Some functional harmony in porcupine temperament

Carl has brought up that we're in the pre-functional stage of regular
temperament. So I sought to make some "functional" comma pumps, with
slight variations.

Let's start with porcupine. Here are three variations of a porcupine
comma pump. The basic chord progression is, in meantone notation -
Cmaj -> A/C# -> Dmaj -> B/D# -> Emaj -> C#/E# -> F#maj -> F#m6 ->
C#maj, except the whole thing is tempered such that the C#maj at the
end is actually the same as the Cmaj that you started at, because
250/243 vanishes. The second progression differs from the first in
that it makes Dmaj and Emaj into Dm and Em, to even further spoof
diatonic harmony. The last progression makes Dmaj into Dm, but keeps
Emaj the same, where if you reframe your perspective you will realize
that it's actually somehow now Ebmaj due to the tempering. This is
more consistent with the porcupine[8] structure, with three generators
going up and the rest going down.

Here's the first set of three:

http://soundcloud.com/mikebattagliamusic/functional-porcupine-no-7

Here's another set with all of the secondary dominants turned into
dominant 7ths (using the approximation to 7/4):

http://soundcloud.com/mikebattagliamusic/functional-porcupine-with-7

This is an 11-limit shredfest that throws higher-limit harmonies into
the mix, all of which flirt loosely with your newly established
porcupine hearing. They flirt with the underlying porcupine structure
in an analogous manner to how the blues, Gershwin, etc flirts loosely
with an underlying meantone structure:

http://soundcloud.com/mikebattagliamusic/functional-porcupine-11-limit

These sound very "functional" to me. I think this says a lot about how
functional harmony works, but I'd like to hear your feedback before I
get into what the "rules" were that I used to write these
progressions.

-Mike

--- Mike Battaglia <battaglia01@...> wrote:

> Here's another set with all of the secondary dominants turned
> into dominant 7ths (using the approximation to 7/4):
> http://soundcloud.com/mikebattagliamusic/functional-porcupine-with-7
[snip]
> http://soundcloud.com/mikebattagliamusic/functional-porcupine-11-limit

Nice!!

> These sound very "functional" to me. I think this says a lot about
> how functional harmony works, but I'd like to hear your feedback
> before I get into what the "rules" were that I used to write these
> progressions.

They sound like good chord progressions. A bit bare bones to
pronounce functional for my money. But definitely interesting.

-Carl

On Fri, Apr 22, 2011 at 7:38 PM, Carl Lumma <carl@...> wrote:
>
> They sound like good chord progressions. A bit bare bones to
> pronounce functional for my money. But definitely interesting.

Does this sound more functional to you?

http://soundcloud.com/mikebattagliamusic/functionalporcupineexcerpt

I hear it moving away from the tonic, then suddenly arriving at 4/3
above the root, and then you can do stuff with that. I'm trying to
generalize how in meantone, you can move to the ii chord as a motion
away from the tonic, but then you can go to the V chord immediately
afterward smoothly, because 81/80 vanishes. That in itself is a tiny
comma pump, all of which I feel are adequately generalized by Petr's
examples.

If you analyze the above as though it were a four part dictation,
there are diminished chords resolving to major chords and the like,
and chords 3/2 up resolving to chords 3/2 down, etc. I designed it so
that there are no awkward 81/80 jumps, but if you ever try to play
this in a temperament where 250/243 doesn't vanish, you're going to
have some awkward 250/243 jumps.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Carl has brought up that we're in the pre-functional stage of regular
> temperament. So I sought to make some "functional" comma pumps, with
> slight variations.

I think I'll put up a page of comma pump examples on the Xenwiki, but I don't think SoundCloud is the ideal place to link to.

Yes. These all work for me. It would be nice if they were given some breathing room, hang out on each chord and arpeggiate it a bit or something so that the tension really gets to build up and release. Great job, though! I even liked the "11-limit shredfest". It was odd and maybe a bit too "fused" for me but it still worked.

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Carl has brought up that we're in the pre-functional stage of regular
> temperament. So I sought to make some "functional" comma pumps, with
> slight variations.
>
> Let's start with porcupine. Here are three variations of a porcupine
> comma pump. The basic chord progression is, in meantone notation -
> Cmaj -> A/C# -> Dmaj -> B/D# -> Emaj -> C#/E# -> F#maj -> F#m6 ->
> C#maj, except the whole thing is tempered such that the C#maj at the
> end is actually the same as the Cmaj that you started at, because
> 250/243 vanishes. The second progression differs from the first in
> that it makes Dmaj and Emaj into Dm and Em, to even further spoof
> diatonic harmony. The last progression makes Dmaj into Dm, but keeps
> Emaj the same, where if you reframe your perspective you will realize
> that it's actually somehow now Ebmaj due to the tempering. This is
> more consistent with the porcupine[8] structure, with three generators
> going up and the rest going down.
>
> Here's the first set of three:
>
> http://soundcloud.com/mikebattagliamusic/functional-porcupine-no-7
>
> Here's another set with all of the secondary dominants turned into
> dominant 7ths (using the approximation to 7/4):
>
> http://soundcloud.com/mikebattagliamusic/functional-porcupine-with-7
>
> This is an 11-limit shredfest that throws higher-limit harmonies into
> the mix, all of which flirt loosely with your newly established
> porcupine hearing. They flirt with the underlying porcupine structure
> in an analogous manner to how the blues, Gershwin, etc flirts loosely
> with an underlying meantone structure:
>
> http://soundcloud.com/mikebattagliamusic/functional-porcupine-11-limit
>
> These sound very "functional" to me. I think this says a lot about how
> functional harmony works, but I'd like to hear your feedback before I
> get into what the "rules" were that I used to write these
> progressions.
>
> -Mike
>

Here's a 15-equal version as well, which retains the functionality of
the original:

http://soundcloud.com/mikebattagliamusic/functionalporcupineexcerpt15tet

The weakest part here is the dom7 chords, which just sound terrible in
15-tet no matter how you slice it. However, this is made up by the
stronger melody in the bass - the chromatic semitone is 80 cents in
this tuning, whereas it's closer to 50 cents in 22-tet. So the voice
leading in the bass works much better.

Plus, on top of that, 15-tet is small and fits nicely into most
compact spaces. So despite that it sounds kind of warbly, it's good
enough if you're ever trapped on a desert island, only have a 15-tet
guitar handy, and are badly in need of a porcupine temperament. Plus,
I think the warbliness gives it kind of an endearing quality,
myself...

But other than that, it is apparently true that common-practice style
functional harmony exists in 15-tet that won't work in 12-tet. Ha!

-Mike

On Fri, Apr 22, 2011 at 8:00 PM, Mike Battaglia <battaglia01@...> wrote:
> On Fri, Apr 22, 2011 at 7:38 PM, Carl Lumma <carl@...> wrote:
>>
>> They sound like good chord progressions. A bit bare bones to
>> pronounce functional for my money. But definitely interesting.
>
> Does this sound more functional to you?
>
> http://soundcloud.com/mikebattagliamusic/functionalporcupineexcerpt
>
> I hear it moving away from the tonic, then suddenly arriving at 4/3
> above the root, and then you can do stuff with that. I'm trying to
> generalize how in meantone, you can move to the ii chord as a motion
> away from the tonic, but then you can go to the V chord immediately
> afterward smoothly, because 81/80 vanishes. That in itself is a tiny
> comma pump, all of which I feel are adequately generalized by Petr's
> examples.
>
> If you analyze the above as though it were a four part dictation,
> there are diminished chords resolving to major chords and the like,
> and chords 3/2 up resolving to chords 3/2 down, etc. I designed it so
> that there are no awkward 81/80 jumps, but if you ever try to play
> this in a temperament where 250/243 doesn't vanish, you're going to
> have some awkward 250/243 jumps.
>
> -Mike
>

On Fri, Apr 22, 2011 at 8:35 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > Carl has brought up that we're in the pre-functional stage of regular
> > temperament. So I sought to make some "functional" comma pumps, with
> > slight variations.
>
> I think I'll put up a page of comma pump examples on the Xenwiki, but I don't think SoundCloud is the ideal place to link to.

What's wrong with SoundCloud? I could put them up on archive.org, but
I really don't feel like uploading every single one of them. Do you
think that just the two mini-compositions in 22-tet and 15-tet would
be okay? They use the 1/1 -> up 10/9 -> up 10/9 -> up 10/9 = up 4/3
comma pump, and then its inverse for the third phrase.

-Mike

On Fri, Apr 22, 2011 at 8:48 PM, cityoftheasleep
<igliashon@...> wrote:
>
> Yes. These all work for me. It would be nice if they were given some breathing room, hang out on each chord and arpeggiate it a bit or something so that the tension really gets to build up and release. Great job, though! I even liked the "11-limit shredfest". It was odd and maybe a bit too "fused" for me but it still worked.

Well, there you go. You've now mentally followed a chord progression
in porcupine temperament. To hell with all of this extensive training
period crap. But, to be honest, none of it really sounds xenharmonic
to me. It basically sounds like common practice music that just so
happens to have chord progressions that connect differently. But at
least this is one step closer.

I guess the next step is to generalize modality, which means I have to
mentally prepare myself for the psychoacoustics war over harmonic
entropy that will surely follow.

-Mike

--- Mike Battaglia <battaglia01@...> wrote:

> Does this sound more functional to you?
> http://soundcloud.com/mikebattagliamusic/functionalporcupineexcerpt

I dunno, but you're kicking some major ass now. What tuning
did you use for the "higher limit extensions" version?

> but if you ever try to play this in a temperament where 250/243
> doesn't vanish, you're going to have some awkward 250/243 jumps.

You should make a version with those! -Carl

I don't want to compare Mike's work and mine since each of us seems to have different intentions or goals here. Mike's progressions certainly are a good idea but 1) they don't demand using as few generators as possible (which is one of the keys for finding the primary harmonic possibilities of a temperament) and 2) they remind me of a fifth-based system, which meantone is and porcupine isn't. OTOH, what I was doing is trying to find progressions based on the unison vector itself -- i.e. progressions characteristic for the particular harmonic system coming from that particular temperament. This is also what has lead me to the concept of diatonic and chromatic steps last year. I'm still not definitely sure about it but after doing a few more experiments and verifications, I'll probably post my comma pump article publicly. To make a long story short, a progression like "C-major, A-major, D-major" uses chromatic steps in many temperaments in the first place, including meantone, porcupine, hanson or others. So this is a different category of comma pumps than mine. Both have their meanings but they can't be compared.

Petr

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

> What's wrong with SoundCloud? I could put them up on archive.org, but
> I really don't feel like uploading every single one of them.

Let's please not use archive.org. I want to directly link to the mp3s, like I will to Petr's ogg files, and the simplest solution would be to ask Chris to host them.

Do you
> think that just the two mini-compositions in 22-tet and 15-tet would
> be okay? They use the 1/1 -> up 10/9 -> up 10/9 -> up 10/9 = up 4/3
> comma pump, and then its inverse for the third phrase.

I'm just interested in the pumps, not in how you can goof up the tuning for them, so just 22 would be fine for that purpose.

I wrote:

> using as few generators as possible (which is one of the keys for finding > the primary harmonic possibilities of a temperament)

More precisely, I should have said "as few generators as possible with major/minor triads" -- i.e. 7 for meantone, 8 for porcupine, 11 for hanson and so on.

What I was trying to say there was that there's a difference between "making progressions similar to the familiar ones" and "learning about a new harmonic system". If your aim is to get better understanding about a new harmonic system, you'll probably get only a small step further if you start exploring these "chromatic" pumps prior to the "non-chromatic" ones.

For example, I could obviously make a progression like "C major, A major, D major, G major, C major" in meantone -- or to go still further, "C major, A major, D minor, G minor, C major". In the former case, there's a chromatic step even within the first two chords. In the latter, the number of generators required is as high as 10, which doesn't make a lot of sense if our primary goal is to understand the basic (or "non-chromatic") properties of a particular harmonic system, which I think should be done first, otherwise the chromatic ones just look like mess -- as if you learned about powers and roots before understanding multiplication and division.

And one more thing. If most European music were based on diaschismatic or negri instead of meantone, then the "most familiar" progressions would certainly be different and we would probably be much more used to things like "Db major, F minor, C major" rather than "D minor, G major, C major". Let's imagine all of us here exploring meantone then in a similar way we're exploring the other 2D temperaments now. Would we then be keen on making meantone pumps full of things like "E minor, B major, D# minor, A# major" even though these are not characteristic for meantone? If we wanted to get close to the familiar ones, we probably would. If we wanted, instead, to learn more about the new harmonic system offered by meantone tempering, well, then we would have to get rid of thinking in the other one first. And that is possible (which is what I've been trying to say here all the time) if you know what's tempered out. Honestly, it took me quite some time before I really admitted that "this or that" *can* indeed work and that it isn't a nonsense -- you know, because I was for all those years stuck in the one harmonic system based on meantone. Once I understood what those unison vectors meant, I was finally able to remove that barrier of meantone thinking. This is also why I'm always trying to explain a new harmonic system using a 2D temperament rather than an EDO.

Petr

On Sat, Apr 23, 2011 at 2:43 AM, Carl Lumma <carl@...> wrote:
>
> --- Mike Battaglia <battaglia01@...> wrote:
>
> > Does this sound more functional to you?
> > http://soundcloud.com/mikebattagliamusic/functionalporcupineexcerpt
>
> I dunno, but you're kicking some major ass now. What tuning
> did you use for the "higher limit extensions" version?

Everything's in 22 except for the one 15-tet example. I'm going to
come up with a higher-limit extension version of the functional
excerpt next.

> > but if you ever try to play this in a temperament where 250/243
> > doesn't vanish, you're going to have some awkward 250/243 jumps.
>
> You should make a version with those! -Carl

Sure, but these are a huge pain in the ass to do, and the
higher-numbered the EDO, the more irritating it is. What if I did one
in 19-equal? Would that satisfy?

-Mike

On Sat, Apr 23, 2011 at 6:44 AM, Petr Parízek <petrparizek2000@...> wrote:
> 1) they don't demand using as few generators as possible (which is
> one of the keys for finding the primary harmonic possibilities of a
> temperament)

I'm not sure I agree with this. I thought so for a long time, but now
I don't anymore.

We often strive for this ideal on here to "get rid of our meantone
training" and learn to think in a new system. But if you limit
yourself to something like this, you're going to end up sticking
mainly to the scale's MOS's, whereas the MODMOS's may be really
harmonically useful as well. This requirement that we stick only to
MOS's and use few generators isn't something I really think is
necessary. The harmonic minor scale gives us a historical context for
why I think this is true.

For example, let's pretend that instead of exploring negri, we're from
a culture that only uses functional harmony in the meantone major
scale, and we're now exploring the minor scale for the first time. How
would we have things resolve? Should we use v -> i now, because
there's no V chord in aeolian? Is the tendency to want to use V -> i
just a learned component of our "major thinking?" Should we force
ourselves to not use it so that we can learn the true nature of the
minor scale? Well, that's not what they decided to do - they decided
they liked V -> i because they just liked it, so they sharpened the
seventh and created the harmonic minor scale. They didn't care about
the chain of fifths or anything, they just knew that V -> i was a
pleasing sound. This requires 10 generators, but it's okay, because by
using things like MODMOS's you can still get it to 7 notes.

This indicates to me that something like Gmaj -> Cmaj, or Gmaj -> Cm,
is just a pleasing sound, probably because of the combined
psychoacoustic payload of the leading tone and the root movement by
3/2. That means that I think that having a major chord (or a 4:5:6:7)
a 3/2 up from the root just "resolves" strongly down to the root in
general, and I don't think that means that we're stuck in meantone
thinking or that we need to ignore this impulse. And even if we want
to ignore it for some theoretical ideal - composers will probably not
care and just go with what sounds "natural" to them. And just as we
designed the harmonic minor scale to have a leading tone in it, I
predict that the same will happen with these scales, so that perhaps
instead of using 4 3 3 3 3 3 3 for porcupine[7], we might start using
4 3 3 3 3 4 2. Or maybe even 4 3 3 3 3 2 4, for dominant chords. These
are first-order MODMOS's of porcupine[7] in the same way that the
harmonic minor scale with its sharpened leading tone is a first-order
MODMOS of meantone[7].

> and 2) they remind me of a fifth-based system, which meantone
> is and porcupine isn't.

That's interesting you say that, because this progression won't work
in meantone, only porcupine :)

But motion by fifth is a valid musical technique - isn't that a pretty
fundamental concept? It's just motion by 3/2 - why should we assume
that we need to "get away from that?" I understand there may also be
some new techniques we haven't discovered as well, but I see no reason
that we should feel uncomfortable moving chords around by 3/2.

If I were doing things like I -> ii -> V -> I, and throwing awkward
81/80 jumps in there and trying to "hide them" in porcupine, then I
could see you telling me that I'm just trying to fake meantone
harmony. But in this case I haven't done anything like that - in
porcupine 81/80 is the same thing as 25/24, so any time I ran up to
81/80 I just treated it as a motion by chromatic semitone and worked
things out that way. This has helped me better understand the
structure of the underlying porcupine logic, so I don't need to worry
about comma jumps anymore.

To my ears what I did sounds very "common practice," which honestly I
find boring, but it does prove theoretically that you can make "common
practice" sounding music around something other than meantone
temperament.

> OTOH, what I was doing is trying to find
> progressions based on the unison vector itself -- i.e. progressions
> characteristic for the particular harmonic system coming from that
> particular temperament.

I think that our main difference is that you're trying to figure out
porcupine "diatonic" harmony, whereas I'm trying to just figure out
the porcupine lattice in a more generalized sense. Certain aspects of
porcupine "diatonic" harmony don't really resolve properly for me, so
I'm altering things to get them to resolve. Although this whole time
I've always thought that maybe I could learn to hear things
differently, now I'm thinking - maybe it's just that Gm->Cmaj doesn't
resolve as strongly as Gmaj->Cmaj, period. If so, what's wrong with
altering the chord so it resolves stronger?

Like I keep saying, when they created the harmonic minor scale, they
didn't assume they'd just have to learn to hear the v->i as
"resolving" the same way as V->i, but instead they used the chords
they wanted and built a new scale around that. Even the melodic minor
scale is built around this principle

So in short, I think that stuff like vii dim -> I, or V -> I, probably
just works for psychoacoustic reasons, and it's best to let the scale
follow the harmony. This isn't what happened historically with the
diatonic scale, as people were using it in a 3-limit context before
they ever discovered 5-limit harmony, but it is what happened with the
harmonic and melodic minor scales.

> This is also what has lead me to the concept of
> diatonic and chromatic steps last year. I'm still not definitely sure about
> it but after doing a few more experiments and verifications, I'll probably
> post my comma pump article publicly. To make a long story short, a
> progression like "C-major, A-major, D-major" uses chromatic steps in many
> temperaments in the first place, including meantone, porcupine, hanson or
> others. So this is a different category of comma pumps than mine. Both have
> their meanings but they can't be compared.

OK, I see what you mean. However - I don't think you should consider
secondary dominants as part of the scale like that. For example -

Take Cmaj -> Dm -> Em -> Fmaj -> Fm -> Cmaj. You can add a secondary
dominant before each chord, for example - Cmaj -> A7 -> Dm -> B7 -> Em
-> C7 -> Fmaj -> Fm -> Cmaj. I view the secondary dominants as being
"out of" the scale, and the real interpretation as Cmaj -> (get ready
for Dm) -> Dm -> (get ready for Em) -> Em -> (get ready for C7) ->
Fmaj -> Fm -> Cmaj. You could replace all of the dominants with
diminished chords, e.g. Cmaj -> C#dim -> Dm -> D#dim -> etc, and you'd
get the same basic result. So the whole thing is very much related to
I -> ii -> iii -> IV and meantone diatonic harmony.

If, however, you decided that you wanted to "only stick to meantone,"
and use as few generators as possible, you would probably do something
like Cmaj -> Am -> Dm -> Bdim -> Em -> Cmaj -> Fmaj -> Fm -> Cmaj.
Well, the Am -> Dm sounds good, but the Bdim -> Em doesn't sound as
good. It certainly sounds "diatonic," but it sounds a little awkward.
If I were playing this in a real song, I'd probably change the Bdim to
B7 but play the B octatonic scale over it. So sometimes "functional"
and "diatonic" aren't the same thing. But even if you alter things the
way I described above, e.g. with dom7 chords before each next diatonic
chord, you can't say that the whole thing "isn't meantone," because it
still goes I -> ii -> iii -> IV.

Likewise, certain aspects of porcupine diatonic harmony sound a bit
awkward (there are 3 diminished chords in a row in porcupine[7]). I'm
trying to create the strongest resolutions possible for porcupine, so
that I can hear how the whole thing lays out. So likewise, you can do
the same thing with porcupine - follow the porcupine diatonic chord
structure, and just put dom7 chords before each chord, and it will
"emphasize" that structure.

Also, the diatonic scale is beautifully constructed - think about it:
you can go from Cmaj to Fmaj, which is a motion "away from" the tonic
to the subdominant. Then you can go from Fmaj to Dm, which retains
much of the feel of the subdominant. But then you can go straight from
Dm -> Gmaj, which is a motion back towards the tonic! You can move
"away from the tonic," and then circularly "come back to the tonic."
They've been saying stuff like this for hundreds of years, but they
never knew they were talking about comma pumps! You can do the same
thing in tetracot -> move "away from the tonic" by going up 10/9, and
then after 4 of those you're at the V chord and you can now circularly
"go back to the tonic!" Comma pumps allow for musical journeys such as
these.

-Mike

On Sat, Apr 23, 2011 at 8:28 AM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > What's wrong with SoundCloud? I could put them up on archive.org, but
> > I really don't feel like uploading every single one of them.
>
> Let's please not use archive.org. I want to directly link to the mp3s, like I will to Petr's ogg files, and the simplest solution would be to ask Chris to host them.

OK, well you can download the MP3 straight from SoundCloud if you'd like.

> I'm just interested in the pumps, not in how you can goof up the tuning for them, so just 22 would be fine for that purpose.

Alright.

-Mike

On Sat, Apr 23, 2011 at 10:21 AM, Petr Parízek
<petrparizek2000@...> wrote:
>
> For example, I could obviously make a progression like "C major, A major, D
> major, G major, C major" in meantone -- or to go still further, "C major, A
> major, D minor, G minor, C major". In the former case, there's a chromatic
> step even within the first two chords. In the latter, the number of
> generators required is as high as 10, which doesn't make a lot of sense if
> our primary goal is to understand the basic (or "non-chromatic") properties
> of a particular harmonic system, which I think should be done first,
> otherwise the chromatic ones just look like mess -- as if you learned about
> powers and roots before understanding multiplication and division.

The number of generators required for the harmonic scale is also 10,
but they liked V -> i so much that they didn't care... :) I say
harmony is what's fundamental, let the scales follow that!

Plus, you can always just use MODMOS's and get a 7 note scale that,
"technically" requires 10 contiguous generators, but so what?

> And one more thing. If most European music were based on diaschismatic or
> negri instead of meantone, then the "most familiar" progressions would
> certainly be different and we would probably be much more used to things
> like "Db major, F minor, C major rather than "D minor, G major, C major".

OK, well we may not have Dm -> Gmaj -> Cmaj, but I think we'd still
definitely have Gmaj -> Cmaj no matter what system we're in.

> Let's imagine all of us here exploring meantone then in a similar way we're
> exploring the other 2D temperaments now. Would we then be keen on making
> meantone pumps full of things like "E minor, B major, D# minor, A# major"
> even though these are not characteristic for meantone?

What do you mean? That sounds like romantic harmony to me.

> If we wanted to get close to the familiar ones, we probably would. If we wanted, instead, to
> learn more about the new harmonic system offered by meantone tempering,
> well, then we would have to get rid of thinking in the other one first. And
> that is possible (which is what I've been trying to say here all the time)
> if you know what's tempered out.

I agree, but I also think that V-I may be universal in appeal no
matter what scale you use. I've wondered a lot if motion down by 5/4,
e.g. Emaj -> Cmaj, could have the same kind of impact that V-I has. I
haven't heard it work yet, but maybe.

> Honestly, it took me quite some time before
> I really admitted that "this or that" *can* indeed work and that it isn't a
> nonsense -- you know, because I was for all those years stuck in the one
> harmonic system based on meantone. Once I understood what those unison
> vectors meant, I was finally able to remove that barrier of meantone
> thinking. This is also why I'm always trying to explain a new harmonic
> system using a 2D temperament rather than an EDO.

Alright, but even within meantone some diatonic relationships sound
awkward, such as Bdim "resolving" to Em, and composers will routinely
change things to make that resolution stronger. I think if you applied
the same principles to something like porcupine[7], strengthening the
relationships when necessary but allowing them to remain unaltered if
not, you'd get really strong resolutions but with a porcupine flavor.

-Mike

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

> > > but if you ever try to play this in a temperament where 250/243
> > > doesn't vanish, you're going to have some awkward 250/243 jumps.
> >
> > You should make a version with those! -Carl
>
> Sure, but these are a huge pain in the ass to do, and the
> higher-numbered the EDO, the more irritating it is. What if I
> did one in 19-equal? Would that satisfy?

Can I get 27?

-Carl

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

> I think that our main difference is that you're trying to figure
> out porcupine "diatonic" harmony, whereas I'm trying to just
> figure out the porcupine lattice in a more generalized sense.
> Certain aspects of porcupine "diatonic" harmony don't really
> resolve properly for me, so I'm altering things to get them to
> resolve. Although this whole time I've always thought that
> maybe I could learn to hear things differently, now I'm
> thinking - maybe it's just that Gm->Cmaj doesn't resolve as
> strongly as Gmaj->Cmaj, period. If so, what's wrong with
> altering the chord so it resolves stronger?

What's wrong with weaker resolutions in some places? And
even though weaker than the strongest known change, maybe
they are still strong enough compared to other changes in
the system. I don't put sugar on broccoli.

-Carl

On Sat, Apr 23, 2011 at 4:13 PM, Carl Lumma <carl@...> wrote:
>
> What's wrong with weaker resolutions in some places? And
> even though weaker than the strongest known change, maybe
> they are still strong enough compared to other changes in
> the system.

I never said anything was "wrong" with them. But there's also nothing
wrong with altering them to be stronger if you want. I note that
historically, we did, in fact, alter scales to make certain
resolutions stronger. So in this case you have the scalar structure
following the desired harmonies, which is a reversal of previous
historical trends (e.g. people "discovering" triadic harmony from the
preexisting diatonic scale).

The system is not limited to porcupine[7] or porcupine[8].

-Mike

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

> > You should make a version with those! -Carl
>
> Sure, but these are a huge pain in the ass to do, and the
> higher-numbered the EDO, the more irritating it is. What if I did one
> in 19-equal? Would that satisfy?

I'd suggest doing it first in pain in the ass JI, and then tempering to anything.

Mike wrote:

> > What's wrong with weaker resolutions in some places? And
> > even though weaker than the strongest known change, maybe
> > they are still strong enough compared to other changes in
> > the system.
>
> I never said anything was "wrong" with them. But there's
> also nothing wrong with altering them to be stronger if
> you want.

Of course.

> I note that historically, we did, in fact, alter scales
> to make certain resolutions stronger.

After making music without doing so for several hundred
years, which is still perfectly listenable today.

-Carl

On Sat, Apr 23, 2011 at 4:35 PM, Carl Lumma <carl@...> wrote:
>
> > I note that historically, we did, in fact, alter scales
> > to make certain resolutions stronger.
>
> After making music without doing so for several hundred
> years, which is still perfectly listenable today.

And yet, in our previous discussions, you refused to call harmonies in
those pieces "functional," insisting that they were instead "modal.".

-Mike

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

> > > historically, we did, in fact, alter scales
> > > to make certain resolutions stronger.
> >
> > After making music without doing so for several hundred
> > years, which is still perfectly listenable today.
>
> And yet, in our previous discussions, you refused to call
> harmonies in those pieces "functional," insisting that they
> were instead "modal.".

Yes. -Carl

On Sat, Apr 23, 2011 at 4:53 PM, Carl Lumma <carl@...> wrote:
>
> > And yet, in our previous discussions, you refused to call
> > harmonies in those pieces "functional," insisting that they
> > were instead "modal.".
>
> Yes. -Carl

So I'm trying to write something "functional" in porcupine temperament.

-Mike

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

> OK, well you can download the MP3 straight from SoundCloud if you'd like.

I don't want to download them, I want to link to them. Would it be OK to download them and get Chris to hoast them?

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

> > > And yet, in our previous discussions, you refused to call
> > > harmonies in those pieces "functional," insisting that they
> > > were instead "modal.".
> >
> > Yes. -Carl
>
> So I'm trying to write something "functional" in porcupine
> temperament.

Functional = using the strongest resolutions is an interesting
hypothesis, but not one I'm prepared to accept just yet.

-Carl

On Sat, Apr 23, 2011 at 5:47 PM, Carl Lumma <carl@...> wrote:
>
> > So I'm trying to write something "functional" in porcupine
> > temperament.
>
> Functional = using the strongest resolutions is an interesting
> hypothesis, but not one I'm prepared to accept just yet.

I never said you have to use the strongest resolutions. I said that
using stronger resolutions can allow you to strengthen the
functionality of the harmony.

As you know, I don't think that functionality has anything to do with
scale structure, I think it has to do with root movements by things
like 3/2 and melodic movements by things like 1\17 or 1\19. All of
these features are handily encapsulated in the diatonic V-I. The whole
package is pleasing, so we like it. Once you play this a thousand
times, you come to hope it happens again.

Much has been made in common practice theory of movements "away from
the tonic" vs those that "return to the tonic." Motion from I-IV is
heard as a movement away from the tonic to the subdominant. Motion
from IV-ii is heard as movement to the submediant downward from IV,
which is heard as being "related" to the IV chord - the relative
minor, specifically. But, only in meantone, can you then move smoothly
from ii-V, which is then heard as a motion "back towards" the tonic.
The same applies in reverse, e.g. I-V-ii-IV-I.

Likewise, in porcupine, you can take advantage of the same concept -
you can move "away from the root" by going down a 10/9, but only in
porcupine do you actually land at the V chord by doing so after 4
iterations of this - which enables you to then immediately "move back
to" the tonic. So here you have an entirely different cycle of
departure and resolution. However, porcupine[7] isn't meantone[7].

Meantone[7] has the great property that it includes a full comma pump,
as well as a V-I (a V7-I, actually) right in the diatonic scale. You
can go I -> IV -> ii -> V -> I, as well as a million other things, and
"depart from" and "return to" the tonic (which is what a comma pump
is) very strongly - without ever having to leave the diatonic scale.
We're not as fortunate with porcupine[7], so to actually traverse an
entire comma pump and still have strong resolutions means you might
have to modulate a bit (although the symmetrical mode has a decent
one).

You might find that the best scale to work with is the major mode of
porcupine[7], but with a flattened 4 and a sharpened 7. This is just a
porcupine tempered 5-limit JI major scale. Then, if you ever need to
shift something by 81/80 - 81/80 = 25/24 in porcupine temperament, and
alteration by 25/24 is something that you can easily functionalize.
The result is something that sounds like common practice music, but
will only work in porcupine temperament.

So yes, some adaptation and "learning" is involved - except instead of
learning to hear weaker resolutions as stronger, you just have to
learn to stick to one scale less. Maybe we'll find a porcupine MODMOS
that has a full comma pump in it, with some kind of decently strong
V-I. Then we'll be able to stick more closely to that scale. But we
don't even stick 100% to the diatonic scale, and generally alter
things when we want to, so I'm not convinced it's that important.

Anyway, my goal was to purposefully take a functional sledgehammer to
the whole thing - not to work out what porcupine "diatonic" harmony
should sound like. I'm pretty happy with the results. But if you think
you can improve on them, I encourage you to make your own examples.

-Mike

What is the distinction being drawn? Modal music can have tonality and functional harmony. What I was taught was all modal indicates is that the music is not major or minor.

The little bit I've read of this thread is that the distinction is functional or not.

Is that correct?

Chris

-----Original Message-----
From: "Carl Lumma" <carl@lumma.org>
Sender: tuning@yahoogroups.com
Date: Sat, 23 Apr 2011 20:53:38
To: <tuning@yahoogroups.com>
Reply-To: tuning@yahoogroups.com
Subject: [tuning] Re: Some functional harmony in porcupine temperament

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

> > > historically, we did, in fact, alter scales
> > > to make certain resolutions stronger.
> >
> > After making music without doing so for several hundred
> > years, which is still perfectly listenable today.
>
> And yet, in our previous discussions, you refused to call
> harmonies in those pieces "functional," insisting that they
> were instead "modal.".

Yes. -Carl

On Sat, Apr 23, 2011 at 5:47 PM, Carl Lumma <carl@...> wrote:
>
> Functional = using the strongest resolutions is an interesting
> hypothesis, but not one I'm prepared to accept just yet.

Actually, you know - wtf? How did we switch sides on this? Weren't you
the one who used to insist that only ionian and aeolian had functional
harmony, because the tritone resolving inward was so strong that it
dwarfed all other functional relationships within the diatonic scale?
And I'd always be the one saying that you can find decent functional
relationships even within dorian and phrygian?

But now we've switched sides? How did this happen?

-Mike

On Sat, Apr 23, 2011 at 6:55 PM, <chrisvaisvil@...> wrote:
>
> What is the distinction being drawn? Modal music can have tonality and functional harmony. What I was taught was all modal indicates is that the music is not major or minor.

I agree, and this is what I was taught as well. But, V-i is a stronger
resolution in minor than v-i. That isn't to say that v-i doesn't have
its own unique flavor, just that you can make the resolution stronger
if you make it V-i.

-Mike

--- Mike Battaglia <battaglia01@...>

> > Functional = using the strongest resolutions is an interesting
> > hypothesis, but not one I'm prepared to accept just yet.
>
> Actually, you know - wtf? How did we switch sides on this? Weren't you
> the one who used to insist that only ionian and aeolian had functional
> harmony, because the tritone resolving inward was so strong that it
> dwarfed all other functional relationships within the diatonic scale?
> And I'd always be the one saying that you can find decent functional
> relationships even within dorian and phrygian?
>
> But now we've switched sides? How did this happen?

I don't remember saying tritones were all-important, I remember saying Paul thought they were important and expressing some skepticism at the idea. I've never used the "characteristic dissonance" concept in my generalized diatonic evals (though I haven't explicitly tried to model functionality either). Classically, only the major and minor modes are considered functional. I don't have an opinion whether funtionality may be achieved in other modes. And while I'm at it, I find the idea that somebody here is gonna listen to a progression in an afternoon and decide it's good for functional music or not pretty laughable.

The kind of thing you're talking about with 3:2 root motions and such is an absolutist view of music perception that necessarily limits how different any other tuning is gonna sound. It may be justified or not, I don't know. Just pointing it out.

-Carl

On Sat, Apr 23, 2011 at 8:15 PM, Carl Lumma <carl@...> wrote:
>
> I don't remember saying tritones were all-important, I remember saying Paul thought they were important and expressing some skepticism at the idea.

LOL, wow, what a relief! I thought you were all about the tritone.

> I've never used the "characteristic dissonance" concept in my generalized diatonic evals (though I haven't explicitly tried to model functionality either). Classically, only the major and minor modes are considered functional.

And minor is generally considered more functional if you morph between
melodic, harmonic, and natural minor as necessary.

