More on Ambiguity

In this message, "best" means "with the lowest error, allowing for
optimal scale stretching".

Here are the lists of EDOs < 100 for which the best equal temperament
does not have the patent val:

5-limit
11, 14, 17, 20, 32, 33, 51, 52, 54, 64, 66, 67, 85, 86, 98

7-limit
3, 7, 8, 11, 13, 14, 17, 20, 23, 28, 30, 33, 34, 38, 39, 54, 61, 64,
66, 67, 69, 71, 79, 85, 86, 92, 96, 98

11-limit
3, 4, 7, 8, 11, 13, 14, 17, 18, 20, 23, 25, 27, 28, 30, 33, 34, 38,
39, 42, 45, 47, 52, 54, 60, 61, 64, 65, 66, 67, 69, 70, 71, 75, 76,
79, 82, 85, 86, 90, 91, 92, 96, 97, 98, 99

13-limit
2, 3, 4, 5, 6, 8, 11, 12, 13, 14, 17, 18, 20, 23, 25, 27, 28, 30, 32,
33, 34, 35, 36, 38, 39, 42, 45, 47, 48, 49, 51, 54, 55, 59, 60, 64,
65, 66, 69, 71, 73, 75, 76, 81, 82, 83, 85, 86, 88, 90, 91, 92, 96,
98, 99

At this point, it's simpler to list those where the best equal
temperament *does* have the patent val:

13-limit
7, 9, 10, 15, 16, 19, 21, 22, 24, 26, 29, 31, 37, 40, 41, 43, 44, 46,
50, 52, 53, 56, 57, 58, 61, 62, 63, 67, 68, 70, 72, 74, 77, 78, 79,
80, 84, 87, 89, 93, 94, 95, 97

17-limit
7, 9, 10, 16, 21, 22, 24, 26, 31, 37, 40, 41, 43, 44, 46, 50, 53, 56,
57, 58, 62, 63, 67, 68, 70, 72, 74, 77, 78, 79, 80, 84, 87, 89, 93, 94

19-limit
7, 9, 16, 21, 24, 26, 31, 36, 37, 40, 41, 43, 44, 46, 48, 50, 53, 56,
57, 62, 63, 67, 68, 72, 74, 77, 80, 84, 87, 89, 93, 94

Some of these are highly ambiguous in that the best two mappings give
similar optimal errors. A test for ambiguity has to consider both
properties.

I ran the tests this morning. If you think I got something wrong I
still have the sessions to check this evening.

Now, here's my list of ambiguous equal temperaments from the website
again. I've added a * for those mappings that are included in the top
10 lists of equal temperaments. These are used to find most of the
best higher rank temperaments. So, where there's high ambiguity (two
ETs for the same EDO in the same top 10) I've listed the higher rank
examples they're involved in.

5-limit 10-cent badness

2p = <2, 3, 5]
2b = <2, 3, 4]

5p = <5, 8, 12]*
5b = <5, 8, 11]

17b = <17, 27, 40]
17p = <17, 27, 39]

5-limit 5-cent badness

17 pair again

7-limit 10-cent badness

7c = <7, 11, 16, 19]
7p = <7, 11, 16, 20]

12p = <12, 19, 28, 34]*
12c = <12, 19, 28, 33]

2bc = <2, 3, 4, 5]
2p = <2, 3, 5, 6]

3c = <3, 5, 7, 9]
3p = <3, 5, 7, 8]

5p = <5, 8, 12, 14]*
5b = <5, 8, 11, 14]

17b = <17, 27, 40, 48]
17p = <17, 27, 39, 48]

7-limit 5-cent badness

7-pair again

34c = <34, 54, 79, 96]
34p = <34, 54, 79, 95]

11-limit 10-cent badness

7c = <7, 11, 16, 19, 24]*
7p = <7, 11, 16, 20, 24]*

10p = <10, 16, 23, 28, 35]*
10d = <10, 16, 23, 28, 34]

3cd = <3, 5, 7, 9, 11]
3p = <3, 5, 7, 8, 10]

4d = <4, 6, 9, 11, 13]
4p = <4, 6, 9, 11, 14]

5p = <5, 8, 12, 14, 17]
5d = <5, 8, 12, 14, 18]

12p = <12, 19, 28, 34, 42]*
12d = <12, 19, 28, 34, 41]

2p = <2, 3, 5, 6, 7]
2bcd = <2, 3, 4, 5, 6]

