8-note Blackjack chords/scales with max consonance

Joseph,

Paul has given an excellent explanation of how to find such things using the slide rule. But I've gone on from there and used computer assistance to find the best (or at least, close to the best). It's possible I have missed some, with my "greedy" heuristic and crude consonance measure.

Try these 8 note chords with the suggested voicings. You will see that they each occur in two positions in Blackjack. Let me know how they sound, and how you would rank them for consonance. This is all theory so far.

C> D[Eb< Ev F Gb^ G> A[Bb< Bv C Db^ D> E[ F<F#v G Ab^ A> B[ C<
M14 s18 m13
M3 sm7 M7 rt s11 A11 P5 m6 M9
+--------------+--+--+-----------+--+--+-----------------+
+--------------+--+--+--------------+--+--+--------------+
+-----------------+--+--+-----------+--+--+--------------+
rt A11 P5 m6 SM6 M9 m10 SM3 Wm7
A18 m13 SM13
C> D[Eb< Ev F Gb^ G> A[Bb< Bv C Db^ D> E[ F<F#v G Ab^ A> B[ C<

Have a look at them on Paul's or Graham's lattice. They rely on lattice-wraparound due to the tempering. They will not work the same in any rational tuning, i.e. they are "magic". The middle one is 15-limit saturated and the outer two are 21-limit saturated.

Notice that you'd have a lot more naturals in these chords using the key I suggested for Alison:

F#v G Ab^ A> B[ C<C#v D Eb^ E> F] G<G#v A Bb^ B> C] D<D#v E F^

Or if you want to have a C natural, use this one:

Bv C Db^ D> E[ F<F#v G Ab^ A> B[ C<C#v D Eb^ E> F] G<G#v A Bb^
M14 s18 m13
M3 sm7 M7 rt s11 A11 P5 m6 M9
+--------------+--+--+-----------+--+--+-----------------+
+--------------+--+--+--------------+--+--+--------------+
+-----------------+--+--+-----------+--+--+--------------+
rt A11 P5 m6 SM6 M9 m10 SM3 Wm7
A18 m13 SM13
Bv C Db^ D> E[ F<F#v G Ab^ A> B[ C<C#v D Eb^ E> F] G<G#v A Bb^

Alison,

Notice that these 8-note chords can also be treated as 8-note _scales_. I suspect the middle one is the best, for both purposes.

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

I deleted my original reply to this. I was totally misreading Dave
K.'s diagrams. My apologies to Dave and to those getting this list in
individual e-mails.

Dave K., are you back?

Now for the "real" response:

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> Joseph,
>
> Paul has given an excellent explanation of how to find such things
using the slide rule. But I've gone on from there and used computer
assistance to find the best (or at least, close to the best). It's
possible I have missed some, with my "greedy" heuristic and crude
consonance measure.
>
> Try these 8 note chords with the suggested voicings. You will see
that they each occur in two positions in Blackjack. Let me know how
they sound, and how you would rank them for consonance. This is all
theory so far.
>
> C> D[Eb< Ev F Gb^ G> A[Bb< Bv C Db^ D> E[ F<F#v G Ab^ A> B[ C<
> M14 s18 m13
> M3 sm7 M7 rt s11 A11 P5 m6 M9
> +--------------+--+--+-----------+--+--+-----------------+
> +--------------+--+--+--------------+--+--+--------------+
> +-----------------+--+--+-----------+--+--+--------------+
> rt A11 P5 m6 SM6 M9 m10 SM3 Wm7
> A18 m13 SM13
> C> D[Eb< Ev F Gb^ G> A[Bb< Bv C Db^ D> E[ F<F#v G Ab^ A> B[ C<

Cool! Here is a voicing of the first chord I think Joseph might want
to try:

A[ E[ C> Gb^ B[ D> G> Db^ or its transposition up a secor,
Bb< F< D[ G> C< E[ A[ D>

The other chords would be trickier to voice well . . .

Dave, did you consider 11-limit intervals? If you did this search
using an octave-equivalent consonance measure, perhaps you should try
an octave-specific one instead (don't use integer limit -- use
product or harmonic entropy).