> I don't have an opinion whether funtionality may be achieved in other modes.

In the middle of a discussion a while ago about the perception of
chord progressions, you pulled out of your hat that only certain modes
are "tonal modes," and then used that to somehow argue against my
point. What was that then?

> And while I'm at it, I find the idea that somebody here is gonna listen to a progression in an afternoon and decide it's good for functional music or not pretty laughable.

LOL, whatever. I'll just leave it at a polite encouragement to make
your own examples and improve on my initial effort. :)

You can assert that it'll take another 23 years of training to outdo
my first 23 and finally hear v-I in porcupine[7] as functional, but
first you'll need to explain why the porcupine[7] MOS is the necessary
scale that we're supposed to be making use of for "true porcupine
harmony." On the contrary, it seems that porcupine[7] is just another
scale that exists in the greater porcupine system. Although it is
theoretically distinguished by being an MOS of the system; e.g. it's a
bunch of porcupine generators stacked on top of one another, I don't
see why that's supposed to be of prime cognitive concern. I really
don't see why we need to elevate that MOS up to such a level where
harmony in porcupine temperament doesn't count as "true porcupine
harmony" unless it matches up to the MOS. Paul got away from MOS's
with Pajara and I think the results were good.

If anything, I think this suggests another scale ideal to work
towards: scales that are built around the minimal comma pump for a
temperament. Scales like that will enable you to move away from the
tonic and then return from it in the same way that I->IV->ii->V7->I
allows you to in meantone. Maybe these won't be the MOS's at all! But
I predict they'll certainly be tonal, and that you'll learn to hear
the cycles of moving away from the tonic -> returning via "pun" as
very "functional" with a bit of training.

> The kind of thing you're talking about with 3:2 root motions and such is an absolutist view of music perception that necessarily limits how different any other tuning is gonna sound. It may be justified or not, I don't know. Just pointing it out.

Like I said to Petr, I don't see why you couldn't create a setup where
5:4 resolves downward as well. I can't hear things that way, but maybe
it'll become easier if you get used to magic temperament. Maybe it'll
be easier 200 years from now. But, we don't live 200 years from now,
we live today, and today, V-I is a pretty nice package to work with.
So one piece of the puzzle, and a piece I'm enjoying solving, is
figuring out how it works in other temperaments. The result to my ears
sounds something like common practice music, but in other
temperaments. I hear diminished chords resolving and V moving to I,
but the whole thing lines up only in porcupine. That's what I was
going for, and so my requirements for functional harmony are
satisfied.

You're talking about radically changing what we consider functional
such that we compulsively make new musical predictions. That's a good
ideal, but historically I think it noteworthy that the harmonic
explosion of the 21st century, which introduced lots of "functional"
music built around modes, was prefaced by hundreds of years of V-I. So
likewise, I think that using existing techniques in new systems is a
good stepping stone to finding new techniques in those systems. The
entire concept of moving away from the root and coming back to it via
comma pump is itself something that probably matters only because of
learning, but why throw the baby out with the bathwater?

Lastly, and while we're at it, why not learn to hear Dorian and
Phrygian as having functional harmony? That's what I claim to have
done, but you always seem to express skepticism over it. If it's
possible to hear functional harmony in, say, the sLsssss mode of
porcupine[7] (and I believe it is, and that my examples can a stepping
stone to hearing it), why the skepticism over hearing functional
harmony in Mixolydian or Dorian?

-Mike

Mike wrote:

> This requirement that we stick only to
> MOS's and use few generators isn't something I really think is
> necessary. The harmonic minor scale gives us a historical context for
> why I think this is true.

This is the first time I'm speaking explicitly against what you're saying -- but I will anyway.
You've ommitted the fact that before exploring scales like the harmonic minor, people were playing in pure diatonic triadic harmony for decades. OTOH, what I'm not sure about is the development of the harmonic minor scale in cultures promoting mostly one-voiced melodic music. But I think these may have been past permutations of ancient chromatic tetrachords in a similar way you get permutations of diatonic tetrachords -- for example, Dorian to Phrygian. In this context, it might be interesting to know how much use there might be (in this day and age) for permutations of the enharmonic tetrachords.

> For example, let's pretend that instead of exploring negri, we're from
> a culture that only uses functional harmony in the meantone major
> scale, and we're now exploring the minor scale for the first time. How
> would we have things resolve? Should we use v -> i now, because
> there's no V chord in aeolian? Is the tendency to want to use V -> i
> just a learned component of our "major thinking?" Should we force
> ourselves to not use it so that we can learn the true nature of the
> minor scale? Well, that's not what they decided to do - they decided
> they liked V -> i because they just liked it, so they sharpened the
> seventh and created the harmonic minor scale. They didn't care about
> the chain of fifths or anything, they just knew that V -> i was a
> pleasing sound. This requires 10 generators, but it's okay, because by
> using things like MODMOS's you can still get it to 7 notes.

Exactly what I meant, only said with different words and from the opposite point of view. Remember that when triadic harmony started to "spread" during the early 1500's or possibly late 1400's, the only chords which were considered "the propper chords" were major chords in the root position. Only much later did minor chords come into use, followed by things like octave inversions and similar stuff.

> This indicates to me that something like Gmaj -> Cmaj, or Gmaj -> Cm,
> is just a pleasing sound, probably because of the combined
> psychoacoustic payload of the leading tone and the root movement by
> 3/2. That means that I think that having a major chord (or a 4:5:6:7)
> a 3/2 up from the root just "resolves" strongly down to the root in
> general, and I don't think that means that we're stuck in meantone
> thinking or that we need to ignore this impulse. And even if we want
> to ignore it for some theoretical ideal - composers will probably not
> care and just go with what sounds "natural" to them.

Which is what they've been doing for centuries. And now look what effect it had on the tuning requirements and what effect the used temperaments had on the composers. After the the great expansion of meantone, they started using progressions which they probably wouldn't otherwise think of if 5-limit JI had been still in use. Then, when well-temperaments came, they started considering enharmonic "puns", mainly in the form of enharmonic modulations. Later, when 12-EDO was widely recognized as the "standard" temperament, musicians became to use chords that might sound strange in the other ones, which is pretty obvious in the octatonic music of Ravel and Messiaen or in jazz, just to name a few examples. Before the wide adoption of 12-EDO, the primary favored chord progressions were still meantone-based and well-temperaments were often viewed as some sort of a "substitute" for the model tuning -- i.e. meantone.

> And just as we
> designed the harmonic minor scale to have a leading tone in it, I
> predict that the same will happen with these scales, so that perhaps
> instead of using 4 3 3 3 3 3 3 for porcupine[7], we might start using
> 4 3 3 3 3 4 2. Or maybe even 4 3 3 3 3 2 4, for dominant chords. These
> are first-order MODMOS's of porcupine[7] in the same way that the
> harmonic minor scale with its sharpened leading tone is a first-order
> MODMOS of meantone[7].

You're connecting things that may not necessarily have to do something with each other.
First of all, the reason why people started using major dominants in minor keys was because major triads were considered more acoustically "aligned" than minor ones because of the correspondence to the harmonic series. For exactly the same reason, many composers also used a major tonic to end a minor-keyed piece (someone called this a "Picardy third" or a "Picardian third"). The concept of tuning the "leading tone" closer to the resolving tone started appearing perhaps in the late 1700's and was at its best, IIRC, during the strongly romantic period which gave a lot of stress on melodic lines and apparently didn't find the topic of high-quality concords very important. Later, some theorists went even as far as to claim that the great Pythagoras advocated this as well -- which we may, with a help of some knowledge, understand in terms of Pythagorean tuning, where a minor second really *is* smaller than 1/12 of a pure octave. But these theorists ommitted the important fact that Pythagorean tuning was not at all designed for chords based on thirds. So you have two things here -- malodic intonation (where minor seconds are a bit smaller and C# is higher than Db) and harmonic intonation (where minor seconds are a bit larger and C# is lower than Db). You can't meet both at the same time since one contradicts the other, therefore it's mathematically impossible to be well in tune in chords and at the same time to get the "romantic expressivity".

> But motion by fifth is a valid musical technique - isn't that a pretty
> fundamental concept? It's just motion by 3/2 - why should we assume
> that we need to "get away from that?" I understand there may also be
> some new techniques we haven't discovered as well, but I see no reason
> that we should feel uncomfortable moving chords around by 3/2.

Okay, try playing around with hanson or semisixths for some time and maybe you'll realize one day that the "limitations" (in the positive sense) of the temperament do indeed affect the harmonic preferences in each particular case.

> If I were doing things like I -> ii -> V -> I, and throwing awkward
> 81/80 jumps in there and trying to "hide them" in porcupine, then I
> could see you telling me that I'm just trying to fake meantone
> harmony. But in this case I haven't done anything like that - in
> porcupine 81/80 is the same thing as 25/24, so any time I ran up to
> 81/80 I just treated it as a motion by chromatic semitone and worked
> things out that way. This has helped me better understand the
> structure of the underlying porcupine logic, so I don't need to worry
> about comma jumps anymore.

Well, probably I'll have to think more of why it reminds me of meantone even though it doesn't work in meantone. Maybe it's because of the modulation in the third phrase or whatever, I'm not sure in fact.

> To my ears what I did sounds very "common practice," which honestly I
> find boring, but it does prove theoretically that you can make "common
> practice" sounding music around something other than meantone
> temperament.

And that's exactly why I didn't want compare my examples to yours because either of those had different expectations and requirements. I wasn't thinking about whether the progressions would or wouldn't sound like common practice in the end.

> I think that our main difference is that you're trying to figure out
> porcupine "diatonic" harmony, whereas I'm trying to just figure out
> the porcupine lattice in a more generalized sense. Certain aspects of
> porcupine "diatonic" harmony don't really resolve properly for me, so
> I'm altering things to get them to resolve. Although this whole time
> I've always thought that maybe I could learn to hear things
> differently, now I'm thinking - maybe it's just that Gm->Cmaj doesn't
> resolve as strongly as Gmaj->Cmaj, period. If so, what's wrong with
> altering the chord so it resolves stronger?

I didn't say I was against doing that. I only said that I like to have some idea about what progressions might be more preferable or less preferable in a particular temperament and that's why I like to explore the "diatonics" first and the chromatic alterations afterwards.

> Like I keep saying, when they created the harmonic minor scale, they
> didn't assume they'd just have to learn to hear the v->i as
> "resolving" the same way as V->i, but instead they used the chords
> they wanted and built a new scale around that. Even the melodic minor
> scale is built around this principle

The melodic major/minor scales probably have their origin in the distant past of tetrachordal classification when it was observed that some musicians felt more comfortable tuning the "middle" tones of the tetrachord slightly higher when going up and slightly lower when going down to add more melodic expressivity.

> So in short, I think that stuff like vii dim -> I, or V -> I, probably
> just works for psychoacoustic reasons, and it's best to let the scale
> follow the harmony. This isn't what happened historically with the
> diatonic scale, as people were using it in a 3-limit context before
> they ever discovered 5-limit harmony, but it is what happened with the
> harmonic and melodic minor scales.

I understand what you mean and an even better example of this would be the harmonic major scale since it has a minor chord on the subdominant. But do you think people would be able to quickly "invent" something like the harmonic major scale without first playing in the diatonic major scale for some time? I don't.

> Take Cmaj -> Dm -> Em -> Fmaj -> Fm -> Cmaj. You can add a secondary
> dominant before each chord, for example - Cmaj -> A7 -> Dm -> B7 -> Em
> -> C7 -> Fmaj -> Fm -> Cmaj. I view the secondary dominants as being
> "out of" the scale, and the real interpretation as Cmaj -> (get ready
> for Dm) -> Dm -> (get ready for Em) -> Em -> (get ready for C7) ->
> Fmaj -> Fm -> Cmaj. You could replace all of the dominants with
> diminished chords, e.g. Cmaj -> C#dim -> Dm -> D#dim -> etc, and you'd
> get the same basic result. So the whole thing is very much related to
> I -> ii -> iii -> IV and meantone diatonic harmony.

This was the beginning of the concept of modulation in classical harmony and people didn't use it even as late as in early Baroque when triadic harmony was still mostly diatonic -- i.e. with a few exceptions later leading to things like the Picardy third or the harmonic minor scale.

> If, however, you decided that you wanted to "only stick to meantone,"
> and use as few generators as possible, you would probably do something
> like Cmaj -> Am -> Dm -> Bdim -> Em -> Cmaj -> Fmaj -> Fm -> Cmaj.
> Well, the Am -> Dm sounds good, but the Bdim -> Em doesn't sound as
> good. It certainly sounds "diatonic," but it sounds a little awkward.
> If I were playing this in a real song, I'd probably change the Bdim to
> B7 but play the B octatonic scale over it. So sometimes "functional"
> and "diatonic" aren't the same thing.

You're talking about todays harmony having its origin in meantone. I'm talking about Baroque harmony having its origin in meantone. Don't forget that in the Baroque era, there was one rule about playing diminished triads -- they could be played as sixth chords -- and that voice leading was considered of great importance. So if I was playing this thing some 300 years ago, all of the major 7th chords which you used would probably be preferred in the 1st inversion (like C#-E-G-A instead of A-C#-E-G) or as diminished chords without the silent tonic (i.e. C#-E-G-Bb with no A). And if I was playing this about 400 years ago, I would probably do it like "C-E-G, C-A-E, D-F-A, D-F-B, E-G-B, E-G-C, F-A-C, F-A-D, C-G-E".

> But even if you alter things the
> way I described above, e.g. with dom7 chords before each next diatonic
> chord, you can't say that the whole thing "isn't meantone," because it
> still goes I -> ii -> iii -> IV.

Possibly; but you can't start explaining these "passing" chords without a thorough understanding of the "anchor" chords first.

> Likewise, certain aspects of porcupine diatonic harmony sound a bit
> awkward (there are 3 diminished chords in a row in porcupine[7]). I'm
> trying to create the strongest resolutions possible for porcupine, so
> that I can hear how the whole thing lays out. So likewise, you can do
> the same thing with porcupine - follow the porcupine diatonic chord
> structure, and just put dom7 chords before each chord, and it will
> "emphasize" that structure.

Same for this.

> Also, the diatonic scale is beautifully constructed - think about it:
> you can go from Cmaj to Fmaj, which is a motion "away from" the tonic
> to the subdominant. Then you can go from Fmaj to Dm, which retains
> much of the feel of the subdominant. But then you can go straight from
> Dm -> Gmaj, which is a motion back towards the tonic! You can move
> "away from the tonic," and then circularly "come back to the tonic."
> They've been saying stuff like this for hundreds of years, but they
> never knew they were talking about comma pumps! You can do the same
> thing in tetracot -> move "away from the tonic" by going up 10/9, and
> then after 4 of those you're at the V chord and you can now circularly
> "go back to the tonic!" Comma pumps allow for musical journeys such as
> these.

And to learn what harmony is all about.

I wrote:

> > And one more thing. If most European music were based on diaschismatic > > or
> > negri instead of meantone, then the "most familiar" progressions would
> > certainly be different and we would probably be much more used to things
> > like "Db major, F minor, C major rather than "D minor, G major, C > > major".
>
> OK, well we may not have Dm -> Gmaj -> Cmaj, but I think we'd still
> definitely have Gmaj -> Cmaj no matter what system we're in.

Pretty far in the triad chain. I'm not guessing how long it would take us to find it if that were the case. -- No, we would have "G major, C minor" first and then "G major, C major" some decades later.

> > Let's imagine all of us here exploring meantone then in a similar way > > we're
> > exploring the other 2D temperaments now. Would we then be keen on making
> > meantone pumps full of things like "E minor, B major, D# minor, A# > > major"
> > even though these are not characteristic for meantone?
>
> What do you mean? That sounds like romantic harmony to me.

These things are "ordinary" from the negri-like perspective but they're considered "highly evolved modulations" from the meantone-like perspective. Does this answer your question?

> I agree, but I also think that V-I may be universal in appeal no
> matter what scale you use. I've wondered a lot if motion down by 5/4,
> e.g. Emaj -> Cmaj, could have the same kind of impact that V-I has. I
> haven't heard it work yet, but maybe.

I'm not sure. I think this has something to do not only with the temperament used but also with the prime limit of the original untempered intervals. For example, since both 6/5 and 5/4 have a higher limit than 4/3 or 3/2, that might be the reason why progressions by thirds may sound less convincing than progressions by fourths/fifths. But this is just my personal hypothesis, I haven't consulted it with anyone.

> Honestly, it took me quite some time before
> I really admitted that "this or that" *can* indeed work and that it isn't > a
> nonsense -- you know, because I was for all those years stuck in the one
> harmonic system based on meantone. Once I understood what those unison
> vectors meant, I was finally able to remove that barrier of meantone
> thinking. This is also why I'm always trying to explain a new harmonic
> system using a 2D temperament rather than an EDO.

> Alright, but even within meantone some diatonic relationships sound
> awkward, such as Bdim "resolving" to Em, and composers will routinely
> change things to make that resolution stronger. I think if you applied
> the same principles to something like porcupine[7], strengthening the
> relationships when necessary but allowing them to remain unaltered if
> not, you'd get really strong resolutions but with a porcupine flavor.

Of course you can. But how sooner than understanding the basics? It's like trying to use coloquialisms without first learning the proper meanings of words.
Okay, let's admit that I'm now still in the stage of getting to know the basics and I haven't manage to move further yet as I want to be sure that I understand enough of that before delving into the "chromatics".
In meantone, 16/15 is a diatonic step and 25/24 is a chromatic step. In porcupine, it's the other way round. So a progression like "C major, A major" can occur in porcupine more frequently than in meantone and a progression like "C major, G major" can occur in meantone more frequently than in porcupine. And there's nothing strange about this fact, the preferences come straight out of the properties of each of the temperaments. And there isn't much of a point in ignoring this fact either.

Petr

--- In Mike Battaglia <battaglia01@...> wrote:

> You can assert that it'll take another 23 years of training
> to outdo my first 23 and finally hear v-I in porcupine[7] as
> functional, but first you'll need to explain why the
> porcupine[7] MOS is the necessary scale that we're supposed
> to be making use of for "true porcupine harmony."

No, that's a straw man. Also, please cease bombarding the
list with your lengthy expositions now. Thanks in advance.

-Carl

On Sat, Apr 23, 2011 at 9:11 PM, Petr Parízek <petrparizek2000@...> wrote:
>
> Mike wrote:
>
> You've ommitted the fact that before exploring scales like the harmonic
> minor, people were playing in pure diatonic triadic harmony for decades.

Yes, but they also used the diatonic scale even before then as well.
After they found what they liked about the diatonic scale, they
generalized it. Gmaj -> Cmaj exists even without the diatonic scale,
it just so happens to be in the diatonic scale. Once people found they
liked that kind of progression, they started putting it everywhere.

> OTOH, what I'm not sure about is the development of the harmonic minor scale
> in cultures promoting mostly one-voiced melodic music.

Yes, but melodic music in porcupine is also a lot easier than harmonic music :)

> But I think these may
> have been past permutations of ancient chromatic tetrachords in a similar
> way you get permutations of diatonic tetrachords -- for example, Dorian to
> Phrygian.

What tetrachordal permutation gives you harmonic minor?

> > They didn't care about
> > the chain of fifths or anything, they just knew that V -> i was a
> > pleasing sound. This requires 10 generators, but it's okay, because by
> > using things like MODMOS's you can still get it to 7 notes.
>
> Exactly what I meant, only said with different words and from the opposite
> point of view.

What do you mean? I thought you were saying the opposite, that we
should stick to MOS's for now.

> Later, when 12-EDO was widely recognized as the "standard" temperament,
> musicians became to use chords that might sound strange in the other ones,
> which is pretty obvious in the octatonic music of Ravel and Messiaen or in
> jazz, just to name a few examples.

OK, so then I predict that when we give composers porcupine[7],
they'll modulate around more within it so they can do things like V-I
when they want, and that more modulation will be necessary than with
the diatonic scale - but I don't think this is "cheating" :)

> First of all, the reason why people started using major dominants in minor
> keys was because major triads were considered more acoustically "aligned"
> than minor ones because of the correspondence to the harmonic series.

Why does it have to be because of some mathematical ideal? Why not
that they just started doing that because they liked the way it
sounded?

> So you have two things here -- malodic intonation
> (where minor seconds are a bit smaller and C# is higher than Db) and
> harmonic intonation (where minor seconds are a bit larger and C# is lower
> than Db). You can't meet both at the same time since one contradicts the
> other, therefore it's mathematically impossible to be well in tune in chords
> and at the same time to get the "romantic expressivity".

Right, but why not just raise the leading tone by a chromatic semitone
when you want it to move back to I? But either way, this is a
different topic from that v-I doesn't work as well as V-I though...

> > I understand there may also be
> > some new techniques we haven't discovered as well, but I see no reason
> > that we should feel uncomfortable moving chords around by 3/2.
>
> Okay, try playing around with hanson or semisixths for some time and maybe
> you'll realize one day that the "limitations" (in the positive sense) of the
> temperament do indeed affect the harmonic preferences in each particular
> case.

I don't think I understand - why would hanson make you feel
uncomfortable to make you move by 3/2? Just because the hanson
generator happens to be a 6/5, and just because it so happens that in
the abelian group structure we have set out, 6/5 is the interval that
generates the other ones, why should that mean that 6/5 itself should
become more important than 3/2 in the greater hanson temperament?

> Well, probably I'll have to think more of why it reminds me of meantone even
> though it doesn't work in meantone. Maybe it's because of the modulation in
> the third phrase or whatever, I'm not sure in fact.

I actually just reworked one of the chords in the third phrase to lead
more smoothly - do you like it better now?

> And that's exactly why I didn't want compare my examples to yours because
> either of those had different expectations and requirements. I wasn't
> thinking about whether the progressions would or wouldn't sound like common
> practice in the end.

I didn't expect it to sound like that, and was surprised when it did.
So now I feel I've learned something about "common practice" harmony.

> I didn't say I was against doing that. I only said that I like to have some
> idea about what progressions might be more preferable or less preferable in
> a particular temperament and that's why I like to explore the "diatonics"
> first and the chromatic alterations afterwards.

Alright, although if anything what all of this has shown me is that
MOS's don't matter. Not only is the meantone diatonic scale an MOS,
but it's set up to have a bunch of comma pumps built right in, so you
can leave the tonic and then return. Maybe we can cleverly devise a
porcupine scale like that, and then that will become the One True
Porcupine Scale.

> The melodic major/minor scales probably have their origin in the distant
> past of tetrachordal classification when it was observed that some musicians
> felt more comfortable tuning the "middle" tones of the tetrachord slightly
> higher when going up and slightly lower when going down to add more melodic
> expressivity.

I don't think it's just an issue of ease of singing. They imply
different harmonies. I tend to think that it's because sharping the
seventh itself creates a certain sound, but then you have the
augmented second, so they just sharpened the sixth as well to fill up
the space.

> I understand what you mean and an even better example of this would be the
> harmonic major scale since it has a minor chord on the subdominant. But do
> you think people would be able to quickly "invent" something like the
> harmonic major scale without first playing in the diatonic major scale for
> some time? I don't.

No, but now that we have centuries of common practice theory behind
us, we don't have to reinvent the wheel all over again :)

> This was the beginning of the concept of modulation in classical harmony and
> people didn't use it even as late as in early Baroque when triadic harmony
> was still mostly diatonic -- i.e. with a few exceptions later leading to
> things like the Picardy third or the harmonic minor scale.

I interpret this to mean that musicians have a long and gradual
history of moving AWAY from the diatonic scale - we found little
things we liked within it, and then we applied those in ways that made
us leave the scale. So now I'm just trying to apply those same things
we like to porcupine.

> And if I was playing this about 400 years ago, I would probably do it like
> "C-E-G, C-A-E, D-F-A, D-F-B, E-G-B, E-G-C, F-A-C, F-A-D, C-G-E".

OK, and the diatonic scale allows for such a smooth resolution like
that. But there are reasons that it sounds smooth other than just that
we're used to it. And I think that porcupine[7] scale doesn't always
allow for smooth resolutions. But maybe you can prove me wrong.

> > OK, well we may not have Dm -> Gmaj -> Cmaj, but I think we'd still
> > definitely have Gmaj -> Cmaj no matter what system we're in.
>
> Pretty far in the triad chain. I'm not guessing how long it would take us to
> find it if that were the case. -- No, we would have "G major, C minor" first
> and then "G major, C major" some decades later.

Assuming that people always started with MOS, yes. But as in the case
of Paul's pentachordal major scale, sometimes the non-MOS's sound more
tonal.

> > What do you mean? That sounds like romantic harmony to me.
>
> These things are "ordinary" from the negri-like perspective but they're
> considered "highly evolved modulations" from the meantone-like perspective.
> Does this answer your question?

OK, I see. Well, as of 2011, to my ears, they sound like beautiful
5-limit progressions, and comma pumps connect different things in the
5-limit together in cool ways. So that's how I'm learning to hear
everything now.

> I'm not sure. I think this has something to do not only with the temperament
> used but also with the prime limit of the original untempered intervals. For
> example, since both 6/5 and 5/4 have a higher limit than 4/3 or 3/2, that
> might be the reason why progressions by thirds may sound less convincing
> than progressions by fourths/fifths. But this is just my personal
> hypothesis, I haven't consulted it with anyone.

I notice that motion -upward- by 5/4 sounds really nice though. Like
Amaj -> C#maj. It sounds like bVI -> I. I guess what it really sounds
like is iv->I.

-Mike

On Sat, Apr 23, 2011 at 10:18 PM, Carl Lumma <carl@...> wrote:
>
> --- In Mike Battaglia <battaglia01@...> wrote:
>
> > You can assert that it'll take another 23 years of training
> > to outdo my first 23 and finally hear v-I in porcupine[7] as
> > functional, but first you'll need to explain why the
> > porcupine[7] MOS is the necessary scale that we're supposed
> > to be making use of for "true porcupine harmony."
>
> No, that's a straw man.

If that's a straw man, then please clarify your argument.

> Also, please cease bombarding the
> list with your lengthy expositions now. Thanks in advance.

I have no intention of stopping anything of the sort. If you're the
kind of person who prefers not to discuss complex ideas, then this is
not the thread for you.

-Mike

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

> If that's a straw man, then please clarify your argument.

I didn't make the argument that the 7-tone MOS of anything
was anything. I've never even said anything close to that.
Where on earth did you get the idea?

> > Also, please cease bombarding the
> > list with your lengthy expositions now. Thanks in advance.
>
> I have no intention of stopping anything of the sort.

Ut oh. -Carl

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

> You're connecting things that may not necessarily have to do
> something with each other.

He does that a lot. -Carl

On Sat, Apr 23, 2011 at 10:49 PM, Carl Lumma <carl@...> wrote:
>
> > If that's a straw man, then please clarify your argument.
>
> I didn't make the argument that the 7-tone MOS of anything
> was anything. I've never even said anything close to that.
> Where on earth did you get the idea?

You said

> What's wrong with weaker resolutions in some places? And
> even though weaker than the strongest known change, maybe
> they are still strong enough compared to other changes in
> the system. I don't put sugar on broccoli.

What is "the system?" I thought you meant porcupine[7].

> > You're connecting things that may not necessarily have to do
> > something with each other.
>
> He does that a lot. -Carl

LOL, what childish behavior. You might do everyone better if you made
some examples that demonstrate how you think things work instead. But
if you're content to rather play the list heckler, go for it.

-Mike

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

> Exactly what I meant, only said with different words and from the opposite
> point of view. Remember that when triadic harmony started to "spread" during
> the early 1500's or possibly late 1400's, the only chords which were
> considered "the propper chords" were major chords in the root position.

Eh? Minor triads and triads, major or minor, in first inversion appear constantly in early Renaissance music.

Gene wrote:

> Eh? Minor triads and triads, major or minor, in first inversion appear > constantly in early Renaissance music.

Please don't mix up the concept of trines and the concept of triadic harmony, these are two different things and this is not a valid argument. A trine of "A-C-F" and a triad of _A-C-F" have different meanings. While a triad of "A-C-F" is understood as an octave inversion of an "F-A-C" triad, an "A-C-F" trine was back then considered an acoustically "destabilized" modification of "F-C-F", similarly to the idea of 7th chords in Baroque harmony. The added "A" was said to have a slightly disturbing effect on the two other tones, therefore it was called a "falso bordone" and you weren't allowed to finish a phrase with a trine containing it. If you try to play this kind of music in strict Pythagorean, you may get a better idea why that was the case.

Petr

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:
>
> Gene wrote:
>
> > Eh? Minor triads and triads, major or minor, in first inversion appear
> > constantly in early Renaissance music.
>
> Please don't mix up the concept of trines and the concept of triadic
> harmony, these are two different things and this is not a valid argument.

I said triads and I meant triads. I don't know where you got your claim from, but it's completely false as simply listening to some early Renaissance music will quickly show.

Gene wrote:

> I said triads and I meant triads. I don't know where you got your claim > from, but it's completely false as simply > listening to some early > Renaissance music will quickly show.

Depends on what we mean by "early Renaissance". When triadic harmony started to appear, octave inversions were tolerated, AFAIK, only in the densest form possible (what some people call "narrow harmony") but usually not in instances where there was a lot of octave doubling. If you think I'm wrong, prove me wrong.

Petr

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

> Depends on what we mean by "early Renaissance". When triadic harmony started
> to appear, octave inversions were tolerated, AFAIK, only in the densest form
> possible (what some people call "narrow harmony") but usually not in
> instances where there was a lot of octave doubling. If you think I'm wrong,
> prove me wrong.

And this shows there were no minor or first-inversion triads how, exactly?

Gene wrote:

> And this shows there were no minor or first-inversion triads how, exactly?

It shows that octave-inverted triads like E4-G4-C5 may have been encountered on its own possibly in the early 1500's but something like E2-E3-C4-G4 came into use later.

Petr

Mike wrote:

> Yes, but they also used the diatonic scale even before then as well.
> After they found what they liked about the diatonic scale, they
> generalized it. Gmaj -> Cmaj exists even without the diatonic scale,
> it just so happens to be in the diatonic scale. Once people found they
> liked that kind of progression, they started putting it everywhere.

Let's imagine the sequence of events. First, they favored 5-limit (or "Didymic") intonation over Pythagorean intonation while still feeling the significance of the 3/2 interval, which means they also felt the significance of 4/3 if they wanted to be able to move both up or down a fifth from the tonic. Later, they started filling up the fifths using thirds (i.e. linearly splitting a fifth into a major and a minor third) and understanding triads as completely new autonomous musical elements. The chord progression you're talking about, therefore, became very favored and popular because of this. Also, when they wanted to play major triads on both sides away from the tonic similarly to playing single tones, the best thing they could do was to add new fifths to the used tones as there already was one chain of 2 fifths there. For example, if you have 3 tones like "G-C-D-G" and split the fifths into thirds, you get "G-B-C-D-E-G", which allows you to play C major and G major. To be able to play another triad from the opposite side of the tonic, the most obvious solution is to add an F to the chain and split the F-C fifth into "F-A-C". After doing all this in strict JI and sorting the intervals from the low C in an ascending order, you get "9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1", which is exactly what Zarlino called the scale of "perfect harmony".
Obviously, a huge disadvantage of this scale is the D which is out of tune with the A and if you get it in tune, then it's out of tune with the G, which is why some theorists suggested diatonic scales with two Ds, one higher and one lower. However, the distance between these two, the syntonic comma, was considered an interval too small for practical application and even people before Zarlino found ways to get rid of it -- the earliest reference known being probably Pietro Aaron's suggestion of 1/4-comma meantone in the 1520's. Not surprisingly, if you look at the single-step intervals in the 7-tone scale of "perfect harmony", you'll find the largest of them (9/8) is only a comma wider than the second largest one (10/9) and therefore if you reduce it by that, you get a scale consisting of 5 10/9s, 2 16/15s, and 3 81/80s. This time, as there are 7 seconds and 3 commas, the easiest way is to stretch each second by 3/7 the size of the comma, which gives you a diatonic scale in 2/7-comma meantone (Zarlino's highly advocated temperament). This just confirms the fact that because musicians had liked fifths for a long time, they invented meantone. Do you get my point?

> Yes, but melodic music in porcupine is also a lot easier than harmonic > music :)

You've lost me.

> What tetrachordal permutation gives you harmonic minor?

I was meaning mainly the upper tetrachord with the "hiatus" in the middle. Probably the two "minor seconds" were not considered of equivalent meaning and equal sizes in the past. I'll explain my idea using Didymus' versions of the tetrachords as Didymus was probably *the* promoter of 5-limit ratios in Ancient Greece.
If I use "/" and "\" for raising or lowering by a syntonic comma as compared to Pythagorean tuning, then the diatonic tetrachord, according to Didymus, is something like "A-G-F/-E" so the one-step intervals are 8/9, 9/10, 15/16. You can use these steps in any order you want and it always adds up to 3/4. For example, Ptolemy's "tense diatonic" swapped the 8/9 and 9/10, making it "A-G/-F/-E". This is a Dorian tetrachord in two variants. If you reorder the steps in such a way that the 15/16 is no longer at the bottom, it still is a diatonic tetrachord but not Dorian anymore. If it's in the middle, you get a Phrygian tetrachord. If it's at the top, you get a Lydian tetrachord.
The chromatic tetrachord, according to Didymus, is "A-F#\-F/-E". So the steps are 5/6, 24/25, 15/16. Again, you can permute this tetrachord in whatever way you wish and it always adds up to 3/4. One possible permutation leading to the tetrachord in question is "A-Ab//-F/-E", another one is "A-G#\-E#\\-E".
Interestingly enough, for the enharmonic tetrachord, Didymus suggested steps of "4/5, 30/31, 31/32" where the two smaller steps are very similar in size and it makes me feel "tempted" to temper out the 961/960. If I do, then I'm left with only two one-step sizes. But it still allows me to have three different permutations of the enharmonic tetrachord.

> What do you mean? I thought you were saying the opposite, that we
> should stick to MOS's for now.

What I was saying is that after a long time playing diatonic music, they took it one step further and started using chromatic alterations because it just sounded better in some situations than if they hadn't used them. But at that time, the primary concept of meantone-based harmony was already very well established.

> OK, so then I predict that when we give composers porcupine[7],
> they'll modulate around more within it so they can do things like V-I
> when they want, and that more modulation will be necessary than with
> the diatonic scale - but I don't think this is "cheating" :)

Do you consider porcupine[7] a porcupine diatonic?

> Why does it have to be because of some mathematical ideal? Why not
> that they just started doing that because they liked the way it
> sounded?

This is what a teacher read to us at a music interpretation class, quoting various authors. Both the major dominants in minor keys and the "Picardian third" seem to have occurred for the same reason.

> Right, but why not just raise the leading tone by a chromatic semitone
> when you want it to move back to I?

This is a melodic way of thinking in the context of harmonic intonation. In that context, a minor triad simply doesn't sound as attractive as a major triad, that's all there is to it.

> But either way, this is a
> different topic from that v-I doesn't work as well as V-I though...

Okay, but how do you know that it's caused by the absence of a minor second?

> I don't think I understand - why would hanson make you feel
> uncomfortable to make you move by 3/2? Just because the hanson
> generator happens to be a 6/5, and just because it so happens that in
> the abelian group structure we have set out, 6/5 is the interval that
> generates the other ones, why should that mean that 6/5 itself should
> become more important than 3/2 in the greater hanson temperament?

