start starting getting serious

I think it's time I started to start getting serious about
composing some microtonal music.

19 (kleismic, negri, 5-limit meantone)
22 (pajara, porcupine)
26 (injera)
31 (7-limit meantone)
41 (schismic, magic)

What am I missing? I'm trying to think of the simplest
9-limit linear temperaments with less Tenney-weighted error
than 12-equal has in the 5-limit, and list the smallest ETs
that support them without pushing the error above the mark.

-Carl

Carl Lumma wrote:
> I think it's time I started to start getting serious about
> composing some microtonal music.
> > 19 (kleismic, negri, 5-limit meantone)
> 22 (pajara, porcupine)
> 26 (injera)
> 31 (7-limit meantone)
> 41 (schismic, magic)
> > What am I missing? I'm trying to think of the simplest
> 9-limit linear temperaments with less Tenney-weighted error
> than 12-equal has in the 5-limit, and list the smallest ETs
> that support them without pushing the error above the mark.

Taking your criteria literally, here's a list:

12 10 2.985 2.572 (2, -4, -4)
5 14 3.445 2.755 (2, 8, 1)
12 19 3.562 1.382 (1, 4, 10)
4 15 3.786 2.583 (6, 5, 3)
10 9 3.816 2.575 (4, -3, 2)
15 12 4.030 2.228 (3, 0, -6)
19 22 4.274 1.074 (5, 1, 12)
15 22 4.291 2.531 (3, 5, -6)
8 14 4.307 2.780 (6, 10, 10)
5 22 4.589 1.917 (1, 9, -2)
19 8 4.631 1.323 (7, 9, 13)
19 26 4.929 2.117 (1, 4, -9)
4 22 5.047 2.483 (8, 6, 6)
27 10 5.138 2.301 (2, -9, -4)
26 5 5.524 1.425 (3, 12, -1)
5 29 5.599 2.853 (2, 13, 1)
29 12 5.618 0.726 (1, -8, -14)

The first two columns are the numbers of notes in the generating ETs. Then the optimal RMS complexity and the Tenney weighted RMS in cents/octave. Then the octave-equivalent mapping multiplied by the number of periods to an octave, which serves to uniquely identify the temperament.

So you have pajara, bug?, meantone, kleismic, negri, augmented, magic, porcupine, a double porcupine, a half orwell, tiny diesic, a meantone variant, 4&22?, 10&27?, a triple meantone, another half-negri, and schismic. There may still be gaps. I used the default seed ETs.

Injera, as in my catalog, has a slightly higher Tenney error than 5-limit 12-equal (3.138 against 3.106) so it doesn't make the list. It all depends on how you define the error. But that bug thing is similar, and the average complexity comes out almost identical.

Taking your question more vaguely, try miracle and orwell.

Graham

>> I think it's time I started to start getting serious about
>> composing some microtonal music.
>>
>> 19 (kleismic, negri, 5-limit meantone)
>> 22 (pajara, porcupine)
>> 26 (injera)
>> 31 (7-limit meantone)
>> 41 (schismic, magic)
>>
>> What am I missing? I'm trying to think of the simplest
>> 9-limit linear temperaments with less Tenney-weighted error
>> than 12-equal has in the 5-limit, and list the smallest ETs
>> that support them without pushing the error above the mark.
>
>Taking your criteria literally, here's a list:
>
>12 10 2.985 2.572 (2, -4, -4)
> 5 14 3.445 2.755 (2, 8, 1)
>12 19 3.562 1.382 (1, 4, 10)
> 4 15 3.786 2.583 (6, 5, 3)
>10 9 3.816 2.575 (4, -3, 2)
>15 12 4.030 2.228 (3, 0, -6)
>19 22 4.274 1.074 (5, 1, 12)
>15 22 4.291 2.531 (3, 5, -6)
> 8 14 4.307 2.780 (6, 10, 10)
> 5 22 4.589 1.917 (1, 9, -2)
>19 8 4.631 1.323 (7, 9, 13)
>19 26 4.929 2.117 (1, 4, -9)
> 4 22 5.047 2.483 (8, 6, 6)
>27 10 5.138 2.301 (2, -9, -4)
>26 5 5.524 1.425 (3, 12, -1)
> 5 29 5.599 2.853 (2, 13, 1)
>29 12 5.618 0.726 (1, -8, -14)
>
>The first two columns are the numbers of notes in the generating ETs.
>Then the optimal RMS complexity and the Tenney weighted RMS in
>cents/octave. Then the octave-equivalent mapping multiplied by the
>number of periods to an octave, which serves to uniquely identify the
>temperament.
>
>So you have pajara, bug?, meantone, kleismic, negri, augmented, magic,
>porcupine, a double porcupine, a half orwell, tiny diesic, a meantone
>variant, 4&22?, 10&27?, a triple meantone, another half-negri, and
>schismic. There may still be gaps. I used the default seed ETs.
>
>Injera, as in my catalog, has a slightly higher Tenney error than
>5-limit 12-equal (3.138 against 3.106) so it doesn't make the list.
>It all depends on how you define the error. But that bug thing is
>similar, and the average complexity comes out almost identical.

