6 -7 -2]
2) meantone <1 4 10]
3) microkleismatic? <6 5 22]
4) orwell <7 -3 -8]
5) garibaldi <1 -8 -14]
6) magic <5 1 12]
7) ? <16 2 5]
8) ? <2 25 13]
9) ennealimmal <18 27 18]
10) ? <5 13 -17]

Obviously ennealimmal has an optimal error much lower than 0.5 cents per octave but it's still good enough to make this list. It reaches the top at 0.3 cents per octave and stays there below 0.01 cents per octave where my floats run out.

But I'm rambling. How about some 11-limit results? Back to Ek=1 cent per octave:

1) miracle <6 -7 -2 15]
2) orwell <7 -3 8 2]
3) valentine <9 5 -3 7]
4) meantone <1 4 10 18]
5) myna <10 9 7 25]
6) magic <5 1 12 -8]
7) meanpop <1 4 10 -13]
8) squares <4 16 9 10]
9) pajara <2 -4 -4 -12]
10) schismatic <1 -8 -14 -18]

Graham

At 09:59 AM 3/29/2008, you wrote:
>I've coded up a search for rank 2 temperament searches using
>the parametric badness measure I've been talking about.
>At this stage there may still be bugs but it seems to be
>working.
>
>The general behavior is that any given parameter is a
>reasonably restrictive search. So there's no parameter
>that'll give the Middle Path temperaments at the top. On
>the flip side, there are few temperament classes that can
>make the number 1 spot for any parameter. Pajara and magic
>don't make it. The best goes straight from dominant to
>meantone to miracle to ennealimmal.
>
>Some benchmarks then. The form Ek as a parameter
>multiplying complexity to get simple badness is easiest to
>understand. Ek is a kind of weighted error. One obvious
>value for it is 1 cent per octave. The top 10 is then (with
>octave equivalent wedgies)
>
>1) meantone <1 4 10]
>2) magic <5 1 12]
>3) miracle <6 -7 -2]
>4) orwell <7 -3 8]
>5) garibaldi <1 -8 -14]
>6) sensisept <7 9 13]
>7) myna <10 9 7]
>8) kleismatic microtemperament? <6 5 22]
>9) pajara <2 -4 -4]
>10) valentine? <9 5 -3]

I think pajara is a much better 7-limit system
than magic. Anyone disagree?

>But I'm rambling. How about some 11-limit results? Back to
>Ek=1 cent per octave:
>
>1) miracle <6 -7 -2 15]
>2) orwell <7 -3 8 2]
>3) valentine <9 5 -3 7]
>4) meantone <1 4 10 18]
>5) myna <10 9 7 25]
>6) magic <5 1 12 -8]
>7) meanpop <1 4 10 -13]
>8) squares <4 16 9 10]
>9) pajara <2 -4 -4 -12]
>10) schismatic <1 -8 -14 -18]

Don't have as much experience here, but I think pajara
is better than meantone.

-Carl

Graham Breed wrote:

> Some benchmarks then. The form Ek as a parameter > multiplying complexity to get simple badness is easiest to > understand. Ek is a kind of weighted error. One obvious > value for it is 1 cent per octave. The top 10 is then (with > octave equivalent wedgies)
> > 1) meantone <1 4 10]
> 2) magic <5 1 12]
> 3) miracle <6 -7 -2]
> 4) orwell <7 -3 8]
> 5) garibaldi <1 -8 -14]
> 6) sensisept <7 9 13]
> 7) myna <10 9 7]
> 8) kleismatic microtemperament? <6 5 22]
> 9) pajara <2 -4 -4]
> 10) valentine? <9 5 -3]

Myna is a little surprising to show up on a 7-limit list, although it's excellent in higher limits. The <6 5 22] is catakleismic.

> That doesn't include ennealimmal, of course. If you want it > then you should try a lower error "cutoff". 0.5 cents per > octave gives the top 10 of
> > 1) miracle <6 -7 -2]
> 2) meantone <1 4 10]
> 3) microkleismatic? <6 5 22]
> 4) orwell <7 -3 -8]
> 5) garibaldi <1 -8 -14]
> 6) magic <5 1 12]
> 7) ? <16 2 5]
hemiwuerschmidt
> 8) ? <2 25 13]
hemififths
> 9) ennealimmal <18 27 18]
> 10) ? <5 13 -17]
amity

I don't think I'd include amity or catakleismic on any top 10 list. Miracle, meantone, orwell, garibaldi, magic, and ennealimmal are fine, but the first list with sensi and myna seems better. On the other hand, valentine wouldn't be in my top 10 either.