Because the further the intervals are in the generator chain, the more complicated it is to get to them. It's similar to making progressions by thirds in meantone. Something like "Ab major, C major, E major" is undeniably much more convenient to approximate in another temperament if we want to base our music on sequences of thirds or if we don't want to make an impression of "wild adventurous modulations with extended enharmonic relationships".

> I actually just reworked one of the chords in the third phrase to lead
> more smoothly - do you like it better now?

Is that the same link?

> I didn't expect it to sound like that, and was surprised when it did.
> So now I feel I've learned something about "common practice" harmony.

I think I know why it did. Because of the "I-V-I" progression combined with the modulation a fourth lower. I believe something like this would sound like common practice even in untempered 7-limit JI.

> Alright, although if anything what all of this has shown me is that
> MOS's don't matter. Not only is the meantone diatonic scale an MOS,
> but it's set up to have a bunch of comma pumps built right in, so you
> can leave the tonic and then return. Maybe we can cleverly devise a
> porcupine scale like that, and then that will become the One True
> Porcupine Scale.

And that scale is called porcupine[8].

> I don't think it's just an issue of ease of singing. They imply
> different harmonies. I tend to think that it's because sharping the
> seventh itself creates a certain sound, but then you have the
> augmented second, so they just sharpened the sixth as well to fill up
> the space.

Hmmm, I'm not sure what view you're taking on it all. On one hand, you're suggesting a melodic way of thinking in the context of clearly chordal music, on the other, you're suggesting a harmonic way of thinking in the context of clearly melodic music. Is that because you're primarily interested in what it sounds like to us today? But the question which is more important than "what do we feel it like today" is "what may people have felt it like when these things came into use". Similarly, you can't say "I'm not interested in how Euler did this or that; he didn't have computers, I can do it better anyway."

> No, but now that we have centuries of common practice theory behind
> us, we don't have to reinvent the wheel all over again :)

A minor correction: "We don't have to reinvent meantone-based harmony, which is common practice harmony."
But we don't have centuries of other-temperament-based harmony behind us. If we want to have a clearer understanding of the new harmonic systems offered by the other temperaments and we don't want to wait that long, we have to take a different attitude on it and thoroughly trace it down theoretically by carefully classifying various temperament properties, based on what we know about the history of temperaments from the distant past up to the present and what we know about the general facts which occur in acoustics, no matter if it's about music or other sounds. Saying that "we've learned to understand meantone-based harmony during the centuries" doesn't suggest at all that "we don't have to learn to understand other-temperament-based harmony because of that". Whether we do it by experience in practice (which may take centuries) or with the help of systematic theory (which may take years) is another matter. Also, you won't say "Okay, I've learned English so I don't have to learn Italian from scratch" even though some words are similar in both languages.

> I interpret this to mean that musicians have a long and gradual
> history of moving AWAY from the diatonic scale - we found little
> things we liked within it, and then we applied those in ways that made
> us leave the scale. So now I'm just trying to apply those same things
> we like to porcupine.

This is a pretty "Vicentinian" attitude. And to be honest with you, some of Vicentino's thoughts turned out to be just "a bit from here, a bit from there" but not much making sense as a whole. His second tuning has some logic but his first one seems to me a bit "questionable".

> OK, and the diatonic scale allows for such a smooth resolution like
> that. But there are reasons that it sounds smooth other than just that
> we're used to it. And I think that porcupine[7] scale doesn't always
> allow for smooth resolutions. But maybe you can prove me wrong.

Do you really consider porcupine[7] a porcupine diatonic?

> Assuming that people always started with MOS, yes. But as in the case
> of Paul's pentachordal major scale, sometimes the non-MOS's sound more
> tonal.

First, what scale do you mean?
Second, not only there are non-MOS tunings, but there are also many non-MOS tunings where you don't temper particular intervals. A tuning like that is not meant to be classified in temperament terms and its development should not be viewed as a consequence of tempering somewhere out there. For example, a scale like "0-386-498-884-996-1200 cents" probably doesn't have an origin in meantone temperament and certainly sounds very harmonious. But you can't claim it to be a "universally valid" system and defend the claim by approximating it in loads of temperaments.

Petr

On 4/23/2011 8:36 PM, Mike Battaglia wrote:

> You can assert that it'll take another 23 years of training to outdo
> my first 23 and finally hear v-I in porcupine[7] as functional, but
> first you'll need to explain why the porcupine[7] MOS is the necessary
> scale that we're supposed to be making use of for "true porcupine
> harmony." On the contrary, it seems that porcupine[7] is just another
> scale that exists in the greater porcupine system. Although it is
> theoretically distinguished by being an MOS of the system; e.g. it's a
> bunch of porcupine generators stacked on top of one another, I don't
> see why that's supposed to be of prime cognitive concern. I really
> don't see why we need to elevate that MOS up to such a level where
> harmony in porcupine temperament doesn't count as "true porcupine
> harmony" unless it matches up to the MOS. Paul got away from MOS's
> with Pajara and I think the results were good.

I think it's easy to keep things straight if you're talking about the diatonic scale vs. meantone temperament, or the blackjack scale vs. miracle temperament, but we don't have easy distinctions like that in most temperaments. As far as I'm concerned, porcupine harmony is anything that treats intervals as equivalent if they differ by 250/243. This can mean comma pumps or simply different usages of the same pitch in different chords. It doesn't need to fit a porcupine MOS; I'm not even sure if my original porcupine comma pump fits into a porcupine[15] MOS (it likely does, but it wouldn't be essential).

I tried to write something in miracle[31] once, but it turned out that I was using a 12-note subset of it. If it turns out that the chord progression actually uses one of the miracle commas, I still think it would make sense to call it a miracle progression. But I think it may have just been a convenient 12-note scale with some 7-limit intervals. Sometimes I pick a temperament just for the sound of its intervals, like the time I wrote in gorgo. Although I didn't use it specifically for its commas, they do have an influence on the kinds of chords and progressions that are possible. So is it fair to say that I was writing gorgo harmony? Without analyzing the progression for commas, I don't really know. But at least I was using the gorgo MOS.

> If anything, I think this suggests another scale ideal to work
> towards: scales that are built around the minimal comma pump for a
> temperament. Scales like that will enable you to move away from the
> tonic and then return from it in the same way that I->IV->ii->V7->I
> allows you to in meantone. Maybe these won't be the MOS's at all! But
> I predict they'll certainly be tonal, and that you'll learn to hear
> the cycles of moving away from the tonic -> returning via "pun" as
> very "functional" with a bit of training.

I think that's an interesting possibility to explore. Even in meantone we don't stick with just the MOS scales, and the fact that an MOS scale happens to have good harmonic progressions could just be a lucky coincidence. With temperaments like myna and w�rschmidt, the MOS scales are more of a hindrance than anything. Maybe we can find uses for vishnu and luna that are actually manageable.

--- In tuning@yahoogroups.com, Herman Miller <hmiller@...> wrote:

> I tried to write something in miracle[31] once, but it turned out that I
> was using a 12-note subset of it.

It would be interesting to know what that subset was.

Petr, sorry for the delayed response. Don't think I'm ignoring you.
Work, life, and 91/90 have basically worn me down for the moment. I'll
come back to this tomorrow.

-Mike

On Sun, Apr 24, 2011 at 12:44 PM, Petr Parízek
<petrparizek2000@...> wrote:
>

Hi Mike,

okay, understood, I'll keep waiting.

Petr

--- Mike Battaglia <battaglia01@...> wrote:

> > I didn't make the argument that the 7-tone MOS of anything
> > was anything. I've never even said anything close to that.
> > Where on earth did you get the idea?
>
> You said
>
> > What's wrong with weaker resolutions in some places? And
> > even though weaker than the strongest known change, maybe
> > they are still strong enough compared to other changes in
> > the system. I don't put sugar on broccoli.
>
> What is "the system?" I thought you meant porcupine[7].

I meant porcupine.

-Carl

On 4/24/2011 6:17 PM, genewardsmith wrote:
>
>
> --- In tuning@yahoogroups.com, Herman Miller<hmiller@...> wrote:
>
>> I tried to write something in miracle[31] once, but it turned out that I
>> was using a 12-note subset of it.
>
> It would be interesting to know what that subset was.

I found the MIDI file; here's the offsets from 12-ET in MIDI pitch units.

C +720
C# -684
D +0
Eb +684
E +24
F +708
F# -696
G -12
Ab +672
A +12
Bb +696
B +1381

On Sun, Apr 24, 2011 at 5:54 PM, Herman Miller <hmiller@...> wrote:
>
> I think it's easy to keep things straight if you're talking about the
> diatonic scale vs. meantone temperament, or the blackjack scale vs.
> miracle temperament, but we don't have easy distinctions like that in
> most temperaments. As far as I'm concerned, porcupine harmony is
> anything that treats intervals as equivalent if they differ by 250/243.
> This can mean comma pumps or simply different usages of the same pitch
> in different chords. It doesn't need to fit a porcupine MOS; I'm not
> even sure if my original porcupine comma pump fits into a porcupine[15]
> MOS (it likely does, but it wouldn't be essential).

I don't know if chromatic MOS's have any use at all. Is porcupine[15]
that important? Why is it more important than porcupine[22]? If we all
grew up playing in 19-tet, would we care about meantone[12]? I dunno,
maybe.

> I think that's an interesting possibility to explore. Even in meantone
> we don't stick with just the MOS scales, and the fact that an MOS scale
> happens to have good harmonic progressions could just be a lucky
> coincidence. With temperaments like myna and würschmidt, the MOS scales
> are more of a hindrance than anything. Maybe we can find uses for vishnu
> and luna that are actually manageable.

This is exactly what I think. This is what history indicates to me: we
started with the diatonic scale for other reasons than harmony, but at
some point discovered 3 and then 5-limit harmony in it. Then we
screwed around with pre-functional harmony. At some point we
discovered that a nice depart-return cycle exists in the diatonic
scale with ii-V-I and its related variants, and the concept of
functional harmony was born, which lets us leave the tonic and
circularly come back to it (read: comma pump). The diatonic scale,
more or less coincidentally, happens to have smooth 5-limit harmony
everywhere and tons of comma pumps built right in, so as we were
figuring things out at first we stuck to that. As our ears expanded,
we started hearing pretty obvious functional relationships that didn't
exist in the diatonic scale, so we went with those and started getting
more and more away from strict diatonic harmony by inventing things
like the harmonic minor scale. The Romantic era saw an increase in
this. To me, this indicates that whatever it is that we liked about
functional harmony isn't actually something that derives from the
diatonic scale, but that the diatonic scale just so happens to contain
a lot of useful functional nuggets that were useful to get us started.
I do, however, think that meantone tempering plays an important role
in diatonic harmony in the sense that it creates the specific
depart/return cycle that we're all used to, and that this is more
fundamental to the meantone "system" than the diatonic scale.

Anyway, then came along the 20th century and the 7-limit explosion, so
we started seeing ratios of 7 thrown on top of the usual functional
harmony, and the synthesis of these styles ended up being all of the
pop standards of the 30s and 40s and such. Mini comma pumps are
everywhere, such as Imaj7 - iim7 - V7 - Imaj7. We also saw the modal
explosion of Debussy and Ravel, which influenced much of the rest of
the century's music, and aimed to find new functional relationships
that we might have ignored (by making use of chord progressions in
different modes and changing the mode much more frequently).
Meanwhile, folks like Stravinsky and others were exploring things like
the octatonic scale, which is diminished[8]. The folks who sought to
functionalize diminished[8] generally did by throwing it over things
like dominant 7 chords, which then resolve a fifth down. And now...

Now we're back to pre-functional harmony again, but in other tuning
systems. The takehome point to me is that we didn't just learn to hear
pre-functional harmony as functional... we instead invented functional
harmony. This indicates that functional harmony is an actual thing
that we discovered, something that the diatonic scale enabled us to
play around with, but that is not limited to the diatonic scale. Thus,
I don't think that porcupine functional harmony will have to do with
porcupine[8] either. I think functional harmony basically just means,
you know, all of the normal harmonic tricks you've ever learned in
12-tet -- from the common practice harmony of Mozart to the hip stuff
that Yes and Zeppelin were doing -- but the thing is, we've been
basing it all around comma pumps that we've internalized our whole
lives. By utilizing the same fundamental chord progressions, e.g. V-I,
iv-I, Dm/F -> Cmaj7 -> Dm7, whatever you want - but basing them around
different depart and return cycles, aka comma pumps, we can write
functional harmony in other tempered systems. This may mean getting
away from the MOS's of the system, as I don't think the brain knows or
cares much about generators, and cares more about simple harmonic
ratios, but hey, I could be wrong. Thus ends my report.

-Mike

Alright, back to this!

On Sun, Apr 24, 2011 at 12:44 PM, Petr Parízek
<petrparizek2000@...> wrote:
>
> > Yes, but they also used the diatonic scale even before then as well.
> > After they found what they liked about the diatonic scale, they
> > generalized it. Gmaj -> Cmaj exists even without the diatonic scale,
> > it just so happens to be in the diatonic scale. Once people found they
> > liked that kind of progression, they started putting it everywhere.
> This time, as there are 7 seconds and 3 commas, the
> easiest way is to stretch each second by 3/7 the size of the comma, which
> gives you a diatonic scale in 2/7-comma meantone (Zarlino's highly advocated
> temperament). This just confirms the fact that because musicians had liked
> fifths for a long time, they invented meantone. Do you get my point?

Yes. But I thought that people started hearing the concept of triadic
harmony in their heads before meantone was discovered - as a general
harmonic "concept" - and then after the fact, people realized that if
we all like thirds, we should just tune the fifths flat to intone them
better. So I thought at the time it was an issue of intonation, not an
issue of consciously trying to create comma pumps. But it's a good
thing that comma pumps are a side effect of all of this.

As another note, I notice a similar thing with dominant 7 chords: no
matter what tuning system I'm in, I find that the best dominant 7th
generally seems to be 16/9 - like I really want to hear the F-E motion
in G7 to C major as 4/3 -> 5/4 over the root. So if you're in 19 or
31-equal, the best dominant 7th will generally be 9/5, which is the
same as 16/9. But, if you're in 22-tet, the best dominant 7th will be
7/4, which is also tempered to be equal to 16/9.

So we can do the same thing that they did with meantone - for purely
intonation-based reasons, and without thinking of comma pumps at all,
we can decide that by making the fourths slightly flat, two of them
can hit the "ideal" of 7/4. Of course, doing so eliminates 64/63, and
creates a whole new world of comma pumps. Thus, I wonder if we've been
making use of 64/63 comma pumps in 12-equal without even realizing
it...? Maybe this is what Cmaj -> Fmaj -> Bb7#11 -> C7 is? I think so
:)

Anyway, this is a tangent.

> I was meaning mainly the upper tetrachord with the "hiatus" in the middle.
> Probably the two "minor seconds" were not considered of equivalent meaning
> and equal sizes in the past. I'll explain my idea using Didymus' versions of
> the tetrachords as Didymus was probably *the* promoter of 5-limit ratios in
> Ancient Greece.

I really don't know much about Didymus, so maybe my above analysis of
history was wrong.

> Interestingly enough, for the enharmonic tetrachord, Didymus suggested steps
> of "4/5, 30/31, 31/32" where the two smaller steps are very similar in size
> and it makes me feel "tempted" to temper out the 961/960. If I do, then I'm
> left with only two one-step sizes. But it still allows me to have three
> different permutations of the enharmonic tetrachord.

I know nothing about enharmonic harmony but this sounds fascinating.
Why not do it and see what pumps emerge.

> > What do you mean? I thought you were saying the opposite, that we
> > should stick to MOS's for now.
>
> What I was saying is that after a long time playing diatonic music, they
> took it one step further and started using chromatic alterations because it
> just sounded better in some situations than if they hadn't used them. But at
> that time, the primary concept of meantone-based harmony was already very
> well established.

What do you think of my response to Herman in the last thread? That's
what I'd write here - I think that the diatonic scale, coincidentally,
happens to have features of "functional" harmony in it, and so when we
were just starting to realize there was something there, it proved to
be useful as a starting point. Then we got into "pre-functional"
harmony. Instead of learning to hear pre-functional harmony as
functional, we invented functional harmony - aka the harmonic minor
scale. I guess you could say that in the 20th century, prog rock bands
started getting back into pre-functional harmony again, which I think
is awesome.

But I don't think that functional harmony as we know it comes out of
the diatonic scale. I think that it comes out of individual atomic
chord progressions like V-I and iv-I, and on a more complex "modal"
note perhaps things like Fmaj7->Cmaj7->Dm7->Fmaj7->Amaj (melody
A-B-C-A-A on top). But I also think that it comes from an internalized
sense of the 81/80 pump. If we keep the same chord progression atoms,
but internalize a different pump, the effect to my ears is that it
sounds oddly familiar, but lines up in ways you don't expect... which
is cool. I'm not saying that it isn't worthwhile to explore
porcupine[7] and porcupine[8], I'm just saying that thinking in terms
of the JI lattice is more fundamental than thinking in terms of the
porcupine generator... in my opinion.

> > OK, so then I predict that when we give composers porcupine[7],
> > they'll modulate around more within it so they can do things like V-I
> > when they want, and that more modulation will be necessary than with
> > the diatonic scale - but I don't think this is "cheating" :)
>
> Do you consider porcupine[7] a porcupine diatonic?

Sure, why not? I love it. Major and minor triads share the same triad class.

> > Right, but why not just raise the leading tone by a chromatic semitone
> > when you want it to move back to I?
>
> This is a melodic way of thinking in the context of harmonic intonation. In
> that context, a minor triad simply doesn't sound as attractive as a major
> triad, that's all there is to it.

So we agree then?

> > But either way, this is a
> > different topic from that v-I doesn't work as well as V-I though...
>
> Okay, but how do you know that it's caused by the absence of a minor second?

I think that v-I is a beautiful sound - they use it in movies and film
scores sometimes. It's just different than V-I. So I shouldn't have
said that it "works" less well, but it's just different. V-I creates a
sense of expectation and then fulfills it. v-I doesn't do that but is
beautiful in a different way. I'm not sure what makes V-I sound the
way it does, but I'm guessing the leading tone. Maybe I'm wrong.

> > I don't think I understand - why would hanson make you feel
> > uncomfortable to make you move by 3/2? Just because the hanson
> > generator happens to be a 6/5, and just because it so happens that in
> > the abelian group structure we have set out, 6/5 is the interval that
> > generates the other ones, why should that mean that 6/5 itself should
> > become more important than 3/2 in the greater hanson temperament?
>
> Because the further the intervals are in the generator chain, the more
> complicated it is to get to them. It's similar to making progressions by
> thirds in meantone. Something like "Ab major, C major, E major" is
> undeniably much more convenient to approximate in another temperament if we
> want to base our music on sequences of thirds or if we don't want to make an
> impression of "wild adventurous modulations with extended enharmonic
> relationships".

Right, but why are the generators important? They're important
mathematically, as the basis vectors for the resulting abelian group
for a particular temperament, but why should they be more musically
important than things like motion by 3/2?

Maybe we just have different goals - you are trying to explore the
MOS's for these temperaments, whereas I'm just trying to explore the
properties of the resulting lattice. I like what you're doing a lot,
if you haven't caught on - I think it's the biggest breakthrough I've
heard on this list since I joined. My only point is that the MOS's of
a temperament, and the temperament itself, are different things -
nothing more nothing less. There may be a better scale for Myna
"tonal" harmony than Myna[9], for example.

> > I actually just reworked one of the chords in the third phrase to lead
> > more smoothly - do you like it better now?
>
> Is that the same link?

Yes, here it is again

http://soundcloud.com/mikebattagliamusic/functionalporcupineexcerpt

> > I didn't expect it to sound like that, and was surprised when it did.
> > So now I feel I've learned something about "common practice" harmony.
>
> I think I know why it did. Because of the "I-V-I" progression combined with
> the modulation a fourth lower. I believe something like this would sound
> like common practice even in untempered 7-limit JI.

That's the point! :) I was trying to fulfill all of the following requirements:

1) Only use common practice chord progressions, such as I-V-I, which
are clearly "functional," hence enabling you to "follow" the harmony
as you follow common practice music
2) Make it so that the progression only works in some other
temperament, not in meantone
3) Avoid awkward jumps by 81/80. In the case of porcupine, awkward
jumps by 81/80 are transformed into manageable jumps by 25/24, so this
isn't a problem

Those were my goals. I met all of those requirements for myself in the
example I posted. Now I want to get away from common practice harmony
and do the above with all the hip modal stuff that I love, which is my
next goal, and one I explored with my recent AXiS improvisation that I
posted. If you didn't see it, it's here:

http://www.youtube.com/watch?v=XSfnyr1MhXE&feature=BFa&list=PL100AF8DBBDE723C4&index=5

This is less "functional," but only works in porcupine. Does this make
you happier? :)

> > Maybe we can cleverly devise a
> > porcupine scale like that, and then that will become the One True
> > Porcupine Scale.
>
> And that scale is called porcupine[8].

Haha, maybe, but when you move up and down the scale, triads turn into
other triads in inversion, which always weirded me out :)

> > I don't think it's just an issue of ease of singing. They imply
> > different harmonies. I tend to think that it's because sharping the
> > seventh itself creates a certain sound, but then you have the
> > augmented second, so they just sharpened the sixth as well to fill up
> > the space.
>
> Hmmm, I'm not sure what view you're taking on it all. On one hand, you're
> suggesting a melodic way of thinking in the context of clearly chordal
> music, on the other, you're suggesting a harmonic way of thinking in the
> context of clearly melodic music. Is that because you're primarily
> interested in what it sounds like to us today?

If you melodically sing a major chord, does it still make you feel
happy? Do melodically singing minor chords still make you feel sad? Do
you think these things were true 400 years ago? This is a serious
question, because sometimes I feel like people tend to answer that
question with "no."

> > No, but now that we have centuries of common practice theory behind
> > us, we don't have to reinvent the wheel all over again :)
>
> A minor correction: "We don't have to reinvent meantone-based harmony, which
> is common practice harmony."

I agree, but I think that you can generalize common practice harmony
to other tuning systems by using the same chord progression atoms as
common practice music, as I mentioned before - V-I, viib5-I, etc - but
do everything in the new tempered system, and make use of new comma
pumps. The net effect is that the result is intelligible to us 21st
century musicians, but still stimulating in that it works out
differently. I think that's a good way to generalize common practice
harmony, and it demystifies the whole thing.

> But we don't have centuries of other-temperament-based harmony behind us. If
> we want to have a clearer understanding of the new harmonic systems offered
> by the other temperaments and we don't want to wait that long, we have to
> take a different attitude on it and thoroughly trace it down theoretically
> by carefully classifying various temperament properties, based on what we
> know about the history of temperaments from the distant past up to the
> present and what we know about the general facts which occur in acoustics,
> no matter if it's about music or other sounds. Saying that "we've learned to
> understand meantone-based harmony during the centuries" doesn't suggest at
> all that "we don't have to learn to understand other-temperament-based
> harmony because of that". Whether we do it by experience in practice (which
> may take centuries) or with the help of systematic theory (which may take
> years) is another matter. Also, you won't say "Okay, I've learned English so
> I don't have to learn Italian from scratch" even though some words are
> similar in both languages.

I agree, but you and I are just taking two different approaches to
doing it. We both agree that you can't just jump into complex extended
porcupine harmony, because it'll be too overwhelming. However, we have
two different ways to simplify it:

Your way of simplifying it is to look at the chain of generators and
the MOS's and try to work out "porcupine diatonic harmony" first, and
then extend it. I think the results are beautiful, although to my ears
it's still kind of overwhelming sometimes.

My way of simplifying it was to ignore generators and instead look at
the JI lattice, and try to work out "porcupine common practice
harmony" first, and then extend it. I think the results are very
"intelligible," and for a lot of my friends, it was the first time
they actually understood what I was trying to do (a few of them who
aren't musicians didn't even know it was microtonal).

There is no problem with the two approaches coexisting, because they
don't disagree with one another. Yours will likely give more poetic
results, although mine may sound more intelligible to western ears...
for now. You're trying to explore the 2D porcupine lattice, and I'm
trying to explore the 3D JI lattice, but from a porcupine-tempered
perspective. Either way, we're moving towards the same thing, it's
just a matter of which way makes for easier learning at first. Surely
some compromise will win out at the end anyway - I'm trying to stick
closer to porcupine[8] harmony as a way of evolving, and even the
Baroque people threw in V chords where the diatonic scale doesn't put
them. So we'll probably reach the same goal.

> > Assuming that people always started with MOS, yes. But as in the case
> > of Paul's pentachordal major scale, sometimes the non-MOS's sound more
> > tonal.
>
> First, what scale do you mean?
> Second, not only there are non-MOS tunings, but there are also many non-MOS
> tunings where you don't temper particular intervals. A tuning like that is
> not meant to be classified in temperament terms and its development should
> not be viewed as a consequence of tempering somewhere out there. For
> example, a scale like "0-386-498-884-996-1200 cents" probably doesn't have
> an origin in meantone temperament and certainly sounds very harmonious. But
> you can't claim it to be a "universally valid" system and defend the claim
> by approximating it in loads of temperaments.

I don't think that scales are really systems in themselves. I think
that JI harmony + comma pumps are the real system, and that you can
construct scales that encapsulate both of these features.

Phew! These posts are getting long... :)

-Mike

Two things here, and I'll leave out quotes because the paragraphs are long.

Firstly, the major and minor scales arose out of chromatic
exploration. You talk vaguely about the diatonic scale. The Ionian
and Aeolian scales weren't there at the Renaissance. They were stuck
on at the last minute as the modal system was dissolving. The major
key started with a chromatic use of Lydian, and the minor key with
Dorian (among others -- don't confuse me with an authority). The
major scale is itself a result of tonal, rather than modal, thinking.

Secondly, "functional harmony" as I understand it means that every
chord has a function: tonic, dominant, or subdominant. It's a way of
analyzing tonal harmony. I think that can be extended outside
meantone, but you may need to break the connection with fifths.

Graham

On Tue, Apr 26, 2011 at 1:23 AM, Graham Breed <gbreed@...> wrote:
>
> Two things here, and I'll leave out quotes because the paragraphs are long.
>
> Firstly, the major and minor scales arose out of chromatic
> exploration. You talk vaguely about the diatonic scale. The Ionian
> and Aeolian scales weren't there at the Renaissance. They were stuck
> on at the last minute as the modal system was dissolving. The major
> key started with a chromatic use of Lydian, and the minor key with
> Dorian (among others -- don't confuse me with an authority). The
> major scale is itself a result of tonal, rather than modal, thinking.

I'm also not an authority, but I think that ionian and things like
harmonic minor took precedence because they contain the features of
functional harmony, not the other way around. This implies that these
features may not just be learned ways of interpreting the diatonic
scale. There are clearly certain facets of functional harmony that
we've learned to like, and there's also some kind of concept that we
keep discovering as time goes on that seems to defy learning. I don't
claim to know how things lay out, but I think a clue can be found in
the higher limit porcupine example I posted.

> Secondly, "functional harmony" as I understand it means that every
> chord has a function: tonic, dominant, or subdominant. It's a way of
> analyzing tonal harmony. I think that can be extended outside
> meantone, but you may need to break the connection with fifths.

Or that you need to break the connection with things like
I->iii->vi->ii->V->I. I'm afraid of turning into a broken record, but
- much has been made in classical theory of movement away from the
tonic and then back to the tonic, in directions like the dominant or
the subdominant like you said. Comma pumps let you create these
musical journeys by departing from the tonic and smoothly and
circularly come back to it, so in the last few days I've started to
realize that this is a good way to generalize this facet of common
practice theory.

If you just replace "dominant" and "subdominant" with 3/2 and 4/3
respectively, I find the whole thing works. The mediant becomes the
minor 5/4 chord, and the submediant becomes the minor 5/3 chord. In
the porcupine example I made, especially the one with the secondary
dominants, and the little ditty I called an "excerpt," I treated
things exactly as that. I went Cmaj -> A/C# -> Dmaj -> B/D# -> Emaj ->
C#/E# -> F#maj, except if you do this in porcupine, you end up at
Fmaj, not F#maj (I wish I had a better way to notate this
progression). Suddenly you've departed from the tonic, and now you're
mysteriously at the subdominant and in a position to return to it,
just like I-iii-vi-ii-V-I suddenly lands you at the dominant in
meantone. And in my example, I only used things like V-I
relationships, secondary dominants, and things like diminished
resolutions, all of which function as you'd expect, it's just that the
whole thing connects differently. It makes sense in some kind of odd
way, however. When I showed my previous experiments, like the
17-etude, to my 12-tet accultured friends, they weren't that thrilled
with it - they thought it was cool, "they guess," but heard it as
mostly out of tune. But the porcupine example above they tended to
pick up on right away, could follow what was going on, but then they
were surprised to end up at the iv chord.

Yes, there may be other cool things you can do as well, and porcupine
is an 11-limit system which can lead to all kinds of exotic sounds,
but is this not worth anything? In terms of generalizing 5-limit
common practice harmony to other tempered systems I think this is a
decent way to go. And, I think if we had started with porcupine[7]
instead of meantone[7], we'd have evolved into a similar functional
harmony than we did with meantone, altering the scale when necessary
just as we altered meantone[7] for the sake of increasing the
functional harmony.

This is getting long, but since I had already written the following I
might as well throw it in as a concrete example: I said earlier that
meantone lets you depart from the harmony by way of
I->iii->vi->ii>V->I. Something like magic can let you do the same
thing, except you depart by way of 5/4 and return to the tonic by 3/2.
If direct motion by 5/4 after 5/4 after 5/4 sounds odd to you, you
have two options - either get used to it, or try to use some common
practice theory techniques by throwing some secondary dominants in
there and stuff like that. This may make it sound more familiar to
western ears (at least it does to my ears: Cmaj -> B7 -> Emaj -> D#7
-> G#maj -> Gbmaj (magic-tempered to be equivalent to Fxmaj) -> Cbmaj
(B#maj) -> Bb7 -> Ebmaj -> D7 -> Gmaj (holy dominant!)-> G7 -> Cmaj.
Is this "better" than the first one? No, not necessarily, although it
may sound more familiar (and boring). But is it "cheating?" No. And it
certainly only works in magic temperament.

Anyway, just a new paradigm I'm exploring, I don't claim to have all
the answers.

-Mike

On 26 April 2011 10:02, Mike Battaglia <battaglia01@...> wrote:

> I'm also not an authority, but I think that ionian and things like
> harmonic minor took precedence because they contain the features of
> functional harmony, not the other way around. This implies that these
> features may not just be learned ways of interpreting the diatonic
> scale. There are clearly certain facets of functional harmony that
> we've learned to like, and there's also some kind of concept that we
> keep discovering as time goes on that seems to defy learning. I don't
> claim to know how things lay out, but I think a clue can be found in
> the higher limit porcupine example I posted.

That may be my point, but technically Ionian never took precedence.
Modes that didn't include Ionian were replaced by the major key.
Ionian is a theoretical construction that tried to explain tonal
practice in modal terms.

The Lydian mode was already seen as deficient in Medieval times, and
accidentals were invented so that its fourth degree could be altered.
Because of the way Boethius misread the Greek texts, they kept
thinking of the alterations relative to Lydian rather than inventing
Ionian (which was the true Lydian . . . but I forget). With tonal
harmony, based on triads related by fifths, IV was more important then
V/V so the alteration was normalized in the major key. It's possible
to think of major keys in terms of diatonic harmony based on the major
scale, but historically the harmony came before the scale.

Minor keys are more fluid. They've always had movable pitches. The
key signature follows the theoretical Aeolian but minor harmony is not
based on the Aeolian scale. The harmonic minor never had any kind of
precedence. It's a theoretical scale used to describe certain pitches
being used for harmonic reasons.

Still, some features of the harmonic evolution are arbitrary. The
idea that V-I is stronger than IV-I is good, but it's a rule that a
lot of blues still manages to ignore. Whether chords need to be
related by fifths, or functional harmony can be based on other
intervals with enough exposure, is still an open question. Meantone
temperament biases us towards fifths and maybe fifths biased us
towards Meantone.

> If you just replace "dominant" and "subdominant" with 3/2 and 4/3
> respectively, I find the whole thing works. <snip>

Yes, and you can assign other functions accordingly. You can settle
your listener with conventional resolutions. But at some point you
can still kick away the ladder and have cadences based on generating
intervals rather than fifths. If the functions have been established,
they should still work. With some temperaments, fifths are relatively
complex, so it isn't so good to string chords out along them.

> This is getting long, but since I had already written the following I
> might as well throw it in as a concrete example: I said earlier that
> meantone lets you depart from the harmony by way of
> I->iii->vi->ii>V->I. Something like magic can let you do the same
> thing, except you depart by way of 5/4 and return to the tonic by 3/2.
> If direct motion by 5/4 after 5/4 after 5/4 sounds odd to you, you
> have two options - either get used to it, or try to use some common
> practice theory techniques by throwing some secondary dominants in
> there and stuff like that. This may make it sound more familiar to
> western ears (at least it does to my ears: Cmaj -> B7 -> Emaj -> D#7
> -> G#maj -> Gbmaj (magic-tempered to be equivalent to Fxmaj) -> Cbmaj
> (B#maj) -> Bb7 -> Ebmaj -> D7 -> Gmaj (holy dominant!)-> G7 -> Cmaj.
> Is this "better" than the first one? No, not necessarily, although it
> may sound more familiar (and boring). But is it "cheating?" No. And it
> certainly only works in magic temperament.

I'm happy with 5/4 movement in Magic. The comma pump is also good,
though, where you build up with 5/4s and release with a 3/2. It
establishes a direction for 5/4 movement, and so assigns 5/4 a
subdominant function.

Graham

On Tue, Apr 26, 2011 at 2:36 AM, Graham Breed <gbreed@...> wrote:
>
> That may be my point, but technically Ionian never took precedence.
> Modes that didn't include Ionian were replaced by the major key.
> Ionian is a theoretical construction that tried to explain tonal
> practice in modal terms.

Alright.

> The Lydian mode was already seen as deficient in Medieval times, and
> accidentals were invented so that its fourth degree could be altered.
> Because of the way Boethius misread the Greek texts, they kept
> thinking of the alterations relative to Lydian rather than inventing
> Ionian (which was the true Lydian . . . but I forget). With tonal
> harmony, based on triads related by fifths, IV was more important then
> V/V so the alteration was normalized in the major key. It's possible
> to think of major keys in terms of diatonic harmony based on the major
> scale, but historically the harmony came before the scale.

OK, that's what I'm saying, I thought you were disagreeing for some
reason. Likewise one can also put harmony above porcupine[7]. That was
the point I was asserting, in opposition to the view that porcupinized
functional harmony should derive from porcupine[8] or something like
that.

> Minor keys are more fluid. They've always had movable pitches. The
> key signature follows the theoretical Aeolian but minor harmony is not
> based on the Aeolian scale. The harmonic minor never had any kind of
> precedence. It's a theoretical scale used to describe certain pitches
> being used for harmonic reasons.

That's also what I'm trying to say. I just threw harmonic minor out
there because I didn't want to say "aeolian." So yes, I agree. Except
I think that this applies to major as well, and that we just happened
to get lucky in that the diatonic scale facilitated much of what we
were looking for with major, so we didn't have to alter much. You
could say the full major scale is C D E F G Ab A Bb B C, to take into
account some interesting Romantic chord progressions (like For No One,
by the Beatles, noted romantic composers), but we instead prefer to
think of modulations for chords like Bb7 in Cmaj7 as deriving from C D
E F G Ab Bb C. We likewise do the same with minor.

