Graham's top 20, with standard vals

This is the same list as before, only this time I included all of the standard vals consistent with the temperament. The good news is this:

(1) 18 out of the 20 systems can be defined by wedging two standard vals

(2) In no case is it neccessary to use vals where the number of divisions of the octave of one is more than twice that of another

(3) The vals could have been filtered. Here are the ets from 30 to 130, with badness less than 1.6:

[31, 36, 37, 41, 43, 44, 46, 50, 53, 56, 58, 62, 63, 68, 70, 72, 77, 79, 80, 87, 94, 103, 111, 113, 118, 121, 130]

We see that this would have more than sufficed to get our 18.

The bad news is we missed two; one had the biggest error and second lowest complexity on the list, and the other had a very small period of 1/36 octave. A separate system for dealing with small periods may be required. The other bad news is that we are skating on thin ice, as many of the temperaments had only the minimum two standard vals.

Mystery
[0, 29, 29, 29, 29, 46, 46, 46, 46, -14, -33, -40, -19, -26, -7]
[[29, 46, 67, 81, 100, 107], [0, 0, 1, 1, 1, 1]]
[41.3793103448276, 15.8257880102396]
[29, 58, 87, 145, 232]
unweighted badness 242.527516 unweighted complexity 22.463303 rms error 2.277984

Hemififth
[2, 25, 13, 5, -1, 35, 15, 1, -9, -40, -75, -95, -31, -51, -22]
[[1, 1, -5, -1, 2, 4], [0, 2, 25, 13, 5, -1]]
[1200., 351.617195107835]
[41, 58]
unweighted badness 203.445092 unweighted complexity 13.364131 rms error 4.164244

Unidec
[12, 22, -4, -6, 4, 7, -40, -51, -38, -71, -90, -72, -3, 26, 36]
[[2, 5, 8, 5, 6, 8], [0, -6, -11, 2, 3, -2]]
[600., 183.225219270142]
[26, 46, 72]
unweighted badness 229.903139 unweighted complexity 17.401149 rms error 3.167218

Diaschismic
[2, -4, -16, -24, -30, -11, -31, -45, -55, -26, -42, -55, -12, -25, -15]
[[2, 3, 5, 7, 9, 10], [0, 1, -2, -8, -12, -15]]
[600., 103.786589235317]
[46, 58]
unweighted badness 251.546815 unweighted complexity 19.682480 rms error 2.880707

Biminortonic
[14, 30, 4, 6, 22, 15, -33, -39, -17, -75, -90, -60, 3, 47, 54]
[[2, 1, 0, 5, 6, 4], [0, 7, 15, 2, 3, 11]]
[600., 185.994500197218]
[26, 58, 84, 110]
unweighted badness 285.227918 unweighted complexity 17.175564 rms error 4.007057

Schismatic
[1, -8, -14, 23, 20, -15, -25, 33, 28, -10, 81, 76, 113, 108, -16]
[[1, 2, -1, -3, 13, 12], [0, -1, 8, 14, -23, -20]]
[1200., 497.872301871492]
[41, 53, 94, 135]
unweighted badness 245.881799 unweighted complexity 19.724350 rms error 2.806870

Nonkleismic
[10, 9, 7, 25, -5, -9, -17, 5, -45, -9, 27, -45, 46, -40, -110]
[[1, -1, 0, 1, -3, 5], [0, 10, 9, 7, 25, -5]]
[1200., 310.312830917046]
[31, 58]
unweighted badness 254.415991 unweighted complexity 15.006665 rms error 4.376411

Acute
[4, 21, -3, 39, 27, 24, -16, 48, 28, -66, 18, -15, 120, 87, -51]
[[1, 0, -6, 4, -12, -7], [0, 4, 21, -3, 39, 27]]
[1200., 475.694618357624]
[53, 58, 111]
unweighted badness 241.523384 unweighted complexity 22.614155 rms error 2.245891

Amity
[5, 13, -17, 9, -6, 9, -41, -3, -28, -76, -24, -62, 84, 46, -54]
[[1, 3, 6, -2, 6, 2], [0, -5, -13, 17, -9, 6]]
[1200., 339.412784647410]
[7, 39, 46, 53, 99]
unweighted badness 268.549368 unweighted complexity 15.295424 rms error 4.489333

