Consistency charts for 5-limit rank 2 temperaments

I decided to write a program to display the consistency limit of a temperament as the tuning is varied (the number of steps in a chain of generators before a better approximation of one of the prime intervals is reached). The x-axis is the larger of the generators (i.e. the period, which generates the octave by itself without any contribution from the other generator) and the y-axis is the smaller generator (less than 1/2 the size of the period). I expected to see a central region of high consistency with the consistency dropping off on all sides, but instead I see that the geometry is much more complex, with most temperaments having more than one high consistency area separated by regions of lower consistency. Here are the results for some 5-limit temperaments:

/tuning-math/files/Rank%202%20Consistency/

The darker areas are higher consistency. These charts might be useful in choosing a tuning, whether you want to find one with the most consistency or the most ambiguity in a particular size range. Some of these charts also have what appears to be lines of small black dots; I believe these are probably lined up on equal temperaments (where there is no possibility of a better match beyond the size of the temperament).

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> I decided to write a program to display the consistency limit of a
> temperament as the tuning is varied (the number of steps in a chain of
> generators before a better approximation of one of the prime intervals
> is reached).

It's great you did this. I thought I'd try to settle the question, but
discovered it was more complex than I had thought it would be. It
bears further investigation, surely.

At 05:32 PM 3/9/2006, you wrote:
>I decided to write a program to display the consistency limit of a
>temperament as the tuning is varied (the number of steps in a chain of
>generators before a better approximation of one of the prime intervals
>is reached).

If you normalized this to the number of steps in the chain it
may be interesting to compare temperaments.

-Carl

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> I decided to write a program to display the consistency limit of a
> temperament as the tuning is varied (the number of steps in a chain
of
> generators before a better approximation of one of the prime
intervals
> is reached). ...
> Here are the results for some 5-limit
> temperaments:
>
> /tuning-math/files/Rank%202%
20Consistency/

Very nice!

Carl Lumma wrote:
> At 05:32 PM 3/9/2006, you wrote:
> >>I decided to write a program to display the consistency limit of a >>temperament as the tuning is varied (the number of steps in a chain of >>generators before a better approximation of one of the prime intervals >>is reached).
> > > If you normalized this to the number of steps in the chain it
> may be interesting to compare temperaments.
> > -Carl

Multiply by the number of periods in an octave? Or did you have something else in mind?

>>>I decided to write a program to display the consistency limit of
>>>a temperament as the tuning is varied (the number of steps in a
>>>chain of generators before a better approximation of one of the
>>>prime intervals is reached).
>>
>>If you normalized this to the number of steps in the chain it
>>may be interesting to compare temperaments.
>
>Multiply by the number of periods in an octave? Or did you have
>something else in mind?

I was thinking: divide the consistency limit by the graham
complexity. The tuning with the largest such score would be
representative of each temperament, with rms error or TOP
damage breaking ties.

I'm still not very comfortable with how the map can be changed
without changing the temperament. I had always advocated using
maps to define temperaments, but people said this wouldn't work
since the map can be refactored. But they then proceeded to
point out what looked like trivial refactorings involving
transpositions by the period. It looks like going over the
consistency limit means changing the temperament. If so, it
might be nice to show this in red or something on your plots.

-Carl

Carl Lumma wrote:
> I was thinking: divide the consistency limit by the graham
> complexity. The tuning with the largest such score would be
> representative of each temperament, with rms error or TOP
> damage breaking ties.

Okay, I redid the charts with this adjustment, and to make it easier to compare temperaments, the charts are all done to the same scale and centered on the TOP tuning. This way you can see which temperaments (such as luna) are more sensitive to tuning, and so on.

> I'm still not very comfortable with how the map can be changed
> without changing the temperament. I had always advocated using
> maps to define temperaments, but people said this wouldn't work
> since the map can be refactored. But they then proceeded to
> point out what looked like trivial refactorings involving
> transpositions by the period. It looks like going over the
> consistency limit means changing the temperament. If so, it
> might be nice to show this in red or something on your plots.

I tried a few different mappings of meantone for comparison (not all to the same scale).

<1, 2, 4], <0, -1, -4]
/tuning-math/files/Rank%202%20Consistency/meantone5a.png

<1, 1, 0], <0, 1, 4]
/tuning-math/files/Rank%202%20Consistency/meantone5b.png

<1, 2, 4], <1, 1, 0]
/tuning-math/files/Rank%202%20Consistency/meantone5c.png

<2, 3, 4], <1, 2, 4]
/tuning-math/files/Rank%202%20Consistency/meantone5d.png

You can see that there's some similarity in the structure, which I guess is to be expected, since they all represent the same temperament.