Anyway, the point is, maybe we can generate "lucky" scales for other
temperaments in the same way, which may in some cases mean departing
from the MOS's.

> Still, some features of the harmonic evolution are arbitrary. The
> idea that V-I is stronger than IV-I is good, but it's a rule that a
> lot of blues still manages to ignore.

Even in the blues, V7 -> I7 is stronger than IV7 -> I7. That doesn't
mean that it's better, or that you always have to use it. That is a
much stronger statement than I was trying to make. So I don't think
it's a rule at all. I hear the V7-IV7-I7 at the end of a 12-bar blues
as being like a 7-limit plagal cadence, and I think that's what they
were going for. Sometimes they strengthen it by making it a iim7 V7 I7
if they want.

> > If you just replace "dominant" and "subdominant" with 3/2 and 4/3
> > respectively, I find the whole thing works. <snip>
>
> Yes, and you can assign other functions accordingly. You can settle
> your listener with conventional resolutions. But at some point you
> can still kick away the ladder and have cadences based on generating
> intervals rather than fifths. If the functions have been established,
> they should still work. With some temperaments, fifths are relatively
> complex, so it isn't so good to string chords out along them.

I never said you can't - I'm an improvisational musician after all,
and before I got into "xenharmonic" music I was trying to find
"xenharmonic music" in 12-equal by doing just that. I'm just making
the point that fifths are fifths, e.g they are what they are. Bright,
pure, (boring) common practice harmony. They won't sound different in
another tuning system just because technically, the basis set for the
abelian group fully generating the system isn't a fifth. The sound of
motion by 3/2 isn't related to its Graham complexity, but its harmonic
complexity. At least that's the only thing that seems sensible to me.
I'm not saying that you can't learn to hear motion by 5/4 as similarly
intelligible, but that it will always have a different sound than
motion by 3/2, because it's a different interval. Likewise, you can
still move around by 5/4 in meantone even if you aren't in magic, but
only in magic will doing so connect you to 3/2.

The question we're arriving at is whether or not there might be a
better way to generate "tonal" scales for a temperament. Rather than
just by stacking generators till you hit an MOS, you could possibly
build scales around the ideal of having the following four features

1) Contains a simple comma pump for that temperament (inspired by
I-iii-vi-ii-V-I, or even just I-IV-ii-V-I, or maybe iim7-V-I)
2) Has as many usable 5-limit triads on as many steps of the pump as
possible (inspired by how porcupine[7] having four diminished chords
in a row sucks)
3) Has a V-I to the root (inspired by how people have historically
altered the minor scale to do this)
4) Is proper and of a manageable size for melody, and ideally the same
number of notes as an MOS of the scale (inspired by how we didn't add
in the extra notes to make a 9-note minor scale, but rather came up
with three proper minor scales)

If you think that #3, the having a V-I to the root isn't necessary,
then feel free to substitute the 3/2 with a 5/4 or something. I'd be
happy to work the mathematics out in general to allow for whatever
variations you'd like.

Doing so should enable you to depart and return to the tonic in small
cycles, but will keep the number of notes manageable - kind of like
the diatonic scale does. It will "dumb down" or "encapsulate" the
essential tonal features for some temperament. As Gene has recently
developed the concept of the minimal comma pump, which I still don't
understand, maybe that'll do the trick.

There may not be a single scale that encapsulates all of those
features for a single temperament. In the case of the minor scale,
there wasn't, so we ended up with generally nine notes that we use,
encapsulated by three proper scales - natural, melodic, and harmonic
minor. We can always do the same here. Keep in mind that you can smush
as many chords into a single scale as you'd like, but the more you do
this, the less it will work for melody.

> I'm happy with 5/4 movement in Magic. The comma pump is also good,
> though, where you build up with 5/4s and release with a 3/2. It
> establishes a direction for 5/4 movement, and so assigns 5/4 a
> subdominant function.

Well, there you go. I still can't hear 5/4 as dominant chords yet, but
hopefully it's possible. If anything, I actually hear movement by 8/5
as sounding stronger in resolution than movement by 5/4. I can imagine
what it would be like to hear 5/4 as dominant chords, but it never
really sounds that way to me in practice. I hope I figure out the
trick, because that would be nice. Maybe it requires a generalization
of ii-V-I to movement by 5/4.

-Mike

Me:
>> I'm happy with 5/4 movement in Magic. The comma pump is also good,
>> though, where you build up with 5/4s and release with a 3/2. It
>> establishes a direction for 5/4 movement, and so assigns 5/4 a
>> subdominant function.

Mike:
> Well, there you go. I still can't hear 5/4 as dominant chords yet, but
> hopefully it's possible. If anything, I actually hear movement by 8/5
> as sounding stronger in resolution than movement by 5/4. I can imagine
> what it would be like to hear 5/4 as dominant chords, but it never
> really sounds that way to me in practice. I hope I figure out the
> trick, because that would be nice. Maybe it requires a generalization
> of ii-V-I to movement by 5/4.

Aren't you agreeing with me again? 5/4 = subdominant.

Graham

On Tue, Apr 26, 2011 at 3:40 AM, Mike Battaglia <battaglia01@...> wrote:
>
> There may not be a single scale that encapsulates all of those
> features for a single temperament. In the case of the minor scale,
> there wasn't, so we ended up with generally nine notes that we use,
> encapsulated by three proper scales - natural, melodic, and harmonic
> minor. We can always do the same here. Keep in mind that you can smush
> as many chords into a single scale as you'd like, but the more you do
> this, the less it will work for melody.

Since this may be confusing, here's a concrete example:

Let's look at porcupine[7]. I haven't worked this out with
porcupine[8] yet, so we'll stick with the 7-note MOS for now.
Porcupine[7] lends itself to an analogous system to
harmonic/melodic/natural minor, except it's the major scale that needs
altering. Let's say the major scale in porcupine is

Natural porcupine major
G A B C D E F G (4 3 3 3 3 3 3 in 22-tet)

One way to alter this is just like they did with natural minor, to
create the leading tone. Let's denote the # and b accidentals to mean
alteration by c=L-s, which is 1\22 in 22-equal. So now you have

Melodic porcupine major
G A B C D E F# G (4 3 3 3 3 4 2 in 22-tet)

Note that the G-C is 11/8. Now, if you were living in an alternate
universe in which common practice harmony had arose around porcupine,
the impulse to move by 4/3 sometimes would probably be pretty natural.
So we can envision they'd have come up with a third scale, which is

Subdominant porcupine major
G A B Cb D E F G (4 3 2 4 3 3 3 in 22-tet)

But maybe they'd like to combine them, so that you end up with a
leading tone and a fourth as well:

Harmonic porcupine major
G A B Cb D E F# G (4 3 2 4 3 4 2 in 22-tet)

This should look mighty familiar, and it is. It's the 5-limit JI major
scale, but porcupine tempered. Obviously, porcupine tempering doesn't
actually affect the JI major scale directly, but once you go a chord
or two out more things start to matter.

Now this may seem obvious to you, and it may seem like cheating: if
you want common practice harmony, alter porcupine[7] so you get the JI
major scale. But I say that there's a greater point here, which is
that this is the same process that people went through, perhaps
subconsciously, when they worked out common practice harmony for minor
- it wasn't "cheating" back then. Lastly, it may very well be the case
that this process, for many 5-limit temperaments that form 7-note
MOS's, land you at a not-obviously-tempered JI scale. Mavila is a
notable example. But it's also true that you can take this same
paradigm and apply it to many higher-limit temperaments, maybe ones
that don't even have 5, and thus gives you a paradigm paralleling that
of common practice harmony to get started with something completely
different.

-Mike

Mike wrote:

> Yes. But I thought that people started hearing the concept of triadic
> harmony in their heads before meantone was discovered - as a general
> harmonic "concept" - and then after the fact, people realized that if
> we all like thirds, we should just tune the fifths flat to intone them
> better. So I thought at the time it was an issue of intonation, not an
> issue of consciously trying to create comma pumps. But it's a good
> thing that comma pumps are a side effect of all of this.

You thought right, of course.
But (and there's the small but, not the big but) if you read my words carefully, it would be clear to you that after stacking fifths and adding thirds and stacking fifths upon the thirds and so on, they soon discovered the "scale of perfect harmony", as Zarlino called it. For one thing, they knew that the D-A fifth was unplayable in that scale as it was a comma narrower. For another thing, they soon understood that the two major seconds were so similar in size that the best thing they could do was to equate them. So yes, the harmony was first and the temperament followed; but the reason why it was meantone and not another 2D temperament is that they started chaining fifths and understood the thirds as "something else that adds a nice pleasant sound and a new dimension to the harmony". This is similar to what you're saying: Start with fifths, hear what they sound like, then add new things and somehow incorporate them into the system we have.

> As another note, I notice a similar thing with dominant 7 chords: no
> matter what tuning system I'm in, I find that the best dominant 7th
> generally seems to be 16/9 - like I really want to hear the F-E motion
> in G7 to C major as 4/3 -> 5/4 over the root. So if you're in 19 or
> 31-equal, the best dominant 7th will generally be 9/5, which is the
> same as 16/9.

Interestingly enough, there were people like Tartini or Fokker who claimed that there might possibly be an exception made to the rule in some situations, by which they meant what they called a "semi-diminished 7th" or a "sesqui-diminished fifth" -- for example, a tetrad like "G-B-D-F<" (where "<" is a half-flat) contains a sesqui-diminished 5th of "B-F<" and a semi-diminished 7th of "G-F<". The problem here is that most of the 17th century was still using meantone, therefore 4:5:6:7 was much more often viewed as "G-B-D-E#", which is how Huygens explained the similarity.

> But, if you're in 22-tet, the best dominant 7th will be
> 7/4, which is also tempered to be equal to 16/9.

Of course. And 12-EDO is some sort of midway between the two.

> So we can do the same thing that they did with meantone - for purely
> intonation-based reasons, and without thinking of comma pumps at all,
> we can decide that by making the fourths slightly flat, two of them
> can hit the "ideal" of 7/4. Of course, doing so eliminates 64/63, and
> creates a whole new world of comma pumps. Thus, I wonder if we've been
> making use of 64/63 comma pumps in 12-equal without even realizing
> it...? Maybe this is what Cmaj -> Fmaj -> Bb7#11 -> C7 is? I think so
> :)

Quite possibly.

> I really don't know much about Didymus, so maybe my above analysis of
> history was wrong.

Not sure which of your past words you're referring to.

> I know nothing about enharmonic harmony but this sounds fascinating.
> Why not do it and see what pumps emerge.

Once we find a good way to deal with the prime of 31.

> What do you think of my response to Herman in the last thread? That's
> what I'd write here - I think that the diatonic scale, coincidentally,
> happens to have features of "functional" harmony in it, and so when we
> were just starting to realize there was something there, it proved to
> be useful as a starting point. Then we got into "pre-functional"
> harmony. Instead of learning to hear pre-functional harmony as
> functional, we invented functional harmony - aka the harmonic minor
> scale. I guess you could say that in the 20th century, prog rock bands
> started getting back into pre-functional harmony again, which I think
> is awesome.

Possibly. But this doesn't contradict what I'm saying. As long as you start with progressions by fifths, sooner or later, the "best compromise" turns out to be meantone anyway. And from this point of view, any other temperament may sound like nothing more than a lower-quality approximation to triadic progressions by fifths.

> But I don't think that functional harmony as we know it comes out of
> the diatonic scale. I think that it comes out of individual atomic
> chord progressions like V-I and iv-I, and on a more complex "modal"
> note perhaps things like Fmaj7->Cmaj7->Dm7->Fmaj7->Amaj (melody
> A-B-C-A-A on top).

It comes out of I'll tell you what. Stacking pure fifths, followed by linearly splitting pure fifths, followed by the desperate need to do something about the dreaded comma which made the D-A fifth unplayable and which was found too small to be practically applicable as a scale step. What else than meantone would you expect to come out of such a sequence of events?

> But I also think that it comes from an internalized
> sense of the 81/80 pump. If we keep the same chord progression atoms,
> but internalize a different pump, the effect to my ears is that it
> sounds oddly familiar, but lines up in ways you don't expect... which
> is cool.

Okay, then I'm afraid there's actually only one more way to go and that has also been explored many times. Schismatic temperament. And if we allow more mistuning, we get superpyth where a 4:5:6 is mapped to C-D#-G. And if we go still further, we get mavila which maps 4:5:6 to C-Eb-G. But the European-style harmony was always treating 5:4 from the meantone-like view.

> I'm not saying that it isn't worthwhile to explore
> porcupine[7] and porcupine[8], I'm just saying that thinking in terms
> > of the JI lattice is more fundamental than thinking in terms of the
> > porcupine generator... in my opinion.

I'm afraid this is similar to what I was trying to do during 2004. The result was that I was unable to view EDOs like 34 as anything else than an approximation to a 3D untempered system called 5-limit JI. The only 2D temperaments I could understand this way were meantone and schismatic. Therefore, EDOs like 19, 31, 26 or 50 were nothing else for me than meantone systems, EDOs like 41, 53 or 65 were nothing else than schismatic systems. Everything else was just an approximation to 5-limit JI which is not 2D. Once I discovered the great possibilities given by tempering out one particular interval within a 3D system, I finally realized that I had been trapped inside one harmonic system, sticking higher dimensions to it like approximating 7/1, rather than exploring various harmonic systems, all of which, including the familiar one, have their origin in 5-limit JI.

> Sure, why not? I love it. Major and minor triads share the same triad > class.

Aha. But for me the primary diatonic step in porcupine is 10/9 and the secondary one is 25/24 (gosh, I really should finish that article about comma pumps so that others could read it as well). And then there are two more things. First, porcupine[8] is the largest scale which doesn't split the 10/9.
Then, porcupine[8] is the smallest scale which makes a triadic comma pump. This is similar to meantone, where meantone[7] is the largest scale which doesn't split the 9/8 or 10/9 and it is also the smallest scale which makes a triadic comma pump -- and where 10/9 is the primary diatonic step and 16/15 is the secondary step.

I wrote:

> > This is a melodic way of thinking in the context of harmonic intonation. > > In
> > that context, a minor triad simply doesn't sound as attractive as a > > major
> > triad, that's all there is to it.
>
> So we agree then?

I don't know. When I say that some books claim the reason for the Picardy third to be the correspondence to the harmonic series, you are in doubt and argue that the major dominants in minor keys have something to do with melodic thinking. My answer is that this type of melodic thinking is beyond the scope of the harmonic context in which most music was understood at the time of triadic harmony formation.

> I think that v-I is a beautiful sound - they use it in movies and film
> scores sometimes. It's just different than V-I. So I shouldn't have
> said that it "works" less well, but it's just different. V-I creates a
> sense of expectation and then fulfills it. v-I doesn't do that but is
> beautiful in a different way. I'm not sure what makes V-I sound the
> way it does, but I'm guessing the leading tone. Maybe I'm wrong.

Well, then we really are in need of a compromise since if our aim is to prefer minor seconds over major ones, then we should start doing it all Pythagorean again.

> Right, but why are the generators important? They're important
> mathematically, as the basis vectors for the resulting abelian group
> for a particular temperament, but why should they be more musically
> important than things like motion by 3/2?

It's not just about the generators. It's the fact that as long as you build your harmonic system on progressions by fifths and want to somehow add thirds to that, you'll always end up with meantone or schismatic. I'm viewing this as a harmonic barrier which has prevented us for centuries to discover new kinds of progressions. It's not that I would approximate progressions by fifths filled with thirds in various temperaments because those were "coming straight from 5-limit JI". Rather, it's that I try to follow what the particular temperament rules imply. And if they imply moving away from the progressions by fifths, then I understand that's a good thing to do.

> Maybe we just have different goals - you are trying to explore the
> MOS's for these temperaments, whereas I'm just trying to explore the
> properties of the resulting lattice. I like what you're doing a lot,
> if you haven't caught on - I think it's the biggest breakthrough I've
> heard on this list since I joined. My only point is that the MOS's of
> a temperament, and the temperament itself, are different things -
> nothing more nothing less. There may be a better scale for Myna
> "tonal" harmony than Myna[9], for example.

Tetracot[14] or semisixth[16] don't make a MOS but they're the smallest scales which can explain the matter of diatonic and chromatic steps with regard to the shortest triadic comma pumps.

> http://www.youtube.com/watch?v=XSfnyr1MhXE&feature=BFa&list=PL100AF8DBBDE723C4&index=5
>
> This is less "functional," but only works in porcupine. Does this make
> you happier? :)

Yes. Not only is it less functional but it's also more "porcupinian" because of the two triads a minor third apart. Your previous example has some similar achievements to Hermans pump at the end of his Porcupine Ouverture.

> Haha, maybe, but when you move up and down the scale, triads turn into
> other triads in inversion, which always weirded me out :)

Well, hanson doesn't have a 7-tone diatonic either. In fact, 7 tones to hanson is essentially what 5 tones is to meantone. And what 7 tones is to meantone, that's 11 tones to hanson.

> If you melodically sing a major chord, does it still make you feel
> happy? Do melodically singing minor chords still make you feel sad? Do
> you think these things were true 400 years ago? This is a serious
> question, because sometimes I feel like people tend to answer that
> question with "no."

We're talking mostly modal music which had its origin in tetrachords. And a tetrachord like E-F-G-A definitely has a slightly "darker" mood than E-F#-G#-A, even 2000 years ago.

> I agree, but I think that you can generalize common practice harmony
> to other tuning systems by using the same chord progression atoms as
> common practice music, as I mentioned before - V-I, viib5-I, etc - but
> do everything in the new tempered system, and make use of new comma
> pumps. The net effect is that the result is intelligible to us 21st
> century musicians, but still stimulating in that it works out
> differently. I think that's a good way to generalize common practice
> harmony, and it demystifies the whole thing.

Only partially, certainly not as much as you seem to think. For example, take the Indian shruti scale. These are essentially 5-limit implications of Pythagorean tuning, leading to effective schismatic equivalents. A progression reminding us of the 5-limit "F minor, C major" would then come out as "C-F-G#, C-Fb-G". If that's the case, then we should actually interpret the 5-limit "B-D-F-Ab" as a schismatic "Cb-D-F-G#". Is this really what we want if it's so far?

> My way of simplifying it was to ignore generators and instead look at
> the JI lattice, and try to work out "porcupine common practice
> harmony" first, and then extend it. I think the results are very
> "intelligible," and for a lot of my friends, it was the first time
> they actually understood what I was trying to do (a few of them who
> aren't musicians didn't even know it was microtonal).

If by "JI lattice" you mean what I think you mean, then I'm afraid you're doing something similar to what I was doing back in 2004, only using 2D temperaments instead of EDOs. Either you're gonna feel satisfied with what comes out, or one day you'll take it one step further and try to see what the temperament in question is asking for.

> There is no problem with the two approaches coexisting, because they
> don't disagree with one another. Yours will likely give more poetic
> results, although mine may sound more intelligible to western ears...
> for now. You're trying to explore the 2D porcupine lattice, and I'm
> trying to explore the 3D JI lattice, but from a porcupine-tempered
> perspective. Either way, we're moving towards the same thing, it's
> just a matter of which way makes for easier learning at first. Surely
> some compromise will win out at the end anyway - I'm trying to stick
> closer to porcupine[8] harmony as a way of evolving, and even the
> Baroque people threw in V chords where the diatonic scale doesn't put
> them. So we'll probably reach the same goal.

Time will tell. :-)

Petr

PS: Have you downloaded those recordings I referred to recently? There it happened:
/tuning/topicId_98337.html#98379

On Tue, Apr 26, 2011 at 3:48 AM, Graham Breed <gbreed@...> wrote:
>
> Me:
>
> >> I'm happy with 5/4 movement in Magic. The comma pump is also good,
> >> though, where you build up with 5/4s and release with a 3/2. It
> >> establishes a direction for 5/4 movement, and so assigns 5/4 a
> >> subdominant function.
>
> Mike:
> > Well, there you go. I still can't hear 5/4 as dominant chords yet, but
> > hopefully it's possible. If anything, I actually hear movement by 8/5
> > as sounding stronger in resolution than movement by 5/4. I can imagine
> > what it would be like to hear 5/4 as dominant chords, but it never
> > really sounds that way to me in practice. I hope I figure out the
> > trick, because that would be nice. Maybe it requires a generalization
> > of ii-V-I to movement by 5/4.
>
> Aren't you agreeing with me again? 5/4 = subdominant.

I thought not, but now this is starting to get confusing, so I'll just
write out how I hear it and you can tell me if this is how you hear
it.

You can play a mind trick with 3/2: Play Cmaj -> Gmaj, but instead of
imagining it as I-V for Cmaj, reframe it as IV-I in Gmaj. So get
yourself to hear it as a plagal cadence in Gmaj, not I-V in Cmaj, aka
change the "key" that it's in mentally.

That is the same way that I hear Cmaj -> Emaj. It sounds like Cmaj was
the "5-limit subdominant" of Emaj, and it's resolving to Emaj as the
tonic by a "5-limit plagal cadence." Something like Cmaj -> Am -> Emaj
might make it obvious in terms of communicating exactly how I hear it,
although that's cheating for now. Is that how you hear it?

-Mike

PS - we may want to start this discussion over from the basics, as
this is getting long and I think we're misunderstanding each other.

On Tue, Apr 26, 2011 at 4:17 AM, Petr Parízek <petrparizek2000@...> wrote:
> This is similar to what you're saying: Start with fifths, hear what they sound like,
> then add new things and somehow incorporate them into the system we have.

I think you misunderstand - I don't say start with fifths. I'm a jazz
musician man - I hate IV V I :)

What I'm saying is that motion by 3/2 is simpler, psychoacoustically,
and that's just how it sounds, period. This isn't because it's the
generator of meantone, but because it's 3/2. The fact that it
technically forms part of the basis set of the rank 2 abelian group
for meantone is not something I think the brain cares about. So
likewise, when we're dealing with porcupine, motion by 10/9 isn't
going to take the place of 3/2 just because now it's the generator.

It -will- take the place of 3/2 in certain mathematical aspects of how
scale modulation will occur, as well as how the modes will lay out -
stuff that has to do with set theory. But it won't take the place of
3/2 in the sense that the feeling that motion by 3/2 causes won't be
replaced by the motion that 10/9 causes or something.

I just think that V-I is a thing that exists, and we shouldn't feel
ashamed to use it when exploring new temperaments. I don't think it's
cheating. That's all I'm saying, no more and no less.

> > I know nothing about enharmonic harmony but this sounds fascinating.
> > Why not do it and see what pumps emerge.
>
> Once we find a good way to deal with the prime of 31.

Oh, haha, whoops. Well, why not fill it in with 128/125 for now?

> Possibly. But this doesn't contradict what I'm saying. As long as you start
> with progressions by fifths, sooner or later, the "best compromise" turns
> out to be meantone anyway. And from this point of view, any other
> temperament may sound like nothing more than a lower-quality approximation
> to triadic progressions by fifths.

I don't think you have to -only- move by fifth, just that motion by
fifth sometimes isn't cheating, even if it requires chromaticism. What
are your thoughts on the post I just made about altering porcupine[7]
like they altered minor? I don't know porcupine[8] so much, so if we
could stick to porcupine[7] for the moment that would be helpful :)

> It comes out of I'll tell you what. Stacking pure fifths, followed by
> linearly splitting pure fifths, followed by the desperate need to do
> something about the dreaded comma which made the D-A fifth unplayable and
> which was found too small to be practically applicable as a scale step. What
> else than meantone would you expect to come out of such a sequence of
> events?

I never disagreed, but I'm saying that I think we "discovered"
functional harmony. I think that the diatonic scale has lots of
"functional" harmony in it. I think that when, in the diatonic scale,
there wasn't functional harmony where we wanted it, we altered the
diatonic scale, like we did with the leading tone in minor. That's all
I'm saying. I'm not saying that motion by 6/5 and 5/4 can't be used in
functional harmony - I used those motions myself in my porcupine
example! I'm just pointing out that we historically decided to alter
aeolian to have a leading tone. What is wrong with that?

> Okay, then I'm afraid there's actually only one more way to go and that has
> also been explored many times. Schismatic temperament. And if we allow more
> mistuning, we get superpyth where a 4:5:6 is mapped to C-D#-G. And if we go
> still further, we get mavila which maps 4:5:6 to C-Eb-G. But the
> European-style harmony was always treating 5:4 from the meantone-like view.

Why should I be limited to only temperaments where the generator is a
fifth? I like motion by fifth because it's 3/2, not because it's a
generator. I think you've misunderstood something that I said
somewhere, maybe I've miscommunicated. Maybe we should start this
over? What do you think about my approach is rooted in misconception,
exactly?

> > I'm not saying that it isn't worthwhile to explore
> > porcupine[7] and porcupine[8], I'm just saying that thinking in terms
> > > of the JI lattice is more fundamental than thinking in terms of the
> > > porcupine generator... in my opinion.
>
> I'm afraid this is similar to what I was trying to do during 2004. The
> result was that I was unable to view EDOs like 34 as anything else than an
> approximation to a 3D untempered system called 5-limit JI.

Yes! But instead of thinking of it as a 3D untempered system called
5-limit JI, I'm thinking of it as a 3D system called 5-limit JI, but
that has chords connecting to and leading into one another in
interesting ways - comma pumps. To me, the paradigm shift from 3D ->
2D still treats 3/2 and 5/4 as fundamental intervals, not 10/9, even
though 10/9 is the generator. But I'm not sure I can communicate this
right.

> The only 2D temperaments I could understand this way were meantone and schismatic.

Why? I feel it's enabled me to understand porcupine really well.
Awkward 81/80 jumps have now turned into manageable 25/24 jumps -
comma shifts by 81/80 sound really awkward, right? But melodic motion
by 25/24 sounds much less awkward! So you can modulate around the
whole lattice however you want, with it never sounding awkward, and
still take advantage of the porcupine comma pumps.

> Therefore, EDOs like 19, 31, 26 or 50 were nothing else for me than meantone
> systems, EDOs like 41, 53 or 65 were nothing else than schismatic systems.
> Everything else was just an approximation to 5-limit JI which is not 2D.
> Once I discovered the great possibilities given by tempering out one
> particular interval within a 3D system, I finally realized that I had been
> trapped inside one harmonic system, sticking higher dimensions to it like
> approximating 7/1, rather than exploring various harmonic systems, all of
> which, including the familiar one, have their origin in 5-limit JI.

I think you can solve one of these in porcupine by learning that 81/80
now becomes 25/24. Did you hear the higher-limit porcupine example I
posted? That isn't going to work in JI, it'll only work in porcupine.
So instead of tacking 7-limit stuff on top of meantone (like Gershwin,
maybe), I've now tacked it on top of porcupine.

There are three lattices here:
- 3D 5-limit JI lattice with 2/1, 3/2, and 5/4 as fundamental axes
- 2D porcupine lattice with 2/1 as one axis and 10/9 as the other
"fundamental" axis
- Warped 3D 5-limit JI lattice with 2/1, 3/2, and 5/4 as fundamentals,
but the lattice being warped so that things lead into one another in
cool ways

You went from #1 to #2, I went from #1 to #3. But I don't feel that #3
is trapping me in meantone, because I understand how the lattice is
warped. And I find it more useful than thinking in terms of 10/9, but
do whatever's easiest for you.

> Then, porcupine[8] is the smallest scale which makes a triadic comma pump.
> This is similar to meantone, where meantone[7] is the largest scale which
> doesn't split the 9/8 or 10/9 and it is also the smallest scale which makes
> a triadic comma pump -- and where 10/9 is the primary diatonic step and
> 16/15 is the secondary step.

I should learn a lot more about porcupine[8].

> I don't know. When I say that some books claim the reason for the Picardy
> third to be the correspondence to the harmonic series, you are in doubt and
> argue that the major dominants in minor keys have something to do with
> melodic thinking. My answer is that this type of melodic thinking is beyond
> the scope of the harmonic context in which most music was understood at the
> time of triadic harmony formation.

No, that isn't what I was saying... I was saying that I think the
Picardy third evolved because it sounded beautiful. I think that when
they said "correspondence to the harmonic series," they understood
that that meant "major" aka "happy."

> > I think that v-I is a beautiful sound - they use it in movies and film
> > scores sometimes. It's just different than V-I. So I shouldn't have
> > said that it "works" less well, but it's just different. V-I creates a
> > sense of expectation and then fulfills it. v-I doesn't do that but is
> > beautiful in a different way. I'm not sure what makes V-I sound the
> > way it does, but I'm guessing the leading tone. Maybe I'm wrong.
>
> Well, then we really are in need of a compromise since if our aim is to
> prefer minor seconds over major ones, then we should start doing it all
> Pythagorean again.

What do you mean? Do you think that v-I vs V-I will actually sound
different in different systems?

> > Right, but why are the generators important? They're important
> > mathematically, as the basis vectors for the resulting abelian group
> > for a particular temperament, but why should they be more musically
> > important than things like motion by 3/2?
>
> It's not just about the generators. It's the fact that as long as you build
> your harmonic system on progressions by fifths and want to somehow add
> thirds to that, you'll always end up with meantone or schismatic. I'm
> viewing this as a harmonic barrier which has prevented us for centuries to
> discover new kinds of progressions. It's not that I would approximate
> progressions by fifths filled with thirds in various temperaments because
> those were "coming straight from 5-limit JI". Rather, it's that I try to
> follow what the particular temperament rules imply. And if they imply moving
> away from the progressions by fifths, then I understand that's a good thing
> to do.

Did I not fulfill all of these requirements with my porcupine
examples? I didn't "only" move by fifth, but I did sometimes. And why
should the porcupine rules imply motion by 10/9 substituting for 3/2?
And lastly, why do you have to be in magic to move by 5/4, why can't
you move by 5/4 in meantone? Why can't you move by 5/4 in 5-limit JI?
I really don't understand.

> Tetracot[14] or semisixth[16] don't make a MOS but they're the smallest
> scales which can explain the matter of diatonic and chromatic steps with
> regard to the shortest triadic comma pumps.

There are MODMOS's as well.

> > http://www.youtube.com/watch?v=XSfnyr1MhXE&feature=BFa&list=PL100AF8DBBDE723C4&index=5
> >
> > This is less "functional," but only works in porcupine. Does this make
> > you happier? :)
>
> Yes. Not only is it less functional but it's also more "porcupinian" because
> of the two triads a minor third apart. Your previous example has some
> similar achievements to Hermans pump at the end of his Porcupine Ouverture.

But in my "functional excerpt" soundcloud example, I have chords
moving upward by 10/9 directly, I've only preceded them with secondary
dominants. Then, finally, I made use of the 10/9 -> 10/9 -> 10/9 = 4/3
pun. What's wrong with that?

> > If you melodically sing a major chord, does it still make you feel
> > happy? Do melodically singing minor chords still make you feel sad? Do
> > you think these things were true 400 years ago? This is a serious
> > question, because sometimes I feel like people tend to answer that
> > question with "no."
>
> We're talking mostly modal music which had its origin in tetrachords. And a
> tetrachord like E-F-G-A definitely has a slightly "darker" mood than
> E-F#-G#-A, even 2000 years ago.

So we're agreeing? I think our conversation has completely broken
down, because a lot of my comments have been based in that I thought
you were saying the opposite. Maybe we should start over and summarize
our main points again, and I will try my best to be clearer. Plus this
is getting long :)

> A progression reminding us of the 5-limit "F minor, C major" would then come
> out as "C-F-G#, C-Fb-G". If that's the case, then we should actually
> interpret the 5-limit "B-D-F-Ab" as a schismatic "Cb-D-F-G#". Is this really
> what we want if it's so far?

I'd say that we don't have to interpret the 5-limit B-D-F-Ab as
anything but a bunch of 5-limit stacked minor thirds. However, in
schismatic temperament, you should have the awareness that, should you
for some reason want, you can get to the third by a bunch of fifths,
except fifths moving in the other direction. And there are other
things you might want to be aware of, other clever ways in which
32805/32768 might manifest. I'd also say that I don't care about the
concept of "far" because I don't care about the concept of
"generators."

> If by "JI lattice" you mean what I think you mean, then I'm afraid you're
> doing something similar to what I was doing back in 2004, only using 2D
> temperaments instead of EDOs. Either you're gonna feel satisfied with what
> comes out, or one day you'll take it one step further and try to see what
> the temperament in question is asking for.

My view of "the temperament in question" does not mean the MOS's of
that temperament, or anything involving a chain of generators. It
means the larger tempered lattice as a whole. Then, I view the MOS's
of that temperament as interesting scales to play with, scales which
we need. But I don't view them as being fundamental to the structure
of the temperament. Mathematically, they are, but cognitively, I don't
think so.

> PS: Have you downloaded those recordings I referred to recently? There it
> happened:
> /tuning/topicId_98337.html#98379

I listened to Run The Whistle Down (I'll listen to the other next) but
I couldn't figure out what the comma pump was. Can you spell it out
for me?

-Mike

On Tue, Apr 26, 2011 at 4:21 AM, Mike Battaglia <battaglia01@...> wrote:
>
> That is the same way that I hear Cmaj -> Emaj. It sounds like Cmaj was
> the "5-limit subdominant" of Emaj, and it's resolving to Emaj as the
> tonic by a "5-limit plagal cadence." Something like Cmaj -> Am -> Emaj
> might make it obvious in terms of communicating exactly how I hear it,
> although that's cheating for now. Is that how you hear it?

I'm talking about this with Graham on gchat now, although I'm not sure
he hears it the same way. So I'll finish this post:

So the question I've always had is, can we get the Cmaj -> Emaj to
flip the other way, in which it still sounds like Cmaj is the "key,"
and Emaj is its 5-limit otonal "dominant?" I dunno, maybe. I have this
sound in my head, but I don't know how to "activate" it in practice. I
note that Gmaj -> Cmaj is made much stronger if you precede the Gmaj
by Fmaj or Dm or Dm7. Maybe we can try some analogies and see if they
work here?

Abmaj -> Emaj -> Cmaj as analogous to IV V I by using only 5-limit
relationships - I dunno, I think I'm kind of hearing something.
G#m -> Emaj -> Cmaj as analogous to ii V I by using only 5-limit
relationships - definitely hearing something interesting now, but more
so in 19-equal than 12-equal. Resolve it like G# B D# -> G# B E -> G C
E
G#m7 -> Emaj -> Cmaj as analogous to iim7 V I - I like the last one better
Bm7 -> Emaj -> Cmaj as analogous to ii V I in that the ii V is a weak
motion by fifth, and the V I is a strong motion by otonality - this is
pretty good in 19-equal, I like it. Best so far was G#m Emaj Cmaj.

And then you have the issue of the dominant 7. Listening tests seem to
suggest that 16/9 is dominant 7 everyone likes. In 31-equal, this
means going with 9/5 over 7/4, but in 22-equal, this means going with
7/4 over 9/5. In my porcupine examples, I used 4:5:6:7 as dominant,
and it sounded better to me than the 9/5 version. Maybe this is
because it mixes the function of IV and V in a sense. Doing this with
5-limit chords gives you E G# Ab B -> C E G C. This sounds terrible
because of the beating, but if you can stomach it long enough to wrap
your head around it, it actually does make sense in that regard: the
Ab-G melodic motion sounds like a 5-limit plagal cadence, but the root
movement is downward to like b6 -> 5. A solution is to get rid of G#
and make it Ab, which is to say that the "dominant" in this case would
be a supermajor chord. The E stays the same, the B moves to C, and the
Ab goes down to a G. The Ab -> G resolution over C, mentally, should
sound like 8/5 -> 3/2 to you.