Thanks Graham!

If you want to play along, can we make the Tenney error cutoff
an even 3.2? And can you include period and generator? And
the smallest ET tuning that doesn't make the error > 3.2? And, uh,
what's RMS complexity?

I thought bug was really high error... hrm.

>Taking your question more vaguely, try miracle and orwell.

Can you include them on the above list? I seem to remember
wondering why orwell is supposed to be so good.

-Carl

Carl Lumma wrote:

> If you want to play along, can we make the Tenney error cutoff
> an even 3.2? And can you include period and generator? And
> the smallest ET tuning that doesn't make the error > 3.2? And, uh,
> what's RMS complexity?

I'm going for a walk now. I'll try and sort something out this evening, so it'll arrive for your morning.

> I thought bug was really high error... hrm.

I'm looking at a no-fives temperament like this and I think it's something to do with bug. But I may be wrong. A half-negri, anyway.

>>Taking your question more vaguely, try miracle and orwell.
> > Can you include them on the above list? I seem to remember
> wondering why orwell is supposed to be so good.

That's easy! I cut them off the list to save space. Orwell isn't that good, but it's good enough to be worth looking at and in the same ball park as schismic. 9&22.

12 10 2.985 2.572 (2, -4, -4)
5 14 3.445 2.755 (2, 8, 1)
12 19 3.562 1.382 (1, 4, 10)
4 15 3.786 2.583 (6, 5, 3)
10 9 3.816 2.575 (4, -3, 2)
15 12 4.030 2.228 (3, 0, -6)
19 22 4.274 1.074 (5, 1, 12)
15 22 4.291 2.531 (3, 5, -6)
8 14 4.307 2.780 (6, 10, 10)
5 22 4.589 1.917 (1, 9, -2)
19 8 4.631 1.323 (7, 9, 13)
19 26 4.929 2.117 (1, 4, -9)
4 22 5.047 2.483 (8, 6, 6)
27 10 5.138 2.301 (2, -9, -4)
26 5 5.524 1.425 (3, 12, -1)
5 29 5.599 2.853 (2, 13, 1)
29 12 5.618 0.726 (1, -8, -14)
9 27 5.678 2.156 (9, 0, 9)
22 29 5.699 2.357 (3, 5, 16)
9 22 5.709 0.748 (7, -3, 8)
22 39 6.230 1.929 (1, -13, -2)
4 27 6.309 0.952 (10, 9, 7)
26 8 6.309 2.448 (10, 14, 14)
31 34 6.412 1.187 (8, 1, 18)
49 12 6.717 1.862 (5, -4, -10)
34 27 6.726 1.890 (4, 9, -8)
15 26 6.747 1.100 (9, 10, -3)
31 15 6.747 0.858 (9, 5, -3)
8 29 6.768 2.599 (9, 15, 19)
31 10 6.800 0.515 (6, -7, -2)
31 14 6.891 1.393 (4, 16, 9)
8 31 6.940 1.244 (11, 13, 17)
19 56 6.940 1.622 (11, 6, 15)
12 46 6.961 1.009 (2, -4, -16)

Graham

Carl Lumma wrote:

> If you want to play along, can we make the Tenney error cutoff
> an even 3.2? And can you include period and generator? And
> the smallest ET tuning that doesn't make the error > 3.2? And, uh,
> what's RMS complexity?