>> I was thinking: divide the consistency limit by the graham
>> complexity. The tuning with the largest such score would be
>> representative of each temperament, with rms error or TOP
>> damage breaking ties.
>
>Okay, I redid the charts with this adjustment, and to make it easier to
>compare temperaments, the charts are all done to the same scale and
>centered on the TOP tuning. This way you can see which temperaments
>(such as luna) are more sensitive to tuning, and so on.

Hey, interesting! I love how blackwood looks like diminished.

They also show why Dave doesn't like temperaments like bug.

Can you supply the actual max cons.lim/grahamcomp for each
temperament as a list of numbers?

-C.

Carl Lumma wrote:
> Hey, interesting! I love how blackwood looks like diminished.
> > They also show why Dave doesn't like temperaments like bug.
> > Can you supply the actual max cons.lim/grahamcomp for each
> temperament as a list of numbers?
> I'm running into difficulties trying to calculate an actual number for max consistency limit in some cases... I'll have to get back to this.

Carl Lumma wrote:

> Hey, interesting! I love how blackwood looks like diminished.
> > They also show why Dave doesn't like temperaments like bug.
> > Can you supply the actual max cons.lim/grahamcomp for each
> temperament as a list of numbers?

I'm beginning to have some doubts about the usefulness of max consistency limit after getting some odd results from trying to approximate the values for certain temperaments. Take a look at this chart, which is zoomed way in on the helmholtz temperament. It looks about as complex as the rings of Saturn...

/tuning-math/files/Rank%202%20Consistency/helmholtz5-closeup.png

The chart shows narrow bands of super high consistency separated by gaps of lower consistency. Tiny adjustments to the size of the generators make large differences in the overall consistency of the tuning. There might be a maximum in all this (somewhere in the thousands), but the tuning accuracy required may not be worth the effort to find it.

With that in mind, I tried to compensate for this effect by checking the complexity measurement at +/- 0.01 cents in both generators, and only considering the cases where all 5 measures agree. This seems somewhat arbitrary, and you could probably get different results with different methods. So I've included the period and generator sizes in this list, in case a better one comes up later on.

Still, I'm not sure that all of these are quite what I was looking for. I was suspicious of the high score of dicot, so I checked it out and found that it was that high because it was close to a stretched 45-ET. It seems the closer you get to an ET, the more the consistency limit approaches that ET. At any rate, it might be worth trying out some of these tunings and seeing if they make any musical sense.

Temperament: father <1, 2, 2], <0, -1, 1]
per = 1180.47 gen = 437.23 limit = 26 lim/c = 13.00

Temperament: bug <1, 2, 3], <0, -2, -3]
per = 1189.62 gen = 250.46 limit = 17 lim/c = 5.67

Temperament: dicot <1, 1, 2], <0, 2, 1]
per = 1211.50 gen = 349.97 limit = 44 lim/c = 22.00

Temperament: meantone <1, 2, 4], <0, -1, -4]
per = 1202.39 gen = 505.27 limit = 187 lim/c = 46.75

Temperament: augmented <3, 5, 7], <0, -1, 0]
per = 398.67 gen = 92.69 limit = 129 lim/c = 43.00

Temperament: mavila <1, 2, 1], <0, -1, 3]
per = 1208.44 gen = 523.07 limit = 66 lim/c = 16.50

Temperament: porcupine <1, 2, 3], <0, -3, -5]
per = 1196.98 gen = 162.08 limit = 93 lim/c = 18.60

Temperament: blackwood <5, 8, 11], <0, 0, 1]
per = 238.66 gen = 164.05 limit = 80 lim/c = 16.00

Temperament: diminished <4, 6, 9], <0, 1, 1]
per = 298.75 gen = 103.82 limit = 88 lim/c = 22.00

Temperament: srutal <2, 3, 5], <0, 1, -2]
per = 599.30 gen = 106.02 limit = 260 lim/c = 43.33

Temperament: magic <1, 0, 2], <0, 5, 1]
per = 1201.98 gen = 379.89 limit = 188 lim/c = 37.60

Temperament: ripple <1, 2, 3], <0, -5, -8]
per = 1201.12 gen = 101.31 limit = 78 lim/c = 9.75