Lastly, exploring other relationships seems to have some parallel as
well. Instead of Fmaj -> Bb9 -> C you can go Abmaj -> Fbmaj -> Cmaj,
which really trips me out. In general, the paradigm where I just
explore 5-limit harmony, but replace 3-limit relationships with
5-limit ones, is blowing my mind right now. Maybe this is what Petr
was talking about.

-Mike

PS Petr - a comment on the above ideas about 5-limit root movements
would be useful. I guess that this is what you might call "magic"
harmony, as I'm trying to hear motion by 5/4 as analogous to a 5-limit
motion by 3/2, but I haven't been thinking about it as part of the
magic system, but rather as a different part of 5-limit JI that I
haven't explored. Then I can work on magic-tempering it. Is that in
line with your views?

-Mike

On Tue, Apr 26, 2011 at 5:16 AM, Mike Battaglia <battaglia01@...> wrote:
> PS - we may want to start this discussion over from the basics, as
> this is getting long and I think we're misunderstanding each other.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
Thus, I wonder if we've been
> making use of 64/63 comma pumps in 12-equal without even realizing
> it...? Maybe this is what Cmaj -> Fmaj -> Bb7#11 -> C7 is? I think so

Looks like a comma pump to me. My comma pump finder says [1/2, 4/3, 4/3, 8/7] which if you unpack it is what you just did.

Other pumps which may strike a chord, so to speak:

126/125: [1/2, 6/5, 6/5, 7/5]

A typical pump progresses up the diminished triad by two 6/5s to a diminished fifth, then drops by the consonant 7/10 interval back to a unison.

225/224: [2, 3/4, 3/4, 5/4, 5/7]

Various ways to work this; one would be for example 3/4-10/7-3/4-5/4.

Mike wrote:

> I think you misunderstand - I don't say start with fifths. I'm a jazz
> musician man - I hate IV V I :)

You can't want to base your harmony on "motion by 3/2" and at the same time
avoid the progression you've just mentioned. Because as time goes by, the
former naturally turns into the latter -- it's a question of evolution. If
you take a Pythagorean "trine" like C-G-C and split the fifth linearly into
two thirds (one major and one minor), you get a nice-sounding C major triad.
If you want to have the possibility of a major triad on G as well, you
obviously only add D and B to the scale as the G is common to both triads
and it's already there. If you then play "G major, C major" in the way you
suggested, the G major works as what I would personally call a "chord of
expectation" and the C major works as a "chord of explanation" -- or, in
other terms, those are like a dependent clause followed by an independent
clause (but not sooner than the C major starts sounding). But if you play
them in another way (for example, you play the G major louder or longer or
whatever), then the roles are swapped so that the G major is now the main
clause and the following C major is the dependent clause, which makes you
expected something more to follow (you wouldn't end up a piece with a
quarter beat of G major and an eighth beat of C major if the ending is to be
convincingly stable). As this effect is equally
attractive as the previously described one, it's highly desirable to also
have an F major triad available in this context. This in turn gives us two
pairs of triads: "C major, F major, G major, C major". It's just the way it
develops, there's not much you can do about that.

> What I'm saying is that motion by 3/2 is simpler, psychoacoustically,
> and that's just how it sounds, period. This isn't because it's the
> generator of meantone, but because it's 3/2. The fact that it
> technically forms part of the basis set of the rank 2 abelian group
> for meantone is not something I think the brain cares about.

I'm totally confused now. On one hand, you say you didn't mean "starting by
fifths". On the other, you're suggesting "motion by 3/2", which inevitably
leads to the two pairs of chords I described. But you said you
disliked these. Then, you seemed to claim it would be perfectly possible to
get one without the other. This is similar to saying that you could do all
your maths using addition without ever needing subtraction.

> So likewise, when we're dealing with porcupine, motion by 10/9 isn't
> going to take the place of 3/2 just because now it's the generator.

Surely it can't. 10/9 is not something you would use in a conventional
triad.
Therefore, temperaments like this will, sooner or later, suggest movements
like "C major, A minor, D major"
or "C minor, F major, D minor". Even though there *is* some "motion by 3/2"
in these pair of chords, they have a totally different meaning than they
have in common practice harmony. The fact that you've used secondary
diatonic alterations in porcupine (i.e. "A major" instead of "A minor") is
just a slight "enhancement" similar to a meantone progression like "C major,
F major, D major, G major, C major" which also gets you outside the MOS and
gives a totally different mood to it than a diatonic one.

> It -will- take the place of 3/2 in certain mathematical aspects of how
> scale modulation will occur, as well as how the modes will lay out -
> stuff that has to do with set theory. But it won't take the place of
> 3/2 in the sense that the feeling that motion by 3/2 causes won't be
> replaced by the motion that 10/9 causes or something.

Agreed.

> I just think that V-I is a thing that exists, and we shouldn't feel
> ashamed to use it when exploring new temperaments. I don't think it's
> cheating. That's all I'm saying, no more and no less.

I agree in one half, I don't agree in the other. Motion by 3/2 is not as
universal as your
interpretation suggests. To me, wanting to reach meaningful conclusions
using motion by 3/2 and at the same time avoiding motion by 9/8 sounds
contradictory and impossible to achieve.

> Oh, haha, whoops. Well, why not fill it in with 128/125 for now?

You mean tempering out 125/124 to equate 128/125 to 32/31?

> I don't think you have to -only- move by fifth, just that motion by
> fifth sometimes isn't cheating, even if it requires chromaticism. What
> are your thoughts on the post I just made about altering porcupine[7]
> like they altered minor? I don't know porcupine[8] so much, so if we
> could stick to porcupine[7] for the moment that would be helpful :)

Let's first clarify the generator numbers. For convenience, I'll use a
falling neutral second for porcupine to be able to avoid negative generators wherever
possible. So, we'll start with porcupine[7] using generators 0 to 6.
Certainly, we can happily alter one of the tones by the "secondary diatonic
step" (this results in a non-contiguous sequence of generators if you alter
anything else than #0 or #6 but that's another matter). This step is +7
generators for 25/24 or -7 generators for 24/25. If your aim is to replace a
minor triad with a major triad or something like this, it'll probably make
you "unintentionally" go to the closer side of the chain -- i.e. to alter
#5, you go for -2 (6/5), while for altering #1, you go for +8 (15/8). This
is similar to the major dominants or the Picardy thirds in meantone -- in an
A minor key, you also raise the tones instead of lowering them (Gb or Cb
wouldn't make much sense there). But you see, first of all, I'm doing all of
this strictly in the context of the scale layout, the altering scale steps,
and the logic behind the triads already available (which has a lot to do
with the generator range, even if I'm not thinking about it that way).
Nowhere in this procedure have I blamed it on 3/2 relationships. And second,
this just confirms the fact that a progression like "C major, G major" is
more preferable than "C major, A major" in porcupine and that it's the other
way round in meantone.

> I never disagreed, but I'm saying that I think we "discovered"
> functional harmony. I think that the diatonic scale has lots of
> "functional" harmony in it. I think that when, in the diatonic scale,
> there wasn't functional harmony where we wanted it, we altered the
> diatonic scale, like we did with the leading tone in minor. That's all
> I'm saying.

Except that you seem to state that there's only one "functional harmony" and
that it's based on 5-limit JI, while I'm saying that there could be more
systems of functional harmony and that the familiar one is evidently based
on meantone. I understand that this can sound like a bold statement but this
was exactly what it sounded like to me back in 2007 or whatever. It was such
an uneasy thing for me to admit but now I'm happy that I've managed to do
that. That's also why I wrote that article about microtonality later (which
still, for the time being, resides in the Tuning Files folder).

> I'm not saying that motion by 6/5 and 5/4 can't be used in
> > functional harmony - I used those motions myself in my porcupine
> > example! I'm just pointing out that we historically decided to alter
> > aeolian to have a leading tone. What is wrong with that?

Nothing. Except that a progression on minor thirds in common practice
harmony and a progression on minor thirds in hanson have a totally different
meaning rather than being approximations of one or the other. -- And to the
"leading tone concept", I think my porcupine example in the next-to-last
paragraph explains what I mean.

> Why should I be limited to only temperaments where the generator is a
> fifth? I like motion by fifth because it's 3/2, not because it's a
> generator.

While you were previously connecting things which may not necessarily have
had something to do with each other, now you're separating things that
aren't so unrelated as you seem to think. The only answer I can give you is
this: Play the way you like to play for some time; and either you will find
the connection yourself and understand my conclusion, or you will end up
piling things on top of each other. Whether that will eventually lead to a
different conclusion or not, noone knows.

> Yes! But instead of thinking of it as a 3D untempered system called
> 5-limit JI, I'm thinking of it as a 3D system called 5-limit JI, but
> that has chords connecting to and leading into one another in
> interesting ways - comma pumps. To me, the paradigm shift from 3D ->
> 2D still treats 3/2 and 5/4 as fundamental intervals, not 10/9, even
> though 10/9 is the generator. But I'm not sure I can communicate this
> right.

This would sound like both of us having similar expectations but we
definitely don't. I'm also trying to learn about a particular temperament
from the 4:5:6:8-like perspective. But not from a common practice harmony
perspective.

> Why? I feel it's enabled me to understand porcupine really well.
> Awkward 81/80 jumps have now turned into manageable 25/24 jumps -
> comma shifts by 81/80 sound really awkward, right? But melodic motion
> by 25/24 sounds much less awkward! So you can modulate around the
> whole lattice however you want, with it never sounding awkward, and
> still take advantage of the porcupine comma pumps.

> Yeah. For me, this is like solving two expressions using one equation. As
> I've said, I'm gonna let others delve into this topic.

> I think you can solve one of these in porcupine by learning that 81/80
> now becomes 25/24. Did you hear the higher-limit porcupine example I
> posted? That isn't going to work in JI, it'll only work in porcupine.
> So instead of tacking 7-limit stuff on top of meantone (like Gershwin,
> maybe), I've now tacked it on top of porcupine.

Same for this.

> There are three lattices here:
> - 3D 5-limit JI lattice with 2/1, 3/2, and 5/4 as fundamental axes
> - 2D porcupine lattice with 2/1 as one axis and 10/9 as the other
> "fundamental" axis
> - Warped 3D 5-limit JI lattice with 2/1, 3/2, and 5/4 as fundamentals,
> but the lattice being warped so that things lead into one another in
> cool ways
>
> You went from #1 to #2, I went from #1 to #3. But I don't feel that #3
> is trapping me in meantone, because I understand how the lattice is
> warped. And I find it more useful than thinking in terms of 10/9, but
> do whatever's easiest for you.

I didn't go from #1 to #2.
May I add one more?
#4 - a 2D lattice with tempered 4/3 in one axis and tempered 5/8 in the other (which makes an 8-tone equal scale with one tone repeated), with possible octave doubling set separately from the tempering system -- i.e. the two axes make the temperament, the octave doubling is used as an "addition" to turn the tempered 1D system into a 2D one.
That's the one you've ommitted.

> No, that isn't what I was saying... I was saying that I think the
> Picardy third evolved because it sounded beautiful. I think that when
> they said "correspondence to the harmonic series," they understood
> that that meant "major" aka "happy."

I don't understand. People like Fogliano or Zarlino were educated enough to describe ratios and I don't know why someone would have argued for the harmonic series concept if that hadn't been the case.

> What do you mean? Do you think that v-I vs V-I will actually sound
> different in different systems?

If we are looking for "strong" leading tones and if they're "the closer the stronger", then there's the question whether we should care more about approximating 5-limit triads or about leading tones. AFAIK, the concept of leading tones getting closer to the resolving tones was disappearing during the 15th century and reappearing *only* towards the end of the 18th century. If you wish to have both the possibility of approximating 5-limit and 3-limit intonation, schismatic seems to beat most other historical tunings -- i.e. you can do "B-D#-G#, A-E-A" as in the earlier ages, or you can do "B-Eb-Ab, A-E-A" as towards the end of the 15th century when the tuning was still Pythagorean but sharps were often intentionally replaced by enharmonic flats.

> > > Right, but why are the generators important? They're important
> > mathematically, as the basis vectors for the resulting abelian group
> > for a particular temperament, but why should they be more musically
> > important than things like motion by 3/2?
>
> It's not just about the generators. It's the fact that as long as you
> build
> your harmonic system on progressions by fifths and want to somehow add
> thirds to that, you'll always end up with meantone or schismatic. I'm
> viewing this as a harmonic barrier which has prevented us for centuries to
> discover new kinds of progressions. It's not that I would approximate
> progressions by fifths filled with thirds in various temperaments because
> those were "coming straight from 5-limit JI". Rather, it's that I try to
> follow what the particular temperament rules imply. And if they imply
> moving
> away from the progressions by fifths, then I understand that's a good
> thing
> to do.

> Did I not fulfill all of these requirements with my porcupine
> examples? I didn't "only" move by fifth, but I did sometimes. And why
> should the porcupine rules imply motion by 10/9 substituting for 3/2?

Not 10/9 substituting 3/2, but rather giving 10/9 a different meaning than it has in common practice harmony.

> And lastly, why do you have to be in magic to move by 5/4, why can't
> you move by 5/4 in meantone?

You can. But if you were trying to use meantone to imitate the harmonic possibilities implied by magic, you would quickly end up with triple accidentals. We don't have individual notation systems for each 2D temperament so I can't very well demonstrate the effect. But magic temperament is more than simply major third progressions. It's an entirely different harmonic system which would require different rules on voice leading and other things. For example, while a fifth-based harmonic system can bring nice results if you avoid parallel fifths, this is absolute nonsense in temperaments like hanson since there, in contrast, you *have* to use them in some situations, otherwise the voice leading sounds like mess. Listen to my hanson pump and you'll understand what I mean.

> Why can't you move by 5/4 in 5-limit JI?

In JI, you can move by whatever you want, with the obvious disadvantage of possibly ending up some tiny distance higher or lower. 5-limit JI encompasses all possible 5-limit harmonic systems in their untempered form. So there's no great reason to prefer one over another when playing in JI.

> But in my "functional excerpt" soundcloud example, I have chords
> moving upward by 10/9 directly, I've only preceded them with secondary
> dominants. Then, finally, I made use of the 10/9 -> 10/9 -> 10/9 = 4/3
> pun. What's wrong with that?

Nothing. But the way you used the "I-V-I" in combination with the modulation a fourth lower may trap one into thinking that you composed it in a fifth-based system, while half of the phrase is actually coming from a non-fifth-based system called porcupine. So it sounds to me as if a part of the phrase was composed in one harmonic system and another part of the phrase was composed in another one, rather than containing "pure porcupinian progressions" (not in "pure" intonation but rather like "pure" metals or some such).

> So we're agreeing? I think our conversation has completely broken
> down, because a lot of my comments have been based in that I thought
> you were saying the opposite.

I thought you were saying that the melodic minor scale was a result of such and such harmonic reasons. To which my answer was that it probably was a result of musicians' preference to play the moddle tones of a tetrachord higher when going up and lower when going down -- i.e. in the context of one-voiced melodies. Therefore, the reasons don't seem to have a lot to do with harmony. For the harmonic minor, it was quite the opposite. You seemed to be saying that it was a result of such and such melodic reasons. To which my answer was that probably was a result of the preference of major triads over minor triads -- i.e. in the context of clearly "vertical" music.

> Maybe we should start over and summarize
> our main points again, and I will try my best to be clearer. Plus this
> is getting long :)

It is, you're right. :-D

> I'd also say that I don't care about the
> concept of "far" because I don't care about the concept of
> "generators."

Again, the concept of generators comes out of lots of experience you get way before you realize this or that has something to do with them. Once you do, you suddenly start finding order in it all.

> My view of "the temperament in question" does not mean the MOS's of
> that temperament, or anything involving a chain of generators. It
> means the larger tempered lattice as a whole. Then, I view the MOS's
> of that temperament as interesting scales to play with, scales which
> we need. But I don't view them as being fundamental to the structure
> of the temperament. Mathematically, they are, but cognitively, I don't
> think so.

Okay, then let me give you another example of the "lattice #4". Let X = 5/3, Y = 2/5, and now we multiply "XYXXYXXYXXYXXY". If you decrease each consecutive pitch by the 13th root of 20000/19683, you get a 14-tone scale, or rather a 13-tone equal scale, where pitch #4 occurs twice and the tonic triad is made of pitches 0-4-9 (if 0 is the lowest pitch and 13 is the highest).

> I listened to Run The Whistle Down (I'll listen to the other next) but
> I couldn't figure out what the comma pump was. Can you spell it out
> for me?

Fb major, Db minor, Bb minor, G minor,
C major, A minor, F# minor, D# minor,
G# major, E# major (same as Fb).

Petr

I wrote:

> Fb major, Db minor, Bb minor, G minor,
> C major, A minor, F# minor, D# minor,
> G# major, E# major (same as Fb).

Sorry, I thought you were asking about the first one.

Okay, so just to let you know:
This is what was in the first one, not in "Whistle".

Petr

Petr,

On Tue, Apr 26, 2011 at 5:42 PM, Petr Parízek <petrparizek2000@yahoo.com> wrote:
>
> > I think you misunderstand - I don't say start with fifths. I'm a jazz
> > musician man - I hate IV V I :)
>
> You can't want to base your harmony on "motion by 3/2" and at the same time
> avoid the progression you've just mentioned. Because as time goes by, the
> former naturally turns into the latter -- it's a question of evolution.

Since I think we're misunderstanding each other a lot, and since this
exchange has gotten very long, I thought it might be best to start
over.

Let's ignore higher limits for now and just stick to the 5-limit. This
is exactly how I view things - I don't claim it's perfect, and maybe
it's wrong. Perhaps you have a better paradigm. But for clarity's
sake, these are my views:

I think we "discovered" functional harmony from meantone, not from the
diatonic scale MOS. When the diatonic scale wasn't functional enough,
we altered it. But what does this mean? I think

1) There is a "proto-functional" harmony that is based in 5-limit JI.
Some of the "rules" differ from meantone, others are the same.
2) All rank-2 5-limit temperaments take this proto-functional harmony
and add their own unique "spin" on it by introducing comma pumps.
3) For example, meantone "meantonizes" proto-functional harmony by
tempering 81/80 out, so that Cmaj -> Em -> Am -> Dm -> Gmaj -> Cmaj
becomes possible.
4) The combination of the rules of proto-functional harmony, plus the
new possibilities afforded by the 81/80 comma pump, form a new
combined system of meantone-functional harmony, or "common practice
harmony."
5) Porcupine, on the other hand, "porcupinizes" proto-functional
harmony by tempering 81/80 out, so that Cmaj -> A7 -> Dmaj -> B7 ->
Emaj -> C#7 -> F#maj -> F#m -> C#maj actually brings you back to Cmaj.
6) The combination of the rules of proto-functional harmony, plus the
new possibilities afforded by the 250/243 comma pump, form
porcupine-functional harmony.

Lastly, I think that proto-functional harmony is not the only harmony
that exists or that sounds good - there's also proto-modal harmony,
which is different from meantone-modal harmony, and is also different
from porcupine-modal harmony. And there are other, new harmonic
relationships that have nothing to do with either functional or modal
harmony, and there will be a proto-version of those, and a meantone
version, a porcupine version, etc.

-Mike

On Tue, Apr 26, 2011 at 8:43 PM, Mike Battaglia <battaglia01@...> wrote:
>
> Lastly, I think that proto-functional harmony is not the only harmony
> that exists or that sounds good - there's also proto-modal harmony,
> which is different from meantone-modal harmony, and is also different
> from porcupine-modal harmony. And there are other, new harmonic
> relationships that have nothing to do with either functional or modal
> harmony, and there will be a proto-version of those, and a meantone
> version, a porcupine version, etc.

Also, I should add that we don't even know what all of the "rules" of
proto-functional or even meantone-functional harmony are yet. There is
clearly a "sound" that people are going for, and that they've been
moving towards, and that's meantone-functional harmony. You can look
at the modal resurgence of the 20th century (like Ravel) as an
abstract extension of meantone-functional harmony, in that it just
finds new harmonic relationships that sound good. It's more functional
than modal Renaissance music, but still not as vanilla-flavored
functional as common practice harmony.

But you seem more interested in getting away from the
already-established rules of proto-functional harmony, which often
include motion by fifth. So I am starting to likewise get a new
concept, one which generalizes common practice harmony to incorporate
new root movements. For example, I think many 5-limit parallels to V-I
and IV-I resolutions exist.

For example, when you play Cmaj -> Gmaj, you can either think of it as
I -> V in Cmaj, or as IV -> I in Gmaj - like a plagal cadence.
Similarly, when I play Cmaj -> Emaj, I hear it the same way - as bVI
-> I in the key of Emaj. It has a similar "feeling" to a plagal
cadence to me - a more complex (and beautiful) 5-limit version of IV
-> I. It's relaxing, not invigorating like V-I. You can say that
Am->Emaj is a double plagal cadence, or that Am6->Emaj (where Am6 is
1/(4:5:6:7)) is a triple plagal cadence or something maybe.

However, you can also hear Cmaj -> Gmaj as I-V in C, not just IV-I in
G. So my question is, is it possible to hear Cmaj -> Emaj as a 5-limit
version of I-V - e.g. where Cmaj is still the key, and Emaj is the
5-limit "V" chord, the one that creates the expectation? I was
experimenting with this in 19-equal, and I think so - try G#m -> Emaj
-> Cmaj as a generalization of ii-V-I, for example. And try Abmaj ->
Fbmaj -> Cmaj in 19-equal as a generalization of IV -> bVII -> I. It
doesn't always work perfectly, but I think there's something there.
Maybe it's just the placebo effect. It works a lot better for me in
19-equal than 12-equal.

Also, since we tend to like 16/9 as the best dominant 7, E G# Ab B
would do the trick - but there's lots of beating. If you can get past
the beating, you can sort of imagine the Ab as resolving to the G in C
by way of something like 8/5 -> 3/2. Or you could just leave out Ab
and do something like E Ab B -> E G C, which exhibits beautiful voice
leading. If 225/224 vanishes this becomes a supermajor chord.

Anyway, the point: if you hear things the same way I do, then these is
just more of the "rules" of proto-functional harmony that we'd never
discovered. It has motion by 5/4 instead of 3/2. Let's call this
5-motion proto-functional harmony, whereas we're more used to 3-motion
proto-functional harmony. So now you can

-Mix magic temperament with 5-motion proto-functional harmony to get
5-motion magic-functional harmony
-Mix meantone temperament with 5-motion proto-functional harmony to
get 5-motion meantone-functional harmony
-Mix magic temperament with 3-motion proto-functional harmony to get
3-motion magic-functional harmony
-Start looking at 6/5-motion functional harmony or 7-motion functional
harmony, etc

etc. Starting with magic in the way you're doing things should lead
you to understand 5-motion functional harmony and magic-tempered comma
pumps all at once, paralleling what we historically did with meantone.
I find it easier to think in "proto-functional" harmony first, learn
to separate it from meantone, and then retemper. Just my opinion.

-Mike

Mike : So the question I've always had is, can we get the Cmaj -> Emaj to
flip the other way, in which it still sounds like Cmaj is the "key,"
and Emaj is its 5-limit otonal "dominant?"

I don't know if this works in Porcupine, but E7 "can be" on the same diminished seventh
as G7, (think G13b9-->C) so throwing away the G leaves a III-->I cadence something like

D E
B C
G# G
E C

Twiddling the voice timing and exploiting the suspensions may make it musical.

Trying to get an analog of secondary dominants and keeping the root motion could be
something like the following.

A# B# B C (Bb to start over)
F## F# G# G
D## D# D E
B# G# E C

Stringing this out forever will get sort of ''augmenty / Giant Steps" in temperaments with 125/64 == 2.
Meantone will drift sharp (B# < C), schismic(?) will drift downwards (B# > C).
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Mike wrote:

> 1) There is a "proto-functional" harmony that is based in 5-limit JI.
> Some of the "rules" differ from meantone, others are the same.

Okay, if you're viewing it from this totally different angle, then let me introduce another minor correction: Some of the rules are only applicable to meantone, others are applicable to both meantone and the parent "protoharmonic" system.

> 2) All rank-2 5-limit temperaments take this proto-functional harmony
> and add their own unique "spin" on it by introducing comma pumps.

This makes the "protoharmonic" system remarcably difficult to understand but let's assume that I know what you mean.

> 3) For example, meantone "meantonizes" proto-functional harmony by
> tempering 81/80 out, so that Cmaj -> Em -> Am -> Dm -> Gmaj -> Cmaj
> becomes possible.

Without the E minor being necessary for the sole point of comma pumps (just a small detail).

> 4) The combination of the rules of proto-functional harmony, plus the
> new possibilities afforded by the 81/80 comma pump, form a new
> combined system of meantone-functional harmony, or "common practice
> harmony."

I see.

> Lastly, I think that proto-functional harmony is not the only harmony
> that exists or that sounds good - there's also proto-modal harmony,
> which is different from meantone-modal harmony, and is also different
> from porcupine-modal harmony. And there are other, new harmonic
> relationships that have nothing to do with either functional or modal
> harmony, and there will be a proto-version of those, and a meantone
> version, a porcupine version, etc.

What do you mean by modal harmony?

> But you seem more interested in getting away from the
> already-established rules of proto-functional harmony, which often
> include motion by fifth. So I am starting to likewise get a new
> concept, one which generalizes common practice harmony to incorporate
> new root movements. For example, I think many 5-limit parallels to V-I
> and IV-I resolutions exist.

See below.

> However, you can also hear Cmaj -> Gmaj as I-V in C, not just IV-I in
> G. So my question is, is it possible to hear Cmaj -> Emaj as a 5-limit
> version of I-V - e.g. where Cmaj is still the key, and Emaj is the
> 5-limit "V" chord, the one that creates the expectation? I was
> experimenting with this in 19-equal, and I think so - try G#m -> Emaj
> -> Cmaj as a generalization of ii-V-I, for example. And try Abmaj ->
> Fbmaj -> Cmaj in 19-equal as a generalization of IV -> bVII -> I. It
> doesn't always work perfectly, but I think there's something there.
> Maybe it's just the placebo effect. It works a lot better for me in
> 19-equal than 12-equal.

I think the best results should come out in JI.

> Anyway, the point: if you hear things the same way I do, then these is
> just more of the "rules" of proto-functional harmony that we'd never
> discovered. It has motion by 5/4 instead of 3/2. Let's call this
> 5-motion proto-functional harmony, whereas we're more used to 3-motion
> proto-functional harmony. So now you can
>
> -Mix magic temperament with 5-motion proto-functional harmony to get
> 5-motion magic-functional harmony
> -Mix meantone temperament with 5-motion proto-functional harmony to
> get 5-motion meantone-functional harmony
> -Mix magic temperament with 3-motion proto-functional harmony to get
> 3-motion magic-functional harmony
> -Start looking at 6/5-motion functional harmony or 7-motion functional
> harmony, etc
>
> etc. Starting with magic in the way you're doing things should lead
> you to understand 5-motion functional harmony and magic-tempered comma
> pumps all at once, paralleling what we historically did with meantone.
> I find it easier to think in "proto-functional" harmony first, learn
> to separate it from meantone, and then retemper. Just my opinion.

What you've just described is almost exactly what I was doing during the first half of 2006. I've uploaded all those examples here for you to hear:
http://dl.dropbox.com/u/8497979/retuned2.rar

The MIDI files and the associated scales go as follows:
m-12: 12-EDO
m-ji: euler
m-jimin: mmswap
m-jiti: trinv
m-jiti2: trinv2
m-jiwh: wh
m-odd: odd12
m-odd2: odd12a

From these experiments, I've learned one important thing.
When your primarry "aproximant" is 4:5:6:8, the basic sequences of triads can obviously go in fifths, in major thirds, or in minor thirds.
As long as you can do it in JI, you're indeed essentially exploring the various "subsystems" of the great "protoharmonic" system.
But (and this is where the two of us start to take different views) depending on which of these you decide to prefer, this eventually leads you to an idea of a temperament which expresses it conveniently.
For example, if you involve lots of movements by minor thirds, soon you get a possible realization of Messiaens octatonic (like 27/25, 6/5, 5/4, 25/18, 3/2, 5/3, 9/5, 2/1). And if you play in it for a while, soon you think: "Hey, this is similar to that and this pitch is close to that one, what about equating them somehow?" And from there, there are just a couple of steps to, guess what, hanson -- 10/9 splits into 250/243 and 27/25, 27/25 splits into 25/24 and 648/625, 25/24 splits into 648/625 and 15625/15552, ... And there you have it.
You see, first you think about the harmonic properties, then you start following the melodic ones. But you can't just "throw the melodic ones away" as if they were unimportant.
Of course, I'm talking about the most obvious cases which you can almost "trace back by hearing". You could also, for example, split the 10/9 into 25/24 and 16/15 and then possibly split the 27/25 into 16/15 and 81/80. But tell me, if you see how close this 27/25 is to twice the size of 25/24, don't you feel tempted to do something about that? Moreover, if you do the latter instead of the former, you may find more irregularities regarding "commas per third" and that kind of stuff. That's because a longer progression by minor thirds is something relatively "courageous" in meantone but it's completely ordinary stuff in hanson.
Similarly, if you prefer progressions by fifths, soon you'll get to Zarlino's scale of "perfect harmony" and trying to render this in hanson looks like a "courageous" thing as well since it's an awkward thing to do in that temperament.
So you could possibly use it as a "moment of surprise" but probably not as a resting part of the piece.
Another example might be a progression by a falling minor third and then a rising fourth, which might in turn suggest making the whole thing porcupine or tetracot, depending on the context.
If you read a bit of what I've written about the "lattice #4" and the hanson parallel fifths in my last letter, you may get a clue why I'm saying this.
If you use, for example, progressions of minor thirds in meantone, you're effectifely mixing properties of two different "subsystems of the protoharmonic system", which always leads to 1) a higher degree of "tension" or "surprise" than working within a single "subsystem" and 2) some amount of harmonic inconsistency, which may be understood as non-diatonic jumps or modulation.

So my suggestion would be:
Whatever you want to compose this way, try to play it in JI. As long as it comes out okay for you in JI, you are indeed working within some sort of protoharmonic system and there's not much of a point in tempering something.
Once you delve into progressions on a repeating interval, you'll realize that one interval might possibly be useful to temper out.

So yes, there might be something like a protoharmonic system. But depending on what kind of progressions you select out of it, soon you may find that an appropriate temperament might be able to "say more about it".
That's why I've always based my pump examples not on some "originally untempered JI progressions" but strictly on the interval which is tempered out -- i.e. 15552/15625 is the same as multipliing "XYXYXYXXYXY" if X = 2/5 and Y = 3/1.

Petr

I wrote:

> If you use, for example, progressions of minor thirds in meantone, you're
> effectifely mixing properties of two different "subsystems of the
> protoharmonic system", which always leads to 1) a higher degree of > "tension"
> or "surprise" than working within a single "subsystem" and 2) some amount > of
> harmonic inconsistency, which may be understood as non-diatonic jumps or
> modulation.

I may also add that your concept of "3-motion protofunctional harmony" and "5-motion protofunctional harmony" doesn't make a lot of sense to me.
If we want to start with 5-limit JI, then there's one parent protoharmonic system which grows in many branches -- i.e. the untempered versions of many possible harmonic systems.
If you then repeat progressions by fifths, then you're exploring one branch of the protoharmonic system, if you repeat progressions like "C major, A minor, D major", you're exploring a different one. Maybe the former is what you called the "3-motion protofunctional harmony" (which eventually calls for meantone or schismatic) but then I have no idea what you would call the latter (which may call for porcupine or tetracot).

Petr

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

> > 2) All rank-2 5-limit temperaments take this proto-functional harmony
> > and add their own unique "spin" on it by introducing comma pumps.
>
> This makes the "protoharmonic" system remarcably difficult to understand but
> let's assume that I know what you mean.

It seems to me it makes it remarkably easy to understand. Here are the major characteristics:

(1) I, IV, V are the mainstays of functional harmony in a major key.

(2) Adding the root of IV to V gives V7; resolving that to I tends to define I as the tonic.

(3) iii serves as an intermediate chord linking I and V, and vi as an intermediate chord linking I and IV.

(4) ii does not exist as such, it is either v/V, or it is a chord related to IV and vi; these are two different chords which should not be confused.

--- In tuning@yahoogroups.com, Petr Parízek <petrparizek2000@...> wrote:
> So my suggestion would be:
> Whatever you want to compose this way, try to play it in JI. As long as it
> comes out okay for you in JI, you are indeed working within some sort of
> protoharmonic system and there's not much of a point in tempering something.
> Once you delve into progressions on a repeating interval, you'll realize
> that one interval might possibly be useful to temper out.
>
> So yes, there might be something like a protoharmonic system. But depending
> on what kind of progressions you select out of it, soon you may find that an
> appropriate temperament might be able to "say more about it".
> That's why I've always based my pump examples not on some "originally
> untempered JI progressions" but strictly on the interval which is tempered
> out -- i.e. 15552/15625 is the same as multipliing "XYXYXYXXYXY" if X = 2/5
> and Y = 3/1.

I'm jumping in to some mighty deep waters here, but I don't see why we have to "start" with JI, or why we need to appeal to JI to explain this stuff in the first place. For one, I think when it comes to chordal motion, talking ratios is silly. Motion by a "3/2" works just as well if it's actually motion by a 17/11 or a 16/11. Okay, it might be ever so slightly weaker, but the point is it's actually kind of hard to tell the difference unless you're listening analytically. When it comes to melodic movement and/or root movement, I'm not at all convinced that the "harmonic series detector" in the brain is being engaged, but even if it is, it seems to tolerate a much greater error, to the point where explaining things in terms of ratios really distorts the picture of what works and what doesn't.

In any case, I also don't see why it's necessary to start in JI. How is that more helpful than starting in the temperament we wish to work out some functional harmony for? Also, I am a firm believer in the idea that functional harmony can "work" to an extent in tunings where the approximations to JI are so far off it doesn't even make sense to call them that. I've got a piece on my next album that uses this scale from 18-EDO:

0
200
333.333
533.333
666.667
866.667
1000
1200

I analyzed the different modes, and there is only one approximate 5/4 in the whole scale, and nothing closer to 3/2 than that 666.67-cent interval. 333.333 cents is the generator, so it's sort of "anti-Dicot". Harmonically, it's pretty damn dirty, but you know what? I got a 5-voice functional harmonic progression out of it that resolves as clean as anything in meantone (although it does resolve to either an octave or another bare dyad, I forget which). Sure, it sounds pretty damn rough, but it still "functions". I don't even know how I would represent it in JI, though. Same goes for mavila and dicot temperaments--I can do functional stuff in those temperaments, no problem, but does it make any sense to represent those progressions in terms of JI? Hardly. Exotemperaments have a lot to teach us about the limitations (and lack thereof) of our ideas about harmony.

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Tue, Apr 26, 2011 at 4:21 AM, Mike Battaglia <battaglia01@...> wrote:

> So the question I've always had is, can we get the Cmaj -> Emaj to
> flip the other way, in which it still sounds like Cmaj is the "key,"
> and Emaj is its 5-limit otonal "dominant?" I dunno, maybe. I have this
> sound in my head, but I don't know how to "activate" it in practice.