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:

/tuning/topicId_30045.html#30045

> Joseph,
>
> Paul has given an excellent explanation of how to find such things
using the slide rule. But I've gone on from there and used computer
assistance to find the best (or at least, close to the best). It's
possible I have missed some, with my "greedy" heuristic and crude
consonance measure.
>

Thanks so much, Dave, for this valuable post... I'll let you know
what I think of these.

I sure have plenty to work with now!

best,

Joseph

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_30045.html#30048

> Now for the "real" response:
>
> --- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> > Joseph,
> >
> > Paul has given an excellent explanation of how to find such
things
> using the slide rule. But I've gone on from there and used computer
> assistance to find the best (or at least, close to the best). It's
> possible I have missed some, with my "greedy" heuristic and crude
> consonance measure.
> >
> > Try these 8 note chords with the suggested voicings. You will see
> that they each occur in two positions in Blackjack. Let me know how
> they sound, and how you would rank them for consonance. This is all
> theory so far.
> >
> > C> D[Eb< Ev F Gb^ G> A[Bb< Bv C Db^ D> E[ F<F#v G Ab^ A> B[ C<
> > M14 s18 m13
> > M3 sm7 M7 rt s11 A11 P5 m6 M9
> > +--------------+--+--+-----------+--+--+-----------------+
> > +--------------+--+--+--------------+--+--+--------------+
> > +-----------------+--+--+-----------+--+--+--------------+
> > rt A11 P5 m6 SM6 M9 m10 SM3 Wm7
> > A18 m13 SM13
> > C> D[Eb< Ev F Gb^ G> A[Bb< Bv C Db^ D> E[ F<F#v G Ab^ A> B[ C<
>
> Cool! Here is a voicing of the first chord I think Joseph might
want
> to try:
>
> A[ E[ C> Gb^ B[ D> G> Db^ or its transposition up a secor,
> Bb< F< D[ G> C< E[ A[ D>
>

Thanks, Paul, for "spelling this out" for me. That was a big "time
saver."

What a chord! That's a great sound. I didn't even want to "leave
anything out..." :)

Joseph

--- In tuning@y..., jpehrson@r... wrote:

> > >
> > > C> D[Eb< Ev F Gb^ G> A[Bb< Bv C Db^ D> E[ F<F#v G Ab^ A> B[
C<
> > > M14 s18 m13
> > > M3 sm7 M7 rt s11 A11 P5 m6 M9
> > > +--------------+--+--+-----------+--+--+-----------------+
> > > +--------------+--+--+--------------+--+--+--------------+
> > > +-----------------+--+--+-----------+--+--+--------------+
> > > rt A11 P5 m6 SM6 M9 m10 SM3 Wm7
> > > A18 m13 SM13
> > > C> D[Eb< Ev F Gb^ G> A[Bb< Bv C Db^ D> E[ F<F#v G Ab^ A> B[ C<
> >
> > Cool! Here is a voicing of the first chord I think Joseph might
> want
> > to try:
> >
> > A[ E[ C> Gb^ B[ D> G> Db^ or its transposition up a secor,
> > Bb< F< D[ G> C< E[ A[ D>
> >
>
> Thanks, Paul, for "spelling this out" for me. That was a big "time
> saver."

Thank Dave, for finding it!

> What a chord! That's a great sound. I didn't even want to "leave
> anything out..." :)

Dave Keenan's _second_ chord should be the real winner, if dyadic
consonance means anything for octads (ogdoads?). It is very
instructive to view it on the lattice, observing all the multiple
places notes appear. Let me suggest a possible voicing:

A[ C> F< Gb^ B[ E[ G> D> or its transposition up a secor,
Bb> D[ F#v G> C< F< A[ E[

Try other voicings and inversions.

Perhaps Dave can improve on this voicing, and also suggest one for
his third chord?

I wrote (regarding Dave Keenan's second Blackjack ogdoad),

> It is very
> instructive to view it on the lattice, observing all the multiple
> places notes appear.

What I see now is that the octad is really an otonal tetrad plus a
utonal tetrad:

A[ E[ C> Gb^ -- 1:3:5:7
F< B[ G> D> -- 1/7:1/5:1/3:1/1

The two chords' fifths are a 16:15 apart from one another when moving
in one direction on the lattice, and a 14:15 apart in the other [the
two results are equivalent because (16:15)/(14:15) = 225:224, which
vanishes in 72-tET and MIRACLE]. There are three 7-limit consonances
connecting the chords in _each_ of the directions (not to mention
many 9-limit 'consonances'). And, as Dave pointed out,
every 'dissonant' interval in the chord approximates a ratio of at
most 15. Incredible.