Subminorsixth
[6, -48, 10, -50, 26, -90, -1, -100, 19, 158, 50, 238, -175, 36, 275]
[[2, 4, -2, 7, 0, 11], [0, -3, 24, -5, 25, -13]]
[600., 166.081391969544]
[36, 94, 130, 224, 318, 354, 542, 578]
unweighted badness 207.849710 unweighted complexity 43.788126 rms error .717323

Minorsemi
[18, 15, -6, 9, 42, -18, -60, -48, 0, -56, -31, 42, 46, 140, 112]
[[3, 6, 8, 8, 11, 14], [0, -6, -5, 2, -3, -14]]
[400., 83.0061237758442]
[15, 72, 87, 159]
unweighted badness 231.510127 unweighted complexity 25.260641 rms error 1.823490

Tricontaheximal
[0, 36, 0, 36, 0, 57, 0, 57, 0, -101, -41, -133, 101, 0, -133]
[[36, 57, 83, 101, 124, 133], [0, 0, 1, 0, 1, 0]]
[33.3333333333333, 15.7721139478765]
[72]
unweighted badness 416.491693 unweighted complexity 25.455844 rms error 3.242837

Spearmint
[2, -4, 30, 22, 16, -11, 42, 28, 18, 81, 65, 52, -42, -66, -26]
[[2, 3, 5, 3, 5, 6], [0, 1, -2, 15, 11, 8]]
[600., 104.895179196541]
[46, 80, 126]
unweighted badness 288.935571 unweighted complexity 18.477013 rms error 3.637922

Supersupermajor
[3, 17, -1, -13, -22, 20, -10, -31, -46, -50, -89, -114, -33, -58, -28]
[[1, 1, -1, 3, 6, 8], [0, 3, 17, -1, -13, -22]]
[1200., 234.480133729376]
[41, 46, 87, 128, 133]
unweighted badness 219.395264 unweighted complexity 18.448577 rms error 2.768745

Suprasubminorsixth
[6, 46, 10, 44, 26, 59, -1, 49, 19, -106, -57, -110, 89, 36, -73]
[[2, 4, 11, 7, 13, 11], [0, -3, -23, -5, -22, -13]]
[600., 165.810532415714]
[58, 94]
unweighted badness 277.161590 unweighted complexity 26.724521 rms error 2.006172

Paraorwell
[11, -6, 10, 7, 15, -35, -15, -27, -17, 40, 37, 57, -15, 5, 26]
[[1, 4, 1, 5, 5, 7], [0, -11, 6, -10, -7, -15]]
[1200., 263.695224647719]
[9, 41, 50, 91]
unweighted badness 271.488360 unweighted complexity 13.234425 rms error 5.638890

[3, 29, 11, 16, 7, 39, 9, 15, 0, -56, -63, -91, 7, -21, -35]
[[1, 3, 16, 8, 11, 7], [0, -3, -29, -11, -16, -7]]
[1200., 565.936121918345]
[53, 70]
unweighted badness 282.926501 unweighted complexity 14.126217 rms error 5.328866

Miracle
[6, -7, -2, 15, 38, -25, -20, 3, 38, 15, 59, 114, 49, 114, 76]
[[1, 1, 3, 3, 2, 0], [0, 6, -7, -2, 15, 38]]
[1200., 116.779511703154]
[41, 72, 113]
unweighted badness 224.267777 unweighted complexity 21.718656 rms error 2.215733

[4, 9, 26, 10, -2, 5, 30, 2, -18, 35, -8, -38, -62, -102, -44]
[[1, 1, 1, -1, 2, 4], [0, 4, 9, 26, 10, -2]]
[1200., 175.873678872191]
[41]
unweighted badness 280.118702 unweighted complexity 13.315405 rms error 5.765149

[4, -37, -3, -19, -31, -68, -16, -44, -64, 97, 84, 65, -43, -76, -37]
[[1, 0, 17, 4, 11, 16], [0, 4, -37, -3, -19, -31]]
[1200., 476.008069832587]
[58, 63, 121, 184]
unweighted badness 298.019307 unweighted complexity 25.845696 rms error 2.268099

Gene Ward Smith wrote:

>This is the same list as before, only this time I included all of the standard vals consistent with the temperament. The good news is this:
>
>(1) 18 out of the 20 systems can be defined by wedging two standard val
>
So these have to be consistent?

>(2) In no case is it neccessary to use vals where the number of divisions of the octave of one is more than twice that of another
>
Now that is interesting..

