The Superpyth Tonality Challenge

Hi all,

We've thrown a lot of interesting ideas about tonality out there in the past
few weeks.
- I had a great conversation with Kalle, where we discussed an evolved form
of Paul's views on Pajara. I changed my mind about a lot of things because
of this.
- Meanwhile, Petr's been continuing the Great Comma Pump Revolution of 2011,
which is related to tonality and was also very influential on my paradigm of
how tonality works.
- Additionally, I outlined some basic observations that might scale upward
into a theory for melody (sequences of notes agglutinate into the sensation
of "arpeggiated chords," the brain reifies in the consonance of the chord
even though VF perception isn't taking place), although it seems to need
some more tweaking to adequately describe everyone's perception.

At the end of the day, there seem to be a few differing ideas floating
around, and I'm now pretty confused and not sure I understand concretely how
tonality works. So I thought I'd issue a challenge to the members of the
list to apply their understanding of tonality to superpyth[7], which has the
potential to be a gold mine in this subfield of study. Superpyth has the
following properties

1) If we're in C ionian, there are four 4:6:7 "major" chords - over D, E, G,
and A.
2) There are three 4:6:7:9 extended "major" chords - over D, G, and A. E has
a flatted 9th.
3) There are two 14:21:27 "minor" chords - over C and F.
4) There is one ~8:11:14 "I don't know what" chord - over B.

I've been playing around in it on my 22-equal guitar and it sounds amazing -
the 4:6:7 chords from this scale in 22-equal just produce an entirely
different sensation than the same notes would in 12-equal. It's like I'm
deriving another hidden musical universe out of the same diatonic notes
we're all used to. But getting away from the nice sensation of there being
lots of random 4:6:7 chords everywhere and moving towards a tonal system
that can communicate information entirely via the logic of the system is
proving trickier.

So my question is, how would you guys on the list (Kalle, Petr, perhaps Carl
or anyone else with some ideas) analyze this scale? What would the strongest
resolutions be? Would the 8:11:14 work as a higher-limit dominant 7th that
resolves to E? Or would the E resolve to A, because the b2 creates an
instability? I have no idea, but would love to hear everyone's thoughts.

-Mike

Mike wrote:

> So my question is, how would you guys on the list (Kalle, Petr, perhaps Carl
> or anyone else with some ideas) analyze this scale? What would the strongest
> resolutions be? Would the 8:11:14 work as a higher-limit dominant 7th that
> resolves to E? Or would the E resolve to A, because the b2 creates an
> instability? I have no idea, but would love to hear everyone's thoughts.

The mirror inversion of 6:7:8 is 21:24:28 so if the primary target chord is 4:6:7, its harmonic inversion might be something like 14:21:24, which is probably what you wanted to say.

My personal problem is that these "minor" 7-limit chords sound so unresolved to me and unconvincing that I probably wouldn't use them as a "resting state" as I would a 5-limit minor triad.
Perhaps I should play around with it some more and maybe I'll learn to hear it differently one day. So far, I've tried superpyth either favoring chords like 6:7:9 or as a 5-limit temperament.

Petr

On Fri, Jul 29, 2011 at 12:32 PM, petrparizek2000
<petrparizek2000@...> wrote:
>
> The mirror inversion of 6:7:8 is 21:24:28 so if the primary target chord is 4:6:7, its harmonic inversion might be something like 14:21:24, which is probably what you wanted to say.

No, I meant 14:21:27, which is a 3/2 with a 9/7 on top. I wasn't too
concerned with inverses and utonality, but with triad classes -
14:21:27 shares a triad class with 4:6:7 in superpyth[7]. Furthermore,
14:21:27 is sort of analogous to 10:12:15 in meantone[7] in that it
consists of 2/3 with some "crap" thrown in there; the 3/2 on the
bottom makes it rooted but the 9/7 on top doesn't fit in any sort of
meaningful way. The outer dyad is around 1150 cents, which whole thing
sounds sort of ambiguous and maximally unresolved, perhaps between
something like 8:12:15 and 2:3:4 at first, but whatever it is it's
certainly more dissonant and "sadder," whatever you want to call it.

> My personal problem is that these "minor" 7-limit chords sound so unresolved to me and unconvincing that I probably wouldn't use them as a "resting state" as I would a 5-limit minor triad.