Raising the cutoff doesn't add much. I'll put a full printout at the end of the message. If you want to play along at home, the code is:

>>> import regular
>>> def complexityBadness(lt, width, error):
... return width
...
>>> ets = regular.getEqualTemperaments(regular.limit7)
>>> regular.getLinearTemperaments(ets, 7, 3.2/1200, badness=complexityBadness)

Your simplest nice ET looks like 19:

>>> def err(et):
... return regular.optimalRMSError(et.weightedPrimes())
...
>>> maxerr = err(regular.Temperament(regular.limit5, 12))
>>> [et[0] for et in ets if err(et) <= maxerr][0]
19

I can guess what RMS complexity would be, but it's not what I use. The complexity here is the max-min Tenney weighted number of steps per prime interval, which we believe to be the same as the maximum Kees complexity. There's a measure that goes better with RMS error: the standard devation instead of max-min.

> I thought bug was really high error... hrm.

Looks like it's the same interval pattern but a different 5-mapping.

Here's that listing:

[2/11

598.859 cents period
106.844 cents generator

mapping by period and generator:
[2, 3, 5, 6]
[0, 1, -2, -2]

mapping by steps:
(12, 19, 28, 34)
(10, 16, 23, 28)

complexity measure: 2.985
RMS weighted error: 2.572 cents/octave
max weighted error: 3.791 cents/octave, 4/19

1203.853 cents period
253.446 cents generator

mapping by period and generator:
[1, 2, 4, 3]
[0, -2, -8, -1]

mapping by steps:
(5, 8, 12, 14)
(14, 22, 32, 39)
complexity measure: 3.445
RMS weighted error: 2.755 cents/octave
max weighted error: 3.853 cents/octave, 2/13

600.683 cents period
94.483 cents generator

mapping by period and generator:
[2, 3, 4, 5]
[0, 1, 4, 4]

mapping by steps:
(14, 22, 32, 39)
(12, 19, 28, 34)

complexity measure: 3.445
RMS weighted error: 3.138 cents/octave
max weighted error: 4.459 cents/octave, 13/31

1201.242 cents period
504.026 cents generator

mapping by period and generator:
[1, 2, 4, 7]
[0, -1, -4, -10]

mapping by steps:
(12, 19, 28, 34)
(19, 30, 44, 53)

complexity measure: 3.562
RMS weighted error: 1.382 cents/octave
max weighted error: 2.206 cents/octave, 5/19

1202.646 cents period
317.170 cents generator

mapping by period and generator:
[1, 0, 1, 2]
[0, 6, 5, 3]

mapping by steps:
(4, 6, 9, 11)
(15, 24, 35, 42)

complexity measure: 3.786
RMS weighted error: 2.583 cents/octave
max weighted error: 4.283 cents/octave, 2/19

1203.503 cents period
125.975 cents generator

mapping by period and generator:
[1, 2, 2, 3]
[0, -4, 3, -2]

mapping by steps:
(10, 16, 23, 28)
(9, 14, 21, 25)

complexity measure: 3.816
RMS weighted error: 2.575 cents/octave
max weighted error: 3.657 cents/octave, 2/9

398.752 cents period
90.460 cents generator

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

mapping by steps:
(15, 24, 35, 42)
(12, 19, 28, 34)

complexity measure: 4.030
RMS weighted error: 2.228 cents/octave
max weighted error: 3.745 cents/octave, 13/41

1201.082 cents period
380.695 cents generator

mapping by period and generator:
[1, 0, 2, -1]
[0, 5, 1, 12]

mapping by steps:
(19, 30, 44, 53)
(22, 35, 51, 62)

complexity measure: 4.274
RMS weighted error: 1.074 cents/octave
max weighted error: 1.487 cents/octave, 5/37

1197.839 cents period
162.587 cents generator

mapping by period and generator:
[1, 2, 3, 2]
[0, -3, -5, 6]

mapping by steps:
(15, 24, 35, 42)
(22, 35, 51, 62)

complexity measure: 4.291
RMS weighted error: 2.531 cents/octave
max weighted error: 3.762 cents/octave, 3/11

599.619 cents period
164.248 cents generator

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

mapping by steps:
(8, 13, 19, 23)
(14, 22, 32, 39)

complexity measure: 4.307
RMS weighted error: 2.780 cents/octave
max weighted error: 4.237 cents/octave, 11/27

1197.067 cents period
488.512 cents generator

mapping by period and generator:
[1, 2, 6, 2]
[0, -1, -9, 2]

mapping by steps:
(5, 8, 12, 14)
(22, 35, 51, 62)

complexity measure: 4.589
RMS weighted error: 1.917 cents/octave
max weighted error: 2.933 cents/octave, 10/27