>(3) The vals could have been filtered. Here are the ets from 30 to 130, with badness less than 1.6:
>
>[31, 36, 37, 41, 43, 44, 46, 50, 53, 56, 58, 62, 63, 68, 70, 72, 77, 79, 80, 87, 94, 103, 111, 113, 118, 121, 130]
>
>We see that this would have more than sufficed to get our 18
>
I misread that, and generated 27 ETs with a cutoff of 0.6 scale step errors.
[17, 26, 27, 29, 31, 34, 36, 41, 43, 46, 50, 53, 58, 60, 62, 63, 65, 72, 77, 80, 87, 94, 103, 104, 111, 113, 121]

They give me all 20 temperaments (it took 5 seconds). And also one more, h26&h104, with a generator of 9.8 and a period of 1/26 octaves. Generator mapping [1 2 0 0 1]. It breaks your rule because 26*2<104.

I'll have missed it before because I wasn't going high enough to get 104-equal. If I'd taken 104 ETs instead of 100 I would have got it. It takes a whole minute to search through 200 ETs.

>The bad news is we missed two; one had the biggest error and second lowest complexity on the list, and the other had a very small period of 1/36 octave. A separate system for dealing with small periods may be required. The other bad news is that we are skating on thin ice, as many of the temperaments had only the minimum two standard vals.
> >
Was that one of them? The error's 2.5 cents (minimax) and the complexity's 52. So it doesn't sound like either. I can't find another one that was missing.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
>
> >This is the same list as before, only this time I included all of the standard vals consistent with the temperament. The good news is this:
> >
> >(1) 18 out of the 20 systems can be defined by wedging two standard val
> >
> So these have to be consistent?

Vals are consistent by definition. What I meant was, wedging the val with the wedgie will give a zero vector.

> >The bad news is we missed two; one had the biggest error and second lowest complexity on the list, and the other had a very small period of 1/36 octave. A separate system for dealing with small periods may be required. The other bad news is that we are skating on thin ice, as many of the temperaments had only the minimum two standard vals.
> >
> >
> Was that one of them? The error's 2.5 cents (minimax) and the
> complexity's 52. So it doesn't sound like either. I can't find another
> one that was missing.

The missing ones are the two on the list with only one standard val; you may not be using only standard vals. I wish you would explain this. By "standard" I mean the mapping is determined by rounding
log2(p) for prime p to the nearest integer.

Gene Ward Smith wrote:

> Vals are consistent by definition. What I meant was, wedging the val with the wedgie will give a zero vector.

Say what? I thought a val was the complement of a vector.

> The missing ones are the two on the list with only one standard val; you may not be using only standard vals. I wish you would explain this. By "standard" I mean the mapping is determined by rounding
> log2(p) for prime p to the nearest integer.

I did explain this -- I'm not only taking nearest prime approximations any more.

Tricontaheximal is h72&h36. I don't see why you missed that. The other one, where you only give 41, requires an alternative mapping of 34-equal. This is actually more accurate than the nearest-prime mapping, and so the only one I use in the search.

[34, 54, 79, 96, 118, 126]

I didn't realize alternative mappings were getting in with such a low cutoff. g104 is another one, so here's the temperament that the unison vectors don't seem to cover:

1/5, 9.8 cent generator

basis:
(0.038461538461538464, 0.0081713150481090846)

mapping by period and generator:
[(26, 0), (41, 1), (60, 2), (73, 0), (90, 0), (96, 1)]

mapping by steps:
[(104, 26), (165, 41), (242, 60), (292, 73), (360, 90), (385, 96)]

highest interval width: 2
complexity measure: 52 (78 for smallest MOS)
highest error: 0.002107 (2.528 cents)
unique

I've updated the scripts at http://x31eq.com/temper/ to use alternative mappings now.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <graham@m...> wrote:
> Gene Ward Smith wrote:
>
> > Vals are consistent by definition. What I meant was, wedging the val with the wedgie will give a zero vector.
>
> Say what? I thought a val was the complement of a vector.

It's the dual of a vector, if by a vector you mean an interval in Monzo notation. Due to the magic of Poincare duality, you can wedge with either a vector or a val.

> > The missing ones are the two on the list with only one standard val; you may not be using only standard vals. I wish you would explain this. By "standard" I mean the mapping is determined by rounding
> > log2(p) for prime p to the nearest integer.
>
> I did explain this -- I'm not only taking nearest prime approximations
> any more.

So what are you taking?

> Tricontaheximal is h72&h36. I don't see why you missed that.

