2.3.7.13 subgroups and "ultrapyth" tunings

Although the supermajor triad that everyone usually talks about is
14:18:21, 10:13:15 has some interesting properties. It's lower in
tenney height, but all of the dyads are higher in Tenney height than
those of 14:18:21. This will probably make for music that has some
interesting quirks, like that triads will be more concordant than
dyads, as opposed to superpyth supermajor where it's the other way
around. I personally find 10:13:15 to be more concordant overall than
14:18:21 - it doesn't just sound like an annoyingly sharp 4:5:6, but
like its own thing. Maybe that's just me.

Another interesting property is that it can be combined with the minor
chord to make 10:12:13:15, which is decently well represented in
19-tet (10:12:13 is better).

Anyway, the utonal inversion of 10:13:15 is 26:30:39, and although
this is a sound I could probably get used to, it's close enough to
6:7:9 to consider tempering. If we temper things so that 26:30:39 and
6:7:9 get equated, and conversely so that 10:13:15 and 14:18:21 get
equated, then we have decent triads both ways. This means that 91/90
vanishes. That the comma vanishing is actually "91/90" makes me happy
for very obvious and shallow reasons.

So let's say we have a 2.3.7.13 subgroup temperament - tempering out
91/90 puts us at rank 3. Another choice to bring us down to rank 2
might be to temper out 64/63, and then we get a rank 2 "ultrapyth"
tuning in which 27-equal seems like a pretty good choice.

Any other ideas based on this train of thought?

-Mike

The last property, which I thought too obvious to mention but should
probably include anyway for the sake of completeness, is that 10:13:15
is roughly equivalent in Tenney height to the 5-limit minor triad.

Furthermore, the 5-limit minor triad also has a "rooted" counterpart
at 16:19:24. There has long been a debate about which one is the
"real" minor triad - since we don't usually play 10:12:15 in
isolation, but with the root doubled several octaves down, it might be
more that the perception of the minor triad is more related to
1:2:4:8:16:19:24 than 5:10:20:40:48:60.

Likewise, the 13-limit ultramajor triad has a rooted counterpart at
16:21:24. The difference between 19/16 and 6/5 is 18 cents; the
difference between 21/16 and 13/10 is even less at 16.6 cents. The
main difference is that 16:21 is closer to the 4:3 field of
attraction, and that it's also generally perceived as a "fourth"; this
would necessitate some clever musical context being used.

Lastly, 10:13:15 is one of a few triads with reasonable Tenney height
that have a 3/2 as an outer dyad and where the lower two dyads are
split relatively evenly; e.g. they are sized something like "a third."
Here's the list:

4:5:6 - 5-limit major triad
6:7:9 - 7-limit subminor triad
10:12:15 - 5-limit minor triad
10:13:15 - **13-limit ultramajor triad**
14:17:21 - **17-limit superminor triad**
14:18:21 - 7-limit supermajor triad
16:19:24 - 19-limit minor triad
16:21:24 - doesn't really belong here, but maybe works as the 7-limit
ultramajor triad if you can manage this via musical context

Anyway, 14:18:21 is already too hard for a lot of people to swallow as
being concordant, so I'll stop it there. The only reason I went up to
16:18:21 is because 16 is a power of 2, so if you're playing octave
equivalent music you get a huge rootedness boost (since you can double
the root up to 4 octaves in the bass without having to double the rest
of the numbers in the triad).

There are only two triads in this list that aren't generally talked
about much, and the lowest is 10:13:15. The next one is 14:17:21,
which should be more absolutely concordant than 14:18:21, and is a
"superminor" triad in the same way that 14:18:21 is a supermajor
triad. Thus in terms of triads with a fifth on the outside, 10:13:15
is the next logical xenharmonic choice to explore after 6:7:9.

This is assuming that you aren't immediately turned off by 10:13:15,
as some have been - I guess it's one of those chords you either love
or hate.

-Mike

On Fri, Jan 7, 2011 at 12:09 AM, Mike Battaglia <battaglia01@...> wrote:
> Although the supermajor triad that everyone usually talks about is
> 14:18:21, 10:13:15 has some interesting properties. It's lower in
> tenney height, but all of the dyads are higher in Tenney height than
> those of 14:18:21. This will probably make for music that has some
> interesting quirks, like that triads will be more concordant than
> dyads, as opposed to superpyth supermajor where it's the other way
> around. I personally find 10:13:15 to be more concordant overall than
> 14:18:21 - it doesn't just sound like an annoyingly sharp 4:5:6, but
> like its own thing. Maybe that's just me.
>
> Another interesting property is that it can be combined with the minor
> chord to make 10:12:13:15, which is decently well represented in
> 19-tet (10:12:13 is better).
>
> Anyway, the utonal inversion of 10:13:15 is 26:30:39, and although
> this is a sound I could probably get used to, it's close enough to
> 6:7:9 to consider tempering. If we temper things so that 26:30:39 and
> 6:7:9 get equated, and conversely so that 10:13:15 and 14:18:21 get
> equated, then we have decent triads both ways. This means that 91/90
> vanishes. That the comma vanishing is actually "91/90" makes me happy
> for very obvious and shallow reasons.
>
> So let's say we have a 2.3.7.13 subgroup temperament - tempering out
> 91/90 puts us at rank 3. Another choice to bring us down to rank 2
> might be to temper out 64/63, and then we get a rank 2 "ultrapyth"
> tuning in which 27-equal seems like a pretty good choice.
>
> Any other ideas based on this train of thought?
>
> -Mike
>

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

> Any other ideas based on this train of thought?