Meanenneadecal is 12p & 7c
Opossum is 15 & 7c
Thrasher is 7c & 12p & 15
Potassium is 9 & 12p & 7c
Calliope is 14 & 12p & 7c
Dominant+ is 12p & 7p -- one off the shortlist

11-limit, 5-cent badness

7 pair again
12 pair again*
5 pair again

19p = <19, 30, 44, 53, 66]*
19d = <19, 30, 44, 53, 65]*

13-limit, 10-cent badness

12e = <12, 19, 28, 34, 42, 45]*
12p = <12, 19, 28, 34, 42, 44]
12d = <12, 19, 28, 34, 41, 44]

15p = <15, 24, 35, 42, 52, 56]*
15e = <15, 24, 35, 42, 52, 55]

10p = <10, 16, 23, 28, 35, 37]*
10d = <10, 16, 23, 28, 34, 37]

7p = <7, 11, 16, 20, 24, 26]*
7c = <7, 11, 16, 19, 24, 26]
7ce = <7, 11, 16, 19, 24, 25]

19p = <19, 30, 44, 53, 66, 70]*
19d = <19, 30, 44, 53, 65, 70]*

14e = <14, 22, 32, 39, 48, 51]*
14p = <14, 22, 32, 39, 48, 52]

5d = <5, 8, 12, 14, 18, 19]*
5p = <5, 8, 12, 14, 17, 19]

22p = <22, 35, 51, 62, 76, 81]
22e = <22, 35, 51, 62, 76, 82]

3cde = <3, 5, 7, 9, 11, 12]
3cd = <3, 5, 7, 9, 11, 11]

Negric is 9 & 19d
Meantone is 12e & 19d
Minerva is 19d & 9 & 12e
15p & 9 & 19d is also listed
(These use neither the best nor patent mapping of 19)
Calliope is 19p & 12e & 7p

13-limit 5-cent badness

31p = <31, 49, 72, 87, 107, 115]*
31e = <31, 49, 72, 87, 107, 114]

12e = <12, 19, 28, 34, 42, 45]*
12p = <12, 19, 28, 34, 42, 44]

19p = <19, 30, 44, 53, 66, 70]*
19d = <19, 30, 44, 53, 65, 70]*

10p = <10, 16, 23, 28, 35, 37]*
10d = <10, 16, 23, 28, 34, 37]

22p = <22, 35, 51, 62, 76, 81]
22e = <22, 35, 51, 62, 76, 82]

7p = <7, 11, 16, 20, 24, 26]
7c = <7, 11, 16, 19, 24, 26]

Sensisept is 19p & 27
Negri is 19p & 9
Meanpop is 31p & 19p
Sensus is 19d & 37 <= lesser approximation of 19
Tripod is 19p & 31p & 41
Krypton is 9 & 19p & 27

13-limit 2-cent badness

31 pair again*
19 pair again**
22 pair again

13-limit 1-cent badness

19 pair again

17-limit 5-cent

22p = <22, 35, 51, 62, 76, 81, 90]*
22e = <22, 35, 51, 62, 76, 82, 90]*

19d = <19, 30, 44, 53, 65, 70, 77]*
19f = <19, 30, 44, 53, 66, 70, 77]
19p = <19, 30, 44, 53, 66, 70, 78]

12e = <12, 19, 28, 34, 42, 45, 49]*
12p = <12, 19, 28, 34, 42, 44, 49]

15f = <15, 24, 35, 42, 52, 56, 62]
15p = <15, 24, 35, 42, 52, 56, 61]

9p = <9, 14, 21, 25, 31, 33, 37]
9f = <9, 14, 21, 25, 31, 33, 36]

Winston+ is 31 & 22e
Pajara+ is 22p & 12e
Srutar+ is 22p & 46
Orwell+ is 31 & 22p
Hedgehog++ is 22p & 8
Jubilee++ is 26 & 22p & 12e
Octarod+++ is 22p & 27 & 19d
Tripod+ is 22e & 31 & 10
22 is fully ambiguous here

17-limit 2-cent

22 pair again*

17-limit 1-cent

118p = <118, 187, 274, 331, 408, 437, 482]
118e = <118, 187, 274, 331, 408, 436, 482]

19-limit 5-cent

9p = <9, 14, 21, 25, 31, 33, 37, 38]*
9f = <9, 14, 21, 25, 31, 33, 36, 38]

15f = <15, 24, 35, 42, 52, 56, 62, 64]*
15p = <15, 24, 35, 42, 52, 56, 61, 64]