> Obviously ennealimmal has an optimal error much lower than > 0.5 cents per octave but it's still good enough to make this > list. It reaches the top at 0.3 cents per octave and stays > there below 0.01 cents per octave where my floats run out.
> > But I'm rambling. How about some 11-limit results? Back to > Ek=1 cent per octave:
> > 1) miracle <6 -7 -2 15]
> 2) orwell <7 -3 8 2]
> 3) valentine <9 5 -3 7]
> 4) meantone <1 4 10 18]
> 5) myna <10 9 7 25]
> 6) magic <5 1 12 -8]
> 7) meanpop <1 4 10 -13]
> 8) squares <4 16 9 10]
> 9) pajara <2 -4 -4 -12]
> 10) schismatic <1 -8 -14 -18]

The last one is the 11-limit subset of cassandra 2 (a good 13-limit temperament, but it doesn't show up on my 11-limit top 3 grades; meanpop beats it in error and complexity, and meanpop in turn is beat out by semififth and magic).

This method seems to do well for the higher complexity range (where logflat badness includes too many complex temperaments), but it misses a lot of good lower complexity temperaments. The gold/silver/bronze lists might be better for low complexity temperaments, along with an error cutoff; the 11-limit gold list includes dominant, injera, augene, hedgehog, and porcupine. Here's a range of the gold list from complexity = 15 to 50:

12&19 [<1, 2, 4, 7, 6], <0, -1, -4, -10, -6]> meanenneadecal
12&17c [<1, 2, 4, 2, 1], <0, -1, -4, 2, 6]> dominant
12&26 [<2, 3, 4, 5, 6], <0, 1, 4, 4, 6]> injera
12&15 [<3, 5, 7, 8, 10], <0, -1, 0, 2, 2]> augene
8d&22 [<2, 4, 6, 7, 8], <0, -3, -5, -5, -4]> hedgehog
7&15 [<1, 2, 3, 2, 4], <0, -3, -5, 6, -4]> porcupine
12&22 [<2, 3, 5, 6, 8], <0, 1, -2, -2, -6]> pajara
12&31 [<1, 2, 4, 7, 11], <0, -1, -4, -10, -18]> meantone
9&22 [<1, 0, 3, 1, 3], <0, 7, -3, 8, 2]> orwell
31&58 [<1, -1, 0, 1, -3], <0, 10, 9, 7, 25]> myna
10&31 [<1, 1, 3, 3, 2], <0, 6, -7, -2, 15]> miracle

But one of the shortcomings of this method is that meanpop doesn't show up until grade 5.

Carl Lumma wrote:
> At 09:59 AM 3/29/2008, you wrote:
>> I've coded up a search for rank 2 temperament searches using >> the parametric badness measure I've been talking about. >> At this stage there may still be bugs but it seems to be >> working.
>>
>> The general behavior is that any given parameter is a >> reasonably restrictive search. So there's no parameter >> that'll give the Middle Path temperaments at the top. On >> the flip side, there are few temperament classes that can >> make the number 1 spot for any parameter. Pajara and magic >> don't make it. The best goes straight from dominant to >> meantone to miracle to ennealimmal.
>>
>> Some benchmarks then. The form Ek as a parameter >> multiplying complexity to get simple badness is easiest to >> understand. Ek is a kind of weighted error. One obvious >> value for it is 1 cent per octave. The top 10 is then (with >> octave equivalent wedgies)
>>
>> 1) meantone <1 4 10]
>> 2) magic <5 1 12]
>> 3) miracle <6 -7 -2]
>> 4) orwell <7 -3 8]
>> 5) garibaldi <1 -8 -14]
>> 6) sensisept <7 9 13]
>> 7) myna <10 9 7]
>> 8) kleismatic microtemperament? <6 5 22]
>> 9) pajara <2 -4 -4]
>> 10) valentine? <9 5 -3]
> > I think pajara is a much better 7-limit system
> than magic. Anyone disagree?