I could do a search for the lowest badness figures for 91/90 temperaments of full rank, or your subgroup of choice. One possibility, for instance, would be to also temper out 169/168, 385/384 and 441/440; 46 works pretty well for this. 11 is the most complex prime in this temperament, and eliminating 11 and going with 2.3.5.7.13 would be logical; commas defining the temperament could then be 91/90, 169/168 and 2048/2025. A 20-note MOS would give plenty of scope for this to operate.

Map: [<2 2 7 6 0 7|, <0 3 -6 -1 0 1|]
POTE generator: 235.027

If you also want 11:

Map: [<2 2 7 6 3 7|, <0 3 -6 -1 10 1|]
POTE generator: 235.088

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

> So let's say we have a 2.3.7.13 subgroup temperament - tempering out
> 91/90 puts us at rank 3.

By the way, though I don't know what you mean by a 2.3.7.13 temperament tempering out 91/90, we can also ditch the 5 from my example, getting this:

Commas: 169/168, 1029/1024
Map: [<2 2 0 6 0 7|, <0 3 0 -1 0 1|]
EDOs: 10, 16, 26, 36, 46, 62, 72
POTE generator: 233.604

On Fri, Jan 7, 2011 at 1:08 AM, genewardsmith
<genewardsmith@...> wrote:
>
> I could do a search for the lowest badness figures for 91/90 temperaments of full rank, or your subgroup of choice. One possibility, for instance, would be to also temper out 169/168, 385/384 and 441/440; 46 works pretty well for this. 11 is the most complex prime in this temperament, and eliminating 11 and going with 2.3.5.7.13 would be logical; commas defining the temperament could then be 91/90, 169/168 and 2048/2025. A 20-note MOS would give plenty of scope for this to operate.
>
> Map: [<2 2 7 6 0 7|, <0 3 -6 -1 0 1|]
> POTE generator: 235.027
>
> If you also want 11:
>
> Map: [<2 2 7 6 3 7|, <0 3 -6 -1 10 1|]
> POTE generator: 235.088

Hi Gene,

Thanks for this! The 11 note MOS is beautiful here, although the 25
cent steps are a little bit weird. I also note that 31 and 41 seems to
support this temperament, which I guess stems because this is in the
gamelan clan?

I'm also confused on how to read the map - I thought the map was
supposed to look something like <12 19 28 34|... ? But I guess this is
the 5&26 temperament, based on the continued fraction approximation to
the generator.

Could we maybe try 2.3.7.13, or perhaps 2.3.7.11.13? It would be
really nice to get away from 5 for a change. I'm also hoping that from
this will spring some kind of diatonic-sized temperament, for tonal
purposes.

-Mike

On Fri, Jan 7, 2011 at 1:20 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > So let's say we have a 2.3.7.13 subgroup temperament - tempering out
> > 91/90 puts us at rank 3.
>
> By the way, though I don't know what you mean by a 2.3.7.13 temperament tempering out 91/90, we can also ditch the 5 from my example, getting this:
>
> Commas: 169/168, 1029/1024
> Map: [<2 2 0 6 0 7|, <0 3 0 -1 0 1|]
> EDOs: 10, 16, 26, 36, 46, 62, 72
> POTE generator: 233.604

Sorry, you're right. I guess I meant to say that it would be nice to
get away from optimizing for 4:5:6 and 10:12:15, and try to optimize
for 10:13:15 and 6:7:9 instead. But obviously 5 is involved. Maybe it
would be best to represent this as some kind of non-prime subgroup
temperament, although at first glance I'm not sure how to do that...

-Mike

On Fri, Jan 7, 2011 at 1:27 AM, Mike Battaglia <battaglia01@...> wrote:
>
> Sorry, you're right. I guess I meant to say that it would be nice to
> get away from optimizing for 4:5:6 and 10:12:15, and try to optimize
> for 10:13:15 and 6:7:9 instead. But obviously 5 is involved. Maybe it
> would be best to represent this as some kind of non-prime subgroup
> temperament, although at first glance I'm not sure how to do that...

I figured it out. I guess it would be something like 2.3.7.13/5. I'm
not really sure what a normal list is, though, or how to put it in
normal list form...

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Thanks for this! The 11 note MOS is beautiful here, although the 25
> cent steps are a little bit weird.

There's no 11 note MOS, as the period is 1/2 octave, so I don't know what you are looking at.

