"Bad" Temperaments

I'm having a discussion off-list with Carl about my experiences working with high-error temperaments like Father[8], and how I tend to find that the TOP-RMS generator is not necessarily the one that produces the most consonant harmonies. I say this because in high-error temperaments (specifically, those with high error in primes 3, 5, and 7, and even more those that give both high error AND high complexity to those primes), I often find that the there are "lurking" consonant chords made up of compound (i.e. non-prime) intervals which can be rendered more in tune if the primes themselves are made more out-of-tune. For instance, in Father, you can get a bunch of 7/6's and 9/8's that are really almost Just if you jack up the 5-limit error to give a generator of around 468 cents, and these intervals will sound considerably nicer than the best 3- or 5-limit harmonies you will ever get out of Father temperament.

So, I'd like to do a little survey of some other "bad" rank-2 temperaments and see if I can't find variations on the TOP-RMS generators that produce better-sounding harmony using optimized compound intervals rather than primes. I'm sure this idea probably doesn't make much sense, as I don't have the best grasp of how TOP-RMS errors are calculated so I'm probably coming at this ass-backwards. It could very well be that what I consider a "better-sounding but non-optimal Father temperament" is really just a different temperament all together, and if so, I'll be glad to be corrected.

All the same, if any of you could provide me generators for some of the worst rank-2 temperaments you know of, in terms of having both high error and high complexity of primes 3, 5, and/or 7, I'd like to examine them and see if I can't find related generators that produce alternative smoother-sounding harmonies, despite increasing the error in the primes.

Thanks!

-Igs

On Thu, Jan 6, 2011 at 5:11 PM, cityoftheasleep <igliashon@...> wrote:
>
> I'm having a discussion off-list with Carl about my experiences working with high-error temperaments like Father[8], and how I tend to find that the TOP-RMS generator is not necessarily the one that produces the most consonant harmonies.

That's ironic, because I'm having a discussion offlist with Paul about
how 20-tet blackwood sounds so much better than 15-tet blackwood
because 180 cents works better than 160 cents.

My current theory is that the reason is that 8:9:10 is better well
represented in 20-tet than 15-tet, and 8 -> 9 -> 10 is a pretty
intuitive melodic motion to make. I'm not sure if the tetradic damage
to 8:9:10:12 is worse in 20-tet or 15-tet, but it might be useful to
start looking at the damage produced to "composite" chords like this
as well, for the sake of melody, which would lead to a different way
of optimizing things.

> I say this because in high-error temperaments (specifically, those with high error in primes 3, 5, and 7, and even more those that give both high error AND high complexity to those primes), I often find that the there are "lurking" consonant chords made up of compound (i.e. non-prime) intervals which can be rendered more in tune if the primes themselves are made more out-of-tune.

I just wrote the above before I read this, so ha.

> For instance, in Father, you can get a bunch of 7/6's and 9/8's that are really almost Just if you jack up the 5-limit error to give a generator of around 468 cents, and these intervals will sound considerably nicer than the best 3- or 5-limit harmonies you will ever get out of Father temperament.

Maybe it's best to analyze Father as a subgroup temperament. I guess
the way you're doing this is treating 9/8 and 7/6 as "faux-primes"
then. Or maybe a better way is to think of them as direct-to-root
consonances, rather than indirect-to-root consonances.

> So, I'd like to do a little survey of some other "bad" rank-2 temperaments and see if I can't find variations on the TOP-RMS generators that produce better-sounding harmony using optimized compound intervals rather than primes. I'm sure this idea probably doesn't make much sense, as I don't have the best grasp of how TOP-RMS errors are calculated so I'm probably coming at this ass-backwards. It could very well be that what I consider a "better-sounding but non-optimal Father temperament" is really just a different temperament all together, and if so, I'll be glad to be corrected.

I'd like to hear what you make of the whitewood scale I sent you. I
like 35-tet for this but maybe 21-tet or 28-tet would be better,
although the error is worse.

> All the same, if any of you could provide me generators for some of the worst rank-2 temperaments you know of, in terms of having both high error and high complexity of primes 3, 5, and/or 7, I'd like to examine them and see if I can't find related generators that produce alternative smoother-sounding harmonies, despite increasing the error in the primes.

How about bug temperament?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> That's ironic, because I'm having a discussion offlist with Paul about
> how 20-tet blackwood sounds so much better than 15-tet blackwood
> because 180 cents works better than 160 cents.