Before you joined these lists, there was considerable discussion of isometries of the 5- and 7- limit symmetrical lattices, which can convert one kind of JI functional harmony into another. This can have a disconcerting effect, judging by the reactions of my family one Christmas in the 70s when I bombarded them with altered versions of Christmas carols, where 8/5 or 5/3 took the place of 3/2.

This can done by multiplying monzos by a suitable matrix, or by just for instance replacing 3 with 16/5 and 5 with 24/5. In this way you get a rotation group of order 3, extendable to a dihedral group (isomorphic to S3) of order 6, which cab be further extended by adding the major-minor flip, and still further with inversion and transposition, which gives the full group of isometries.

On Thu, Apr 28, 2011 at 10:03 AM, Petr Parízek
<petrparizek2000@...> wrote:
>
> Mike wrote:
>
> > 1) There is a "proto-functional" harmony that is based in 5-limit JI.
> > Some of the "rules" differ from meantone, others are the same.
>
> Okay, if you're viewing it from this totally different angle, then let me
> introduce another minor correction: Some of the rules are only applicable to
> meantone, others are applicable to both meantone and the parent
> "protoharmonic" system.

That's what I think I was trying to say. In what way do you mean your
statement to differ from mine?

> > 2) All rank-2 5-limit temperaments take this proto-functional harmony
> > and add their own unique "spin" on it by introducing comma pumps.
>
> This makes the "protoharmonic" system remarcably difficult to understand but
> let's assume that I know what you mean.

For example, I think that the fact that V-I works so well derives from
proto-functional harmony, not from meantone. However, meantone adds
new possibilities, such as I-IV-ii-V-I, which don't exist in
proto-functional harmony. If you try to use them, you're going to be
stuck with awkward comma jumps.

As a side note, motion by 81/80 itself might not be so bad, if you
learn how to "use" it; i.e. there's no reason I can see why 81/80
should be some kind of cursed and damned interval that is unusable.
However, if you try to take a meantone chord progression and try to
put it back in JI, it'll probably just sound awkward. But in
porcupine, 81/80 becomes equated with 25/24, so assuming you
understand proto-functional 25/24 logic, you can now create a new
logic that enables you to handle porcupine functional harmony. Just
turn every 81/80 shift into a 25/24, and now you can modulate around
the entire porcupine lattice and do whatever you like. Yes!

Gene may have explained it better in the post following yours.

> > Lastly, I think that proto-functional harmony is not the only harmony
> > that exists or that sounds good - there's also proto-modal harmony,
> > which is different from meantone-modal harmony, and is also different
> > from porcupine-modal harmony. And there are other, new harmonic
> > relationships that have nothing to do with either functional or modal
> > harmony, and there will be a proto-version of those, and a meantone
> > version, a porcupine version, etc.
>
> What do you mean by modal harmony?

I just meant that all of this doesn't have to apply to standard, old
school, vanilla ice cream common practice harmony. In the 20th
century, people started getting back into finding chord progressions
and resolutions that stuck closer to other modes than just major and
minor, which is a trend that probably started with the French school
of composition. The Beatles wrote lots of songs that you'd probably
call "modal" (like Come Together, which is predominantly Dorian, or
Norwegian Wood, which is Mixolydian in the verse and Dorian on the
chorus). In the 90's, Kurt Cobain made almost a career out of
utilizing borrowed chords to such an extent that it doesn't really
make sense to think of him basing his songs around the major or minor
scale at all. For example

http://www.youtube.com/watch?v=BJr-iFh1OZk - The major resolution in
this song is IV7 -> i, which is a very "Dorian" sound
Sublime got down on the modal action too, like with mixolydian in this
song: http://www.youtube.com/watch?v=qAAXPOlZy9A

For a while, hip hop dominated non-standard, but still oddly
"functional" harmony back to front, since that's what makes for good
beats that don't get boring when you play them for 3 minutes in a row:
http://www.youtube.com/watch?v=HU_4pf8BSQw
Or this 128/125 comma pump: http://www.youtube.com/watch?v=Zyf0YwUJcqk (lol)

Or if 90s rock or hip hop isn't your thing, the whole thing was
preceded by another modal explosion in the 60s and 70s -
ELP - http://www.youtube.com/watch?v=3epPMa5rq0U - Much of the verse
is based around i -> IVmajadd4, which is Dorian, then shifts to
Aeolian
Yes - http://www.youtube.com/watch?v=BsRdT9hwqGs&t=0m31s - this
switches through modes a lot, starting on G#aeolian, moving to Dorian,
then landing on F#maj/G# as a common chord pivot back to Aeolian
Pretty much everything Carlos Santana has ever written is in Dorian -
http://www.youtube.com/watch?v=8NsJ84YV1oA

Lots of stuff in the 80s too -
Van Halen - http://www.youtube.com/watch?v=Bl4dEAtxo0M
The Police - http://www.youtube.com/watch?v=UbQd3jxth5k (verse has a
bVII -> I resolution, which you could say is Mixolydian)
Rick James - http://www.youtube.com/watch?v=wsXzDMRFWkk (Dorian all over this)

Or maybe electronic is what you're more into
Royksopp - http://www.youtube.com/watch?v=klxai9aaxUw (not really
modal, but definitely non-standard)
Jon Kennedy - http://www.youtube.com/watch?v=km099Nkni8c (Beautiful
chord progression, Dmaj -> Bbmaj/C -> Gadd6/9)

Anyway, point is, all of these sounds aren't really "modal" at all.
They usually don't stick to any single mode, but they make use enough
of borrowed chords that they aren't sticking to things like Ionian or
Aeolian either. Sometimes they deliberately use borrowed chords as
stopping points in interesting ways, which makes this all perfectly
"functional." I don't think there's any real distinction, but people
tend to reserve the term "functional" for the strongest level of
common-practice resolutions, so I left it at that. They happen to make
predominant use of chord progressions that you might not pay attention
to if you focus on Ionian and Aeolian and harmonic minor and so on.

Chord progressions like these make up a large part of the harmonic
innovations of the 20th century, but the "classical" world has been
slow to catch on, except for maybe Steve Reich:

http://www.youtube.com/watch?v=UA-iDNxKeco
http://www.youtube.com/watch?v=egwXKQDYcvc

The point is that even these interesting, colorful, modern, "modal"
progressions have their basis in a combination of JI harmony and 81/80
comma pumps, and that similarly colorful progressions could exist in
porcupine too, but sound like nothing we've ever heard.

> > etc. Starting with magic in the way you're doing things should lead
> > you to understand 5-motion functional harmony and magic-tempered comma
> > pumps all at once, paralleling what we historically did with meantone.
> > I find it easier to think in "proto-functional" harmony first, learn
> > to separate it from meantone, and then retemper. Just my opinion.
>
> What you've just described is almost exactly what I was doing during the
> first half of 2006. I've uploaded all those examples here for you to hear:
> http://dl.dropbox.com/u/8497979/retuned2.rar
>
> The MIDI files and the associated scales go as follows:
> m-12: 12-EDO
> m-ji: euler
> m-jimin: mmswap
> m-jiti: trinv
> m-jiti2: trinv2
> m-jiwh: wh
> m-odd: odd12
> m-odd2: odd12a

These are cool! I think the "wh" one was the one like what I was
describing. There's still something that needs to be done though,
before the Emaj -> Cmaj sounds anything like Gmaj -> Cmaj, but I don't
know what it is.

> From these experiments, I've learned one important thing.
> When your primarry "aproximant" is 4:5:6:8, the basic sequences of triads
> can obviously go in fifths, in major thirds, or in minor thirds.
> As long as you can do it in JI, you're indeed essentially exploring the
> various "subsystems" of the great "protoharmonic" system.
> But (and this is where the two of us start to take different views)
> depending on which of these you decide to prefer, this eventually leads you
> to an idea of a temperament which expresses it conveniently.

I don't disagree with this at all, and I've had lots of arguments with
Paul Erlich about this. For example, I think we went from meantone to
12-tet because we heard that 16/15 and 25/24, which should probably be
more aptly described as the diatonic and chromatic semitones, were
about the same size and roughly of equal discordance. I never said
otherwise.

But I also think that temperaments exist that are really useful that
you might not "naturally" come to discover. For example, if you tend
to move around by 3/2 a lot, it's pretty natural that you'll discover
meantone. Or if you tend to move around by 5/4 a lot, it's pretty
natural that you'll discover magic. But if you tend to move around by
5/4 a lot, it may not be quite as natural that you'll discover
blackwood, or mavila, or whitewood. But these temperaments are still
awesome (especially blackwood), and you can still make use of them to
create interesting comma pumps.

> And from there, there are just a couple of
> steps to, guess what, hanson -- 10/9 splits into 250/243 and 27/25, 27/25
> splits into 25/24 and 648/625, 25/24 splits into 648/625 and 15625/15552,
> ... And there you have it.

That's an interesting analysis. We should make a chart with this kind
of stuff. I've noticed a similar pattern with the following 5-limit
ratios:

16/15 -> 25/24 -> 128/125 -> 3125/3072
Father -> Dicot -> Augmented -> Magic
5-equal -> 7-equal -> 12-equal -> 19-equal (if mixed with meantone)

Do you see some kind of linear splitting pattern here?

> You see, first you think about the harmonic properties, then you start
> following the melodic ones. But you can't just "throw the melodic ones away"
> as if they were unimportant.
> Of course, I'm talking about the most obvious cases which you can almost
> "trace back by hearing". You could also, for example, split the 10/9 into
> 25/24 and 16/15 and then possibly split the 27/25 into 16/15 and 81/80. But
> tell me, if you see how close this 27/25 is to twice the size of 25/24,
> don't you feel tempted to do something about that?

What specifically are you suggesting? Eliminating 648/625?

> Moreover, if you do the latter instead of the former, you may find more irregularities regarding
> "commas per third" and that kind of stuff. That's because a longer
> progression by minor thirds is something relatively "courageous" in meantone
> but it's completely ordinary stuff in hanson.

I'd say that a more general perspective is that it's "courageous" in
proto-functional harmony, and becomes less courageous in a temperament
that involves a quick comma pump based on repeated 6/5's. Thus
diminished makes it barely courageous at all, and hanson slightly more
courageous, etc.

> Similarly, if you prefer progressions by fifths, soon you'll get to
> Zarlino's scale of "perfect harmony" and trying to render this in hanson
> looks like a "courageous" thing as well since it's an awkward thing to do in
> that temperament.

I think a longer progression by 3/2's is pretty courageous even in
meantone. Cmaj -> Gmaj -> Dmaj -> Amaj -> Emaj -> Bmaj... where does
it end? You're in outer space now. But, if you're in Blackwood and you
temper out 256/243, it's not courageous at all, because you're back
where you started. Likewise if you're in Whitewood, which tempers out
2187/2048 instead, you're also back where you started.

> So you could possibly use it as a "moment of surprise" but probably not as a
> resting part of the piece.
> Another example might be a progression by a falling minor third and then a
> rising fourth, which might in turn suggest making the whole thing porcupine
> or tetracot, depending on the context.
> If you read a bit of what I've written about the "lattice #4" and the hanson
> parallel fifths in my last letter, you may get a clue why I'm saying this.

I did see lattice #4, but how was that different from my "warped 3D"
lattice? It looked like you were saying the same thing I was.

> If you use, for example, progressions of minor thirds in meantone, you're
> effectifely mixing properties of two different "subsystems of the
> protoharmonic system", which always leads to 1) a higher degree of "tension"
> or "surprise" than working within a single "subsystem" and 2) some amount of
> harmonic inconsistency, which may be understood as non-diatonic jumps or
> modulation.

I viewed it as meaning just that if you have a minor third progression
that works in the protoharmonic system with no inconsistency, it will
work in meantone as well, but that one should also be aware that there
will be additional things you can do in a temperament like hanson that
you can't do in meantone with modulation by minor thirds.

> So yes, there might be something like a protoharmonic system. But depending
> on what kind of progressions you select out of it, soon you may find that an
> appropriate temperament might be able to "say more about it".
> That's why I've always based my pump examples not on some "originally
> untempered JI progressions" but strictly on the interval which is tempered
> out -- i.e. 15552/15625 is the same as multipliing "XYXYXYXXYXY" if X = 2/5
> and Y = 3/1.

Alright, but it's also true that there are ways to do that than to
stick to the temperament's MOS's, or think about whatever the
generating interval technically is. For example, let's say you're in
porcupine, and you decide to extend the system to the 13-limit by
tempering out some insane comma that, technically, causes the period
to become 1/2-oct instead of 1 oct, similar to how Hedgehog does it.
Would this fundamentally change the properties of the system, just
because this one stupid 13-limit comma has unfortunately caused this
to happen? I don't think so.

-Mike

On Thu, Apr 28, 2011 at 10:38 AM, Petr Parízek
<petrparizek2000@...> wrote:
>
> I may also add that your concept of "3-motion protofunctional harmony" and
> "5-motion protofunctional harmony" doesn't make a lot of sense to me.
> If we want to start with 5-limit JI, then there's one parent protoharmonic
> system which grows in many branches -- i.e. the untempered versions of many
> possible harmonic systems.
> If you then repeat progressions by fifths, then you're exploring one branch
> of the protoharmonic system, if you repeat progressions like "C major, A
> minor, D major", you're exploring a different one. Maybe the former is what
> you called the "3-motion protofunctional harmony" (which eventually calls
> for meantone or schismatic) but then I have no idea what you would call the
> latter (which may call for porcupine or tetracot).

By "protofunctional harmony" I meant that common practice "functional
harmony" derives, ultimately, from some kind of protoharmonic system
that we keep discovering as time goes on. Although the diatonic scale
was the catalyst for this discovery, it isn't the cause of the
characteristically functional "sound." The way in which meantone,
common practice harmony differs from this proto-functional harmony is
that it also enables you to run around in circles by adding the 81/80
pun into the mix.

By 3-motion vs 5-motion I didn't mean anything about stereotyping the
chord progressions of different temperaments - I meant more in the
sense of generalizing the concept of the "dominant" and the
"subdominant." So for example, the following chord progression, which
is protoharmonic

Cmaj -> Gmaj -> Cmaj

You're moving by 3/2 to the dominant (V chord), which causes an
expectation that you'll move back to the Cmaj, which you then fulfill.
Or

Cmaj -> Fmaj -> Cmaj

You're moving by 4/3 to the subdominant (IV chord), which instead of
causing an expectation that you'll move back to the I chord, simply
causes a "relaxing" sensation when you do the IV-I - a plagal cadence.

I have noticed some limited symmetry with 5-limit ratios in this
regard. For example, take the following chord progression:

Cmaj -> Gmaj

This could be either I -> V in the key of C, or it could be IV -> I in
the key of G. You can flip your brain around to hear it either way.
Now check out the 5-limit equivalent:

Cmaj -> Emaj

To my ears, this sounds like bVI -> I in the key of Emaj. In that
respect, it sounds like a 5-limit "plagal" cadence - it doesn't sound
like the invigorating "expectation -> fulfillment" of V->I, but rather
the relaxing and calming plagal cadence of IV->I, except a more exotic
5-limit version of it. It sounds similar to

Am -> Emaj

But, I can't flip my brain around to hear Cmaj -> Emaj as a 5-limit
I-V -- such that Cmaj -> Emaj -> Cmaj sounds like a 5-limit I-V-I. So
I tried some tricks to get it to work - mixing the 5-limit IV and V
chord by changing the G# in Emaj to Ab (generalization of the dominant
7), preceding the Emaj with a G#m (generalization of ii-V-I), etc. I'm
seeing some limited success with some of these, and some of the rest
sound confusing.

So my point is that this has never been done before in history, at
least not that I know. But, it's still a property of the protoharmonic
system. Then, if you want, you can magic temper it to create awesomely
ridiculous comma pumps, but at first I'm trying to understand the
protoharmonic properties of the concept.

-Mike

--- "cityoftheasleep" <igliashon@...> wrote:

> For one, I think when it comes to chordal motion, talking
> ratios is silly.

It's not, for 2 reasons:

1. If your chords contain ratios (like 3/2) then root motion
by those ratios produce common tones and common dyads between
chords, and this is a highly audible condition.

2. Root motion is melodic motion, and 3-limit intervals
(at least) seem to be recognizable melodically.

> When it comes to melodic movement and/or root movement, I'm
> not at all convinced that the "harmonic series detector" in
> the brain is being engaged, but even if it is, it seems to
> tolerate a much greater error,

Melodic thresholds are looser than harmonic ones, true.
But root motions by 40/27 are noticeable vs 3/2 to me and
others.

> In any case, I also don't see why it's necessary to start
> in JI. How is that more helpful than starting in the
> temperament we wish to work out some functional harmony for?

I think Petr and Mike were asking what motivates temperament.
Tuning in JI is a sure way to discover that.

> Also, I am a firm believer in the idea that functional
> harmony can "work" to an extent in tunings where the
> approximations to JI are so far off it doesn't even make
> sense to call them that. I've got a piece on my next
> album that uses this scale from 18-EDO:
> 0
> 200
> 333.333
> 533.333
> 666.667
> 866.667
> 1000
> 1200
> I analyzed the different modes, and there is only one
> approximate 5/4 in the whole scale, and nothing closer to
> 3/2 than that 666.67-cent interval. 333.333 cents is the
> generator, so it's sort of "anti-Dicot". Harmonically,
> it's pretty damn dirty, but you know what? I got a 5-voice
> functional harmonic progression out of it that resolves as
> clean as anything in meantone (although it does resolve
> to either an octave or another bare dyad, I forget which).
> Sure, it sounds pretty damn rough, but it still "functions".

I can't wait to hear it. I'd even be interesting in hearing
the progression by itself and seeing it in ASCII notation.
I'm not sure what the ultimate requirements for "function"
are. Probably you need a set of chords (maybe as few as 2)
with some being more tense than others. I dunno.

> I don't even know how I would represent it in JI, though.
> Same goes for mavila and dicot temperaments--I can do
> functional stuff in those temperaments, no problem, but
> does it make any sense to represent those progressions in
> terms of JI?

Yes.

-Carl

On Thu, Apr 28, 2011 at 12:32 PM, genewardsmith
<genewardsmith@...> wrote:
> >
> > This makes the "protoharmonic" system remarcably difficult to understand but
> > let's assume that I know what you mean.
>
> It seems to me it makes it remarkably easy to understand. Here are the major characteristics:
>
> (1) I, IV, V are the mainstays of functional harmony in a major key.
>
> (2) Adding the root of IV to V gives V7; resolving that to I tends to define I as the tonic.
>
> (3) iii serves as an intermediate chord linking I and V, and vi as an intermediate chord linking I and IV.
>
> (4) ii does not exist as such, it is either v/V, or it is a chord related to IV and vi; these are two different chords which should not be confused.

Exactly. I guess some minutiae to add for rigor

(1) is what I was calling "3-motion" protofunctional harmony, in that
the idea is that you're surround the root by triads a 3-limit jump or
two away.

The point that I was making is that when people jump into magic
tempering, the first thing that they do is start exploring motion by
5/4, which is what I was calling 5-motion harmony. But there's nothing
stopping you from exploring motion by 5/4 in meantone, or negri, or
hanson, or 5-limit JI, which is the protoharmonic system. Motion by
5/4 isn't somehow fundamental to magic just because 5/4 is the
generator, although if you like to take advantage of that comma pump
you might end up using it a lot. The MOS's of magic are not the same
as the greater magic lattice, and just because 5/4 is technically the
"generator" doesn't mean anything in terms of music cognition. Or if
it does, I don't see how.

(2) I think has some far-reaching consequences on music cognition that
I don't think I've ever seen adequately expressed.

(3) and (4) are obviously true, but I guess to be pedantic it should
be noted that even this is going to change in an inconsistent system
where 9'/9 doesn't vanish. Or perhaps a system in which you have a
(6/5)' or something like that, although I'm not sure why this would
ever come up. So not even 5-limit JI is the protoharmonic system.
Although that's definitely overkill for now.

-Mike

On Thu, Apr 28, 2011 at 12:49 PM, cityoftheasleep
<igliashon@...> wrote:
>
> I'm jumping in to some mighty deep waters here, but I don't see why we have to "start" with JI, or why we need to appeal to JI to explain this stuff in the first place. For one, I think when it comes to chordal motion, talking ratios is silly. Motion by a "3/2" works just as well if it's actually motion by a 17/11 or a 16/11. Okay, it might be ever so slightly weaker, but the point is it's actually kind of hard to tell the difference unless you're listening analytically. When it comes to melodic movement and/or root movement, I'm not at all convinced that the "harmonic series detector" in the brain is being engaged, but even if it is, it seems to tolerate a much greater error, to the point where explaining things in terms of ratios really distorts the picture of what works and what doesn't.

Here's a test for you - arpeggiate the following three triads:

C-E-G
E-G-C
G-C-E

Do any of these triads evoke the sense of there being a "root?" If so,
does the root in each case sound like it's a C? Do they sound more
rooted and consonant than a sung C-Eb-Gb? If so, then that's
significant, and should not be ignored. It doesn't necessarily mean
that subliminal virtual pitches are firing left and right (although
that's always possible), but it may very well mean that we have
learned, through years of exposure, to make sense of isolated
fragments of harmonically-related but sequential notes by relating
them to a set of culturally-acquired or learned harmonic templates.
There's a concept called "preattentive f0 estimation" that I'm reading
about in the literature which addresses much of this now, although I
have nothing concrete to throw out there yet.

It's also probably much of the reason why melodic intonation matters
so much less than harmonic intonation (the other being the lack of
beating). In fact, I'd wager it's also part of the reason why it's so
difficult to learn to sing in another tuning system at first - because
your brain just keeps firing at you "this is a major third" or "that's
a minor third" when in actuality you're singing neutral thirds.

> In any case, I also don't see why it's necessary to start in JI. How is that more helpful than starting in the temperament we wish to work out some functional harmony for? Also, I am a firm believer in the idea that functional harmony can "work" to an extent in tunings where the approximations to JI are so far off it doesn't even make sense to call them that.
>
> Same goes for mavila and dicot temperaments--I can do functional stuff in those temperaments, no problem, but does it make any sense to represent those progressions in terms of JI? Hardly. Exotemperaments have a lot to teach us about the limitations (and lack thereof) of our ideas about harmony.

I'd say it makes sense to call them that if they work, cognitively, to
substitute for the simpler intervals that you're trying to substitute
for. But you could also set up a cognitive structure where a
magic-tempered 5/4, which will be narrow, is a "different thing" than
an augmented-tempered 5/4, which will be larger, but that they both
share the "5/4 property." Or that a mavila fifth is a "different
thing" than a superpyth fifth, but that both share the 3/2 property.
Or that a dicot 5/4 is a different thing from a meantone 5/4, but that
both share the 5/4 property, and that dicot's 5/4 also shares the 6/5
property with the meantone minor third.

Since you're fond of the idea that the generator "encodes" the
information about the temperament it generates, you might like this
paradigm better, since this "encoding" is basically an awareness of
how the tempering and size of the generator leads to new intervals
down the road. But either way, you can then still make the claim that
root movements by "intervals that have the 3/2 property," of which the
mavila, father, blackwood, and meantone fifths are examples, all tend
to share certain characteristics. Or, if you really want to strengthen
all of this, you can make the claim that root movements by an interval
that have the 3/2 property, and only in which you as a Westerner have
learned that those intervals actually do have that property, will
behave a certain way. But this is pretty wordy, so I find it easier to
just think in terms of JI and then tempering.

-Mike

On Thu, Apr 28, 2011 at 1:32 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > On Tue, Apr 26, 2011 at 4:21 AM, Mike Battaglia <battaglia01@...> wrote:
>
> > So the question I've always had is, can we get the Cmaj -> Emaj to
> > flip the other way, in which it still sounds like Cmaj is the "key,"
> > and Emaj is its 5-limit otonal "dominant?" I dunno, maybe. I have this
> > sound in my head, but I don't know how to "activate" it in practice.
>
> Before you joined these lists, there was considerable discussion of isometries of the 5- and 7- limit symmetrical lattices, which can convert one kind of JI functional harmony into another. This can have a disconcerting effect, judging by the reactions of my family one Christmas in the 70s when I bombarded them with altered versions of Christmas carols, where 8/5 or 5/3 took the place of 3/2.

Ha! Do you still have a copy of this? I'd love to hear it.

> This can done by multiplying monzos by a suitable matrix, or by just for instance replacing 3 with 16/5 and 5 with 24/5. In this way you get a rotation group of order 3, extendable to a dihedral group (isomorphic to S3) of order 6, which cab be further extended by adding the major-minor flip, and still further with inversion and transposition, which gives the full group of isometries.

So if you do that, 4:5:6 becomes... 1/1 6/5 8/5, I think, so then
inverting and transposing it should get you back to 4:5:6 again.

A related idea I had was, what if we came up with a way to take a
common-practice melody, perhaps one that's protoharmonic and doesn't
make use of any comma pumps, come up with a generalized imprint of it,
and translate it into a tuning around another base chord? For example,
let's say

C-E-G -> B-D-G -> C-E-G -> C-F-A -> E-G-C -> D-G-B -> C-G-C

And then turn that into a generalization based around 7:9:11 chords
instead of 4:5:6 ones. It sounds easy, but it ends up being trickier
than it sounds, because the 7:9:11 equivalent of the major scale
(three 7:9:11 triads separated by 3/2) ends up yielding 9 notes
instead of 7, meaning melodies get a little tricky. You could solve
this problem by making it three 7:9:11 triads separated by 11/7, but
I'm not sure that would end up sounding the same. Secondly, sometimes
movement by leading tone is more important than any harmonic ratio, so
you'd have to work that into the "imprint" you create.

The result I'm trying to get is that you can take a little fragment of
common practice harmony and then get something that oddly enough
sounds "the same," except it's all based around 7:9:11 or 5:6:7 or
whatever instead of 4:5:6. You could probably come up with a
matrix-based approach for this as well, but it would require an
understanding of the fact that motion by 3/2 is more important than
motion by the outer dyad of the target chord.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> Here's a test for you - arpeggiate the following three triads:
>
> C-E-G
> E-G-C
> G-C-E
>
> Do any of these triads evoke the sense of there being a "root?" If so,
> does the root in each case sound like it's a C?

In 12-TET? I dunno. I can't really answer this question honestly, because when I was initially learning to play guitar, I heard all of these as different chords. In a way, I still kinda do. But I remember I used to play a chord like E-C-E-G and call it some kind of E, as if treating the C as a B#, hearing the whole thing as a sort of Em(#5) or something. It wasn't until I met a classically-trained pianist in high school that I learned it was actually an inverted C chord. So in other words, my naive/intuitive grasp of the concept of "root" was always whatever the lowest note in the chord was. The idea that a minor 6th and a major 3rd were related was big news to me, because they sound so totally different. However, now that I've "learned" better, I can recognize inverted chords as such, and know to call the that. So if you ask me what I'm hearing as the "root" I know to tell you all those chords are rooted on C, but I don't know that I hear C as the root in any fundamental psychoacoustic sense.

> Do they sound more
> rooted and consonant than a sung C-Eb-Gb?

More rooted--no. More consonant? Sure. Unless you mean consonant as a synonym for concordance, in which case I can't answer that question.

> If so, then that's
> significant, and should not be ignored. It doesn't necessarily mean
> that subliminal virtual pitches are firing left and right (although
> that's always possible), but it may very well mean that we have
> learned, through years of exposure, to make sense of isolated
> fragments of harmonically-related but sequential notes by relating
> them to a set of culturally-acquired or learned harmonic templates.
> There's a concept called "preattentive f0 estimation" that I'm reading
> about in the literature which addresses much of this now, although I
> have nothing concrete to throw out there yet.

What does this have to do with JI and temperament?

> It's also probably much of the reason why melodic intonation matters
> so much less than harmonic intonation (the other being the lack of
> beating). In fact, I'd wager it's also part of the reason why it's so
> difficult to learn to sing in another tuning system at first - because
> your brain just keeps firing at you "this is a major third" or "that's
> a minor third" when in actuality you're singing neutral thirds.

I have a hard enough time singing in my "native" tuning system. I'm pretty sure I accidentally sing neutral intervals all the dang time.

> I'd say it makes sense to call them that if they work, cognitively, to
> substitute for the simpler intervals that you're trying to substitute
> for.

What do you mean, "if they work", or "substitute"? I don't think you can "substitute" JI for something Mavila-tempered or Dicot-tempered without totally corrupting how the chords move. How would you tune a Dicot progression in 5-limit JI? 25/24 is such a huge comma, it would sound like nonsense!

> But you could also set up a cognitive structure where a
> magic-tempered 5/4, which will be narrow, is a "different thing" than
> an augmented-tempered 5/4, which will be larger, but that they both
> share the "5/4 property."

How do you set up a cognitive structure?

> Or that a mavila fifth is a "different
> thing" than a superpyth fifth, but that both share the 3/2 property.
> Or that a dicot 5/4 is a different thing from a meantone 5/4, but that
> both share the 5/4 property, and that dicot's 5/4 also shares the 6/5
> property with the meantone minor third.

But what is it about these shared properties that ties them both to 5/4? What evidence is there that that ratio explains anything about how we hear dicot?

> Since you're fond of the idea that the generator "encodes" the
> information about the temperament it generates, you might like this
> paradigm better, since this "encoding" is basically an awareness of
> how the tempering and size of the generator leads to new intervals
> down the road. But either way, you can then still make the claim that
> root movements by "intervals that have the 3/2 property," of which the
> mavila, father, blackwood, and meantone fifths are examples, all tend
> to share certain characteristics.

I just don't see why calling it a "3/2" property is sensible. Especially considering that 16/11 and 17/11 have this property too, more or less. Heck, it might be even as wide as 13/9 to 14/9...the more time I spend in 11-EDO, the more I start to hear 763.64 cents as a "fifth" (in terms of movement, definitely NOT in terms of harmony, in case Michael S. is reading this and wants to say "Aha! Someone else thinks of 14/9 as a fifth too!"), and in 13-EDO, that 646-cent interval seems to work about as well as the 738-cent one.

> Or, if you really want to strengthen
> all of this, you can make the claim that root movements by an interval
> that have the 3/2 property, and only in which you as a Westerner have
> learned that those intervals actually do have that property, will
> behave a certain way. But this is pretty wordy, so I find it easier to
> just think in terms of JI and then tempering.

I get that JI is a useful abstraction here, at least for most temperaments. But have you ever tried to represent dicot or father in terms of JI, like on a lattice? It doesn't make any damn sense. I mean, in dicot, going up a 6/5 gets you to the same place as going up a 5/4. In father, going up a 5/4 gets you to the same place as going down a 3/2. It's insane. If there's any such thing as a 3/2 property or a 5/4 property or a 6/5 property, it makes zero sense to speak of one interval having multiple numbers of these properties. Because if one interval has the "proto-functional" (or whatever) properties of two intervals, you can prove from that the two intervals are actually one interval through the transitive property of identity. And I don't think you want to say that 3/2 equals 8/5 or that 5/4 equals 6/5.

You might say, "but in father temperament, 3/2 DOES equal 8/5", and I will respond that we don't need to invoke the idea of father temperament to produce an interval of 750 cents, we can get an interval that size out of many other temperaments where 16/15 is not tempered out, and guess what--it'll sound the same no matter how we mathematically construct it because an interval of 750 cents always sounds like an interval of 750 cents. And if it's the case in father temperament that 750 cents has the properties of 8/5 and the properties of 3/2, it has those properties outside of father temperament too.

-Igs

On Thu, Apr 28, 2011 at 10:28 PM, cityoftheasleep
<igliashon@...> wrote:
>
> > C-E-G
> > E-G-C
> > G-C-E
> >
> > Do any of these triads evoke the sense of there being a "root?" If so,
> > does the root in each case sound like it's a C?
>
> In 12-TET? I dunno. I can't really answer this question honestly, because when I was initially learning to play guitar, I heard all of these as different chords. In a way, I still kinda do. But I remember I used to play a chord like E-C-E-G and call it some kind of E, as if treating the C as a B#, hearing the whole thing as a sort of Em(#5) or something. It wasn't until I met a classically-trained pianist in high school that I learned it was actually an inverted C chord. So in other words, my naive/intuitive grasp of the concept of "root" was always whatever the lowest note in the chord was. The idea that a minor 6th and a major 3rd were related was big news to me, because they sound so totally different. However, now that I've "learned" better, I can recognize inverted chords as such, and know to call the that. So if you ask me what I'm hearing as the "root" I know to tell you all those chords are rooted on C, but I don't know that I hear C as the root in any fundamental psychoacoustic sense.

Are you sure you weren't hearing it like Em(b6), e.g. like the x-files
theme song or something? Either way, when I hear those chords I each
think C as the root, so perhaps this is more evidence of learning
being a factor in the harmonic identification of melodic figures.

> > Do they sound more
> > rooted and consonant than a sung C-Eb-Gb?
>
> More rooted--no. More consonant? Sure. Unless you mean consonant as a synonym for concordance, in which case I can't answer that question.

Concordance has gotten to be a fuzzy word lately, but I meant
consonant and said it that way on purpose.

> > There's a concept called "preattentive f0 estimation" that I'm reading
> > about in the literature which addresses much of this now, although I
> > have nothing concrete to throw out there yet.
>
> What does this have to do with JI and temperament?

When you hear a melodic figure, you tend to assign a harmonic identity
to it. If I sing an arpeggiated major triad, most people can tell it's
a major triad and that it sounds happy. So f0 estimation is still
involved in some respect when you hear melodies, albeit not as
strongly as with concurrent harmonies and in a way that may be more
strongly influenced by learning.

> > It's also probably much of the reason why melodic intonation matters
> > so much less than harmonic intonation (the other being the lack of
> > beating). In fact, I'd wager it's also part of the reason why it's so
> > difficult to learn to sing in another tuning system at first - because
> > your brain just keeps firing at you "this is a major third" or "that's
> > a minor third" when in actuality you're singing neutral thirds.
>
> I have a hard enough time singing in my "native" tuning system. I'm pretty sure I accidentally sing neutral intervals all the dang time.

But your brain probably doesn't tell you that you're doing it, right?
As far as you know, you're singing the right notes, but maybe you have
some vague awareness that people don't hear things the same way.

> > I'd say it makes sense to call them that if they work, cognitively, to
> > substitute for the simpler intervals that you're trying to substitute
> > for.
>
> What do you mean, "if they work", or "substitute"? I don't think you can "substitute" JI for something Mavila-tempered or Dicot-tempered without totally corrupting how the chords move. How would you tune a Dicot progression in 5-limit JI? 25/24 is such a huge comma, it would sound like nonsense!

I meant in terms of individual root movements, as in V-I or something
like that. We've been talking about the larger picture of how
progressively linked chains of root movements work differently in
different temperaments this whole time, so I'm not sure why you think
I'm suddenly saying otherwise...

> > But you could also set up a cognitive structure where a
> > magic-tempered 5/4, which will be narrow, is a "different thing" than
> > an augmented-tempered 5/4, which will be larger, but that they both
> > share the "5/4 property."
>
> How do you set up a cognitive structure?

