The revised list of new 2D temperaments

Hi again.

Gene has suggested that if I want to use higher limit extensions of an originally 5-limit 2D system, I should list the possible higher limit mappings instead of just the 5-limit ones. Therefore, I have revised my list of new 2D temperaments and I'm presenting it here as pairs of EDOs. Note that for many of these temperaments the 5-limit intervals are much closer to JI than the higher limit ones while for others it's the other way round. For this reason, in most cases, I like to choose such a generator size that the least mistuned primes are as close to JI as possible, possibly disregarding the others entirely. For example, while finding an appropriate generator for the 37&50 temperament, I don't care about 7/1 as it's remarcably more mistuned than all the other primes although it's perfectly okay to use it. It's a similar situation to temperaments like hanson or semisixths where 7-limit intervals are still usable although the 5-limit ones are much more in tune and therefore we can harmlessly choose a generator size based on those.

BTW: This is probably the last time I'm listing temperaments by EDO pairs; I think I'll switch to wedgies pretty soon. The reason why I'm using EDO pairs here is that there are a couple of people very much used to it.

--------------------

84&99, 7-limit (nessafof?)

84&56, full 13-limit

43&54, full 13-limit

77&87, 13-limit (possibly without 11/1)

99&103, 7-limit

50&61, full 13-limit

31&49, full 13-limit semisept (which one of the two?)

34&16, full 13-limit vishnu

34&26, full 13-limit fifive

37&50, full 13-limit

37&53, full 13-limit

37&72, full 13-limit

50&53, full 13-limit

53&61d, full 13-limit (7/1 more mistuned than the rest)

53&67, full 13-limit

53&77, full 13-limit hemischis (11/1 more mistuned)

61&87, full 13-limit

87&84, full 13-limit mutt

87&78, full 13-limit

87&71, full 13-limit

22&74, 11-limit quazy

85&128d, full 13-limit

55&65, full 13-limit

53&55, full 13-limit

65&67, full 13-limit

47&34d, full 13-limit

47&37, full 13-limit

161&183, full 13-limit

--------------------

Petr

I wrote:

> 85&128d, full 13-limit

Again, only now am I realizing that 43&85 may sum it up in a more understandable way.

Petr

I wrote:

> Therefore, I have revised my list > of new 2D temperaments and I'm presenting it here as pairs of EDOs.

And one more:
41&43, full 13-limit

Petr

--- In tuning@yahoogroups.com, Petr Pařízek <petrparizek2000@...> wrote:

> 77&87, 13-limit (possibly without 11/1)

Name?

> 37&50, full 13-limit

Name?

> 37&53, full 13-limit

Name?

> 37&72, full 13-limit

Name?

> 50&53, full 13-limit

Name?

> 53&61d, full 13-limit (7/1 more mistuned than the rest)

Name?

> 53&77, full 13-limit hemischis (11/1 more mistuned)

Name?

> 61&87, full 13-limit

Name?

> 87&78, full 13-limit

Name?

> 87&71, full 13-limit

Name?

--- In tuning@yahoogroups.com, Petr Pařízek <petrparizek2000@...> wrote:
>
> I wrote:
>
> > Therefore, I have revised my list
> > of new 2D temperaments and I'm presenting it here as pairs of EDOs.
>
> And one more:
> 41&43, full 13-limit

Name?

2011/8/18 Petr Pařízek <petrparizek2000@...>
>
> Hi again.
>
> Gene has suggested that if I want to use higher limit extensions of an
> originally 5-limit 2D system, I should list the possible higher limit
> mappings instead of just the 5-limit ones. Therefore, I have revised my list
> of new 2D temperaments and I'm presenting it here as pairs of EDOs. Note
> that for many of these temperaments the 5-limit intervals are much closer to
> JI than the higher limit ones while for others it's the other way round. For
> this reason, in most cases, I like to choose such a generator size that the
> least mistuned primes are as close to JI as possible, possibly disregarding
> the others entirely. For example, while finding an appropriate generator for
> the 37&50 temperament, I don't care about 7/1 as it's remarcably more
> mistuned than all the other primes although it's perfectly okay to use it.
> It's a similar situation to temperaments like hanson or semisixths where
> 7-limit intervals are still usable although the 5-limit ones are much more
> in tune and therefore we can harmlessly choose a generator size based on
> those.