1199.714 cents period
443.277 cents generator
mapping by period and generator:
[1, -1, -1, -2]
[0, 7, 9, 13]

mapping by steps:
(19, 30, 44, 53)
(8, 13, 19, 23)

complexity measure: 4.631
RMS weighted error: 1.323 cents/octave
max weighted error: 2.014 cents/octave, 7/18

601.479 cents period
232.661 cents generator

mapping by period and generator:
[2, 2, 5, 6]
[0, 3, -1, -1]

mapping by steps:
(26, 41, 60, 73)
(10, 16, 23, 28)

complexity measure: 4.647
RMS weighted error: 3.200 cents/octave
max weighted error: 4.986 cents/octave, 19/45

1203.646 cents period
507.759 cents generator

mapping by period and generator:
[1, 2, 4, -1]
[0, -1, -4, 9]

mapping by steps:
(19, 30, 44, 53)
(26, 41, 60, 73)

complexity measure: 4.929
RMS weighted error: 2.117 cents/octave
max weighted error: 3.646 cents/octave, 7/13

600.047 cents period
325.744 cents generator

mapping by period and generator:
[2, 1, 3, 4]
[0, 4, 3, 3]

mapping by steps:
(4, 6, 9, 11)
(22, 35, 51, 62)

complexity measure: 5.047
RMS weighted error: 2.483 cents/octave
max weighted error: 3.850 cents/octave, 11/37

1196.642 cents period
354.908 cents generator

mapping by period and generator:
[1, 1, 5, 4]
[0, 2, -9, -4]

mapping by steps:
(27, 43, 63, 76)
(10, 16, 23, 28)

complexity measure: 5.138
RMS weighted error: 2.301 cents/octave
max weighted error: 3.358 cents/octave, 6/31

1200.937 cents period
232.375 cents generator

mapping by period and generator:
[1, 1, 0, 3]
[0, 3, 12, -1]

mapping by steps:
(26, 41, 60, 73)
(5, 8, 12, 14)

complexity measure: 5.524
RMS weighted error: 1.425 cents/octave
max weighted error: 2.457 cents/octave, 7/34

1201.478 cents period
247.860 cents generator

mapping by period and generator:
[1, 2, 5, 3]
[0, -2, -13, -1]
mapping by steps:
(5, 8, 12, 14)
(29, 46, 67, 81)

complexity measure: 5.599
RMS weighted error: 2.853 cents/octave
max weighted error: 4.364 cents/octave, 17/41

1200.125 cents period
497.967 cents generator

mapping by period and generator:
[1, 2, -1, -3]
[0, -1, 8, 14]

mapping by steps:
(29, 46, 67, 81)
(12, 19, 28, 34)

complexity measure: 5.618
RMS weighted error: 0.726 cents/octave
max weighted error: 1.165 cents/octave, 1/4

133.029 cents period
40.408 cents generator

mapping by period and generator:
[9, 14, 21, 25]
[0, 1, 0, 1]

mapping by steps:
(9, 14, 21, 25)
(27, 43, 63, 76)

complexity measure: 5.678
RMS weighted error: 2.156 cents/octave
max weighted error: 3.142 cents/octave, 7/51

1200.087 cents period
164.424 cents generator

mapping by period and generator:
[1, 2, 3, 5]
[0, -3, -5, -16]

mapping by steps:
(22, 35, 51, 62)
(29, 46, 67, 81)

complexity measure: 5.699
RMS weighted error: 2.357 cents/octave
max weighted error: 3.520 cents/octave, 7/31

1200.021 cents period
271.513 cents generator

mapping by period and generator:
[1, 0, 3, 1]
[0, 7, -3, 8]

mapping by steps:
(9, 14, 21, 25)
(22, 35, 51, 62)

complexity measure: 5.709
RMS weighted error: 0.748 cents/octave
max weighted error: 1.176 cents/octave, 25/61

1196.995 cents period
490.530 cents generator

mapping by period and generator:
[1, 2, -3, 2]
[0, -1, 13, 2]

mapping by steps:
(22, 35, 51, 62)
(39, 62, 91, 110)

complexity measure: 6.230
RMS weighted error: 1.929 cents/octave
max weighted error: 3.005 cents/octave, 8/33

