white key scales (Dan, Miracle folk)

> Dan says
>
> Yes I agree with all of this, excellent points! But I think this
> underscores some of the problems as well. The "miracle scale" is not a
> scale paradigm at all in the diatonic sense where there's a symbiotic
> correspondence between a horizontal and a vertical structure.
>
> There's a lot of unanswered (but fascinating) questions beyond the
> 5-limit. One of which is what kind of paradigm shift might be
> necessary to accommodate a horizontal concept that'll accompany higher
> harmonics in their readymade horizontal sense... whether by the weight
> of acculturation or some innate psychoacoustic threshold, clearly a
> "white key" concept collapses pretty quickly once the scale expands
> very much beyond 7.
>
> What could a non generalized diatonic diatonic be... any ideas?
>

The 'scale tree' idea can lead one up from a "lotta notes" to
something "white key" like.

for instance (pause for quick 'grep')...

+-------------------Blackjack-21-MOS-improper-CS------------+
2 9 16 23 30 37 44 51 58 65 0 7 14 21 28 35 42 49 56 63 70

2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2
0 2 7 9 14 16 21 23 28 30 35 37 42 44 49 51 56 58 63 65 70

Grahams 10 note scale

7 7 7 7 7 7 7 7 7 7 2
0 7 14 21 28 35 41 49 56 63 70

or try to push it up in a different direction and come across
this gem, the "neutral ionian" (perhaps this was named something
else as I didn't see it pop up all by its lonesome like this).

9 12 9 12 9 12 9
0 150 350 500 700 850 1050 1200 (cents now)

and lots of others...

(growing out of a septimal pentatonic!)

16 12 16 12 16
7 9 12 7 9 12 7 9
2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2

There are a lot of ways to play this game and some of them might
lead somewhere. It all gets back (in my small head at least) that
thinking melodically (white keys, dulcimer frets, holes on a
flute) is a linear structure that should have some nice structure
to appreciate tunings with the way lattices provide it for harmonic
resources. Scale trees may provide a way to lead up from 'black
notes' to 'white notes' ("ebony, and ivory...")

Bob Valentine

In-Reply-To: <200105241023.NAA56962@ius281.iil.intel.com>
Bob Valentine wrote:

> or try to push it up in a different direction and come across
> this gem, the "neutral ionian" (perhaps this was named something
> else as I didn't see it pop up all by its lonesome like this).
>
> 9 12 9 12 9 12 9
> 0 150 350 500 700 850 1050 1200 (cents now)

I call this "anti-Dorian" because each tetrachord is like Dorian but with
large and small intervals reversed.

Incidentally, 94-equal works as a neutral-third scale, although the fifth
isn't the best 3:2.

> There are a lot of ways to play this game and some of them might
> lead somewhere. It all gets back (in my small head at least) that
> thinking melodically (white keys, dulcimer frets, holes on a
> flute) is a linear structure that should have some nice structure
> to appreciate tunings with the way lattices provide it for harmonic
> resources. Scale trees may provide a way to lead up from 'black
> notes' to 'white notes' ("ebony, and ivory...")

I'm currently working on a program that can take pairs of consistent
15-limit temperaments, find out where they are on the scale tree, work out
a linear temperament to cover both of them, and step backwards to find
some "white notes" for notation. Once that's working, it should be
possible to adapt it for any set of target pitches. So find some
consonances in an inharmonic timbre, and then generate the temperaments
that work well with it.

This must be similar to what Dave Keenan's generator-finding program does.

Graham

--- In tuning@y..., Robert C Valentine <BVAL@I...> wrote:
> Grahams 10 note scale
>
> 7 7 7 7 7 7 7 7 7 7 2
> 0 7 14 21 28 35 41 49 56 63 70

You've shown the 11 note scale above. Of course it should have been.

Miracle-10

7 7 7 7 7 7 7 7 7 9
0 7 14 21 28 35 41 49 56 63

> or try to push it up in a different direction and come across
> this gem, the "neutral ionian" (perhaps this was named something
> else as I didn't see it pop up all by its lonesome like this).
>
> 9 12 9 12 9 12 9
> 0 150 350 500 700 850 1050 1200 (cents now)

I think Graham first described is simply as the 7-tone neutral-thirds
MOS. It's in the Scala archive. Apparently the Arabs call this
particular mode "mohajira".

> (growing out of a septimal pentatonic!)
>
> 16 12 16 12 16
> 7 9 12 7 9 12 7 9
> 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2 5 2
Decimal note names
D D> N N> Eo E> Fo F> Go G> H H> A< Ao B< Bo C< Co L< L D< D
4 4> 5 5> 6 6> 7 7> 8 8> 9 9> 0< 0 1< 1 2< 2 3< 3 4< 4

Legend
Symbol Pronunciation Meaning
------------------------------------
> "super" 2/72 oct up
o "decimal" natural
< "sub" 2/72 oct down

Now that's interesting! Melodically we can take the 7-step steps and
9-step steps to be the same size. And in fact you can make a choice
for each of the 16-step steps in the septimal pentatonic, whether to
split it into 7+9 or 9+7. So this octatonic scale is melodically
similar to 22322322 (a mode of 18-EDO). Strictly proper, but not MOS.
As 11211211 it would be a mode of 10-EDO.

I can't find anything like it in the Scala archive.

In decimal Miracle notation it's
D,(N or N>),E>,Go,(H or H>),Ao,C<,(L< or L) or
4,(5 or 5>),6>,8 ,(9 or 9>),0 ,2<,(3< or 3)

On a chain of Miracle generators (in this case Blackjack) it looks
like.
4> 5> 6> 7> 8> 9> 0 1 2 3 4 5 6 7 8 9 0< 1< 2< 3< 4<
N--+--------H--+--------L--+--N--------+--H--------+--L

where you choose one of the N's, one of the H's and one of the L's at
any given time. We can see that it occurs in up to 5 positions within
Blackjack (Miracle-21).

It will be interesting to see what chords are available in it with the
various choices for N,H,L.

Maybe "Valentine's mutable octatonic" deserves to be called "the"
white-note scale for Miracle?

-- Dave Keenan

--- In tuning@y..., graham@m... wrote:
> This must be similar to what Dave Keenan's generator-finding program
does.

No. It works by brute force. It's an Excel spreadsheet. It simply does
the same calculation for every generator between 0 to 600c in 0.1c
increments. It calculates how many generators in a chain (up to some
maximum) give the best (octave equivalent) approximation for each of
1:3, 1:5, 1:7, 1:11 and what the errors are. From those it calculates
the hexad-width and the MA and RMS errors over all 11-limit intervals
(consistency is guaranteed). Then it flags those that meet the set
criteria for hexad-width and errors.

The only possible cleverness is in keeping it to one row per
generator.

-- Dave Keenan

Unfortunately Bob Valentine's mutable octatonic in
Miracle-21/Blackjack has very few "consonant" chords. So I guess it's
not "the" white note scale of Miracle.

It has:
1 1-3-5 triad in every mutation
2 3-5-7 triads in every mutation
1 3-7-9 triad in some mutations
1 otonal and 1 utonal 3-9-11 in some mutations
and a few additional bare dyads

The problem is that in all mutations it has too many of what might be
considered a characteristic dyadic dissonance of Miracle (and Wonder)
chains, which consists of 4 Miracle generators (or two Wonder
generators) at 467 cents. Its inversion is 733 cents. As Graham
pointed out, these are the meantone wolf fourth and fifth. They can
also be called the sub-fourth and super-fifth.

-- Dave Keenan