What's Paul's perspective?

> Maybe it's best to analyze Father as a subgroup temperament. I guess
> the way you're doing this is treating 9/8 and 7/6 as "faux-primes"
> then. Or maybe a better way is to think of them as direct-to-root
> consonances, rather than indirect-to-root consonances.

There are no primes I can think of which, when given TOP-RMS optimization for Father, produce the generator I like. But what I've been discussing with Carl is the possibility of a rather ornery-looking calculation involving optimizing temperaments for various subsets of the set of low-TH intervals, calculating the badness for each subset, and then figuring out for which subset the temperament gives the least badness. I can almost guarantee that for ANY temperament, there will be a subset of no fewer than two low-TH intervals (not including 1/1 and 2/1) for which the optimally-tuned temperament will have a very low badness.

> I'd like to hear what you make of the whitewood scale I sent you. I
> like 35-tet for this but maybe 21-tet or 28-tet would be better,
> although the error is worse.

Well, it's a 14-note scale, so I'd say regardless of the potential harmonies within it, it's going to be somewhat bad just based on complexity. But it's also going to be pretty rife with harmonic possibilities. Not sure what else to say about it, though--it's not a very bad temperament.

> How about bug temperament?

What's the TOP-RMS optimal generator and period?

-Igs

On Thu, Jan 6, 2011 at 6:20 PM, cityoftheasleep <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> > That's ironic, because I'm having a discussion offlist with Paul about
> > how 20-tet blackwood sounds so much better than 15-tet blackwood
> > because 180 cents works better than 160 cents.
>
> What's Paul's perspective?

He doesn't think blackwood sounds better in 20-tet. We're going in
endless circles about it. He seems to think I'm saying that 160 cents
is entirely unusable for a melodic interval in -any- context, which
isn't what I think. I think that it's not the best 9:10, and 9:10 is
pretty important in this regard

> > Maybe it's best to analyze Father as a subgroup temperament. I guess
> > the way you're doing this is treating 9/8 and 7/6 as "faux-primes"
> > then. Or maybe a better way is to think of them as direct-to-root
> > consonances, rather than indirect-to-root consonances.
>
> There are no primes I can think of which, when given TOP-RMS optimization for Father, produce the generator I like. But what I've been discussing with Carl is the possibility of a rather ornery-looking calculation involving optimizing temperaments for various subsets of the set of low-TH intervals, calculating the badness for each subset, and then figuring out for which subset the temperament gives the least badness. I can almost guarantee that for ANY temperament, there will be a subset of no fewer than two low-TH intervals (not including 1/1 and 2/1) for which the optimally-tuned temperament will have a very low badness.
>
> > I'd like to hear what you make of the whitewood scale I sent you. I
> > like 35-tet for this but maybe 21-tet or 28-tet would be better,
> > although the error is worse.
>
> Well, it's a 14-note scale, so I'd say regardless of the potential harmonies within it, it's going to be somewhat bad just based on complexity. But it's also going to be pretty rife with harmonic possibilities. Not sure what else to say about it, though--it's not a very bad temperament.

I think that although it has a lot of notes per octave, it's not
really that complex to comprehend because the period is 1/7 of an
octave, and because it's omnitetrachordal, and because it's infinitely
transposable.

Although I guess I don't understand in what sense high complexity is
"bad." High complexity is bad for some temperaments because it leads
to certain primes not appearing until you have a ton of notes, which
means that you don't get a lot of harmonic efficiency in a compact
scale. So in hanson temperament, for example, you don't get to 3/1 for
the first time until you've stacked a bunch of minor thirds on top of
each other, which means you aren't going to be getting very many
fifths.

This doesn't apply to this scale, because there's major thirds and
fifths all over the place. Is there some other sense in which
complexity is bad?

> > How about bug temperament?
>
> What's the TOP-RMS optimal generator and period?

