Eikosany Tetrads

Looking at Kraig's 'tree toad' article there's a section where alternate
complete harmonic and subharmonic tetrads are formed. The harmonic ones
I can see at at glance, eg : -

3-5-11, 1-3-5, 3-5-7, 3-5-9 - by factoring out 3 and 5 you get 1 - 7 -
9 - 11.

What I can't see at a glance is how the following gives a subharmonic
tetrad : -

1-5-7, 5-7-9, 1-7-9, 1-5-9.

Any clues as to how to spot the subharmonic tetrad from the CPS or do I
have to go to the ratios and work it out from there?

While I'm on I'd like to ask anyone who has worked with the Eikosany
(Kraig mainly) if they find it easier to work at it thinking in terms of
CPS, ie 1-3-5, etc. or by using note names ie, A, A+ etc. I find the
former easier as with familiarity I get faster at seeing the common
factors, but I'm wondering if it's a better investment of time to learn
structures as named notes.

Many thanks in anticipation

Kind Regards
a.m.

Alison Monteith <alison.monteith3@w...> wrote...
>Looking at Kraig's 'tree toad' article there's a section where
>alternate complete harmonic and subharmonic tetrads are formed.

We don't usually call them complete, because they each lack
two of the identities used to define the tonespace in which
the CPS is embedded (usually for the eikosany, it's 11-limit
JI [1 3 5 7 9 11], and we only have tetrads).

>What I can't see at a glance is how the following gives a
>subharmonic tetrad
>
> 1-5-7, 5-7-9, 1-7-9, 1-5-9.

Here's the long-division way, applicable to both harmonic
and subharmonic tetrads...

- Remove all factors common to each point.
(there aren't any)
- Multiply the remaining terms together for each point
and sort to express the tetrad as a harmonic series segment.
(35 45 63 315)
- If the numbers are too big it may be more convenient
to express each point as a ratio.
(1/1 9/7 9/5 9/8)

For subharmonic tetrads, you can just pick one of the points,
call it 1/1, and find the ratios between it and the other
points in your head...

- The ratios for the other points all have the same numerator.
It's the one factor they all have in common.

- The denominator for a given other point is the factor in
your 1/1 point that doesn't appear in that other point.

-Carl

"Carl Lumma " wrote:

> Alison Monteith <alison.monteith3@w...> wrote...
> >Looking at Kraig's 'tree toad' article there's a section where
> >alternate complete harmonic and subharmonic tetrads are formed.
>
> We don't usually call them complete, because they each lack
> two of the identities used to define the tonespace in which
> the CPS is embedded (usually for the eikosany, it's 11-limit
> JI [1 3 5 7 9 11], and we only have tetrads).

Thanks - I didn't know that. 'Complete' is the term used in the article.

> >What I can't see at a glance is how the following gives a
> >subharmonic tetrad
> >
> > 1-5-7, 5-7-9, 1-7-9, 1-5-9.
>
> Here's the long-division way, applicable to both harmonic
> and subharmonic tetrads...
>
> - Remove all factors common to each point.
> (there aren't any)
> - Multiply the remaining terms together for each point
> and sort to express the tetrad as a harmonic series segment.
> (35 45 63 315)
> - If the numbers are too big it may be more convenient
> to express each point as a ratio.
> (1/1 9/7 9/5 9/8)
>
> For subharmonic tetrads, you can just pick one of the points,
> call it 1/1, and find the ratios between it and the other
> points in your head...
>
> - The ratios for the other points all have the same numerator.
> It's the one factor they all have in common.
>
> - The denominator for a given other point is the factor in
> your 1/1 point that doesn't appear in that other point.
>
> -Carl

Thanks again Carl but I wondered if it was possible to tell if the tetrad is subharmonic just from
looking at the four CPS numbers, ie 1-3-5 etc on the fly as it were, without any further
calculation. Probably not it seems, until I familiarise myself more with my Eikosany marimba.