I hear both 14:21:24 and 14:21:27 as sounding resolved. The second one
is a bit more dissonant and "ambiguous" at first, to my ear between
8:12:15 and 2:3:4, but I found that after some exposure to the chord I
started to perceive this ambiguous sensation as being just what the
chord "is" - the chord itself is neither of those, but a dissonant
entity that doesn't really resolve either way. This is when it started
to take on the "sad" gestalt.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Hi all,
>
> We've thrown a lot of interesting ideas about tonality out there in the past
> few weeks.
> - I had a great conversation with Kalle, where we discussed an evolved form
> of Paul's views on Pajara. I changed my mind about a lot of things because
> of this.
> - Meanwhile, Petr's been continuing the Great Comma Pump Revolution of 2011,
> which is related to tonality and was also very influential on my paradigm of
> how tonality works.
> - Additionally, I outlined some basic observations that might scale upward
> into a theory for melody (sequences of notes agglutinate into the sensation
> of "arpeggiated chords," the brain reifies in the consonance of the chord
> even though VF perception isn't taking place), although it seems to need
> some more tweaking to adequately describe everyone's perception.
>
> At the end of the day, there seem to be a few differing ideas floating
> around, and I'm now pretty confused and not sure I understand concretely how
> tonality works. So I thought I'd issue a challenge to the members of the
> list to apply their understanding of tonality to superpyth[7], which has the
> potential to be a gold mine in this subfield of study. Superpyth has the
> following properties
>
> 1) If we're in C ionian, there are four 4:6:7 "major" chords - over D, E, G,
> and A.
> 2) There are three 4:6:7:9 extended "major" chords - over D, G, and A. E has
> a flatted 9th.
> 3) There are two 14:21:27 "minor" chords - over C and F.
> 4) There is one ~8:11:14 "I don't know what" chord - over B.
>
> I've been playing around in it on my 22-equal guitar and it sounds amazing -
> the 4:6:7 chords from this scale in 22-equal just produce an entirely
> different sensation than the same notes would in 12-equal. It's like I'm
> deriving another hidden musical universe out of the same diatonic notes
> we're all used to. But getting away from the nice sensation of there being
> lots of random 4:6:7 chords everywhere and moving towards a tonal system
> that can communicate information entirely via the logic of the system is
> proving trickier.
>
> So my question is, how would you guys on the list (Kalle, Petr, perhaps Carl
> or anyone else with some ideas) analyze this scale? What would the strongest
> resolutions be? Would the 8:11:14 work as a higher-limit dominant 7th that
> resolves to E? Or would the E resolve to A, because the b2 creates an
> instability? I have no idea, but would love to hear everyone's thoughts.

Some observations:

In my ears the ~8:11 of B-F strongly wants to contract into ~9:7 of
C-E. Try D B F A - A C E A for example.

I can't hear your "minor chords" mentioned in 3) as consonant.

Kalle

On Fri, Aug 12, 2011 at 2:21 PM, Kalle Aho <kalleaho@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Some observations:
>
> In my ears the ~8:11 of B-F strongly wants to contract into ~9:7 of
> C-E. Try D B F A - A C E A for example.

So what would be an example dominant -> tonic resolution? Would D B F
A - A C E A be strongest? How about G-F-B -> C-E-C, and hence resolve
to 7:9:14?

> I can't hear your "minor chords" mentioned in 3) as consonant.

You can get used to them, I really like them now. I wonder what minor
chords would sound like if we had only played major chords for all of
our lives.

Lastly, what about mavila[9]?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Aug 12, 2011 at 2:21 PM, Kalle Aho <kalleaho@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > Some observations:
> >
> > In my ears the ~8:11 of B-F strongly wants to contract into ~9:7 of
> > C-E. Try D B F A - A C E A for example.
>
> So what would be an example dominant -> tonic resolution? Would D B F
> A - A C E A be strongest? How about G-F-B -> C-E-C, and hence resolve
> to 7:9:14?

I believe it depends on which chord you want to act as the tonic. The
chord that then functions as the dominant (in one generalized sense)
will be a chord that requires the tonic in order to resolve in the
most satisfactory way. Obviously there might not be a dominant for
every tonic. There are probably both psychoacoustic reasons and
reasons based on custom for hearing a chord as the dominant. These
are intentionally left out of the definition.

One problem with this idea is that we don't think of suspensions as
having a dominant function although in common practice they resolve in
extremely predictable patterns. But one possible response to this is
that in common practice they are not really independent chords but
rather contrapuntal simultaneities resulting from voice leading. This
is supported by the fact that the quartal chords of popular music
indeed move more freely.

> > I can't hear your "minor chords" mentioned in 3) as consonant.
>
> You can get used to them, I really like them now. I wonder what minor
> chords would sound like if we had only played major chords for all of
> our lives.

I think that there is more to the minor triad than the fifth + crap
principle. All the dyads are 5-limit and I really believe this makes
it more concordant than other fifth + crap triads.

> Lastly, what about mavila[9]?

That'll have to wait for another time. :)

Kalle

--- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@...> wrote:

> I think that there is more to the minor triad than the fifth + crap
> principle. All the dyads are 5-limit and I really believe this makes
> it more concordant than other fifth + crap triads.

What do you think of fifth + 7/6, which is otonal?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Kalle Aho" <kalleaho@> wrote:
>
> > I think that there is more to the minor triad than the fifth + crap
> > principle. All the dyads are 5-limit and I really believe this makes
> > it more concordant than other fifth + crap triads.
>
> What do you think of fifth + 7/6, which is otonal?