Pajara has a TOP-RMS error of about 2.6 cent/oct so it won't come out so well if you search around 1 cent/oct. Search for 2.6 cent/oct and you get:

12&19 3.975 <1 4 10] meantone
12&10 4.375 <2 -4 -4] pajara
5&12 4.834 <1 4 -2] dominant
15&12 4.898 <3 0 -6] augene
10&19 5.030 <4 -3 2] negrisept
19&14 5.059 <2 8 1] semaphore
19&22 5.060 <5 1 12] magic
4&12 5.089 <4 4 4] dimisept
19&15 5.117 <6 5 3] keemun
12&9 5.473 <3 0 6] august

>> But I'm rambling. How about some 11-limit results? Back to >> Ek=1 cent per octave:
>>
>> 1) miracle <6 -7 -2 15]
>> 2) orwell <7 -3 8 2]
>> 3) valentine <9 5 -3 7]
>> 4) meantone <1 4 10 18]
>> 5) myna <10 9 7 25]
>> 6) magic <5 1 12 -8]
>> 7) meanpop <1 4 10 -13]
>> 8) squares <4 16 9 10]
>> 9) pajara <2 -4 -4 -12]
>> 10) schismatic <1 -8 -14 -18]
> > Don't have as much experience here, but I think pajara
> is better than meantone.

How so? The complexities are similar but meantone has a much smaller error by most measures.

Graham

>>> 1) miracle <6 -7 -2 15]
>>> 2) orwell <7 -3 8 2]
>>> 3) valentine <9 5 -3 7]
>>> 4) meantone <1 4 10 18]
>>> 5) myna <10 9 7 25]
>>> 6) magic <5 1 12 -8]
>>> 7) meanpop <1 4 10 -13]
>>> 8) squares <4 16 9 10]
>>> 9) pajara <2 -4 -4 -12]
>>> 10) schismatic <1 -8 -14 -18]
>>
>> Don't have as much experience here, but I think pajara
>> is better than meantone.
>
>How so? The complexities are similar but meantone has a
>much smaller error by most measures.

You're right.

-Carl

Herman Miller wrote:
> Graham Breed wrote:
> >> Some benchmarks then. The form Ek as a parameter >> multiplying complexity to get simple badness is easiest to >> understand. Ek is a kind of weighted error. One obvious >> value for it is 1 cent per octave. The top 10 is then (with >> octave equivalent wedgies)
>>
>> 1) meantone <1 4 10]
>> 2) magic <5 1 12]
>> 3) miracle <6 -7 -2]
>> 4) orwell <7 -3 8]
>> 5) garibaldi <1 -8 -14]
>> 6) sensisept <7 9 13]
>> 7) myna <10 9 7]
>> 8) kleismatic microtemperament? <6 5 22]
>> 9) pajara <2 -4 -4]
>> 10) valentine? <9 5 -3]
> > Myna is a little surprising to show up on a 7-limit list, although it's > excellent in higher limits. The <6 5 22] is catakleismic.

What else is there with a weighted error around 1 cent/oct?

>> That doesn't include ennealimmal, of course. If you want it >> then you should try a lower error "cutoff". 0.5 cents per >> octave gives the top 10 of
>>
>> 1) miracle <6 -7 -2]
>> 2) meantone <1 4 10]
>> 3) microkleismatic? <6 5 22]
>> 4) orwell <7 -3 -8]
>> 5) garibaldi <1 -8 -14]
>> 6) magic <5 1 12]
>> 7) ? <16 2 5]
> hemiwuerschmidt
>> 8) ? <2 25 13]
> hemififths
>> 9) ennealimmal <18 27 18]
>> 10) ? <5 13 -17]
> amity
> > I don't think I'd include amity or catakleismic on any top 10 list. > Miracle, meantone, orwell, garibaldi, magic, and ennealimmal are fine, > but the first list with sensi and myna seems better. On the other hand, > valentine wouldn't be in my top 10 either.

You probably don't want to target 0.5 cent/oct then.

Valentine has a significantly lower TOP-RMS than TOP-max error so it'll tend to do well in these lists. I don't have any experience with it to say if that's justified.

>> But I'm rambling. How about some 11-limit results? Back to >> Ek=1 cent per octave:
>>
>> 1) miracle <6 -7 -2 15]
>> 2) orwell <7 -3 8 2]
>> 3) valentine <9 5 -3 7]
>> 4) meantone <1 4 10 18]
>> 5) myna <10 9 7 25]
>> 6) magic <5 1 12 -8]
>> 7) meanpop <1 4 10 -13]
>> 8) squares <4 16 9 10]
>> 9) pajara <2 -4 -4 -12]
>> 10) schismatic <1 -8 -14 -18]
> > The last one is the 11-limit subset of cassandra 2 (a good 13-limit > temperament, but it doesn't show up on my 11-limit top 3 grades; meanpop > beats it in error and complexity, and meanpop in turn is beat out by > semififth and magic).