22eg = <22, 35, 51, 62, 76, 82, 90, 94]
22p = <22, 35, 51, 62, 76, 81, 90, 93]
22g = <22, 35, 51, 62, 76, 81, 90, 94]

10g = <10, 16, 23, 28, 35, 37, 41, 43]
10p = <10, 16, 23, 28, 35, 37, 41, 42]

8cg = <8, 13, 19, 23, 28, 30, 33, 35]
8c = <8, 13, 19, 23, 28, 30, 33, 34]

19-limit 2-cent

72p = <72, 114, 167, 202, 249, 266, 294, 306]*
72g = <72, 114, 167, 202, 249, 266, 294, 305]

41p = <41, 65, 95, 115, 142, 152, 168, 174]*
41f = <41, 65, 95, 115, 142, 152, 167, 174]

53p = <53, 84, 123, 149, 183, 196, 217, 225]*
53f = <53, 84, 123, 149, 183, 196, 216, 225]

Every higher limit temperament in the shortlist
except Lizard++ uses 72p. None uses 72g.

19-limit 1-cent

72 pair again*

Every higher limit temperament in the shortlist
except Buzzard++ uses 72p. None uses 72g.

Graham

On 4/13/2011 3:24 AM, Graham Breed wrote:

> Meanenneadecal is 12p& 7c
> Opossum is 15& 7c

I have opossum as 7d&15. Are we talking about the same thing?
[<1 2 3 4 4|, <0 -3 -5 -9 -4|]

7c&15 would be this:
[<1 2 1 2 4|, <0 -3 10 6 -4|]

It looks like 7d also for meanenneadecal, although that can also be labeled as 12&19 (or 12p&19p).

Actually, it looks like all of these labels are one letter off from what I've been using. What you're calling 12p is what I would originally have labeled as 12a, but these days just plain 12. Your 12d is my 12e, and so on.

Herman Miller <hmiller@IO.COM> wrote:

> Actually, it looks like all of these labels are one
> letter off from what I've been using. What you're calling
> 12p is what I would originally have labeled as 12a, but
> these days just plain 12. Your 12d is my 12e, and so on.

Oh, yes, I misread your original message. But that's
what's now live on my website so you'll have to relearn all
the names :-P

http://x31eq.com/cgi-bin/rt.cgi?ets=7c_15&limit=11

I've reserved "o" for when the octave isn't the equivalence
interval. Also, letters from q to z for prime intervals
that aren't prime numbers. A lot of things are working.
Approximations other than the best or second best of
each prime aren't supported yet. I'll accept bug reports
for anything else.

Graham

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

I'll accept bug reports
> for anything else.

My first bug report is that your front end doesn't seem to accept input like "7c"

"genewardsmith" <genewardsmith@sbcglobal.net> wrote:

> My first bug report is that your front end doesn't seem
> to accept input like "7c"

Which front end, why did you expect it to work, and how is
it failing?

Graham

On 4/14/2011 1:43 PM, Graham Breed wrote:
> Herman Miller<hmiller@IO.COM> wrote:
>
>> Actually, it looks like all of these labels are one
>> letter off from what I've been using. What you're calling
>> 12p is what I would originally have labeled as 12a, but
>> these days just plain 12. Your 12d is my 12e, and so on.
>
> Oh, yes, I misread your original message. But that's
> what's now live on my website so you'll have to relearn all
> the names :-P
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=7c_15&limit=11
>
> I've reserved "o" for when the octave isn't the equivalence
> interval. Also, letters from q to z for prime intervals
> that aren't prime numbers. A lot of things are working.
> Approximations other than the best or second best of
> each prime aren't supported yet. I'll accept bug reports
> for anything else.

I've been doing it this way for years, so I consider it a bug that you're using an incompatible standard for labeling. I'm not about to change all my old documents and notes for what looks like a simple mistake.

Herman Miller <hmiller@IO.COM> wrote:

> I've been doing it this way for years, so I consider it a
> bug that you're using an incompatible standard for
> labeling. I'm not about to change all my old documents
> and notes for what looks like a simple mistake.

Sorry, then. I've fixed it now. You can treat the other
naming as a bad dream.

Graham

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> "genewardsmith" <genewardsmith@...> wrote:
>
> > My first bug report is that your front end doesn't seem
> > to accept input like "7c"
>
> Which front end, why did you expect it to work, and how is
> it failing?

This front end, and I wasn't expecting anything, I was just checking.

http://x31eq.com/temper/