Temperament: hanson <1, 0, 1], <0, 6, 5]
per = 1201.53 gen = 316.63 limit = 148 lim/c = 24.67

Temperament: negri <1, 2, 2], <0, -4, 3]
per = 1202.50 gen = 126.30 limit = 215 lim/c = 30.71

Temperament: tetracot <1, 1, 1], <0, 4, 9]
per = 1198.22 gen = 176.65 limit = 156 lim/c = 17.33

Temperament: superpyth <1, 2, 6], <0, -1, -9]
per = 1196.61 gen = 487.78 limit = 148 lim/c = 16.44

Temperament: helmholtz <1, 2, -1], <0, -1, 8]
per = 1199.97 gen = 497.96 limit = 155 lim/c = 17.22

Temperament: sensi <1, -1, -1], <0, 7, 9]
per = 1200.00 gen = 442.64 limit = 129 lim/c = 14.33

Temperament: passion <1, 2, 2], <0, -5, 4]
per = 1197.81 gen = 98.27 limit = 190 lim/c = 21.11

Temperament: wuerschmidt <1, -1, 2], <0, 8, 1]
per = 1198.18 gen = 387.84 limit = 172 lim/c = 21.50

Temperament: compton <12, 19, 28], <0, 0, -1]
per = 100.06 gen = 16.21 limit = 444 lim/c = 37.00

Temperament: amity <1, 3, 6], <0, -5, -13]
per = 1198.43 gen = 339.04 limit = 139 lim/c = 10.69

Temperament: orson <1, 0, 3], <0, 7, -3]
per = 1201.80 gen = 271.93 limit = 134 lim/c = 13.40

Temperament: vishnu <2, 4, 5], <0, -7, -3]
per = 600.77 gen = 71.90 limit = 220 lim/c = 15.71

Temperament: luna <1, 4, 2], <0, -15, 2]
per = 1198.76 gen = 193.01 limit = 116 lim/c = 6.82

>> Can you supply the actual max cons.lim/grahamcomp for each
>> temperament as a list of numbers?
>
>I'm beginning to have some doubts about the usefulness of max
>consistency limit after getting some odd results from trying to
>approximate the values for certain temperaments. Take a look at this
>chart, which is zoomed way in on the helmholtz temperament. It looks
>about as complex as the rings of Saturn...
>The chart shows narrow bands of super high consistency separated by gaps
>of lower consistency. Tiny adjustments to the size of the generators
>make large differences in the overall consistency of the tuning.
>With that in mind, I tried to compensate for this effect by checking the
>complexity measurement at +/- 0.01 cents in both generators, and only
>considering the cases where all 5 measures agree. This seems somewhat
>arbitrary, and you could probably get different results with different
>methods. So I've included the period and generator sizes in this list,
>in case a better one comes up later on.

Thanks for trying! I hadn't expected this... is it a problem even
for simpler temperaments like meantone?

>Still, I'm not sure that all of these are quite what I was looking for.
>I was suspicious of the high score of dicot, so I checked it out and
>found that it was that high because it was close to a stretched 45-ET.
>It seems the closer you get to an ET, the more the consistency limit
>approaches that ET. At any rate, it might be worth trying out some of
>these tunings and seeing if they make any musical sense.

Below I've sorted it by lim/c. It's kinda interesting... it
penalizes too-simple temperaments like bug, but meantone, still
a fairly simple temperament, tops the list. Still, anything
so sensitive to tuning can't be good.