> Let me suggest a possible voicing:
>
(1) > A[ C> F< Gb^ B[ E[ G> D> or its transposition up a secor,
(2) > Bb> D[ F#v G> C< F< A[ E[

Both otonal and utonal, huh? What would the chord sound upside-down
(that is, reflected about C)?

(1u) Bb< F< A[ D[ F#v G> C< E[
(2u) A[ E[ G> C> F< Gb^ B[ D>

(2u) is just a voicing of (1), and (1u) is just a voicing of (2) . . .
The chord is symmetrically _between_ otonal and utonal!

As an exercise, you should come up with a voicing that is perfectly
symmetrical (interval sequence upwards = interval sequence downward).

Since 225:224 vanishes in 22-tET and 31-tET, one can try this chord
in those tunings too . . . too bad I have only 6 or 7 strings on each
of my guitars, and fewer fingers on my left hand . . . :(

>Dave K., are you back?

No. I will probably never be "back" like I was before.

I wrote:

Try these 8 note chords with the suggested voicings [which I've changed since first posting them].

C> D[Eb< Ev F Gb^ G> A[Bb< Bv C Db^ D> E[ F<F#v G Ab^ A> B[ C<
sm14 M14 s18 A18
M3 sm7 M7 rt s11 A11 P5 M9
+--------------+--+--+-----------+--+--+-----------------+
+--------------+--+--+--------------+--+--+--------------+
+-----------------+--+--+-----------+--+--+--------------+
rt P5 m13 SM6 M9 m10 SM3 Wm7
SM13 M16 m17 Wm14
C> D[Eb< Ev F Gb^ G> A[Bb< Bv C Db^ D> E[ F<F#v G Ab^ A> B[ C<

Here's how to read this admittedly cryptic diagram. It seems even Paul didn't understand how I was indicating possible voicings on it.

Take the first chord, represented by the top line of +--+... The suggested root is marked "rt", look above to see that this is A[. The next highest note is marked M3 for major third. It is C>. Then we have a perfect fifth, P5 at E[. Next we have a choice. We could have both subminor and major sevenths but more likely we will want to move one of them up an octave to become a 14th. And so on with major 9th, sub 11th, augmented 11th etc.

Actually I don't think all possible combinations of these will be useful, so I should just spell them out:

A[ C> E[ Gb^ B[ Db^ G> D>
rt M3 P5 sm7 M9 s11 M14 A18

A[ C> E[ G> B[ D> Gb^ Db^
rt M3 P5 M7 M9 A11 sm14 s18

A[ C> E[ Gb^ G> B[ Db^ D>
rt M3 P5 sm7 M7 M9 s11 A11

and here's Paul's voicing with a tiny change (E[ and C> switched) to make it more compact.
A[ C> E[ Gb^ B[ D> G> Db^
rt M3 P5 sm7 M9 A11 M14 s18

>The other chords would be trickier to voice well . . .

Yes.

>Dave, did you consider 11-limit intervals?

Yes but only one came out. A 6:11 in the second chord.

>If you did this search
>using an octave-equivalent consonance measure,

You guessed it. I used odd limit.

>perhaps you should try
>an octave-specific one instead (don't use integer limit -- use
>product or harmonic entropy).

Good idea, but I don't have time. The search also assumed o/u duality, which is wrong.

>Joseph:
>> What a chord! That's a great sound. I didn't even want to "leave
>> anything out..." :)

Good. Every note is a member of at least one otonality (at least 3 notes) with an odd limit no greater than 9.

>Dave Keenan's _second_ chord should be the real winner, if dyadic
>consonance means anything for octads (ogdoads?).

It may lose because it doesn't have the above otonal property. Although it has one more 11-limit interval than the first, the 6:11 is very difficult to voice as such, without screwing up other intervals. It tends to come out as an 11:12 or 11:24 (the latter being preferred since its wide span will make it less dissonant).

I also try to find voicings (for all these chords) where the four 1 secor intervals are voiced as 8:15 M7 or failing that, 7:15 m9, rather than 14:15 or 15:16 m2.

