Some questions about regular temperaments

I think the regular mapping paradigm <http://x31eq.com/paradigm.html>
(RMP) is without doubt one of the greatest achievements for microtonal
music theory (my appreciation to the people who contributed to it!), and
I definitely want to get a better understanding of it!

So here are two questions, both related to the "Temperament class from
ETs" option in Graham's temperament finder <http://x31eq.com/temper/> :

1.) There are often letters added to the ETs, like 'b', 'c', 'd' or 'p'.
My understanding is that 'p' means "use patent vals from this ET", while
'a', 'b', 'c' and 'd' mean "use the val for 2:1, 3:1, 5:1 and 7:1,
resp., that is slightly worse than the patent val". Also, 'd', 'dd' and
'ddd' mean second-best, third-best and fourth-best approximation, resp.
. Is this correct? If so, then what happens if the next-best
approximation is ambigous, like with "12a"?

2) Quoted from http://x31eq.com/paradigm.html#keyuv :

"A suitable set of unison vectors contains the same information as the
mapping. There will usually be different unison vectors you could
choose, the same way that there are different lists of numbers that
could denote the mapping. We have mathematical algorithms to convert
between mappings and sets of unison vectors so the two concepts are
nearly equivalent."

Now there are some special cases, like 22&84
<http://x31eq.com/cgi-bin/rt.cgi?ets=22%2684&limit=7> (compare orwell =
22&31 <http://x31eq.com/cgi-bin/rt.cgi?ets=31_22&limit=7> ), or 12&41&50
<http://x31eq.com/cgi-bin/rt.cgi?ets=12%2641%2650&limit=7> (compare
marvel = 19&31&41 <http://x31eq.com/cgi-bin/rt.cgi?ets=41_31_19&limit=7>
).

According to the temperament finder, those temperaments have the same
list of unison vectors as another temperament, but they don't share the
same name. The mapping is not completely different (e.g. the octave
generator 1\1 is replaced by 1\2), but different enough that (judging
from my limited knowledge) I think it should be considered a different
temperament.

So, does that mean a regular temperament is not fully defined by a list
of unison vectors? Or maybe those examples aren't regular temperaments?
Or, is there a bug in the temperament finder, and the list of unison
vectors is computed incorrectly? Or should temperaments like 22&84 be
called "orwell", despite the different mapping (and generators)?

Thanks in advance!

-Gedankenwelt

On Sat, Apr 20, 2013 at 9:58 AM, gedankenwelt94
<gedankenwelt94@...> wrote:
>
> 1.) There are often letters added to the ETs, like 'b', 'c', 'd' or 'p'. My understanding is that 'p' means "use patent vals from this ET", while 'a', 'b', 'c' and 'd' mean "use the val for 2:1, 3:1, 5:1 and 7:1, resp., that is slightly worse than the patent val". Also, 'd', 'dd' and 'ddd' mean second-best, third-best and fourth-best approximation, resp. . Is this correct? If so, then what happens if the next-best approximation is ambigous, like with "12a"?

I'll let Herman or Graham answer this, as I don't know exactly how the
notation is supposed to work in these cases. I haven't seen "a" used
too much.

> So, does that mean a regular temperament is not fully defined by a list of unison vectors? Or maybe those examples aren't regular temperaments? Or, is there a bug in the temperament finder, and the list of unison vectors is computed incorrectly? Or should temperaments like 22&84 be called "orwell", despite the different mapping (and generators)?

Congratulations, you've stumbled on the royal pain in the ass that is
the contorsion problem. Here's a simpler example:

http://x31eq.com/cgi-bin/rt.cgi?ets=12_14&limit=5

This temperament is not meantone, but it has the same unison vectors
as meantone (81/80). What's going on here? What's going on is that
it's meantone, with one of the generators split in half: the perfect
fifth is now comprised of two "neutral thirds." These neutral thirds
don't have any JI interval mapping to them - they're "unmapped" (or,
if you like, they map only to things like sqrt(3/2)). Temperaments
which have this property are called "contorted."