That schismatic is in the 11-limit list you gave me. I don't think it makes my top 3 for any parameter either. Or top 9 for that matter. What's semififth?

> This method seems to do well for the higher complexity range (where > logflat badness includes too many complex temperaments), but it misses a > lot of good lower complexity temperaments. The gold/silver/bronze lists > might be better for low complexity temperaments, along with an error > cutoff; the 11-limit gold list includes dominant, injera, augene, > hedgehog, and porcupine. Here's a range of the gold list from complexity > = 15 to 50:

IT does well for the higher complexity range because I set it to the higher complexity range. If you raise the target error you'll get lower complexities. Let's try 4 cent/oct

12&9 6.459 <3 0 6 6] august?
19&7 6.763 <1 4 10 6] meanenneadecal
5&12 6.826 <1 4 -2 6] dominant variant
8&4 6.829 <4 4 4 0] dimisept?
8&15 6.841 <3 5 9 4] porcupine variant
15&12 6.940 <3 0 -6 -6] augene
12&10 7.113 <2 -4 -4 0] pajara variant
14&7 7.125 <0 0 7 0] ?
5&17 7.183 <1 4 -2 -6] dominant
14&12 7.199 <2 8 8 12] injera

That's in "let it all hang out" territory. Let's try 2 cent/oct.

12&31 4.724 <1 4 10 18] meantone
9&31 4.731 <7 -3 8 2] orwell
15&27 4.804 <3 0 -6 -6] augene
12&22 4.907 <2 -4 -4 -12] pajara
15&22 5.079 <3 5 -6 4] porcupine
31&15 5.199 <9 5 -3 7] valentine
12&19 5.298 <1 4 10 6] meanenneadecal
22&14 5.314 <6 10 10 8] hedgehog
14&12 5.438 <2 8 8 12] injera
31&14 5.445 <4 16 9 10] squares

> 12&19 [<1, 2, 4, 7, 6], <0, -1, -4, -10, -6]> meanenneadecal

Silver medallist from 3.5 to 4.1 cents per octave but never a winner.

> 12&17c [<1, 2, 4, 2, 1], <0, -1, -4, 2, 6]> dominant

Doesn't do that well at all. Whereas the <0, -1, -4, 2, -6] variant is a silver medallist.

> 12&26 [<2, 3, 4, 5, 6], <0, 1, 4, 4, 6]> injera

Gets to 6th place around 2.9 cent/oct.

> 12&15 [<3, 5, 7, 8, 10], <0, -1, 0, 2, 2]> augene

Gold medallist from 2.2 to 3.2 cent/oct

> 8d&22 [<2, 4, 6, 7, 8], <0, -3, -5, -5, -4]> hedgehog

That's not what I have as hedgehog from the list you gave me. Doesn't do very well.

> 7&15 [<1, 2, 3, 2, 4], <0, -3, -5, 6, -4]> porcupine

Scrapes into 4th place at 2.5 cent/oct

> 12&22 [<2, 3, 5, 6, 8], <0, 1, -2, -2, -6]> pajara

Silver medallist from 2.4 to 2.7 cent/oct

> 12&31 [<1, 2, 4, 7, 11], <0, -1, -4, -10, -18]> meantone

Gold medal at 2.0-2.1 cent/oct.

> 9&22 [<1, 0, 3, 1, 3], <0, 7, -3, 8, 2]> orwell

Gold medal from 1.1 to 1.9 cent/oct.

> 31&58 [<1, -1, 0, 1, -3], <0, 10, 9, 7, 25]> myna

Bronze from 0.5 to 0.7 cent/oct. Orwell still has the silver here.

> 10&31 [<1, 1, 3, 3, 2], <0, 6, -7, -2, 15]> miracle

Gold medal from 0.2 to 1.0 cent/oct.

> But one of the shortcomings of this method is that meanpop doesn't show > up until grade 5.

7th place at 0.8 cent/oct.

Graham

Graham Breed wrote:
> Herman Miller wrote:
>> Graham Breed wrote:
>>
>>> Some benchmarks then. The form Ek as a parameter >>> multiplying complexity to get simple badness is easiest to >>> understand. Ek is a kind of weighted error. One obvious >>> value for it is 1 cent per octave. The top 10 is then (with >>> octave equivalent wedgies)
>>>
>>> 1) meantone <1 4 10]
>>> 2) magic <5 1 12]
>>> 3) miracle <6 -7 -2]
>>> 4) orwell <7 -3 8]
>>> 5) garibaldi <1 -8 -14]
>>> 6) sensisept <7 9 13]
>>> 7) myna <10 9 7]
>>> 8) kleismatic microtemperament? <6 5 22]
>>> 9) pajara <2 -4 -4]
>>> 10) valentine? <9 5 -3]
>> Myna is a little surprising to show up on a 7-limit list, although it's >> excellent in higher limits. The <6 5 22] is catakleismic.
> > What else is there with a weighted error around 1 cent/oct?