By the way, these interval names and abbreviations are given in
http://dkeenan.com/Music/Miracle/MiracleIntervalNaming.txt

>It is very
>instructive to view it on the lattice, observing all the multiple
>places notes appear. Let me suggest a possible voicing:
>
>A[ C> F< Gb^ B[ E[ G> D>
...
>Perhaps Dave can improve on this voicing,

Here's one with the 6:11 intact and all the 1 secor intervals as 8:15 or 7:15, but it's a bit weird otherwise. It essentially consists of a stack of fourths and seconds, but no step is narrower than a 7:8 SM2 and none is wider than a 5:7 A4.

G> C> Gb^ A[ D> F< B[ E[
rt P4 M7 m9 P12 SM13 m17 m20

These two are probably more consonant but I don't know if they are any better than Paul's.

F< A[ C> E[ G> B[ D> Gb^
rt M3 A5 M7 sm10 A11 sm14 N16

F< A[ C> E[ Gb^ B[ D> G>
rt M3 A5 M7 N9 A11 sm14 sm17

>and also suggest one for his third chord?

Actually I now think the third chord is crap. Too utonal. After all it is simply the dual of the first chord. You take the dual on the slide rule by flipping the pattern horizontally, the mirror image. But you can read some suggested voicings off the diagram at the top if you're keen to try it.

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_30045.html#30068

> > >
> > > Cool! Here is a voicing of the first chord I think Joseph might
> > want to try:
> > >
> > > A[ E[ C> Gb^ B[ D> G> Db^ or its transposition up a secor,
> > > Bb< F< D[ G> C< E[ A[ D>
> > >
> >

> > Thanks, Paul, for "spelling this out" for me. That was a
big "time saver."
>
> Thank Dave, for finding it!
>

[JP:]

Of course! That's an incredible job. I was, initially, a little
mesmerized by the chart he presented. I was encouraged to hear,
Paul, that it momentarily confused even *you*!

However, it wasn't really difficult at all... I just had to look at
it a couple of times, and it was too late the first evening...

> > What a chord! That's a great sound. I didn't even want
to "leave anything out..." :)
>
> Dave Keenan's _second_ chord should be the real winner, if dyadic
> consonance means anything for octads (ogdoads?). It is very
> instructive to view it on the lattice, observing all the multiple
> places notes appear.

[JP:]

It's really "wild" on the lattice, yes? Things are all connected but
in somewhat "unpredictable" ways, yes?

>Let me suggest a possible voicing:
>
> A[ C> F< Gb^ B[ E[ G> D> or its transposition up a secor,
> Bb> D[ F#v G> C< F< A[ E[
>
> Try other voicings and inversions.
>
> Perhaps Dave can improve on this voicing, and also suggest one for
> his third chord?

I've been noticing on all the blackjack chords that voicing makes a
*tremendous* difference... much more than moving inversions around
with traditional 12-tET chords... That's important to keep in mind.

An "algo" composer could almost write an entire piece by varying the
various blackjack pitches by octaves! There would be substantial
variety in that... (not my particular style of composing, though...)

JP

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_30045.html#30072

> I wrote (regarding Dave Keenan's second Blackjack ogdoad),
>
> > It is very
> > instructive to view it on the lattice, observing all the multiple
> > places notes appear.
>
> What I see now is that the octad is really an otonal tetrad plus a
> utonal tetrad:
>
> A[ E[ C> Gb^ -- 1:3:5:7
> F< B[ G> D> -- 1/7:1/5:1/3:1/1

[JP:]
Oh sure... we can put that in the category "Why didn't *I* see that,
too?.." It looked like it was "all over the place" at first, but
here they are, nicely right next to each other!

>
> The two chords' fifths are a 16:15 apart from one another when
moving in one direction on the lattice, and a 14:15 apart in the
other [the two results are equivalent because (16:15)/(14:15) =
225:224, which vanishes in 72-tET and MIRACLE].

[JP:]
Got it! I actually understand what you mean here...