Kind regards
a.m.

>Thanks again Carl but I wondered if it was possible to tell
>if the tetrad is subharmonic just from looking at the four CPS
>numbers, ie 1-3-5 etc on the fly as it were, without any further
>calculation. Probably not it seems, until I familiarise myself
>more with my Eikosany marimba.

Well, you discovered a way. The points of a harmonic tetrad
will share two factors in common, the points of a subharmonic
tetrad won't share any.

If you want faster than that, you can look at the treetoad
while playing.

-Carl

--- In tuning@yahoogroups.com, Alison Monteith
<alison.monteith3@w...> wrote:
> Looking at Kraig's 'tree toad' article there's a section where
alternate
> complete harmonic and subharmonic tetrads are formed. The harmonic
ones
> I can see at at glance, eg : -
>
> 3-5-11, 1-3-5, 3-5-7, 3-5-9 - by factoring out 3 and 5 you get 1 -
7 -
> 9 - 11.
>
> What I can't see at a glance is how the following gives a
subharmonic
> tetrad : -
>
> 1-5-7, 5-7-9, 1-7-9, 1-5-9.
>
> Any clues as to how to spot the subharmonic tetrad from the CPS

great question alison! yes, indeed there is an easy way! notice that
there are a total of only four different factors here: 1, 5, 7, and
9. this is something you should always be on the lookout for. now to
get the four pitches in question, all you have to do is divide
1*5*7*9 by each of the four factors:

1*5*7*9/9 = 1*5*7
1*5*7*9/7 = 1*5*9
1*5*7*9/5 = 1*7*9
1*5*7*9/1 = 5*7*9

so the chord is a 1/9:1/7:1/5:1/1 chord, i.e., subharmonic!

"Carl Lumma " wrote:

> >Thanks again Carl but I wondered if it was possible to tell
> >if the tetrad is subharmonic just from looking at the four CPS
> >numbers, ie 1-3-5 etc on the fly as it were, without any further
> >calculation. Probably not it seems, until I familiarise myself
> >more with my Eikosany marimba.
>
> Well, you discovered a way. The points of a harmonic tetrad
> will share two factors in common, the points of a subharmonic
> tetrad won't share any.
>
> If you want faster than that, you can look at the treetoad
> while playing.
>
> -Carl

Thanks again for your reply Carl. I'll work away till I hit another brick wall....

Regards
a.m.

"wallyesterpaulrus " wrote:

> --- In tuning@yahoogroups.com, Alison Monteith
> <alison.monteith3@w...> wrote:
> > Looking at Kraig's 'tree toad' article there's a section where
> alternate
> > complete harmonic and subharmonic tetrads are formed. The harmonic
> ones
> > I can see at at glance, eg : -
> >
> > 3-5-11, 1-3-5, 3-5-7, 3-5-9 - by factoring out 3 and 5 you get 1 -
> 7 -
> > 9 - 11.
> >
> > What I can't see at a glance is how the following gives a
> subharmonic
> > tetrad : -
> >
> > 1-5-7, 5-7-9, 1-7-9, 1-5-9.
> >
> > Any clues as to how to spot the subharmonic tetrad from the CPS
>
> great question alison! yes, indeed there is an easy way! notice that
> there are a total of only four different factors here: 1, 5, 7, and
> 9. this is something you should always be on the lookout for. now to
> get the four pitches in question, all you have to do is divide
> 1*5*7*9 by each of the four factors:
>
> 1*5*7*9/9 = 1*5*7
> 1*5*7*9/7 = 1*5*9
> 1*5*7*9/5 = 1*7*9
> 1*5*7*9/1 = 5*7*9
>
> so the chord is a 1/9:1/7:1/5:1/1 chord, i.e., subharmonic!

Thank you Paul. Yet another way of looking at the problem and as always a most elegant solution. I
tried this with the others and can figure them out on sight which is what I need.

Kind Regards
a.m.