Well, there's not much with better error than meantone in that complexity range, so if you're looking for temperaments better than meantone this makes sense. Myna has a lower complexity than I expected (around the same as miracle).

> That schismatic is in the 11-limit list you gave me. I > don't think it makes my top 3 for any parameter either. Or > top 9 for that matter. What's semififth?

One of those unfortunate names. I always have to look it up because "semififth" (or "semififths") and "hemififths" are two entirely different things.

[<1, 1, 0, 6, 2], <0, 2, 8, -11, 5]>
TOP-Max P = 1201.698520, G = 348.782195
TOP-RMS P = 1201.165244, G = 348.815068

>> This method seems to do well for the higher complexity range (where >> logflat badness includes too many complex temperaments), but it misses a >> lot of good lower complexity temperaments. The gold/silver/bronze lists >> might be better for low complexity temperaments, along with an error >> cutoff; the 11-limit gold list includes dominant, injera, augene, >> hedgehog, and porcupine. Here's a range of the gold list from complexity >> = 15 to 50:
> > IT does well for the higher complexity range because I set > it to the higher complexity range. If you raise the target > error you'll get lower complexities. Let's try 4 cent/oct
> > 12&9 6.459 <3 0 6 6] august?
> 19&7 6.763 <1 4 10 6] meanenneadecal
> 5&12 6.826 <1 4 -2 6] dominant variant
> 8&4 6.829 <4 4 4 0] dimisept?
I'd just call it diminished, I guess.

> 8&15 6.841 <3 5 9 4] porcupine variant
I've seen this called "opossum".

> 15&12 6.940 <3 0 -6 -6] augene
> 12&10 7.113 <2 -4 -4 0] pajara variant
"Pajaric". (As opposed to "pajarous", which is <2, -4, -4, 10].)

> 14&7 7.125 <0 0 7 0] ?
I have the name for this as "septimal", but it's the same mapping and TOP-Max error as jamesbond.

> 5&17 7.183 <1 4 -2 -6] dominant
> 14&12 7.199 <2 8 8 12] injera

This looks like a pretty good list for the way-low complexity range. They're not that great for the most part, but many of them are actually on the gold list.

> That's in "let it all hang out" territory. Let's try 2 > cent/oct.
> > 12&31 4.724 <1 4 10 18] meantone
> 9&31 4.731 <7 -3 8 2] orwell
> 15&27 4.804 <3 0 -6 -6] augene
> 12&22 4.907 <2 -4 -4 -12] pajara
> 15&22 5.079 <3 5 -6 4] porcupine
> 31&15 5.199 <9 5 -3 7] valentine
> 12&19 5.298 <1 4 10 6] meanenneadecal
> 22&14 5.314 <6 10 10 8] hedgehog
> 14&12 5.438 <2 8 8 12] injera
> 31&14 5.445 <4 16 9 10] squares

That's more like it. I guess the thing about having a free parameter is you can adjust it according to your preferences.

>> 8d&22 [<2, 4, 6, 7, 8], <0, -3, -5, -5, -4]> hedgehog
> > That's not what I have as hedgehog from the list you gave > me. Doesn't do very well.

Well, there may be an error on the list; none of the other 11-limit extensions of hedgehog are very good.

[<2, 4, 6, 7, 5], <0, -3, -5, -5, 7]>
[<2, 4, 6, 7, 10], <0, -3, -5, -5, -11]>

Or do you mean that I didn't label it as "8d&22"? I've also called it "8d&14c". (Most temperaments have more than one possible combination of ET's that can produce them.)

Herman Miller wrote:

> That's more like it. I guess the thing about having a free parameter is > you can adjust it according to your preferences.

Yes!

>>> 8d&22 [<2, 4, 6, 7, 8], <0, -3, -5, -5, -4]> hedgehog
>> That's not what I have as hedgehog from the list you gave >> me. Doesn't do very well.
> > Well, there may be an error on the list; none of the other 11-limit > extensions of hedgehog are very good.

Sorry, my mistake. It gets to number 7 for 2.3 cent/oct. No higher. No other hedgehogs of note.

Graham