>Temperament: bug <1, 2, 3], <0, -2, -3]
>per = 1189.62 gen = 250.46 limit = 17 lim/c = 5.67
>
>Temperament: luna <1, 4, 2], <0, -15, 2]
>per = 1198.76 gen = 193.01 limit = 116 lim/c = 6.82
>
>Temperament: ripple <1, 2, 3], <0, -5, -8]
>per = 1201.12 gen = 101.31 limit = 78 lim/c = 9.75
>
>Temperament: amity <1, 3, 6], <0, -5, -13]
>per = 1198.43 gen = 339.04 limit = 139 lim/c = 10.69
>
>Temperament: father <1, 2, 2], <0, -1, 1]
>per = 1180.47 gen = 437.23 limit = 26 lim/c = 13.00
>
>Temperament: orson <1, 0, 3], <0, 7, -3]
>per = 1201.80 gen = 271.93 limit = 134 lim/c = 13.40
>
>Temperament: sensi <1, -1, -1], <0, 7, 9]
>per = 1200.00 gen = 442.64 limit = 129 lim/c = 14.33
>
>Temperament: vishnu <2, 4, 5], <0, -7, -3]
>per = 600.77 gen = 71.90 limit = 220 lim/c = 15.71
>
>Temperament: blackwood <5, 8, 11], <0, 0, 1]
>per = 238.66 gen = 164.05 limit = 80 lim/c = 16.00
>
>Temperament: superpyth <1, 2, 6], <0, -1, -9]
>per = 1196.61 gen = 487.78 limit = 148 lim/c = 16.44
>
>Temperament: mavila <1, 2, 1], <0, -1, 3]
>per = 1208.44 gen = 523.07 limit = 66 lim/c = 16.50
>
>Temperament: helmholtz <1, 2, -1], <0, -1, 8]
>per = 1199.97 gen = 497.96 limit = 155 lim/c = 17.22
>
>Temperament: tetracot <1, 1, 1], <0, 4, 9]
>per = 1198.22 gen = 176.65 limit = 156 lim/c = 17.33

>Temperament: porcupine <1, 2, 3], <0, -3, -5]
>per = 1196.98 gen = 162.08 limit = 93 lim/c = 18.60
>
>Temperament: passion <1, 2, 2], <0, -5, 4]
>per = 1197.81 gen = 98.27 limit = 190 lim/c = 21.11
>
>Temperament: wuerschmidt <1, -1, 2], <0, 8, 1]
>per = 1198.18 gen = 387.84 limit = 172 lim/c = 21.50
>
>Temperament: diminished <4, 6, 9], <0, 1, 1]
>per = 298.75 gen = 103.82 limit = 88 lim/c = 22.00
>
>Temperament: dicot <1, 1, 2], <0, 2, 1]
>per = 1211.50 gen = 349.97 limit = 44 lim/c = 22.00
>
>Temperament: hanson <1, 0, 1], <0, 6, 5]
>per = 1201.53 gen = 316.63 limit = 148 lim/c = 24.67
>
>Temperament: negri <1, 2, 2], <0, -4, 3]
>per = 1202.50 gen = 126.30 limit = 215 lim/c = 30.71
>
>Temperament: compton <12, 19, 28], <0, 0, -1]
>per = 100.06 gen = 16.21 limit = 444 lim/c = 37.00
>
>Temperament: magic <1, 0, 2], <0, 5, 1]
>per = 1201.98 gen = 379.89 limit = 188 lim/c = 37.60
>
>Temperament: augmented <3, 5, 7], <0, -1, 0]
>per = 398.67 gen = 92.69 limit = 129 lim/c = 43.00
>
>Temperament: srutal <2, 3, 5], <0, 1, -2]
>per = 599.30 gen = 106.02 limit = 260 lim/c = 43.33
>
>Temperament: meantone <1, 2, 4], <0, -1, -4]
>per = 1202.39 gen = 505.27 limit = 187 lim/c = 46.75

-Carl

Have the following temperaments ever been called anything
else?

dicot, passion, helmholtz, orson, ripple, luna

?

-Carl

Carl Lumma wrote:

> Thanks for trying! I hadn't expected this... is it a problem even
> for simpler temperaments like meantone?

Meantone has narrow bands, but not nearly as complex as those of helmholtz. I added a meantone5-closeup chart showing the area around (1202, 505), which seems to be one of the more complex areas of the chart. It definitely seems to be the more complex temperaments that have issues with super-narrow bands.

Carl Lumma wrote:
> Have the following temperaments ever been called anything
> else?
> > dicot, passion, helmholtz, orson, ripple, luna
> > ?

Passion and ripple are/were also known as "subchrome" and "superchrome", among others. Helmholtz is 5-limit schismatic (or schismic), and "orson" is the 5-limit version of orwell. I don't know about luna.

>> Have the following temperaments ever been called anything
>> else?
>>
>> dicot, passion, helmholtz, orson, ripple, luna
>>
>> ?
>
>Passion and ripple are/were also known as "subchrome" and "superchrome",
>among others. Helmholtz is 5-limit schismatic (or schismic), and "orson"
>is the 5-limit version of orwell. I don't know about luna.

Thanks! -C.

Ps-
/metatuning/topicId_10480.html#10482

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> Have the following temperaments ever been called anything
> else?
>
> dicot, passion, helmholtz, orson, ripple, luna

Of course. Helmholtz even has a Wikipedia article under the name
schismatic.