Because I was using a program which gave me a val from a wedgie plus a number, and for 36 it gave me [36, 57, 83, 101, 124, 133], which also works. The right way to do it would be to wedge with the standard val and see if you get a zero vector.

>>Say what? I thought a val was the complement of a vector.
>
>It's the dual of a vector, if by a vector you mean an interval
>in Monzo notation. Due to the magic of Poincare duality, you
>can wedge with either a vector or a val.

I may be dangerously close to understanding vals. I've read
the definition in monz's dictionary. Anybody care to give
an example? Maybe Paul could shed some light on a layman's
definition.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >>Say what? I thought a val was the complement of a vector.
> >
> >It's the dual of a vector, if by a vector you mean an interval
> >in Monzo notation. Due to the magic of Poincare duality, you
> >can wedge with either a vector or a val.
>
> I may be dangerously close to understanding vals. I've read
> the definition in monz's dictionary. Anybody care to give
> an example?

the 5-limit standard val for 12 is [12, 19, 28] since that's how many
semitones are in a 2:1, a 3:1, and a 5:1. a non-standard val for 12
would be [12, 19, 27] since you'd be using 27 semitones, instead of
28 semitones, to approximate 5:1.

> Maybe Paul could shed some light on a layman's
> definition.
>
> -Carl

>the 5-limit standard val for 12 is [12, 19, 28] since that's how
>many semitones are in a 2:1, a 3:1, and a 5:1.

Ah, ok. THANKS PAUL.

>a non-standard val for 12 would be [12, 19, 27] since you'd be
>using 27 semitones, instead of 28 semitones, to approximate 5:1.

What's a non-standard val? Why?

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >the 5-limit standard val for 12 is [12, 19, 28] since that's how
> >many semitones are in a 2:1, a 3:1, and a 5:1.
>
> Ah, ok. THANKS PAUL.
>
> >a non-standard val for 12 would be [12, 19, 27] since you'd be
> >using 27 semitones, instead of 28 semitones, to approximate 5:1.
>
> What's a non-standard val? Why?
>
> -Carl

gene defined "standard val" (not something i'd be particularly
interested in) as where each prime is mapped to its best
approximation in N-equal -- in this case, 12-equal.

> gene defined "standard val" (not something i'd be particularly
> interested in) as where each prime is mapped to its best
> approximation in N-equal -- in this case, 12-equal.

??? Could you explain your reasoning here?

-C.

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> > gene defined "standard val" (not something i'd be particularly
> > interested in) as where each prime is mapped to its best
> > approximation in N-equal -- in this case, 12-equal.
>
> ??? Could you explain your reasoning here?
>
> -C.

my reasoning in not being particularly interested in this? it's that,
like graham, i don't think the standard val is necessarily the best
val for any ET. try 64-equal in the 5-limit.

>my reasoning in not being particularly interested in this?
>it's that, like graham, i don't think the standard val is
>necessarily the best val for any ET. try 64-equal in the
>5-limit.

This wouldn't happen in a linear temperament, though, right?

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>"
<clumma@y...> wrote:
> >my reasoning in not being particularly interested in this?
> >it's that, like graham, i don't think the standard val is
> >necessarily the best val for any ET. try 64-equal in the
> >5-limit.
>
> This wouldn't happen in a linear temperament, though, right?

i don't know what you mean. you need two vals to define a linear
temperament, just like you need two commas to define a temperament
with "codimension 2" (number of primes minus dimensionality of tuning
equals two).

--- In tuning-math@yahoogroups.com, "Carl Lumma <clumma@y...>" <clumma@y...> wrote:

> > gene defined "standard val" (not something i'd be particularly
> > interested in) as where each prime is mapped to its best
> > approximation in N-equal -- in this case, 12-equal.
>
> ??? Could you explain your reasoning here?

My reasoning is that the defintion is straightforward, and in effect other people use it.

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:

> my reasoning in not being particularly interested in this? it's that,
> like graham, i don't think the standard val is necessarily the best
> val for any ET. try 64-equal in the 5-limit.

Do you have an alternative definition?

--- In tuning-math@yahoogroups.com, "Gene Ward Smith
<genewardsmith@j...>" <genewardsmith@j...> wrote:
> --- In tuning-math@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
>
> > my reasoning in not being particularly interested in this? it's
that,
> > like graham, i don't think the standard val is necessarily the
best
> > val for any ET. try 64-equal in the 5-limit.
>
> Do you have an alternative definition?

how about the choice that minimizes the rms error?