Okay, yes, 6:7:9 is another chord that is more than just fifth + crap
even when 6 is heard as the root. It's hard to say which one is more
concordant, 10:12:15 or 6:7:9. Perhaps they are equally concordant
but in a different way. There is something very concordant about
otonal chords low in the harmonic series that has nothing to do
with their perceived root.

Kalle

On Sat, Aug 13, 2011 at 12:37 PM, Kalle Aho <kalleaho@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > So what would be an example dominant -> tonic resolution? Would D B F
> > A - A C E A be strongest? How about G-F-B -> C-E-C, and hence resolve
> > to 7:9:14?
>
> I believe it depends on which chord you want to act as the tonic. The
> chord that then functions as the dominant (in one generalized sense)
> will be a chord that requires the tonic in order to resolve in the
> most satisfactory way. Obviously there might not be a dominant for
> every tonic. There are probably both psychoacoustic reasons and
> reasons based on custom for hearing a chord as the dominant. These
> are intentionally left out of the definition.

And you're again defining "dominant" as any chord that sets you up to
"resolve" to another chord?

> > > I can't hear your "minor chords" mentioned in 3) as consonant.
> >
> > You can get used to them, I really like them now. I wonder what minor
> > chords would sound like if we had only played major chords for all of
> > our lives.
>
> I think that there is more to the minor triad than the fifth + crap
> principle. All the dyads are 5-limit and I really believe this makes
> it more concordant than other fifth + crap triads.

There is more to the minor triad than the fifth + crap principle, but
there is a sense in which the chords I listed above start to take on a
sort of "sad" quality in superpyth[7]. No, it doesn't have every
single quality that minor triads do, most of all not the sort of
polytonal setup where you can hear more than one root in 10:12:15,
which no doubt determines a large percentage of the minor "sound," but
they do sound "sad" in the proper context and that's worth something
to me.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Aug 13, 2011 at 12:37 PM, Kalle Aho <kalleaho@...> wrote:
> >
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > > So what would be an example dominant -> tonic resolution? Would D B F
> > > A - A C E A be strongest? How about G-F-B -> C-E-C, and hence resolve
> > > to 7:9:14?
> >
> > I believe it depends on which chord you want to act as the tonic. The
> > chord that then functions as the dominant (in one generalized sense)
> > will be a chord that requires the tonic in order to resolve in the
> > most satisfactory way. Obviously there might not be a dominant for
> > every tonic. There are probably both psychoacoustic reasons and
> > reasons based on custom for hearing a chord as the dominant. These
> > are intentionally left out of the definition.
>
> And you're again defining "dominant" as any chord that sets you up to
> "resolve" to another chord?

It has to be an unstable chord that clearly prefers the tonic as its'
resolution. It can resolve to something else as in a deceptive cadence
but only the tonic chord completely removes all the tension it
evokes.

Kalle

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> So my question is, how would you guys on the list (Kalle, Petr, perhaps Carl
> or anyone else with some ideas) analyze this scale? What would the strongest
> resolutions be? Would the 8:11:14 work as a higher-limit dominant 7th that
> resolves to E? Or would the E resolve to A, because the b2 creates an
> instability? I have no idea, but would love to hear everyone's thoughts.

First of all, in the past when I've played in superpyth I've mostly stuck to a neo-Gothic, only-3-is-consonant paradigm a la Margo Schulter. ( http://www.medieval.org/emfaq/harmony/pyth.html ) Listen to some Perotin if you want to hear great examples of this kind of harmony. So it's really interesting to use it in a different way where 7 and 9 are consonant too.

When I'm trying to accept 4:6:7 and 4:6:7:9 as consonant chords and avoid always resolving to fifths and 2:3:4 "trines", I think I really prefer roots moving by seconds rather than by fourths or fifths, if only because of the better voice leading. For example, in your B -> E progression, what voice is supposed to move to the D of the E chord? That note kind of has to come out of nowhere. B -> A makes a lot more sense to me. Similarly moving between A and G makes more sense than trying to use D as a "subdominant" for A. I mean, all of them sound good, but the ones that move by seconds seem like they're stronger or work out better.

I guess you don't really worry about these things as much on a guitar, because you just play a big chord and don't worry about what the individual "voices" are doing. But the idea of "roots moving by seconds" might still be interesting to you.

Lastly, there are some wonderful chords waiting just a little outside the 7-note "diatonic" MOS. For example G-B-Eb is a great 7:9:11 chord that I hear as "resolving" to A in a way, even though both chords are part of the same harmonic series. I just love that neutral second between Eb and E. (I guess I'm really using the 2.3.7.11 temperament "supra" here, not superpyth, because I like that B-Eb is definitely a neutral third and not a 6/5. In 22-equal they're the same interval, but maybe the thing I'm talking about still works.)

http://x31eq.com/cgi-bin/rt.cgi?ets=5+12&limit=2.3.7.11

Keenan