I'm just suggesting that this may be a better paradigm for you to
think about things, and is probably closer to what I'm trying to say.
When I talk about "a 5/4 in this temperament" vs "a 5/4 in that
temperament" I'm aware that the two 5/4's are going to have different
properties, but in (most) various harmonic contexts they will still
function as an otonal, concordant, third-like interval - which is a
major third. Knowsur's album was a pretty masterful effort in getting
the dicot third to take on "major" or "minor" characteristics in
different cases.

> > Or that a mavila fifth is a "different
> > thing" than a superpyth fifth, but that both share the 3/2 property.
> > Or that a dicot 5/4 is a different thing from a meantone 5/4, but that
> > both share the 5/4 property, and that dicot's 5/4 also shares the 6/5
> > property with the meantone minor third.
>
> But what is it about these shared properties that ties them both to 5/4? What evidence is there that that ratio explains anything about how we hear dicot?

I certainly hear shades of majorness and minorness in knowsur's album.

> > But either way, you can then still make the claim that
> > root movements by "intervals that have the 3/2 property," of which the
> > mavila, father, blackwood, and meantone fifths are examples, all tend
> > to share certain characteristics.
>
> I just don't see why calling it a "3/2" property is sensible. Especially considering that 16/11 and 17/11 have this property too, more or less.

They certainly do.

> Heck, it might be even as wide as 13/9 to 14/9...the more time I spend in 11-EDO, the more I start to hear 763.64 cents as a "fifth" (in terms of movement, definitely NOT in terms of harmony, in case Michael S. is reading this and wants to say "Aha! Someone else thinks of 14/9 as a fifth too!"), and in 13-EDO, that 646-cent interval seems to work about as well as the 738-cent one.

So how is this different from what I'm saying? With melodic intervals,
categorical perception dominates. You hear the motion by 763.64 as
being in some sense "the same" as motion by 3/2 in JI. Although there
are differences, they resemble each other in a certain way. Motion
upward by the 545 cent interval in 11-equal isn't going to sound the
same as motion upward by the 763 cent interval - if anything it'll
probably sound like the opposite.

> > Or, if you really want to strengthen
> > all of this, you can make the claim that root movements by an interval
> > that have the 3/2 property, and only in which you as a Westerner have
> > learned that those intervals actually do have that property, will
> > behave a certain way. But this is pretty wordy, so I find it easier to
> > just think in terms of JI and then tempering.
>
> I get that JI is a useful abstraction here, at least for most temperaments. But have you ever tried to represent dicot or father in terms of JI, like on a lattice? It doesn't make any damn sense. I mean, in dicot, going up a 6/5 gets you to the same place as going up a 5/4. In father, going up a 5/4 gets you to the same place as going down a 3/2. It's insane.

This idea of proto-functional harmony is an idea I just had like a few
days ago (although I'm clearly not the first to think about things
like that), so no, I can't answer your questions about how to usefully
analyze high-error temperaments like that. In the temperaments we've
been working with, I've been thinking of things in which there are
three axes - 2/1, 3/2, and 5/4 - and they end up warping into another
after a few steps, which you can use to make a comma pump. In father,
two of the axes themselves become the same.

I would suggest either or both of the following as a way to handle it

1) You can use certain contextual cues to clue the ~460 cents into
being heard as a type of 5/4 or a type of 4/3, which is a phenomenon I
note I've personally seen at least for myself
2) You will probably end up learning to hear the 460 cent interval as
a new thing entirely, either not 5/4 or 4/3, or perhaps a blend of
both in a single interval, cognitively encapsulating and merging the
both of them
3) I expect the 4/3 axis will take on mixtures of some properties of
the 5/4 and the 4/3 JI axes, as well as perhaps other higher-limit
intervals in between, since your brain doesn't care what "limit"
you're in
4) Other parts of the father lattice that aren't as severely fludged
up will probably function similarly to the protoharmonic one
5) Other than that, I have no idea, I just figured out how to write
functional harmony in porcupine temperament, I haven't moved onto
father yet

> If there's any such thing as a 3/2 property or a 5/4 property or a 6/5 property, it makes zero sense to speak of one interval having multiple numbers of these properties. Because if one interval has the "proto-functional" (or whatever) properties of two intervals, you can prove from that the two intervals are actually one interval through the transitive property of identity. And I don't think you want to say that 3/2 equals 8/5 or that 5/4 equals 6/5.

I'd say that a father tempered fifth has characteristics of 3/2 and
8/5, but I wouldn't say that 3/2 itself equals 8/5, no.

> You might say, "but in father temperament, 3/2 DOES equal 8/5", and I will respond that we don't need to invoke the idea of father temperament to produce an interval of 750 cents, we can get an interval that size out of many other temperaments where 16/15 is not tempered out, and guess what--it'll sound the same no matter how we mathematically construct it because an interval of 750 cents always sounds like an interval of 750 cents. And if it's the case in father temperament that 750 cents has the properties of 8/5 and the properties of 3/2, it has those properties outside of father temperament too.

I'd say that a 750 cent interval has characteristics of 3/2 and 8/5
even if you're in 16-equal and better approximations to both 5/4 and
3/2 exist in the system. So what?

-Mike

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

> Are you sure you weren't hearing it like Em(b6), e.g. like the x-files
> theme song or something? Either way, when I hear those chords I each
> think C as the root, so perhaps this is more evidence of learning
> being a factor in the harmonic identification of melodic figures.

The thing in identifying the root in an arpeggio is that the identification comes after the melody is finished. You start with an E, you think "okay, this could be any chord with an E in it." Then you hear the G and you think, "okay, probably an E minor", and then the C hits and you go, "oh, that's a C, I just heard an E and a G, so this must be a C chord". You have to put the pieces together. If the C is followed by a D, or a B, or an F# or something, you might even stop thinking of it as an arpeggio and just hear it as a monophonic line. And if you're a naive listener in the first place, you might not even be thinking in terms of "roots". The whole idea of chord having a "root" is something that has to be learned. The distinction between an arpeggio and a melody is something that plenty of listeners aren't even aware of.

> When you hear a melodic figure, you tend to assign a harmonic identity
> to it. If I sing an arpeggiated major triad, most people can tell it's
> a major triad and that it sounds happy.

Only if they know what a major triad is. Otherwise they might just tell you it sounds happy. Unless you're playing "Taps", in which case they'll tell you it sounds sad.

> So f0 estimation is still
> involved in some respect when you hear melodies, albeit not as
> strongly as with concurrent harmonies and in a way that may be more
> strongly influenced by learning.

I don't really know what f0 estimation is, but okay.

> But your brain probably doesn't tell you that you're doing it, right?
> As far as you know, you're singing the right notes, but maybe you have
> some vague awareness that people don't hear things the same way.

Nope, I can tell how off I am. I can acutely hear every wrong interval I sing, I just don't seem to have the ability to correct it. Bad ear-voice coordination, maybe. It's like...I can match a pitch or harmonize in a given way if I have a few seconds to find the right resonance, but in going from one note to the next, I don't know how far to "jump" in pitch so I over-shoot or under-shoot the mark, unless I *really* know the melody and/or have someone to sing along with. And it varies from key to key, as well. I sing better in D minor than in Db minor, for instance.

> I meant in terms of individual root movements, as in V-I or something
> like that. We've been talking about the larger picture of how
> progressively linked chains of root movements work differently in
> different temperaments this whole time, so I'm not sure why you think
> I'm suddenly saying otherwise...

Right, I don't think you're saying differently.

> I'm just suggesting that this may be a better paradigm for you to
> think about things, and is probably closer to what I'm trying to say.
> When I talk about "a 5/4 in this temperament" vs "a 5/4 in that
> temperament" I'm aware that the two 5/4's are going to have different
> properties, but in (most) various harmonic contexts they will still
> function as an otonal, concordant, third-like interval - which is a
> major third.

Aha...this is what I'm getting at. It's that "otonal, concordant, third-like interval" that names the property more accurately than "5/4". 5/4 is an example of an interval that has these properties, but it's not the definition of what this property is.

> Knowsur's album was a pretty masterful effort in getting
> the dicot third to take on "major" or "minor" characteristics in
> different cases.

AFAIK, his album utilized a lot of 14-TET, at least in the melodic lines, and that may have had some influence. OTOH, I tried playing a I-vi-IV-ii-vii-iii-V-I progression in 7-EDO the other day and sure enough I heard minors where I expected to hear them and majors where I expected to hear them. In fact I rendered that progression in the other 6 heptatonic MOS scales (with an L of 3 steps and and s of 2) and remarkably little seemed to change overall. I should post them and see what everyone else thinks, it was pretty trippy. I think there were a few where I heard diminished fifths as being "regular" fifths, even.

> > But what is it about these shared properties that ties them both to 5/4? What evidence > > is there that that ratio explains anything about how we hear dicot?
>
> I certainly hear shades of majorness and minorness in knowsur's album.

You didn't answer the question. If dicot shows us anything, it's that the interval itself isn't telling us whether it's 5/4 or 6/5, but that in fact that information is coming from somewhere else entirely(!), since (as you say) it flip-flops. Think on THAT for a bit.

> So how is this different from what I'm saying? With melodic intervals,
> categorical perception dominates. You hear the motion by 763.64 as
> being in some sense "the same" as motion by 3/2 in JI. Although there
> are differences, they resemble each other in a certain way. Motion
> upward by the 545 cent interval in 11-equal isn't going to sound the
> same as motion upward by the 763 cent interval - if anything it'll
> probably sound like the opposite.

Yes, I agree with this. What I don't agree with is giving 3/2 semantic privilege as the identifier. Nor do I agree that it's the psychoacoustic simplicity of the ratio that explains why motion by that interval is so powerful--precisely because much more complex ratios nearby in the pitch-range evoke the same power-of-motion.

> This idea of proto-functional harmony is an idea I just had like a few
> days ago (although I'm clearly not the first to think about things
> like that), so no, I can't answer your questions about how to usefully
> analyze high-error temperaments like that.

I'm not expecting you to know the answers, but these are important questions that should probably be addressed in the evolution of this idea.

> In the temperaments we've
> been working with, I've been thinking of things in which there are
> three axes - 2/1, 3/2, and 5/4 - and they end up warping into another
> after a few steps, which you can use to make a comma pump. In father,
> two of the axes themselves become the same.

Yes...and yet we don't hear it that way. We don't hear simultaneous interval properties, we tend to hear alternating/switching. Or something else entirely.

> 1) You can use certain contextual cues to clue the ~460 cents into
> being heard as a type of 5/4 or a type of 4/3, which is a phenomenon I
> note I've personally seen at least for myself

Right, but what are those cues, and what explains them?

> 2) You will probably end up learning to hear the 460 cent interval as
> a new thing entirely, either not 5/4 or 4/3, or perhaps a blend of
> both in a single interval, cognitively encapsulating and merging the
> both of them

At 460 cents, I tend not to hear any "thirdness" at all. At 450 cents, though, it's still there.

> 3) I expect the 4/3 axis will take on mixtures of some properties of
> the 5/4 and the 4/3 JI axes, as well as perhaps other higher-limit
> intervals in between, since your brain doesn't care what "limit"
> you're in

Try to draw it out sometime, and dicot too. It's a trip, definitely a worthwhile exercise.

> 4) Other parts of the father lattice that aren't as severely fludged
> up will probably function similarly to the protoharmonic one

Right, because two Father-sized fifths give you something which sounds an awful lot like a 6/5.

> 5) Other than that, I have no idea, I just figured out how to write
> functional harmony in porcupine temperament, I haven't moved onto
> father yet

I tell you what: it's easier in 13-EDO than 8-EDO but it involves higher-limit interpretations, which basically means looking at the same scale as a different temperament, which maybe even means Father temperament doesn't actually exist.

> I'd say that a father tempered fifth has characteristics of 3/2 and
> 8/5, but I wouldn't say that 3/2 itself equals 8/5, no.

> I'd say that a 750 cent interval has characteristics of 3/2 and 8/5
> even if you're in 16-equal and better approximations to both 5/4 and
> 3/2 exist in the system. So what?

Say you hear 750 cents and you think "that's working like an 8/5". That means you're hearing it "as" an 8/5, right? Then you hear it in another context or with another cue or whatever and you think "that's working like a 3/2". So first you say "750 cents is an 8/5", then you say "750 cents is a 3/2", so what you're really saying is "that 8/5 I just heard is actually (or also) a 3/2". If a=b, and b=c, a=c. And yet in a sense, I think there *is* a way that 8/5 is like 3/2. Especially if you're doing something like Gene did with those christmas carols. So who knows?

What I do know is a ratio defines an exact relationship, but the ear is not exact and there are fuzzy regions of the interval spectrum where the brain seems to flip-flop in its categorization--or give up, or invent something new--based on other cues, so why use a name that defines an exact rational frequency relationship to describe a broad (and maybe undefined) range of frequency relationships? The real reason I hate thinking in terms of JI is that it sort of ignores the fuzziness and ambiguity that happens in the real world. I mean, I still do think in terms of JI because it has a certain convenience about it, but really I'm desperate for an alternative.

-Igs

In a few hours, I'll be heading somewhere else and there will probably be very poor internet connectivity there, which means I'll read possible replies on Sunday when I get home.
But anyway:

Mike wrote:

> However, if you try to take a meantone chord progression and try to
> put it back in JI, it'll probably just sound awkward. But in
> porcupine, 81/80 becomes equated with 25/24, so assuming you
> understand proto-functional 25/24 logic, you can now create a new
> logic that enables you to handle porcupine functional harmony. Just
> turn every 81/80 shift into a 25/24, and now you can modulate around
> the entire porcupine lattice and do whatever you like. Yes!

Phew, ... I think I'll really leave this view to you and other people; to me it sounds like mixing "apples and oranges". I think I have already spent quite a few years trying to understand the concept of a protoharmonic system, then it lead me to the idea of finding new individual harmonic system with the help of non-fifth temperaments, and now I don't see much of a point in getting back to something which I think I have "beaten" back in 2006.

> I just meant that all of this doesn't have to apply to standard, old
> school, vanilla ice cream common practice harmony. In the 20th
> century, people started getting back into finding chord progressions
> and resolutions that stuck closer to other modes than just major and
> minor, which is a trend that probably started with the French school
> of composition. The Beatles wrote lots of songs that you'd probably
> call "modal" (like Come Together, which is predominantly Dorian, or
> Norwegian Wood, which is Mixolydian in the verse and Dorian on the
> chorus).

Sure. But the concept of "5l * 2m * 2s = 2/1" (where l=10/9, m=16/15, s=81/80) is equally valid for all of them. Try loading the "parizek_syndiat.scl" from Manuel's scale archive and playing in it for a while.

I'll listen to the examples when I get back.

> The point is that even these interesting, colorful, modern, "modal"
> progressions have their basis in a combination of JI harmony and 81/80
> comma pumps, and that similarly colorful progressions could exist in
> porcupine too, but sound like nothing we've ever heard.

Agreed. I wasn't saying anything against this.

> These are cool! I think the "wh" one was the one like what I was
> describing. There's still something that needs to be done though,
> before the Emaj -> Cmaj sounds anything like Gmaj -> Cmaj, but I don't
> know what it is.

Finding the right harmonic context -- i.e. "appropriate surrounding progressions.

> But I also think that temperaments exist that are really useful that
> you might not "naturally" come to discover. For example, if you tend
> to move around by 3/2 a lot, it's pretty natural that you'll discover
> meantone. Or if you tend to move around by 5/4 a lot, it's pretty
> natural that you'll discover magic. But if you tend to move around by
> 5/4 a lot, it may not be quite as natural that you'll discover
> blackwood, or mavila, or whitewood. But these temperaments are still
> awesome (especially blackwood), and you can still make use of them to
> create interesting comma pumps.

I think there are three "primary" ways to discover these temperaments -- someone may suggest otherwise.
One is determining two approximations using different numbers of the same steps -- i.e. 1 step of "something" for 4/3, 3 steps of the same "something" for 5/2.
Another way is to temper something out of a 3D melodic system by splitting consecutive intervals of a scale into smaller ones. For example, start with 4:5:6:8 and view it as "5/4, 6/5, 4/3". The largets one is 4/3 and let's say we want to "reduce it" by the minor third. So we get one 5/4, two 6/5s and one 10/9. Now we want to get rid of the 5/4 so we can either reduce it by 6/5, which will bring a new factor of 25/24, or by 10/9, which brings a new factor of 9/8. And so on. Once you do as many reductions as you wish, you can then exclude the smallest interval from the melodic steps by tempering it out -- for example, if you start with 2:3:4 and you get to the stage where you have 12 occurrences of 256/243 and 5 Pythagorean commas, you can temper out the comma by widening each minor second by 5/12 of the comma and get the same 1200-cent octave with a 1D system.
The third way is to split the prime exponents of the unison vector into two pairs, which can tell you what is a range for possible generator sizes for the particular temperament. For example, if the "comma" is 78732/78125 whose prime exponents are "2, 9, -7", one possible solution is "9, 9, 0" and "-7, 0, -7", which tells you that 6/1 is approximated by 7 steps of "something" and 10/1 is approximated by 9 steps of the same "something".

> 16/15 -> 25/24 -> 128/125 -> 3125/3072
> Father -> Dicot -> Augmented -> Magic
> 5-equal -> 7-equal -> 12-equal -> 19-equal (if mixed with meantone)
>
> Do you see some kind of linear splitting pattern here?

There definitely has to be something to it ... Will look at it later.

> What specifically are you suggesting? Eliminating 648/625?

No. Since 26/25 is close to twice the size of 25/24, it's similar to saying that 648/625 is close to the size of 25/24. They differ by 15625/15552 and if you temper out this "kleisma", you do essentially no harm to the sound.

> I think a longer progression by 3/2's is pretty courageous even in
> meantone. Cmaj -> Gmaj -> Dmaj -> Amaj -> Emaj -> Bmaj... where does
> it end? You're in outer space now. But, if you're in Blackwood and you
> temper out 256/243, it's not courageous at all, because you're back
> where you started. Likewise if you're in Whitewood, which tempers out
> 2187/2048 instead, you're also back where you started.

You seem to get my point.

> I did see lattice #4, but how was that different from my "warped 3D"
> lattice? It looked like you were saying the same thing I was.

No, there's a huge difference. Your warped 3D lattice doesn't imply some progressions or triads being more easily reachable and others being less easily reachable in a particular harmonic system. Obviously, it doesn't matter in "your context" because you don't probably don't find these implications as important as I do. Just because I treated hanson according to my "lattice #4" rather than your "lattice #3" did I finally arrive at the particular comma pump which I used on the recording. I simply wanted to hear a piece of music which is equally "ordinary" in the minor-third-based harmonic system as is the "C major, A minor, D minor, G major, C major" progression in the fifth-based common practice harmonic system.

> Alright, but it's also true that there are ways to do that than to
> stick to the temperament's MOS's, or think about whatever the
> generating interval technically is. For example, let's say you're in
> porcupine, and you decide to extend the system to the 13-limit by
> tempering out some insane comma that, technically, causes the period
> to become 1/2-oct instead of 1 oct, similar to how Hedgehog does it.
> Would this fundamentally change the properties of the system, just
> because this one stupid 13-limit comma has unfortunately caused this
> to happen? I don't think so.

A couple of years ago, I recorded an improvization in hedgehog before I even knew it was hedgehog. And, what's more, before I actually knew about porcupine. :-D
Later, I recorded another piece, not an improv, in porcupine.
What I discovered was that as soon as I started throwing the 7-limit harmonies in, these were essentially two different harmonic systems which had a few things in common (like dividing a minor sixth into 5 equal steps, for example).
If you wish to listen on your own, here they are both of them:
http://dl.dropbox.com/u/8497979/PPImprovX.mp3
http://dl.dropbox.com/u/8497979/AmongOtherThings2.mp3

Okay, I'm leaving now.
Will get back in two days.

Petr

I wrote:

> If you wish to listen on your own, here they are both of them:
> http://dl.dropbox.com/u/8497979/PPImprovX.mp3
> http://dl.dropbox.com/u/8497979/AmongOtherThings2.mp3

I should add that the porcupine recording is actually made of two parts. One
uses the temperament according to your "lattice #3", the other uses it
according to my "lattice #4".
Do you really think the first part sounds "porcupinian" just because it is
in porcupine?
To me, the first part sounds more like some sort of "screwed up common practice harmony" while
the second part sounds to me like porcupinian harmony.
OTOH, taking it from the porcupinian point of view, the first part sounds to me like a bunch of wild chromatic modulations, similarly to a progression by thirds in common practice harmony. That's why I better like to view it as a 3D system with some minor pitch alterations, which perfectly corresponds to you "lattice #3" concept.

Petr

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

> > Before you joined these lists, there was considerable discussion of isometries of the 5- and 7- limit symmetrical lattices, which can convert one kind of JI functional harmony into another. This can have a disconcerting effect, judging by the reactions of my family one Christmas in the 70s when I bombarded them with altered versions of Christmas carols, where 8/5 or 5/3 took the place of 3/2.
>
> Ha! Do you still have a copy of this? I'd love to hear it.

Ha ha! No, the work I did on HP desktop computers was all lost. It's not generally appreciated that with such models as the 9835 and 9845, HP had gone and invented the personal computer before it officially was introduced into the world by Apple and IBM. My brother Robin the chess whiz would bring these home from work, which was encouraged because HP had a strange idea people would use them for working. In fact, we hooked it up to a device Robin made which took the output of a crystal oscillator, a very stable vibration in the megahertz range, counted off any integer number of vibrations, and triggered the change of a square wave. So I had an infernal device which had very accurate and stable tuning, but which would only produce at most four voices of square waves. Perfect for the crazed experimenter in the realm of music.

Sadly, a few years later these computers were no longer available, but I did manage to explore some aspects of xenharmonic composition as well as do goofy things like ruin Christmas.

--- Petr Parizek <petrparizek2000@...> wrote:

> > http://dl.dropbox.com/u/8497979/AmongOtherThings2.mp3
>
> I should add that the porcupine recording is actually made
> of two parts. One uses the temperament according to your
> "lattice #3", the other uses it according to my "lattice #4".
> Do you really think the first part sounds "porcupinian" just
> because it is in porcupine?
> To me, the first part sounds more like some sort of "screwed
> up common practice harmony" while the second part sounds to
> me like porcupinian harmony.

Hopefully you'll get this when you return from your trip.
(I see I picked a bad moment to post my longer exposition
on the subject.)

If I understand the point you mean in the above mp3, I agree
that the progression before sounds like a screwy meantone
progression, whereas the second part sounds more natural.
I hesitate to use the word "functional". But at least, the
first part does NOT sound more functional than the second.

-Carl

On Wed, Apr 27, 2011 at 3:57 AM, Valentine, Bob <bob.valentine@...> wrote:
>
> I don’t know if this works in Porcupine, but E7 “can be” on the same diminished seventh
>
> as G7, (think G13b9àC) so throwing away the G leaves a IIIàI cadence something like
> Twiddling the voice timing and exploiting the suspensions may make it musical.

This is pretty commonly used as a polychord over the G7, e.g.
G7b9add13. You'd play diminished[8] over this in 12-equal. This chord
has always baffled me, so maybe what's going on is that it mixes the
5-limit and the 3-limit dominant chords in one polychord. You can
likewise mix the 5-limit and the 3-limit subdominant chords to get
something like Fm -> C -- if the 3-limit subdominant is Fmaj -> C,
then the 5-limit subdominant is Abmaj -> C, so mixing them together
gets you Fm -> C. I'm going to voice it Fm/C -> C, because I think
it's a better way to illustrate the progression that follows.
Throwing the 7-limit subdominant in there gets you Fm6/C -> C.
Throwing the 9-limit subdominant in there gets you Fm6sus/C -> C.
Throwing the 11-limit subdominant in there gets you Fm6sus(n2)/C -> C
(beautiful sound!)
Throwing the 13-limit subdominant in there gets you Fm6sus(n2,n7)/C ->
C (also beautiful sound!)

So you can see that this approach basically constructs utonalities
under C. While 13-limit utonal chords usually don't sound all that
great, I think they actually do sound good if used as described above.

Even if you just look at Abmaj -> C vs Fmaj -> C, there does seem to
be some utility in the otonal -> authentic and utonal -> plagal
approach. Both of these are utonal, "subdominant" resolutions, and
they both sound "calming and relaxing" (plagal, IV-I), not exciting
and invigorating (authentic, V-I). Getting the authentic side of the
equation to work would really make the symmetry apparent.

> Stringing this out forever will get sort of ‘’augmenty / Giant Steps” in temperaments with 125/64 ==  2.

It works in one sense, but still doesn't sound like there's a strong
sense of "key" to me. I think what needs to be done is for us to find
the one magic 5-limit resolution that does the trick. Perhaps in this
case it stems from the fact that ii-V-I is related to IV-V-I, but the
ii is the mediant of the IV. So Fm->Emaj->Cmaj might do the trick, and
it sounds pretty decent to me:

F Ab F C -> B G# E B -> C G E C

If you make the Emaj an E7, perhaps B G# D B (which is also a
diminished chord voicing), it works even better. So I'm not sure if
what I'm hearing here is the diminished chord, or if this is perhaps
part of how the diminished chord itself works. I was also
experimenting with G#m -> Emaj -> Cmaj before, but this one might be a
little bit better.

-Mike

On Fri, Apr 29, 2011 at 4:23 AM, Petr Parízek <petrparizek2000@...> wrote:
>
> Phew, ... I think I'll really leave this view to you and other people; to me
> it sounds like mixing "apples and oranges". I think I have already spent
> quite a few years trying to understand the concept of a protoharmonic
> system, then it lead me to the idea of finding new individual harmonic
> system with the help of non-fifth temperaments, and now I don't see much of
> a point in getting back to something which I think I have "beaten" back in
> 2006.

Alright. Well, from my perspective, I've been stuck in generator-land
since I discovered what a generator is, so this is a liberating new
paradigm. Let's see if we meet in the middle.

> > These are cool! I think the "wh" one was the one like what I was
> > describing. There's still something that needs to be done though,
> > before the Emaj -> Cmaj sounds anything like Gmaj -> Cmaj, but I don't
> > know what it is.
>
> Finding the right harmonic context -- i.e. "appropriate surrounding
> progressions.

I think Fm -> Emaj -> Cmaj does the trick. Fm is the submediant to
Abmaj, so the whole thing sounds like IV V I, kind of. Also there's

||: Bb A D F -> D G C E -> D F# C D :||

That sounds like a pretty good 5-limit resolution to me.

> One is determining two approximations using different numbers of the same
> steps -- i.e. 1 step of "something" for 4/3, 3 steps of the same "something"
> for 5/2.

Right.

> Another way is to temper something out of a 3D melodic system by splitting
> consecutive intervals of a scale into smaller ones... Once you do as many reductions as you wish, you
> can then exclude the smallest interval from the melodic steps by tempering
> it out -- for example, if you start with 2:3:4 and you get to the stage
> where you have 12 occurrences of 256/243 and 5 Pythagorean commas, you can
> temper out the comma by widening each minor second by 5/12 of the comma and
> get the same 1200-cent octave with a 1D system.

OK, that makes sense.

> The third way is to split the prime exponents of the unison vector into two
> pairs, which can tell you what is a range for possible generator sizes for
> the particular temperament. For example, if the "comma" is 78732/78125 whose
> prime exponents are "2, 9, -7", one possible solution is "9, 9, 0" and "-7,
> 0, -7", which tells you that 6/1 is approximated by 7 steps of "something"
> and 10/1 is approximated by 9 steps of the same "something".

What about linear splitting? For example, take 4:5:6 - you split it
and get 8:9:10:11:12. Equating 9/8 -> 10/9 and 12/11 -> 10/9 gets you
something like mohajira. Equating 12/11 and 11/10, and 11/10 and 10/9,
gets you something like porcupine.

Also, there's now a fourth option - go to Graham's temperament finder
and see what comes out. I'm serious - I can't imagine anyone would
have ever discovered Blackwood before, but it's one of my favorite
tunings ever. I guess some culture could have started with the
pentatonic scale and evened it out, and discovered Blackwood that way,
but I dunno.

> > What specifically are you suggesting? Eliminating 648/625?
>
> No. Since 26/25 is close to twice the size of 25/24, it's similar to saying
> that 648/625 is close to the size of 25/24. They differ by 15625/15552 and
> if you temper out this "kleisma", you do essentially no harm to the sound.

OK, I see.

> > I did see lattice #4, but how was that different from my "warped 3D"
> > lattice? It looked like you were saying the same thing I was.
>
> No, there's a huge difference. Your warped 3D lattice doesn't imply some
> progressions or triads being more easily reachable and others being less
> easily reachable in a particular harmonic system.

To make a minor correct, it does imply some progressions of triads
being more easily reachable or less reachable in a particular harmonic
system's MOS's, but I just don't want to mix the MOS's with the larger
5-limit picture.

> Obviously, it doesn't
> matter in "your context" because you don't probably don't find these
> implications as important as I do. Just because I treated hanson according
> to my "lattice #4" rather than your "lattice #3" did I finally arrive at the
> particular comma pump which I used on the recording. I simply wanted to hear
> a piece of music which is equally "ordinary" in the minor-third-based
> harmonic system as is the "C major, A minor, D minor, G major, C major"
> progression in the fifth-based common practice harmonic system.

OK, but I tend to like how Coltrane did it in Giant Steps: you move up
by major thirds, but you preface everything with fifth based ii-V's.
I've been exploring a diminished[12] harmonization in that fashion,
even in 12-equal. Check out

||: Am7 -> Em7 | F#m7 -> C#m7 | Ebm7 -> Bbm7 | Cm7 -> Gm7 :||

It's its own sound - since you're using diminished[12], you're using
the full 12-equal set but it's definitely completely different than
other 12-equal progressions that use also use the full 12-equal set.
It's its own sound. It sounds a lot different than just stuff in
diminished[8], because we've added a simple iv-i progression over each
root, which "sweetens" it a bit. This creates a certain sound, and I
think that sound is beautiful. I wouldn't have found it if I had
limited myself to diminished[8], but because I'm now free to think in
terms of any chord progression I want, I found it. What happens if you
do the same thing, but you move down by Hanson generators? You get
something that sounds awesome, but is still uniquely Hanson (you get
the opportunity to connect it all together at Dm7 -> Am7). That's how
I think of it. I'll make some examples of this.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Apr 28, 2011 at 12:32 PM, genewardsmith
> <genewardsmith@...> wrote:

> > (2) Adding the root of IV to V gives V7; resolving that to I tends
> > to define I as the tonic.

> (2) I think has some far-reaching consequences on music cognition that
> I don't think I've ever seen adequately expressed.

Are you (or Gene?) implying that you could do the corresponding thing
with 5-relations etc. and get another way to define a tonic? Of course
you get V7 by adding the root of IV but is that really the reason why
V7 works?

Kalle

On Fri, Apr 29, 2011 at 4:42 AM, Petr Parízek <petrparizek2000@...> wrote:
>
> I wrote:
>
> > If you wish to listen on your own, here they are both of them:
> > http://dl.dropbox.com/u/8497979/PPImprovX.mp3
> > http://dl.dropbox.com/u/8497979/AmongOtherThings2.mp3
>
> I should add that the porcupine recording is actually made of two parts. One
> uses the temperament according to your "lattice #3", the other uses it
> according to my "lattice #4".
> Do you really think the first part sounds "porcupinian" just because it is
> in porcupine?
> To me, the first part sounds more like some sort of "screwed up common
> practice harmony" while
> the second part sounds to me like porcupinian harmony.

I assume both of these are in reference to "Among Other Things 2." In
my mind, the part with the solo harp sounds like you're sticking
mainly with "porcupine diatonic harmony," generator-based thinking,
etc. The second part, where the whole band comes in, sounds more like
porcupinized common practice harmony, which I think is an awesome
sound. There were a few times I heard some screwy 81/80 shifts, but on
the whole it seems to work. So I hear it the opposite as what you've
described.

-Mike

On Fri, Apr 29, 2011 at 8:23 PM, Kalle Aho <kalleaho@...> wrote:
>
> Are you (or Gene?) implying that you could do the corresponding thing
> with 5-relations etc. and get another way to define a tonic?

I'm noting that if you had to categorize the following chord progression:

Cmaj -> Emaj (in the key of Emaj, so that it's bVI -> I in Emaj)

as being in line with one of the following concepts:

1) Plagal, IV-I (calming, relaxing, serene)
2) Authentic, V-I (invigorating, energetic, etc)

Which would it be? I hear it clearly fitting in line with #1. Check my
post in response to Bob Valentine for the full train of thought:
/tuning/topicId_98428.html#98723

> Of course you get V7 by adding the root of IV but is that really the reason why
> V7 works?

There seems to be something to it. Carl mentioned a listening test,
and one which I remember stumbling on myself before joining the list,
in which people seemed to express a preference for 16/9 as the
dominant 7 over 7/4 or 9/5. In 31-equal, things seem to be in line
with this, so that 9/5 ends up being a better dominant 7th than 7/4.
In 22-equal, however, it's reversed, so that 7/4 ends up working
better than 9/5. So I think there's something to it.

For example, in this excerpt, I use 4:5:6:7 chords as dominant 7 chords

http://soundcloud.com/mikebattagliamusic/functionalporcupineexcerpt

This is a slowed down version of the main "depart from -> return to
tonic" cycle, which by now should translate to "comma pump"

http://soundcloud.com/mikebattagliamusic/functional-porcupine-with-7

If you turn those dominant 7th into 9/5's in 22-equal, say G-B-D-F ->
C-C-C-E, instead of the F-E going 4/3 -> 5/4 over the root, it goes
11/8 -> 5/4 over the root. This is cool, I guess, but not really the
dominant 7th sound.

-Mike

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

> Aha...this is what I'm getting at. It's that "otonal, concordant, >third-like interval" that names the property more accurately than >"5/4". 5/4 is an example of an interval that has these properties, >but it's not the definition of what this property is.

Neither "otonal" nor "concordant" are necessary for an interval to be perceived of as a "major third", as you know. A difference, on the content level rather than the context level, between the perceived "major third" and a contrasting "minor third" may not be necessary, either, as you guys note here:
>
> > Knowsur's album was a pretty masterful effort in getting
> > the dicot third to take on "major" or "minor" characteristics in
> > different cases.
>
> AFAIK, his album utilized a lot of 14-TET, at least in the melodic >lines, and that may have had some influence. OTOH, I tried playing a >I-vi-IV-ii-vii-iii-V-I progression in 7-EDO the other day and sure >enough I heard minors where I expected to hear them and majors where >I expected to hear them.

The only reasonable explanation we have is that some intervallic qualities are at least as much about the motion as the meat.

> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > Are you sure you weren't hearing it like Em(b6), e.g. like the x-files
> > theme song or something? Either way, when I hear those chords I each
> > think C as the root, so perhaps this is more evidence of learning
> > being a factor in the harmonic identification of melodic figures.

You gave three inversions of the "C Major" chord. The voicing with the fourth in the bass (G-C-E) was not considered properly rooted in "common practice" and when used was a mild dissonance- as a predominant for example. If you play it in a low register, and you haven't been too thoroughly conditioned to scan for theoretical roots in triadic formation, you can hear that it is indeed less stable and "properly" rooted than the other versions, and so hear the reason for the common-practice "rule".

On Fri, Apr 29, 2011 at 12:59 AM, cityoftheasleep
<igliashon@...> wrote:
>
> > When you hear a melodic figure, you tend to assign a harmonic identity
> > to it. If I sing an arpeggiated major triad, most people can tell it's
> > a major triad and that it sounds happy.
>
> Only if they know what a major triad is. Otherwise they might just tell you it sounds happy. Unless you're playing "Taps", in which case they'll tell you it sounds sad.

