magic[10]

Has anyone noticed there's a 8:12:15 or 3:4:5 chord
in every mode of magic[10] with the scale pattern 1-6-9?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Has anyone noticed there's a 8:12:15 or 3:4:5 chord
> in every mode of magic[10] with the scale pattern 1-6-9?

Except for that one 6:9:10 chord. Right?

Keenan

On Apr 11, 2012, at 1:31 PM, Carl Lumma <carl@...> wrote:

Has anyone noticed there's a 8:12:15 or 3:4:5 chord
in every mode of magic[10] with the scale pattern 1-6-9?

-Carl

I'm not at my comp to check, but that can't be right - the upper dyad in
both cases is 5/4, corresponding to 3\magic[10], but this interval class
must come in two sizes. In general, all "triad classes" in an MOS come in
three specific types.

-Mike

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > Has anyone noticed there's a 8:12:15 or 3:4:5 chord
> > in every mode of magic[10] with the scale pattern 1-6-9?
>
> Except for that one 6:9:10 chord. Right?
>
> Keenan

Quite right. Does the mode where that chord occurs on 2-7-10
sound the most stable to you? It does to me.

-Carl

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

> In general, all "triad classes" in an MOS come in
> three specific types.

The upper and lower intervals will each have two sizes,
for a total of three possible combinations (ss, Ls, and LL).
But can you prove that the cycles will never align such
that one of these combinations does not occur?

L s s
s L s

-Carl

On Thu, Apr 12, 2012 at 2:22 PM, Carl Lumma <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > In general, all "triad classes" in an MOS come in
> > three specific types.
>
> The upper and lower intervals will each have two sizes,
> for a total of three possible combinations (ss, Ls, and LL).
> But can you prove that the cycles will never align such
> that one of these combinations does not occur?
>
> L s s
> s L s

I don't understand the question. What interval classes exactly do the
triples of intervals above represent? In L s s, what interval classes
exactly are each L and s?

If it helps, there's exception to the rule I listed - you have to make
sure that none of the intervals contained in the triad class are equal
to the period, which only comes in 1 size.

Also, if it helps, all three of your combinations can't appear in the
scale. ss, Ls, and LL imply three different sizes for the outer dyad,
but the outer dyad must only come in two sizes like everything else.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > The upper and lower intervals will each have two sizes,
> > for a total of three possible combinations (ss, Ls, and LL).
> > But can you prove that the cycles will never align such
> > that one of these combinations does not occur?
> >
> > L s s
> > s L s
>
> I don't understand the question. What interval classes exactly do the
> triples of intervals above represent? In L s s, what interval classes
> exactly are each L and s?

It doesn't matter.

> Also, if it helps, all three of your combinations can't appear in the
> scale. ss, Ls, and LL imply three different sizes for the outer dyad,
> but the outer dyad must only come in two sizes like everything else.

Quite right, it's order that matters (sL and Ls). And so to prove the triad classes thing, we just need to show that both orders will appear across the modes.

-Carl (greetings from a meeting)

I wrote:
> Quite right, it's order that matters (sL and Ls). And so to prove the triad classes
> thing, we just need to show that both orders will appear across the modes.

Howabout this: the two pairings can't be ss and LL, because then the two
sizes of the outer interval would vary by more than a comma.
And therefore, the two chains for the inner intervals must not line up, and
since they don't line up, both orders are produced...

-Carl

On Apr 12, 2012, at 6:06 PM, Carl Lumma <carl@...> wrote:

I wrote:
> Quite right, it's order that matters (sL and Ls). And so to prove the
triad classes
> thing, we just need to show that both orders will appear across the modes.

Howabout this: the two pairings can't be ss and LL, because then the two
sizes of the outer interval would vary by more than a comma.
And therefore, the two chains for the inner intervals must not line up, and
since they don't line up, both orders are produced...

Right. If it's LL and ss, then the outer dyad would differ by 2c, which is
impossible. So it has to be only one of those, Ls, and sL.

It also couldn't be just Ls and LL or something, because that would mean
the lower dyad would only come in one size, which is "L" - if this happens,
you've screwed up and set your lower dyad to the period (which is what
happens if you look for 4:5:6 triad class patterns in, say, augmented[9] or
something).

So that's another useful property that MOS's have. In addition to
generalizing there being "major" and "minor" interval classes, it also
generalizes there being "major," "minor," and something like dim or aug for
triad classes.

-Mike

Very good. To summarize: The variety of any triad class in a MOS is 3.
Proof:

* Both inner intervals must have variety 2, making legal pairings
1. ss, sL, Ls, LL
2. ss, sL, Ls
3. ss, sL, LL
4. ss, Ls, LL
5. sL, Ls, LL
6. ss, LL
7. sL, Ls

* Pairings 1, 3, 4 & 6 are ruled out because the varieties of the outer
interval would differ by more than a comma.

* Pairing 7 is ruled out because the variety of the outer interval
would be 1.

* That leaves pairings 2 & 5 (ss, sL, Ls and sL, Ls, LL), both of which
give triadic variety 3.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> It also couldn't be just Ls and LL or something, because that would mean
> the lower dyad would only come in one size, which is "L" - if this happens,
> you've screwed up and set your lower dyad to the period (which is what
> happens if you look for 4:5:6 triad class patterns in, say, augmented[9] or
> something).
>
> So that's another useful property that MOS's have. In addition to
> generalizing there being "major" and "minor" interval classes, it also
> generalizes there being "major," "minor," and something like dim or aug for
> triad classes.
>
> -Mike
>