1203.817 cents period
290.948 cents generator

mapping by period and generator:
[1, 4, 4, 4]
[0, -10, -7, -5]

mapping by steps:
(29, 46, 67, 81)
(4, 6, 9, 11)

complexity measure: 6.309
RMS weighted error: 3.166 cents/octave
max weighted error: 3.817 cents/octave, 8/31

1199.344 cents period
309.976 cents generator

mapping by period and generator:
[1, -1, 0, 1]
[0, 10, 9, 7]

mapping by steps:
(4, 6, 9, 11)
(27, 43, 63, 76)

complexity measure: 6.309
RMS weighted error: 0.952 cents/octave
max weighted error: 1.496 cents/octave, 4/17

599.616 cents period
140.259 cents generator

mapping by period and generator:
[2, 2, 3, 4]
[0, 5, 7, 7]

mapping by steps:
(26, 41, 60, 73)
(8, 13, 19, 23)

complexity measure: 6.309
RMS weighted error: 2.448 cents/octave
max weighted error: 4.080 cents/octave, 21/65

1199.979 cents period
387.376 cents generator

mapping by period and generator:
[1, -1, 2, -3]
[0, 8, 1, 18]

mapping by steps:
(31, 49, 72, 87)
(34, 54, 79, 96)

complexity measure: 6.412
RMS weighted error: 1.187 cents/octave
max weighted error: 1.848 cents/octave, 5/61

1197.047 cents period
97.912 cents generator

mapping by period and generator:
[1, 2, 2, 2]
[0, -5, 4, 10]

mapping by steps:
(49, 78, 114, 138)
(12, 19, 28, 34)

complexity measure: 6.717
RMS weighted error: 1.862 cents/octave
max weighted error: 2.953 cents/octave, 9/61

1196.800 cents period
176.731 cents generator

mapping by period and generator:
[1, 1, 1, 4]
[0, 4, 9, -8]

mapping by steps:
(34, 54, 79, 96)
(27, 43, 63, 76)

complexity measure: 6.726
RMS weighted error: 1.890 cents/octave
max weighted error: 3.200 cents/octave, 3/46

1200.077 cents period
77.869 cents generator

mapping by period and generator:
[1, 1, 2, 3]
[0, 9, 5, -3]

mapping by steps:
(31, 49, 72, 87)
(15, 24, 35, 42)

complexity measure: 6.747
RMS weighted error: 0.858 cents/octave
max weighted error: 1.372 cents/octave, 11/41

1200.768 cents period
322.136 cents generator

mapping by period and generator:
[1, 4, 5, 2]
[0, -9, -10, 3]

mapping by steps:
(15, 24, 35, 42)
(26, 41, 60, 73)

complexity measure: 6.747
RMS weighted error: 1.100 cents/octave
max weighted error: 1.652 cents/octave, 14/37

1199.357 cents period
454.205 cents generator

mapping by period and generator:
[1, 5, 8, 10]
[0, -9, -15, -19]

mapping by steps:
(8, 13, 19, 23)
(29, 46, 67, 81)

complexity measure: 6.768
RMS weighted error: 2.599 cents/octave
max weighted error: 4.407 cents/octave, 4/41

1200.822 cents period
116.755 cents generator

mapping by period and generator:
[1, 1, 3, 3]
[0, 6, -7, -2]

mapping by steps:
(31, 49, 72, 87)
(10, 16, 23, 28)
complexity measure: 6.800
RMS weighted error: 0.515 cents/octave
max weighted error: 0.822 cents/octave, 16/45

1201.255 cents period
426.387 cents generator

mapping by period and generator:
[1, 3, 8, 6]
[0, -4, -16, -9]

mapping by steps:
(31, 49, 72, 87)
(14, 22, 32, 39)

complexity measure: 6.891
RMS weighted error: 1.393 cents/octave
max weighted error: 2.358 cents/octave, 5/39

1199.619 cents period
154.529 cents generator

mapping by period and generator:
[1, 3, 4, 5]
[0, -11, -13, -17]

mapping by steps:
(8, 13, 19, 23)
(31, 49, 72, 87)

complexity measure: 6.940
RMS weighted error: 1.244 cents/octave
max weighted error: 1.844 cents/octave, 4/75

1200.730 cents period
64.034 cents generator

mapping by period and generator:
[1, 1, 2, 2]
[0, 11, 6, 15]

mapping by steps:
(19, 30, 44, 53)
(56, 89, 130, 157)

complexity measure: 6.940
RMS weighted error: 1.622 cents/octave
max weighted error: 2.443 cents/octave, 5/29

599.447 cents period
103.585 cents generator

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

mapping by steps:
(12, 19, 28, 34)
(46, 73, 107, 129)

complexity measure: 6.961
RMS weighted error: 1.009 cents/octave
max weighted error: 1.615 cents/octave]

>> If you want to play along, can we make the Tenney error cutoff
>> an even 3.2? And can you include period and generator? And
>> the smallest ET tuning that doesn't make the error > 3.2? And, uh,
>> what's RMS complexity?
>
>Raising the cutoff doesn't add much.