Not as sad as if I played it in minor.

> > So f0 estimation is still
> > involved in some respect when you hear melodies, albeit not as
> > strongly as with concurrent harmonies and in a way that may be more
> > strongly influenced by learning.
>
> I don't really know what f0 estimation is, but okay.

F0 estimation is the actual term for the process by which harmonic
entropy comes to exist at all. It's your brain's harmonic series
detector doing its thing, as you put it.

> > But your brain probably doesn't tell you that you're doing it, right?
> > As far as you know, you're singing the right notes, but maybe you have
> > some vague awareness that people don't hear things the same way.
>
> Nope, I can tell how off I am. I can acutely hear every wrong interval I sing, I just don't seem to have the ability to correct it. Bad ear-voice coordination, maybe. It's like...I can match a pitch or harmonize in a given way if I have a few seconds to find the right resonance, but in going from one note to the next, I don't know how far to "jump" in pitch so I over-shoot or under-shoot the mark, unless I *really* know the melody and/or have someone to sing along with. And it varies from key to key, as well. I sing better in D minor than in Db minor, for instance.

Alright, well maybe you're better at it than I am. Try to sing an
orwell chain of 7/6's leading up to the tritave and see how easy it
is. Not too easy for me.

> > I'm aware that the two 5/4's are going to have different
> > properties, but in (most) various harmonic contexts they will still
> > function as an otonal, concordant, third-like interval - which is a
> > major third.
>
> Aha...this is what I'm getting at. It's that "otonal, concordant, third-like interval" that names the property more accurately than "5/4". 5/4 is an example of an interval that has these properties, but it's not the definition of what this property is.

The property that you are referring to is the "perceived as 5/4"
property. A just 5/4 may not always be perceived as 5/4. Other, more
complex JI intervals, on the other hand, might still be perceived as
5/4. It's like when you were talking about mavila fourths - you said
they sounded like drunken, wobbly fourths, not just crappy fourths.
OK, but they still sound like fourths. They don't sound rooted. And
furthermore, they don't sound like strong, resonant, non-wobbly
27/20's or something like that. They sound like wobbly fourths, so
your brain is hearing them as detuned, drunken wobbly 4/3's. They have
the 4/3 property.

> > Knowsur's album was a pretty masterful effort in getting
> > the dicot third to take on "major" or "minor" characteristics in
> > different cases.
>
> AFAIK, his album utilized a lot of 14-TET, at least in the melodic lines, and that may have had some influence. OTOH, I tried playing a I-vi-IV-ii-vii-iii-V-I progression in 7-EDO the other day and sure enough I heard minors where I expected to hear them and majors where I expected to hear them. In fact I rendered that progression in the other 6 heptatonic MOS scales (with an L of 3 steps and and s of 2) and remarkably little seemed to change overall. I should post them and see what everyone else thinks, it was pretty trippy. I think there were a few where I heard diminished fifths as being "regular" fifths, even.

Please do. I made a post in response to Carl about some techniques
that Knowsur uses to signal major vs minor, but you didn't respond so
I don't know if you caught it.

> > > But what is it about these shared properties that ties them both to 5/4? What evidence > > is there that that ratio explains anything about how we hear dicot?
> >
> > I certainly hear shades of majorness and minorness in knowsur's album.
>
> You didn't answer the question. If dicot shows us anything, it's that the interval itself isn't telling us whether it's 5/4 or 6/5, but that in fact that information is coming from somewhere else entirely(!), since (as you say) it flip-flops. Think on THAT for a bit.

The "somewhere else" you are referring to, I think, are pre-learned
ways to bias the harmonic series detector in the brain. That's it. The
harmonic entropy curve gives each interval a Gaussian-shaped error
curve, but in real life it doesn't work like that. In real life, the
harmonic series detector more than likely ends up getting jerked
around to pre-look in advance for 5/4's and 6/5's in different cases.
So you can play a mavila 5/4, but to you it might sound like a 6/5. So
for you, subjectively, at that point in time, a mavila 5/4 would have
the 6/5 property.

> What I don't agree with is giving 3/2 semantic privilege as the identifier. Nor do I agree that it's the psychoacoustic simplicity of the ratio that explains why motion by that interval is so powerful--precisely because much more complex ratios nearby in the pitch-range evoke the same power-of-motion.

I disagree. The fact that you hear these different complex ratios
nearby as concordant and otonal still suggests to me that 3/2 is
involved in some level.

> > 1) You can use certain contextual cues to clue the ~460 cents into
> > being heard as a type of 5/4 or a type of 4/3, which is a phenomenon I
> > note I've personally seen at least for myself
>
> Right, but what are those cues, and what explains them?

See the other thread.

> > I'd say that a 750 cent interval has characteristics of 3/2 and 8/5
> > even if you're in 16-equal and better approximations to both 5/4 and
> > 3/2 exist in the system. So what?
>
> Say you hear 750 cents and you think "that's working like an 8/5". That means you're hearing it "as" an 8/5, right? Then you hear it in another context or with another cue or whatever and you think "that's working like a 3/2". So first you say "750 cents is an 8/5", then you say "750 cents is a 3/2", so what you're really saying is "that 8/5 I just heard is actually (or also) a 3/2".

Or that it sounded like an 8/5 the first time and a 3/2 the second time.

> If a=b, and b=c, a=c.

More like a(1 sec) = b, a(2 sec) = c.

> What I do know is a ratio defines an exact relationship, but the ear is not exact and there are fuzzy regions of the interval spectrum where the brain seems to flip-flop in its categorization--or give up, or invent something new--based on other cues, so why use a name that defines an exact rational frequency relationship to describe a broad (and maybe undefined) range of frequency relationships?

I don't personally find it confusing. If you load up a synth and start
screwing with some complex near-3/2 ratios, and set the timbre equal
to two sines playing the ratio, the virtual note will pop out as
though it were a 3/2. So there's one meaning that you can assign to a
ratio, which is how it ends up being perceived at the end of the day,
and another meaning, which is how it's tuned. So 81/64, in one sense,
will evoke a sense of "sharp 5/4," and in another sense, it's also
just 81/64.

> The real reason I hate thinking in terms of JI is that it sort of ignores the fuzziness and ambiguity that happens in the real world. I mean, I still do think in terms of JI because it has a certain convenience about it, but really I'm desperate for an alternative.

I'm the one who's talking about fuzzy 3/2's, and you're the one who's
telling me that we can't call them fuzzy 3/2's because some of them
are un-fuzzy 13/9's! Who's the one ignoring fuzziness?

-Mike

On Sat, Apr 30, 2011 at 4:09 AM, lobawad <lobawad@...> wrote:
>
> > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > > Are you sure you weren't hearing it like Em(b6), e.g. like the x-files
> > > theme song or something? Either way, when I hear those chords I each
> > > think C as the root, so perhaps this is more evidence of learning
> > > being a factor in the harmonic identification of melodic figures.
>
> You gave three inversions of the "C Major" chord. The voicing with the fourth in the bass (G-C-E) was not considered properly rooted in "common practice" and when used was a mild dissonance- as a predominant for example. If you play it in a low register, and you haven't been too thoroughly conditioned to scan for theoretical roots in triadic formation, you can hear that it is indeed less stable and "properly" rooted than the other versions, and so hear the reason for the common-practice "rule".

That's how I'm saying I hear it.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Apr 23, 2011 at 10:21 AM, Petr Parízek
> <petrparizek2000@...> wrote:

> > Let's imagine all of us here exploring meantone then in a similar way we're
> > exploring the other 2D temperaments now. Would we then be keen on making
> > meantone pumps full of things like "E minor, B major, D# minor, A# major"
> > even though these are not characteristic for meantone?
>
> What do you mean? That sounds like romantic harmony to me.
>

But romantic harmony came centuries after meantone first appeared, and meantone may not be the best temperament for plenty of romantic harmony. So starting with romantic harmony in meantone isn't the most sensible thing to do, which is what I think Petr is saying. Surely with meantone we'd start with the "raison d'etre" of the thing (4th root of 5, 81/80...) And so we should first establish "main ingredient" commatic statements with new temperaments- I agree.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Apr 30, 2011 at 4:09 AM, lobawad <lobawad@...> wrote:
> >
> > > --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> > >
> > > > Are you sure you weren't hearing it like Em(b6), e.g. like the x-files
> > > > theme song or something? Either way, when I hear those chords I each
> > > > think C as the root, so perhaps this is more evidence of learning
> > > > being a factor in the harmonic identification of melodic figures.
> >
> > You gave three inversions of the "C Major" chord. The voicing with the fourth in the bass (G-C-E) was not considered properly rooted in "common practice" and when used was a mild dissonance- as a predominant for example. If you play it in a low register, and you haven't been too thoroughly conditioned to scan for theoretical roots in triadic formation, you can hear that it is indeed less stable and "properly" rooted than the other versions, and so hear the reason for the common-practice "rule".
>
> That's how I'm saying I hear it.
>
> -Mike
>

Okay, I must have misread, as I thought you were implying that you heard "C" as root throughout. Which is a "legitimate" response in its own way, too, of course, as long as you're not thinking that this learned response is somehow psychoacoustically "true". An exceptional musician I know has, in my estimation, picked up this mistake from playing jazz, and I suspect that this may be one of things holding him back as far as creativity (absolute conviction that there's always some "natural inevitable correct root" to any chord, which is to a degree like being convinced there's a right answer to Rorschach tests).

On Sat, Apr 30, 2011 at 5:48 AM, lobawad <lobawad@...> wrote:
>
> > What do you mean? That sounds like romantic harmony to me.
> >
>
> But romantic harmony came centuries after meantone first appeared, and meantone may not be the best temperament for plenty of romantic harmony. So starting with romantic harmony in meantone isn't the most sensible thing to do, which is what I think Petr is saying. Surely with meantone we'd start with the "raison d'etre" of the thing (4th root of 5, 81/80...) And so we should first establish "main ingredient" commatic statements with new temperaments- I agree.

When you say it may not be the best temperament for romantic harmony,
do you mean to retune existing romantic harmony, or that the romantic
"sound" may lend itself to lots of other temperaments than just
meantone?

> > > You gave three inversions of the "C Major" chord. The voicing with the fourth in the bass (G-C-E) was not considered properly rooted in "common practice" and when used was a mild dissonance- as a predominant for example. If you play it in a low register, and you haven't been too thoroughly conditioned to scan for theoretical roots in triadic formation, you can hear that it is indeed less stable and "properly" rooted than the other versions, and so hear the reason for the common-practice "rule".
> >
> > That's how I'm saying I hear it.
>
> Okay, I must have misread, as I thought you were implying that you heard "C" as root throughout. Which is a "legitimate" response in its own way, too, of course, as long as you're not thinking that this learned response is somehow psychoacoustically "true". An exceptional musician I know has, in my estimation, picked up this mistake from playing jazz, and I suspect that this may be one of things holding him back as far as creativity (absolute conviction that there's always some "natural inevitable correct root" to any chord, which is to a degree like being convinced there's a right answer to Rorschach tests).

I hear G-C-E specifically as having some kind of mixture of G and C as
the root, with G being the temporary root for the moment, but then
strongly implying C. The cluster chord G-B-C-D-E really fuses them
together. E-G-C is easier to flip into sounding like a kind of E minor
than G-C-E.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sat, Apr 30, 2011 at 5:48 AM, lobawad <lobawad@...> wrote:
> >
> > > What do you mean? That sounds like romantic harmony to me.
> > >
> >
> > But romantic harmony came centuries after meantone first appeared, and meantone may not be the best temperament for plenty of romantic harmony. So starting with romantic harmony in meantone isn't the most sensible thing to do, which is what I think Petr is saying. Surely with meantone we'd start with the "raison d'etre" of the thing (4th root of 5, 81/80...) And so we should first establish "main ingredient" commatic statements with new temperaments- I agree.
>
> When you say it may not be the best temperament for romantic harmony,
> do you mean to retune existing romantic harmony, or that the romantic
> "sound" may lend itself to lots of other temperaments than just
> meantone?

I think many temperaments are implied in romantic music. As I've said here many many times, if you listen to pre-war recordings of late romantic music, you'll hear tons of "microtonality". As far as tuning, I find that 34 equal is very reminiscent of actual pre-war tuning practice.

>
> > > > You gave three inversions of the "C Major" chord. The voicing with the fourth in the bass (G-C-E) was not considered properly rooted in "common practice" and when used was a mild dissonance- as a predominant for example. If you play it in a low register, and you haven't been too thoroughly conditioned to scan for theoretical roots in triadic formation, you can hear that it is indeed less stable and "properly" rooted than the other versions, and so hear the reason for the common-practice "rule".
> > >
> > > That's how I'm saying I hear it.
> >
> > Okay, I must have misread, as I thought you were implying that you heard "C" as root throughout. Which is a "legitimate" response in its own way, too, of course, as long as you're not thinking that this learned response is somehow psychoacoustically "true". An exceptional musician I know has, in my estimation, picked up this mistake from playing jazz, and I suspect that this may be one of things holding him back as far as creativity (absolute conviction that there's always some "natural inevitable correct root" to any chord, which is to a degree like being convinced there's a right answer to Rorschach tests).
>
> I hear G-C-E specifically as having some kind of mixture of G and C as
> the root, with G being the temporary root for the moment, but then
> strongly implying C.

Which fits in nicely with its common practice usage. That's how I hear it too, but that may be learned from exposure to context. Anyway the ambiguity of root is part of its identity, and its place in "functional" harmony.

>The cluster chord G-B-C-D-E really fuses them
> together.

Yes I think that's pretty open as far as how you could root it, therefore good for uprooting as well.

> E-G-C is easier to flip into sounding like a kind of E minor
> than G-C-E.

I agree but I don't know how much conditioning is involved there, as the C could so easily be suspended (resolving by half-step to B, voila e minor)

On Sat, Apr 30, 2011 at 8:33 AM, lobawad <lobawad@...> wrote:
>
> I think many temperaments are implied in romantic music. As I've said here many many times, if you listen to pre-war recordings of late romantic music, you'll hear tons of "microtonality". As far as tuning, I find that 34 equal is very reminiscent of actual pre-war tuning practice.

Can you give an example of a temperament implied? When you say
temperament, do you mean regular temperament? Because now all I hear
when I listen to classical music in general is that there are 81/80
comma pumps everywhere, and that I have no idea how you'd ever retune
any of it to anything else.

> > I hear G-C-E specifically as having some kind of mixture of G and C as
> > the root, with G being the temporary root for the moment, but then
> > strongly implying C.
>
> Which fits in nicely with its common practice usage. That's how I hear it too, but that may be learned from exposure to context.

I guess so, but I don't see why psychoacoustics couldn't also have
something to do with it.

-Mike

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

> But romantic harmony came centuries after meantone first appeared, and meantone may not be the best temperament for plenty of romantic harmony. So starting with romantic harmony in meantone isn't the most sensible thing to do, which is what I think Petr is saying.

Bah. Meantone is the correct tuning fot "At The Hop". It doesn't matter what tuning system they thought they were writing in--they were high school students. What did they know?

Seriously, decisions about what tuning is or is not appropriate is something it makes sense to do on a case-by-case basis, unless you have a fetish for historical accuracy which trumps how good the result sounds. As you say, meantone isn't best choice for "plenty" of the romantic repertoire, but sometimes it works just fine.

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

> Can you give an example of a temperament implied? When you say
> temperament, do you mean regular temperament? Because now all
> I hear when I listen to classical music in general is that there
> are 81/80 comma pumps everywhere, and that I have no idea how
> you'd ever retune any of it to anything else.

Much romantic music relies also on 648/625, which is already
rank 1 in the 5-limit.

-Carl

On Sat, Apr 30, 2011 at 8:13 PM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > Can you give an example of a temperament implied? When you say
> > temperament, do you mean regular temperament? Because now all
> > I hear when I listen to classical music in general is that there
> > are 81/80 comma pumps everywhere, and that I have no idea how
> > you'd ever retune any of it to anything else.
>
> Much romantic music relies also on 648/625, which is already
> rank 1 in the 5-limit.

That's true, and while we're at it this is why I generally prefer to
stick to rank 1 anyways.

-Mike

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
>
> > Can you give an example of a temperament implied? When you say
> > temperament, do you mean regular temperament? Because now all
> > I hear when I listen to classical music in general is that there
> > are 81/80 comma pumps everywhere, and that I have no idea how
> > you'd ever retune any of it to anything else.
>
> Much romantic music relies also on 648/625, which is already
> rank 1 in the 5-limit.
>
> -Carl
>

Traditionally we distinguish Classical (which basically assumes quarter-comma or something like 1/5th-comma meantone, as far as I know) from Romantic. Augmented chords are a signature of Romantic, so if 128/125 isn't tempered out, we'd be looking at nearly quarter-tone shifts as early as Liszt. Of course, as Gene said, there isn't necessarily "one" "right" way. It's possible that a hyper-Romantic performance would temper on paper but not in practice. The resemblance to tuning in 34-edo I perceive in old orchestral recordings might be caused by NOT tempering out 128/125, for example- certainly the pitch does move about. I imagine that what was going on (in practice, whatever was written on paper) was a state of numerous conditional, contextual temperaments. The heavy vibrato and portamento of the time must have facilitated this as well.

On Sun, May 1, 2011 at 12:52 AM, lobawad <lobawad@...> wrote:
>
> Traditionally we distinguish Classical (which basically assumes quarter-comma or something like 1/5th-comma meantone, as far as I know) from Romantic. Augmented chords are a signature of Romantic, so if 128/125 isn't tempered out, we'd be looking at nearly quarter-tone shifts as early as Liszt. Of course, as Gene said, there isn't necessarily "one" "right" way. It's possible that a hyper-Romantic performance would temper on paper but not in practice. The resemblance to tuning in 34-edo I perceive in old orchestral recordings might be caused by NOT tempering out 128/125, for example- certainly the pitch does move about. I imagine that what was going on (in practice, whatever was written on paper) was a state of numerous conditional, contextual temperaments. The heavy vibrato and portamento of the time must have facilitated this as well.

Very well put. I'm not sure why I've argued with you over this so much
in the past if this is how you see it.

But what about comma pumps? 81/80 comma pumps are so fundamental to
almost everything I've ever heard in all of Western music that I can't
imagine Romantic music has suddenly stopped using them. If a Romantic
composition makes use of chord progressions that don't make sense
unless 81/80 vanishes, then it doesn't matter if they adaptively
retune the constituent chords to be closer to JI, because the whole
thing relies on an underlying meantone logic in terms of how
individual root movements take you away from the tonic and return you
to it.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, May 1, 2011 at 12:52 AM, lobawad <lobawad@...> wrote:
> >
> > Traditionally we distinguish Classical (which basically assumes quarter-comma or something like 1/5th-comma meantone, as far as I know) from Romantic. Augmented chords are a signature of Romantic, so if 128/125 isn't tempered out, we'd be looking at nearly quarter-tone shifts as early as Liszt. Of course, as Gene said, there isn't necessarily "one" "right" way. It's possible that a hyper-Romantic performance would temper on paper but not in practice. The resemblance to tuning in 34-edo I perceive in old orchestral recordings might be caused by NOT tempering out 128/125, for example- certainly the pitch does move about. I imagine that what was going on (in practice, whatever was written on paper) was a state of numerous conditional, contextual temperaments. The heavy vibrato and portamento of the time must have facilitated this as well.
>
> Very well put. I'm not sure why I've argued with you over this so much
> in the past if this is how you see it.
>
> But what about comma pumps? 81/80 comma pumps are so fundamental to
> almost everything I've ever heard in all of Western music that I can't
> imagine Romantic music has suddenly stopped using them. If a Romantic
> composition makes use of chord progressions that don't make sense
> unless 81/80 vanishes, then it doesn't matter if they adaptively
> retune the constituent chords to be closer to JI, because the whole
> thing relies on an underlying meantone logic in terms of how
> individual root movements take you away from the tonic and return you
> to it.
>
> -Mike
>

Yes this is something I wonder about too- on paper, it's all more or less 1/4-comma meantone for the simple reason that our notation is.

But, for example, in more than one of the many recorded versions of Scheherezade I've owned since childhood, 81/80 (or another comma of about the same size, or multiple commas adding up...???? hmmm...) is NOT tempered out. You can hear it plain as day in the highly exposed solo violin part- the pitch (an E iirc) shifts, but logically. I noticed this long before I had more than the vaguest idea of what a "comma" is.

But I must say that my opinion that Romantic music is tempering out other commas is coming from the fact that I work mostly (by far) with tunings that don't temper out 81/80, and I could swear that there is more than a passing resemblance to the sound of olde-schoole performances of Romantic music in the harmonic feelings. (up to my 20's probably half my listening was to old Melodiya records, unfortunately all stolen. I'm not to good with material possessions. :-) )

This whole area bears detailed study of course, I pine for the day I'll have the time to do so. Just reading a little study of the Faust intro (Liszt) that doesn't assume 12-tET would be balsam for the soul.

On Sun, May 1, 2011 at 1:12 AM, lobawad <lobawad@...> wrote:
>
> Yes this is something I wonder about too- on paper, it's all more or less 1/4-comma meantone for the simple reason that our notation is.
>
> But, for example, in more than one of the many recorded versions of Scheherezade I've owned since childhood, 81/80 (or another comma of about the same size, or multiple commas adding up...???? hmmm...) is NOT tempered out. You can hear it plain as day in the highly exposed solo violin part- the pitch (an E iirc) shifts, but logically. I noticed this long before I had more than the vaguest idea of what a "comma" is.

There are plenty of times I notice this well, but in another sense it
is tempered out, because I guarantee you whatever version of
Shcerherhaerhzeraizerde (I don't know this guy really) that you're
listening to is utilizing some kind of pump around 81/80. And in that
sense 81/80 certainly is tempered, because you end up back at the
pitch you start with.

> But I must say that my opinion that Romantic music is tempering out other commas is coming from the fact that I work mostly (by far) with tunings that don't temper out 81/80, and I could swear that there is more than a passing resemblance to the sound of olde-schoole performances of Romantic music in the harmonic feelings. (up to my 20's probably half my listening was to old Melodiya records, unfortunately all stolen. I'm not to good with material possessions. :-) )

I imagine that part of that sound is the improved intonation, which
may well be closer to 34-ET than anything else, but as you've noticed
comma pumps create their own sound - that of the temperament turning
in on itself. You don't even notice it until you get accustomed to JI
- the fact that we can build these long and drawn out chord
progressions like

Gmaj -> Cmaj -> A7 -> Dmaj -> B7 -> Em

How unbearably and soul-crushingly tense! The sheer humanity of it is
enough to reduce anyone to tears! But in meantone, you can go

Em -> Am/C -> Dmaj -> Gmaj

Ah, how refreshing! Now you're back to the root. Actually, it kind of
cheapens it a bit, doesn't it?

-Mike

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

> Augmented chords are a signature of Romantic,

Augmented triads? I'm trying to think of examples... -Carl

Hi again - two days later. Back home.

Mike wrote:

> Alright. Well, from my perspective, I've been stuck in generator-land
> since I discovered what a generator is, so this is a liberating new
> paradigm. Let's see if we meet in the middle.

Okay.

> I think Fm -> Emaj -> Cmaj does the trick. Fm is the submediant to
> Abmaj, so the whole thing sounds like IV V I, kind of.

Possibly.

> What about linear splitting? For example, take 4:5:6 - you split it
> and get 8:9:10:11:12. Equating 9/8 -> 10/9 and 12/11 -> 10/9 gets you
> something like mohajira. Equating 12/11 and 11/10, and 11/10 and 10/9,
> gets you something like porcupine.

I've tried similar things as well.
In most cases, I like to use linear splitting mainly for the starting chord and do the rest in an "exponential" way. For example, if I decide to use a 2/1 equivalence interval, the most obvious idea is, of course, to split it into 3/2 and 4/3. Then if I decide, for instance, that I want to make a 2D temperament, then I want to add another interval whose size (logarithmically, like in cents) is not a rational fraction of 3/2 or 5/4. Again, the most obvious solution is to either split the 3/2 into 5/4 and 6/5 or to split the 4/3 into 7/6 and 8/7. So now we have 3 intervals which add up to an octave. From then on, I usually split the larger ones by the smaller ones in a different way -- i.e. dividing their factors. So 4/3 can be split either into 5/4 and 16/15 or into 6/5 and 10/9. And so on, until the smallest of the three interval is so small that you feel like tempering it out. This procedure also gives you the ability to find an amount of tempering for making octaves slightly mistuned while some other two particular JI intervals and their combinations are pure. For example, if you get to steps of "10/9, 16/15, 81/80", this procedure can tell you that widening every major or minor second by 4/9-comma makes a version of meantone which has both 5/4 and 6/5 only 1/9-comma narrower and whose octave is 1/9-comma wider than pure. The downside of this particular tuning is that fourths are 1/3-comma wider, which may sound too excited for a "resting" interval in some situations.

> I'm serious - I can't imagine anyone would
> have ever discovered Blackwood before, but it's one of my favorite
> tunings ever. I guess some culture could have started with the
> pentatonic scale and evened it out, and discovered Blackwood that way,
> but I dunno.

I've discovered blackwood by the process I've just described. When you get to the stage where you have steps of "16/15, 135/128, 256/243", then the limma is the smallest interval there (albeit still pretty large) and if you temper it out, blackwood comes out.

> I assume both of these are in reference to "Among Other Things 2." In
> my mind, the part with the solo harp sounds like you're sticking
> mainly with "porcupine diatonic harmony," generator-based thinking,
> etc. The second part, where the whole band comes in, sounds more like
> porcupinized common practice harmony, which I think is an awesome
> sound. There were a few times I heard some screwy 81/80 shifts, but on
> the whole it seems to work. So I hear it the opposite as what you've
> described.

Because I was inexact in labeling the parts. I wasn't talking about the diatonic intro on the solo harp but rather about what follows.

Lastly, thanks for those music examples -- Reich's Desert Music, together with his Sextet, belonged to my unbeatable favs even back in 2000 or perhaps 2001 when I first heard it.
Royksopp was completely new to me.
If you're interested, I can, for an exception, let you hear music in a genre which I almost never compose. :-D
In July 1998, I made a piece called "One Two Three" based on a four-chord progression. Two years later, I shifted the thing down by a "supermajor" second, completely changed the orchestration, but I never recorded any vocals to it. Another change was that the older version was all in 12-EDO while the newer version was alternating between 12-EDO and JI and that it had a "jungle-style" loop allover it.
Anyway, as you were talking about non-standard modality, I'll do the exception this time and send a link to the unfinished recording which is full of these things:
http://dl.dropbox.com/u/8497979/One_two_three_unfinished_background_2000.mp3

Petr

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, May 1, 2011 at 1:12 AM, lobawad <lobawad@...> wrote:
> >
> > Yes this is something I wonder about too- on paper, it's all more or less 1/4-comma meantone for the simple reason that our notation is.
> >
> > But, for example, in more than one of the many recorded versions of Scheherezade I've owned since childhood, 81/80 (or another comma of about the same size, or multiple commas adding up...???? hmmm...) is NOT tempered out. You can hear it plain as day in the highly exposed solo violin part- the pitch (an E iirc) shifts, but logically. I noticed this long before I had more than the vaguest idea of what a "comma" is.
>
> There are plenty of times I notice this well, but in another sense it
> is tempered out, because I guarantee you whatever version of
> Shcerherhaerhzeraizerde (I don't know this guy really) that you're
> listening to is utilizing some kind of pump around 81/80. And in that
> sense 81/80 certainly is tempered, because you end up back at the
> pitch you start with.

You surely do know Korsakov's Sheherezade, it's been used and ripped off in countless movies, probably one of the most influential pieces in movie music ever. Korsakov is grossly underrated, probably because he's so wildly popular, like, you can play him at Mai Tai parties- noone even mentions that he was Stravinsky's teacher (I didn't know til I heard Rite of Spring and said, that's obviously turbocharged Korsakov, and looked up the connection).

>
> > But I must say that my opinion that Romantic music is tempering out other commas is coming from the fact that I work mostly (by far) with tunings that don't temper out 81/80, and I could swear that there is more than a passing resemblance to the sound of olde-schoole performances of Romantic music in the harmonic feelings. (up to my 20's probably half my listening was to old Melodiya records, unfortunately all stolen. I'm not to good with material possessions. :-) )
>
> I imagine that part of that sound is the improved intonation, which
> may well be closer to 34-ET than anything else, but as you've noticed
> comma pumps create their own sound - that of the temperament turning
> in on itself.

Well we all noticed that a good long time ago of course. And comma shifts, which are a trip if you don't know what's causing them- "how come we end up a quartertone lower every time, and the better we sing it, the more obvious the problem?"

>You don't even notice it until you get accustomed to JI
> - the fact that we can build these long and drawn out chord
> progressions like
>
> Gmaj -> Cmaj -> A7 -> Dmaj -> B7 -> Em
>
> How unbearably and soul-crushingly tense! The sheer humanity of it is
> enough to reduce anyone to tears! But in meantone, you can go
>
> Em -> Am/C -> Dmaj -> Gmaj
>
> Ah, how refreshing! Now you're back to the root. Actually, it kind of
> cheapens it a bit, doesn't it?
>
> -Mike
>

Well even without knowing the why's, musicians noticed these things even a nominally 12-tET world. In my case it was usually with the sinking feeling that the director or teacher actually had no idea what the hell they were doing, but I didn't have the knowledge to articulate why. It was a relief to learn that I'd been clearly hearing the conflict between good intonation and 12-tET all along.

Anyway as I mentioned before, there's no reason with instruments of flexible pitch not to have more than one kind of temperament, flexing and even paradoxically coexisting.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "lobawad" <lobawad@> wrote:
>
> > Augmented chords are a signature of Romantic,
>
> Augmented triads? I'm trying to think of examples... -Carl
>

You're kidding, right?

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

> Korsakov is grossly underrated,

R-K definitely deserves a shout-out.

> > Augmented chords are a signature of Romantic,
> >
> > Augmented triads? I'm trying to think of examples...
>
> You're kidding, right?

No... -Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "lobawad" <lobawad@> wrote:
>
> > Korsakov is grossly underrated,
>
> R-K definitely deserves a shout-out.
>
> > > Augmented chords are a signature of Romantic,
> > >
> > > Augmented triads? I'm trying to think of examples...
> >
> > You're kidding, right?
>
> No... -Carl
>

http://www.youtube.com/watch?v=T4L8zV5uQAE

"Textbook" example (I think it actually was, in one my textbooks). After this there is a great deal of the augmented in Romantic music, not referring to the functional aug. 6 here. There is some theory school that in my opinion puts too much weight on the importance of the augmented triad in Romantic music (I don't think root movement by thirds automatically means that the chords implied are the most important structural element, and IMO there's too much post-mortem speculation about chords and scales when reality is probably more about counterpoint and color based on "bitchen'" ), but whatever, it's just a historical fact that there's buttloads (the official term) of augmented chords in Romantic music. Liszt, Wagner, Richard Strauss, etc., ad laudanum.

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

> http://www.youtube.com/watch?v=T4L8zV5uQAE
>
> "Textbook" example (I think it actually was, in one
> my textbooks). After this there is a great deal of the
> augmented in Romantic music, not referring to the
> functional aug. 6 here.

Thanks. Hm
http://en.wikipedia.org/wiki/Augmented_triad#In_tonal_music

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "lobawad" <lobawad@> wrote:
>
> > http://www.youtube.com/watch?v=T4L8zV5uQAE
> >
> > "Textbook" example (I think it actually was, in one
> > my textbooks). After this there is a great deal of the
> > augmented in Romantic music, not referring to the
> > functional aug. 6 here.
>
> Thanks. Hm
> http://en.wikipedia.org/wiki/Augmented_triad#In_tonal_music
>
> -Carl
>

Dug my ragged old Kostka and Payne out of storage recently so let me see- the Liszt examples illustrate omnibus and chromatic mediants. But Faust is the "textbook" example, and I'm not suprised to see it on Wikipedia, as the mainstream textbook music theory stuff seems to be mostly copied from... mainstream textbooks. And I've found the mainstream music theory stuff on Wikipedia very good, with only the occaisional bit "original research", which I delete once in a while, when it's painfully obvious and bad.

Hmm... eh, see, there's that idea I referred to: "organizing many peices by descending major thirds". I'm leary of this idea- it's related to the ideas of the symmetry buffs. You know, Scriabin and Bartok based everything on symmetries, Perl, Lendavi, others. Really I wouldn't be suprised if the story turned out to be that the symmetry guys retrofitted this (symmetrical! at least in 12-tET...) concept to the immediate forbearers of Bartok and Scriabin, in order to establish credibility to the idea of symmetry as a strong organizing principle.

On Sun, May 1, 2011 at 2:56 PM, Petr Parízek <petrparizek2000@...> wrote:
> > I assume both of these are in reference to "Among Other Things 2." In
> > my mind, the part with the solo harp sounds like you're sticking
> > mainly with "porcupine diatonic harmony," generator-based thinking,
> > etc. The second part, where the whole band comes in, sounds more like
> > porcupinized common practice harmony, which I think is an awesome
> > sound. There were a few times I heard some screwy 81/80 shifts, but on
> > the whole it seems to work. So I hear it the opposite as what you've
> > described.
>
> Because I was inexact in labeling the parts. I wasn't talking about the
> diatonic intro on the solo harp but rather about what follows.

OK, I see.

> Lastly, thanks for those music examples -- Reich's Desert Music, together
> with his Sextet, belonged to my unbeatable favs even back in 2000 or perhaps
> 2001 when I first heard it.
> Royksopp was completely new to me.

Royksopp is awesome, I just discovered them myself...

> If you're interested, I can, for an exception, let you hear music in a genre
> which I almost never compose. :-D
> In July 1998, I made a piece called "One Two Three" based on a four-chord
> progression. Two years later, I shifted the thing down by a "supermajor"
> second, completely changed the orchestration, but I never recorded any
> vocals to it. Another change was that the older version was all in 12-EDO
> while the newer version was alternating between 12-EDO and JI and that it
> had a "jungle-style" loop allover it.
> Anyway, as you were talking about non-standard modality, I'll do the
> exception this time and send a link to the unfinished recording which is
> full of these things:
> http://dl.dropbox.com/u/8497979/One_two_three_unfinished_background_2000.mp3

Nice! Does this have 13-limit JI in it? I thought I was hearing
8:12:13 at one point. Very cool, you should do some comma pump based
stuff in this style as well!

-Mike

Mike wrote:

> Nice! Does this have 13-limit JI in it? I thought I was hearing
> 8:12:13 at one point.

From the "higher E" or "lower F" or whatever you want to call that strange pitch, the scale is the same as "parizek.scl" in Manuel's archive.

> Very cool, you should do some comma pump based
> stuff in this style as well!

Wow, then I'll probably go 3D-tempered or something ... You see, plain triads obviously don't work here. :-D

Petr

I wrote:

> From the "higher E" or "lower F" or whatever you want to call that strange > pitch, the scale is the same as "parizek.scl" in Manuel's archive.

Oops, my "Linear Level Tuning 1997" seems to have been stripped out in the newer version.
Fortunately, exactly the same scale is there under the name of "carlos_harm.scl".
My goodness, I had absolutely no idea that Carlos got to that as well.

Petr