Here's an even simpler example - this temperament has the same commas
as 5-limit 12p: http://x31eq.com/cgi-bin/rt.cgi?ets=24&limit=5

The whole thing is a pain in the ass. There are a few ways of dealing
with this: one is to declare such things "not temperaments at all,"
deeming them pathological; there are a few people here who insist that
the use of the word "temperament" in describing these entities is a
misstep. Another is to declare them to be the same as their
non-contorted equivalents. The first is probably the predominant view
right now.

What I know is that the more I dive into subgroup temperaments, the
more annoying they are.

Mike

On 4/20/2013 10:17 AM, Mike Battaglia wrote:
> On Sat, Apr 20, 2013 at 9:58 AM, gedankenwelt94
> <gedankenwelt94@...> wrote:
>>
>> 1.) There are often letters added to the ETs, like 'b', 'c', 'd' or 'p'. My understanding is that 'p' means "use patent vals from this ET", while 'a', 'b', 'c' and 'd' mean "use the val for 2:1, 3:1, 5:1 and 7:1, resp., that is slightly worse than the patent val". Also, 'd', 'dd' and 'ddd' mean second-best, third-best and fourth-best approximation, resp. . Is this correct? If so, then what happens if the next-best approximation is ambigous, like with "12a"?
>
> I'll let Herman or Graham answer this, as I don't know exactly how the
> notation is supposed to work in these cases. I haven't seen "a" used
> too much.

I formerly used "a" for what Graham uses "p", but otherwise "a" doesn't get used. If you take something like a 12-EDO and substitute one of the less accurate approximations to 2/1, you get something with 11 or 13 steps, so instead of 12a it'll be something like 11bbbcccc...

Where "a" would be useful is if you're dealing with non-octave scales like equal divisions of 3/1. (In that case you won't use "b".)

> Congratulations, you've stumbled on the royal pain in the ass that is
> the contorsion problem. Here's a simpler example:
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=12_14&limit=5
>
> This temperament is not meantone, but it has the same unison vectors
> as meantone (81/80). What's going on here? What's going on is that
> it's meantone, with one of the generators split in half: the perfect
> fifth is now comprised of two "neutral thirds." These neutral thirds
> don't have any JI interval mapping to them - they're "unmapped" (or,
> if you like, they map only to things like sqrt(3/2)). Temperaments
> which have this property are called "contorted."

Thanks, that helped a lot! :)

So, 12&14c is basically an impossible 5-limit subgroup version of 7-limit injera, where the 1\2 period is still applied, even though 50/49 isn't a unison vector anymore? Well, art and paradoxes are not necessarily mutually exclusive, so I guess that's fine. ^^

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:
> I formerly used "a" for what Graham uses "p", but otherwise "a" doesn't
> get used. If you take something like a 12-EDO and substitute one of the
> less accurate approximations to 2/1, you get something with 11 or 13
> steps, so instead of 12a it'll be something like 11bbbcccc...
>
> Where "a" would be useful is if you're dealing with non-octave scales
> like equal divisions of 3/1. (In that case you won't use "b".)

I see. So 'a' is redundant, and you can express the same using other letters.

Thanks for the explanation, Herman!

Best
-Gedankenwelt

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Here's a simpler example:
>
> http://x31eq.com/cgi-bin/rt.cgi?ets=12_14&limit=5
>
> This temperament is not meantone, but it has the same unison vectors
> as meantone (81/80). What's going on here? What's going on is that
> it's meantone, with one of the generators split in half: the perfect
> fifth is now comprised of two "neutral thirds."

Wait, what do you mean with "the perfect fifth is now comprised of two "neutral thirds." "? If we're talking about 5-limit 12&14c, it's the octave that is split in half.

Hi Mike,

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
The whole thing is a pain in the ass. There are a few ways of dealing
> with this: one is to declare such things "not temperaments at all,"
> deeming them pathological; there are a few people here who insist that
> the use of the word "temperament" in describing these entities is a
> misstep. Another is to declare them to be the same as their
> non-contorted equivalents. The first is probably the predominant view
> right now.

I think the best approach would be somewhere in the middle: They're something between temperaments and what you've provisional called "ARTS" here:

http://xenharmonic.wikispaces.com/Generated+Tone+Systems

How about calling them "regular semi-temperaments"? They're still regular (as in "regular temperament / tuning system"), and I think "semi-temperament" is very fitting, since only a part of the intervals is mapped.

Best
- Gedankenwelt