I was just trying to add injera.

>I'll put a full printout at the
>end of the message. If you want to play along at home, the code is:
>
> >>> import regular
> >>> def complexityBadness(lt, width, error):
>... return width
>...
> >>> ets = regular.getEqualTemperaments(regular.limit7)
> >>> regular.getLinearTemperaments(ets, 7, 3.2/1200,
>badness=complexityBadness)

Looks like I'd need regular.

>Your simplest nice ET looks like 19:
>
> >>> def err(et):
>... return regular.optimalRMSError(et.weightedPrimes())
>...
> >>> maxerr = err(regular.Temperament(regular.limit5, 12))
> >>> [et[0] for et in ets if err(et) <= maxerr][0]
>19

I'm not sure what this is doing, but I meant finding the smallest
ET compatible with each linear temperament such that the resulting
tuning wouldn't raise the weighted error above the cutoff. So
like, find linear temperaments with optimal tunings below some
cutoff, and then tune them in ETs without going above the cutoff.

>I can guess what RMS complexity would be, but it's not what I use.

I was asking you. "Then the optimal RMS complexity" made it sound
like it's what you use.

>The complexity here is the max-min Tenney weighted number of steps
>per prime interval, which we believe to be the same as the maximum
>Kees complexity.

Yes, I recall a thread about all that. A proof would be interesting.
Did you ever give one?

>Here's that listing:
>
>[2/11
>
> 598.859 cents period
> 106.844 cents generator
>
>mapping by period and generator:
>[2, 3, 5, 6]
>[0, 1, -2, -2]
>
>mapping by steps:
>(12, 19, 28, 34)
>(10, 16, 23, 28)
>
>complexity measure: 2.985
>RMS weighted error: 2.572 cents/octave
>max weighted error: 3.791 cents/octave,

Great, so this is pajara. What's 2/11 mean?
Is that the ET key? I thought you were giving 10&12 before.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> I think it's time I started to start getting serious about
> composing some microtonal music.

Yay! Go team go.

> 19 (kleismic, negri, 5-limit meantone)
> 22 (pajara, porcupine)
> 26 (injera)
> 31 (7-limit meantone)
> 41 (schismic, magic)

It's also not too late to try your hand in the 46-et project.

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

> I thought bug was really high error... hrm.

I don't see what I would call bug on the list, which would be
(2, 3, 1), or <<2 3 1 0 -4 -6| for the full wedgie. It tempers out
21/20, 27/25 and 36/35, and isn't as accurate as 12-et.

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

> I'm not sure what this is doing, but I meant finding the smallest
> ET compatible with each linear temperament such that the resulting
> tuning wouldn't raise the weighted error above the cutoff.

How do you propose to measure the error, and what is the cuttoff?
Something to do with 12-et?

>> I think it's time I started to start getting serious about
>> composing some microtonal music.
>
>Yay! Go team go.
>
>> 19 (kleismic, negri, 5-limit meantone)
>> 22 (pajara, porcupine)
>> 26 (injera)
>> 31 (7-limit meantone)
>> 41 (schismic, magic)
>
>It's also not too late to try your hand in the 46-et project.

Oh yeah. What LTs work with that again? Semisixths and Valentine?

-Carl

>> I'm not sure what this is doing, but I meant finding the smallest
>> ET compatible with each linear temperament such that the resulting
>> tuning wouldn't raise the weighted error above the cutoff.
>
>How do you propose to measure the error,

I was hoping to use the error formula in Paul's latest paper,
but I haven't thought through how that applies to the 9-limit
yet (ie if its 7-limit and 9-limit answers are the same).

>and what is the cuttoff? Something to do with 12-et?

Yeah, whatever the formula gives for 12-et in the 5-limit.