Not sure what the TOP-RMS optimal generator is. It tempers out 27/25.
You can find the woolhouse-optimal generator here:

/tuning/database

-Mike

On Thu, Jan 6, 2011 at 6:34 PM, Mike Battaglia <battaglia01@...> wrote:
> He doesn't think blackwood sounds better in 20-tet. We're going in
> endless circles about it. He seems to think I'm saying that 160 cents
> is entirely unusable for a melodic interval in -any- context, which
> isn't what I think. I think that it's not the best 9:10, and 9:10 is
> pretty important in this regard

This sentence got cut off. The end is, 9:10 is pretty important in
this regard because we're talking about 5-limit harmony. 8 -> 9 -> 10
is a pretty important melodic motion to have, so it's probably a good
bet to start looking at how well things like 8:9:10 or 16:19:20 (or
16:17:19:20) are represented or what not. Meantone has great
approximations to the first one, Diminished has great approximations
to the second one. Blackwood in 15-tet doesn't have good
approximations to either. In 20-tet, the first one starts to really
pop out, and the scale starts to sound bright and amazing. I'm trying
to convince Paul of this, but I guess he likes blackwood in 15-tet
better.

>> There are no primes I can think of which, when given TOP-RMS optimization for Father, produce the generator I like. But what I've been discussing with Carl is the possibility of a rather ornery-looking calculation involving optimizing temperaments for various subsets of the set of low-TH intervals, calculating the badness for each subset, and then figuring out for which subset the temperament gives the least badness. I can almost guarantee that for ANY temperament, there will be a subset of no fewer than two low-TH intervals (not including 1/1 and 2/1) for which the optimally-tuned temperament will have a very low badness.

I also forgot to respond to this. Subgroup temperaments don't have to
involve just primes - Gene was talking about subgroups of 2, 3, and
7/5 before. So you'd be talking about it as a 2.5.7/6.9 subgroup, I
guess.

-Mike

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

> I also forgot to respond to this. Subgroup temperaments don't have to
> involve just primes - Gene was talking about subgroups of 2, 3, and
> 7/5 before. So you'd be talking about it as a 2.5.7/6.9 subgroup, I
> guess.

If you follow my recommendation of using normal lists to give a unique identifier to subgoups, this would be 2.5.9.21.

Hi Igs,
Considering the precise zone of 468 c. you mention, a few generators
have their proper very specific acoustic performances :
One is Phidiam or (Phi + 1)/2 = 466.180592713419 c.,
whose scales can be found from any Fibonacci sequence in which you
take 1 term every 2 :
3 8 21 55 144 377 987 ...
(observe in here that x^2 = 3x - 1 does not need the intermediary
Fibonaccis)
Another is Sotpae = 1.3108847630996616 = 468.64904051016 c. and is
even more directly connected to slendro models ; it is convergent in
the reverse way (the fifths cycle) as you can see in this sequence :
24 36 55 84 128 391 298 455 ...
where 2x^3 = 4x + 1
Not too far away you might also want to try the very slendroic
Mrejdaïrim = 1.3145962122768 = 473.54368002696 c.
convergent idem in the fifths direction :
16 24 36 56 84 32 49 74 113 172 261...
where x^3 = x + 2
- - - - - - -
Jacques

Igs wrote :

> I'm having a discussion off-list with Carl about my experiences
> working with high-error temperaments like Father[8], and how I tend
> to find that the TOP-RMS generator is not necessarily the one that> produces the most consonant harmonies. I say this because in high-
> error temperaments (specifically, those with high error in primes
> 3, 5, and 7, and even more those that give both high error AND high
> complexity to those primes), I often find that the there are
> "lurking" consonant chords made up of compound (i.e. non-prime)
> intervals which can be rendered more in tune if the primes
> themselves are made more out-of-tune. For instance, in Father, you
> can get a bunch of 7/6's and 9/8's that are really almost Just if
> you jack up the 5-limit error to give a generator of around 468
> cents, and these intervals will sound considerably nicer than the
> best 3- or 5-limit harmonies you will ever get out of Father
> temperament.
>
> So, I'd like to do a little survey of some other "bad" rank-2
> temperaments and see if I can't find variations on the TOP-RMS
> generators that produce better-sounding harmony using optimized
> compound intervals rather than primes. I'm sure this idea probably
> doesn't make much sense, as I don't have the best grasp of how TOP-
> RMS errors are calculated so I'm probably coming at this ass-
> backwards. It could very well be that what I consider a "better-
> sounding but non-optimal Father temperament" is really just a
> different temperament all together, and if so, I'll be glad to be
> corrected.
>
> All the same, if any of you could provide me generators for some of
> the worst rank-2 temperaments you know of, in terms of having both
> high error and high complexity of primes 3, 5, and/or 7, I'd like
> to examine them and see if I can't find related generators that
> produce alternative smoother-sounding harmonies, despite increasing
> the error in the primes.
>
> Thanks!
>
> -Igs