There are three 7-limit consonances
> connecting the chords in _each_ of the directions (not to mention
> many 9-limit 'consonances'). And, as Dave pointed out,
> every 'dissonant' interval in the chord approximates a ratio of at
> most 15. Incredible.
>
> > Let me suggest a possible voicing:
> >
> (1) > A[ C> F< Gb^ B[ E[ G> D> or its transposition up a
secor,

> (2) > Bb> D[ F#v G> C< F< A[ E[
>

> Both otonal and utonal, huh? What would the chord sound upside-down
> (that is, reflected about C)?
>
> (1u) Bb< F< A[ D[ F#v G> C< E[
> (2u) A[ E[ G> C> F< Gb^ B[ D>
>
> (2u) is just a voicing of (1), and (1u) is just a voicing of
(2) . . .The chord is symmetrically _between_ otonal and utonal!

[JP:]
No comprendo.

How does one "reflect it around C, again??" Is this an "inversional"
process??

Since the chord is made up of both otonal and utonal tetrads,
wouldn't one expect the symmetry? I must be missing something else
that makes it even more "exciting..."...

>
> As an exercise, you should come up with a voicing that is perfectly
> symmetrical (interval sequence upwards = interval sequence
downward).
>

[JP:]
Isn't that hard to do with all the possible permutations of the eight
elements??

Joseph

My second 8-note chord can be viewed as 9-limit otonal and utonal pentads sharing a common 5:9 (Wm7, wide minor seventh).

Bv C Db^ D> E[ F<F#v G Ab^ A> B[ C<C#v D Eb^ E> F] G<G#v A Bb^
C> D[Eb< Ev F Gb^ G> A[Bb< Bv C Db^ D> E[ F<F#v G Ab^ A> B[ C<
+--------------+--+--+--------------+--+--+--------------+
1/9---------------1/3---------------1/1---1/7------------1/5
5--------------7-----1-----------------3-----------------9

Gb^
A[/
/
C>----G>----D>
/ \ / \ F</
/Gb^\ / \ /
A[----E[----B[
/
/ D>
F<

I've found an excellent voicing for this chord. I've looked pretty hard and I think this is the best you can do. It's only problem is that it's rather wide; just short of 3 octaves. It has the same property as the first chord; the property that every note is a member of at least one 9-odd-limit otonality. But where the first chord could be voiced so none of these otonalities had an integer-limit greater than 9, with this one we are forced to go to an integer-limit of 10. The three otonalities are connected, and in this voicing they are all in root position (each has its lowest note as a 4 identity). All the dissonances are wider than an octave. There are no steps smaller than a 6:7, and it just happens to be symmetrical. I hope it sounds as delicious as the numbers make it look.

The lower half is a supermajor 9th (no 7th). The upper half is a subminor 7th 9th (no 3rd).

C> F< G> D> A[ E[ Gb^ B[
rt SM3 P5 M9 m13 m17 A18 Wm21
4-------6---9 (sd19)
4-------7--10
4---6---7---9
7---9
6---7
7-diss-15
7-diss-15
5-----------7
5-----------7
5--------------16
4-----diss-----15
4-----diss-----15
5--------------16
5------------------24
11------diss--------48
5------------------24
5----------------------28
5----------------------28
5--------------------------36

The lack of octave equivalence is a property of JI chords, particularly beyond the 5-limit, not specific to Blackjack.

Regards,
-- Dave Keenan
Brisbane, Australia
http://dkeenan.com

Paul Erlich wrote:

>
>
> Are you back, David? Your consonance measure must not account well
> for the dissonance peaks surrounding 1/1. These chords would be quite
> tricky to voice well, since each of them has two 33� intervals, one
> 67� interval, two 83� intervals, and two 117� intervals. In fact,
> they may sound rather like triads played on an out-of-tune accordion.

All accordians are out of tune.

Regards

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:

/tuning/topicId_30045.html#30086

>
> C> F< G> D> A[ E[ Gb^ B[
> rt SM3 P5 M9 m13 m17 A18 Wm21
> 4-------6---9 (sd19)
> 4-------7--10
> 4---6---7---9
> 7---9
> 6---7
> 7-diss-15
> 7-diss-15
> 5-----------7
> 5-----------7
> 5--------------16
> 4-----diss-----15
> 4-----diss-----15
> 5--------------16
> 5------------------24
> 11------diss--------48
> 5------------------24
> 5----------------------28
> 5----------------------28
> 5--------------------------36
>

Thanks, Dave. You've really found an exceptional chord here... I am
certain to work with it!