Are these all patent vals?

-Mike

2011/8/18 Petr Pařízek <petrparizek2000@...>
>
> Hi again.

Nm, I see that some of them have warts so I'll just assume that you
mean the patent val otherwise. A few comments -

> 84&99, 7-limit (nessafof?)

This one's interesting, period is 1/3-octave, but the period isn't
mapped to 5/4.

> 84&56, full 13-limit

This temperament has a period of 1/28th of an octave, and the
generator is about 1/3 of that, and 2 generators will get you the
entire tonality diamond. So this is more or less 84-equal.

> 55&65, full 13-limit

I think Gene figured this one out a while ago, yes? Tempers out
245/243 and something else.

-Mike

Gene wrote:

> Name?

Before I start trying to find names for these weird temperaments, I'd like to find a way to filter out those whose prime approximations may be "too far from JI" with regard to the high complexity. I don't know if there's already an "established" method for doing that and what possible kinds of cutoffs need to be taken into account.
But an obvious example of a temperament that I would probably prefer using only as a non-octave equal tuning rather than a 2D one is 43&85, which is effectively nothing more than 54 equal divisions of 12/5 with octaves "tacked on". Having to use 54 generators for an ordinary minor tenth means that a 43-tone MOS doesn't include it (and 43 is already a lot of tones).

The 37&50 temperament is an excellent 2|5|11|13-limit choice, which is exactly how I used it in the recording from March 2008 (as I've said a few days ago regarding the 11/8-like generators). The 3/1 approximation makes it much more complex but it's pretty close to JI as well.

As to the other temperaments, I've only partially studied the 84&99 comma pump in the 5-limit context but so far I haven't tried anything in the other ones yet.

Petr

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> > 55&65, full 13-limit
>
> I think Gene figured this one out a while ago, yes? Tempers out
> 245/243 and something else.

I don't recall it, but 245/242, not 245/243, is one of the commas it tempers out; 686/675, 16807/16384 and 385/384 are others. But you really need to go to 13 to take advantage of whatever merits it possesses.

Mike wrote:

> Are these all patent vals?

Unless there's a letter following it, they are -- except for semisept, which I'm still not decided which one of the two is better.

Petr

> --- In tuning@yahoogroups.com, Mike Battaglia
> <battaglia01@...> wrote:
>
> > > 55&65, full 13-limit
> >
> > I think Gene figured this one out a while ago, yes?
> > Tempers out 245/243 and something else.

You mean they're both choosing 55p&65p over 55f&65f?

"genewardsmith" <genewardsmith@...> wrote:
>
> I don't recall it, but 245/242, not 245/243, is one of
> the commas it tempers out; 686/675, 16807/16384 and
> 385/384 are others. But you really need to go to 13 to
> take advantage of whatever merits it possesses.

Neither 245:242 nor 245:243 work for me. The other ratios
work, for the 11-limit case common to the two simple 55&65
options.

These are the 11-limit unison-vector ratios Pari gives:

385/384, 243/242, 675/686

These for 55p&65p:

104/105, 143/144, 351/352, 675/686

And these for 55f&65f:

91/90, 195/196, 385/384, 2420/2457

They're supposed to be Tenney-LLL reduced. I don't know why
2420:2457 is still there when it could be replaced with
something simpler from the 11-limit basis.

Graham

Graham wrote:

> You mean they're both choosing 55p&65p over 55f&65f?

At least I was.

Petr

I wrote:

> Before I start trying to find names for these weird temperaments, I'd like
> to find a way to filter out those whose prime approximations may be "too > far
> from JI" with regard to the high complexity. I don't know if there's > already
> an "established" method for doing that and what possible kinds of cutoffs
> need to be taken into account.

And I'm unfamiliar with those badnesses and epimericity and ET consistency and all that stuff that some of you seem to be using for making these cutoffs and filters.

> The 37&50 temperament is an excellent 2|5|11|13-limit choice, which is
> exactly how I used it in the recording from March 2008 (as I've said a few
> days ago regarding the 11/8-like generators). The 3/1 approximation makes > it
> much more complex but it's pretty close to JI as well.

A similar situation seems to occur with the aforementioned 53&61d, which is most effectively applicable as a 2|3|5|13-limit temperament since both 7/1 and 11/1 are more mistuned than the other primes.

Petr