-C.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> I'm not sure what this is doing, but I meant finding the smallest
> >> ET compatible with each linear temperament such that the resulting
> >> tuning wouldn't raise the weighted error above the cutoff.
> >
> >How do you propose to measure the error,
>
> I was hoping to use the error formula in Paul's latest paper,
> but I haven't thought through how that applies to the 9-limit
> yet (ie if its 7-limit and 9-limit answers are the same).

I thought Paul just used TOP.

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> I think it's time I started to start getting serious about
> >> composing some microtonal music.
> >
> >Yay! Go team go.
> >
> >> 19 (kleismic, negri, 5-limit meantone)
> >> 22 (pajara, porcupine)
> >> 26 (injera)
> >> 31 (7-limit meantone)
> >> 41 (schismic, magic)
> >
> >It's also not too late to try your hand in the 46-et project.
>
> Oh yeah. What LTs work with that again? Semisixths and Valentine?

Yeah, not to mention diaschismic, shrutar, rodan and even amity.

>> >> I'm not sure what this is doing, but I meant finding the smallest
>> >> ET compatible with each linear temperament such that the resulting
>> >> tuning wouldn't raise the weighted error above the cutoff.
>> >
>> >How do you propose to measure the error,
>>
>> I was hoping to use the error formula in Paul's latest paper,
>> but I haven't thought through how that applies to the 9-limit
>> yet (ie if its 7-limit and 9-limit answers are the same).
>
>I thought Paul just used TOP.

Yes, but I thought there was disagreement about what TOP was.
I never got the end of that 'what's the TOP damage of TOP
meantone' (or similar) thread.

-Carl

Carl Lumma wrote:
>>>If you want to play along, can we make the Tenney error cutoff
>>>an even 3.2? And can you include period and generator? And
>>>the smallest ET tuning that doesn't make the error > 3.2? And, uh,
>>>what's RMS complexity?
>>
>>Raising the cutoff doesn't add much.
> > I was just trying to add injera.

You get a few more complex temperaments as well.

> Looks like I'd need regular.

It's part of the source code zip file on my website.

>>Your simplest nice ET looks like 19:
>>
>>
>>>>>def err(et):
>>
>>... return regular.optimalRMSError(et.weightedPrimes())
>>...
>>
>>>>>maxerr = err(regular.Temperament(regular.limit5, 12))
>>>>>[et[0] for et in ets if err(et) <= maxerr][0]
>>
>>19
> > I'm not sure what this is doing, but I meant finding the smallest
> ET compatible with each linear temperament such that the resulting
> tuning wouldn't raise the weighted error above the cutoff. So
> like, find linear temperaments with optimal tunings below some
> cutoff, and then tune them in ETs without going above the cutoff.

Oh, well, that's more tricky. Perhaps you can hack it manually for each temperament. The full list of candidate ETs below the cutoff might help. You take the [0] off the previous statement.

[19, 22, 27, 31, 34, 39, 41, 43, 46, 49, 50, 53, 56, 58, 60, 63, 65, 65, 68, 72]

So if you see one of them it's below the cutoff. The search algorithm should return the simplest pair or ETs for each LT. If their both too simple, you can try walking the scale tree until you get something on the list. But you can't be sure it's the right mapping.

>>I can guess what RMS complexity would be, but it's not what I use.
> > > I was asking you. "Then the optimal RMS complexity" made it sound
> like it's what you use.

Oh, did I say that? :P

>>The complexity here is the max-min Tenney weighted number of steps
>>per prime interval, which we believe to be the same as the maximum
>>Kees complexity.
> > Yes, I recall a thread about all that. A proof would be interesting.
> Did you ever give one?

No, but I remember Gene being confident about it.

> > >>Here's that listing:
>>
>>[2/11
>>
>> 598.859 cents period
>> 106.844 cents generator
>>
>>mapping by period and generator:
>>[2, 3, 5, 6]
>>[0, 1, -2, -2]
>>
>>mapping by steps:
>>(12, 19, 28, 34)
>>(10, 16, 23, 28)
>>
>>complexity measure: 2.985
>>RMS weighted error: 2.572 cents/octave
>>max weighted error: 3.791 cents/octave,
> > > Great, so this is pajara. What's 2/11 mean?
> Is that the ET key? I thought you were giving 10&12 before.

Yes, it's 10&12, see the mapping by steps. 2/11 is the generator/period ratio. For 10&12, the whole scale has 22 steps. The generator is 2 steps and the period is a half octave, so 2/11.

Graham