> The lack of octave equivalence is a property of JI chords,
particularly beyond the 5-limit, not specific to Blackjack.
>

Oh, of course... I guess my confusion is this:

We generally think of Blackjack as a *tempered* scale with octave
equivalence, yes?? Even though the generators would create an
*untempered* scale in their "raw" form, which would go on forever as
a JI scale, that's not the current definition of Blackjack...

Is this correct??

Maybe Paul can answer that if you don't get back to it...

Thanks for all the help!

Joseph

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>
> /tuning/topicId_30045.html#30072
>
> > I wrote (regarding Dave Keenan's second Blackjack ogdoad),
> >
> > > It is very
> > > instructive to view it on the lattice, observing all the
multiple
> > > places notes appear.
> >
> > What I see now is that the octad is really an otonal tetrad plus
a
> > utonal tetrad:
> >
> > A[ E[ C> Gb^ -- 1:3:5:7
> > F< B[ G> D> -- 1/7:1/5:1/3:1/1
>
>
> [JP:]
> Oh sure... we can put that in the category "Why didn't *I* see
that,
> too?.." It looked like it was "all over the place" at first, but
> here they are, nicely right next to each other!
>
>
> >
> > The two chords' fifths are a 16:15 apart from one another when
> moving in one direction on the lattice, and a 14:15 apart in the
> other [the two results are equivalent because (16:15)/(14:15) =
> 225:224, which vanishes in 72-tET and MIRACLE].
>
> [JP:]
> Got it! I actually understand what you mean here...
>
>
> There are three 7-limit consonances
> > connecting the chords in _each_ of the directions (not to mention
> > many 9-limit 'consonances'). And, as Dave pointed out,
> > every 'dissonant' interval in the chord approximates a ratio of
at
> > most 15. Incredible.
> >
> > > Let me suggest a possible voicing:
> > >
> > (1) > A[ C> F< Gb^ B[ E[ G> D> or its transposition up a
> secor,
>
> > (2) > Bb> D[ F#v G> C< F< A[ E[
> >
>
> > Both otonal and utonal, huh? What would the chord sound upside-
down
> > (that is, reflected about C)?
> >
> > (1u) Bb< F< A[ D[ F#v G> C< E[
> > (2u) A[ E[ G> C> F< Gb^ B[ D>
> >
> > (2u) is just a voicing of (1), and (1u) is just a voicing of
> (2) . . .The chord is symmetrically _between_ otonal and utonal!
>
>
> [JP:]
> No comprendo.
>
> How does one "reflect it around C, again??" Is this
an "inversional"
> process??

Yes. Blackjack pitches go

0(72) <-> 0(72)
2 <-> 70
7 <-> 65
9 <-> 63

etc. or

C <-> C
C< <-> C>
Db^ <-> Bv
D[ <-> B[

etc.

>
> Since the chord is made up of both otonal and utonal tetrads,
> wouldn't one expect the symmetry?

Yes.

> I must be missing something else
> that makes it even more "exciting..."...

I didn't expect it because my brain was tired.

> > As an exercise, you should come up with a voicing that is
perfectly
> > symmetrical (interval sequence upwards = interval sequence
> downward).
> >
>
> [JP:]
> Isn't that hard to do with all the possible permutations of the
eight
> elements??
>
If you're clever, it won't be that hard.

--- In tuning@y..., jpehrson@r... wrote:

> Thanks, Dave. You've really found an exceptional chord here... I
am
> certain to work with it!

I think Dave just gave away the answer to my inversionally-invariant-
voicing puzzle! :)

> We generally think of Blackjack as a *tempered* scale with octave
> equivalence, yes?? Even though the generators would create an
> *untempered*

Not untempered.

> scale in their "raw" form, which would go on forever as
> a JI scale,

Not JI.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:

/tuning/topicId_30045.html#30111

> --- In tuning@y..., jpehrson@r... wrote:
>
> > Thanks, Dave. You've really found an exceptional chord here... I
> am certain to work with it!
>
> I think Dave just gave away the answer to my inversionally-
invariant-voicing puzzle! :)
>
> > We generally think of Blackjack as a *tempered* scale with octave
> > equivalence, yes?? Even though the generators would create an
> > *untempered*
>
> Not untempered.
>
> > scale in their "raw" form, which would go on forever as
> > a JI scale,
>
> Not JI.

Whoopie.... Dumbo. Sorry. That was *obviously* a post-midnight-post!

JP