> Could we maybe try 2.3.7.13, or perhaps 2.3.7.11.13?

Important 2.3.7.13 commas are 64/63, 169/168, 1029/1024, 729/728 and 16848/16807, and you get a rank two temperament from any pair; if you drop 64/63 these are even all good ones. Does "could we try" mean you want a scale, or what, exactly?

On Fri, Jan 7, 2011 at 2:13 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > Thanks for this! The 11 note MOS is beautiful here, although the 25
> > cent steps are a little bit weird.
>
> There's no 11 note MOS, as the period is 1/2 octave, so I don't know what you are looking at.

I saw that the generator was 235.027, so I just plugged it into Scala
and let er fly. I didn't realize the period was a half octave - is
that somehow represented in the map?

> > Could we maybe try 2.3.7.13, or perhaps 2.3.7.11.13?
>
> Important 2.3.7.13 commas are 64/63, 169/168, 1029/1024, 729/728 and 16848/16807, and you get a rank two temperament from any pair; if you drop 64/63 these are even all good ones. Does "could we try" mean you want a scale, or what, exactly?

I guess that after thinking about it, I meant 2.3.7.13/5, although
like I said I'm still not sure how to represent that as a normal list.
10.13.15.35, I guess? Not really sure. What are some good 10.13.15.35
commas?

-Mike

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

> I figured it out. I guess it would be something like 2.3.7.13/5. I'm
> not really sure what a normal list is, though, or how to put it in
> normal list form...

2.3.7.13/5 is a normal list, but it doesn't need to be to stick it into Graham's temperament finder. These come complete with Scala scale files these days, though the choice of scale sizes is often a little goofy.

On Fri, Jan 7, 2011 at 2:25 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > I figured it out. I guess it would be something like 2.3.7.13/5. I'm
> > not really sure what a normal list is, though, or how to put it in
> > normal list form...
>
> 2.3.7.13/5 is a normal list, but it doesn't need to be to stick it into Graham's temperament finder. These come complete with Scala scale files these days, though the choice of scale sizes is often a little goofy.

Ah, thanks, I didn't realize I was able to put subgroups that weren't
integers in. I'll do some more research.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I saw that the generator was 235.027, so I just plugged it into Scala
> and let er fly. I didn't realize the period was a half octave - is
> that somehow represented in the map?

Yes, the first val is the period map, and the second is the generator map. The "2" as the first number of the period map tells you the period is 1/2 octave.

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

> Ah, thanks, I didn't realize I was able to put subgroups that weren't
> integers in. I'll do some more research.

If you want to temper 1-13/10-3/2 triads, the first order of business is to forget 7 for the moment and look at 2.3.13/5 commas: small fractions that can be written as 2^i 3^j (13/5)^k. Important commas of this type are 40/39, 676/675 and 2250/2197. 676/675 is especially interesting because it combines low complexity with high accuracy.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

676/675 is especially interesting because it combines low complexity with high accuracy.
>

If you want to stick completely to 2.3.13/5, then using 676/675 you get a very accurate temperament of low complexity for your triads. From 676/675, two 15/13 generators give 4/3; the Graham complexity of a triad is 3, less than the 4 of meantone, and the accuracy is much better. The 5, 9 or 14 note MOS should be just the ticket. What do you say to a pentatonic scale with a generator of 248.889 cents?

On Fri, Jan 7, 2011 at 3:54 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> 676/675 is especially interesting because it combines low complexity with high accuracy.
> >
>
> If you want to stick completely to 2.3.13/5, then using 676/675 you get a very accurate temperament of low complexity for your triads. From 676/675, two 15/13 generators give 4/3; the Graham complexity of a triad is 3, less than the 4 of meantone, and the accuracy is much better. The 5, 9 or 14 note MOS should be just the ticket. What do you say to a pentatonic scale with a generator of 248.889 cents?

I say that this is such an obvious choice that I don't know why I ever
missed this. There's a 9-note MOS here that is absolutely beautiful.
It's like gamelan meets diatonic or something. I wonder what happens
if we crossbreed this with 64/63...

The harmonies sound almost "hollow," in a sense, I guess because we
really want 5-limit stuff in there but it's not.

-Mike

On Fri, Jan 7, 2011 at 4:26 AM, Mike Battaglia <battaglia01@...> wrote:
>
> I say that this is such an obvious choice that I don't know why I ever
> missed this. There's a 9-note MOS here that is absolutely beautiful.
> It's like gamelan meets diatonic or something. I wonder what happens
> if we crossbreed this with 64/63...

Sorry, I didn't mean 64/63, I meant 91/90. I guess that would
eliminate 49/48 as well, so the fourths would be pretty flat and the
fifths sharp.

There's also a really beautiful 14-note MOS that I need to mess around
with more. Seems that it would be a good option to eliminate 81/80
here, if one wanted to start bringing 5 back in. I guess that 81/80
and 91/90 both vanishing wouldn't be ideal, because then the generator
gets pulled in two opposite directions.

-Mike