Re: 7 limit adaptive puzzle

Hi John,

> [Robert wrote:]
> >Here's the smaller adaptive tuning puzzle.

> >http://members.tripod.com/~robertinventor/tunes/tunes.htm#7_limit_adaptive_puzzle

> Hi, Robert! Sorry this took me so long.

That's all right,

> >It turned out to be 7 limit. (I expect that will please you :-)).

> Interesting! I'll try attacking it both in 5-lim and 7-lim, & see
> what happens.

I'll be interested to hear.

Yes, I intended to do a 5-lim one, but on setting to, got interested in
idea of doing it 7-lim. Could be interesting to see what happens to it
in 5-lim. I might also try it again, as a 5-lim one also short like this.

> >Has an Ab and a G#, but both turn out to be 49/32!

> Now I'm confused! There's only one Ab/G# in the piece, at 18.002
> seconds, and it's tuned +27.88 cents relative to 12-tET. Lessee: 49/32
> would be 737.65 cents from root. I'll bet you're using a G note for
> this, are you not? That's a serious problem for my program, because I
> begin by stripping out your bends, then evaluate the scale degrees.
> (I don't yet have code to recognize input bends as _tuning_. But in
> any case, for purposes of this "puzzle", that would be cheating, yes?)

Should say before going any further, I did this all by ear,
and some looking at the ratios that I play, and the analysis is
after the event.

Yes, it's a G, - I choose midi notes to minimise the pitch bend, so
that if a note needs to be bent while playing, one's got maximum range free
for the pitch glissando.

The score showed the G# as an Ab in Ab minor but I think it is really a G# in
G# minor, and I've corrected the scorre.

At end of bar 6, the 49/64 ~64/21~ 7/3 sounded nice for the fifth,
even though it is a wolf fifth at 32/21 = 729.2 cents; and I liked it.
Wouldn't work as a final chord of course.

Could easily be made pure by putting in a root control at G# at that point,
which leaves the C# at the same pitch as for the C# root.

The Ab you found is in fact the A natural in the scale of E major at the
start of bar 7, which is shifted down in pitch so much at this point that
it is a lot closer to an Ab.

Root at this point is the C# which is 28/27

The sequence of chords in bars 7 to 8 is rather complex with some impure intervals,
and will be interesting to see what happens to it when adaptively tuned.

For example, has an 11/9 A major resolving to an A minor (as shown
on the score) which is really an A major at that point with a
third of 243/196 = 372.1 cents.

I've added a text file with an analysis of the progression to the html page,
with all the j.i. intervals and the note names, and comments.

> >Uses David Canright's 7-limit j.i. scale.

> Which I'm sure I ought to be familiar with, but I'm not. However, I
> can guess that Bb is 7/4, if Ab is 49/32. Wow, then I'd expect D to
> be 35/32, and F to be 21/16; can _that_ be right???

The scale is:

1/1 28/27 9/8 7/6 5/4 4/3 45/32 3/2 14/9 5/3 7/4 15/8 2/1

You can read about it here:
http://www.mbay.net/~anne/david/piano/index.htm

"This offers the chance to contrast, for example, a standard minor scale
(on 5/4) with a septimal minor on 1/1: 1:1 9:8 7:6 4:3 3:2 14:9 7:4."

The septimal minor also gives a nice septimal pentatonic scale, which I use in this piece
C Eb F Ab Bb C
=
1/1 7/6 4/3 14/9 7/4 2/1

steps
7/6 8/7 7/6 9/8 8/7

SCALA info:

Interval class, Number of incidences, Size:
1: 1 9/8 203.910 cents major whole tone
1: 2 8/7 231.174 cents septimal whole tone
1: 2 7/6 266.871 cents septimal minor third
2: 1 9/7 435.084 cents septimal major third, BP third
2: 1 21/16 470.781 cents narrow fourth
2: 3 4/3 498.045 cents perfect fourth
3: 3 3/2 701.955 cents perfect fifth
3: 1 32/21 729.219 cents wide fifth
3: 1 14/9 764.916 cents septimal minor sixth
4: 2 12/7 933.129 cents septimal major sixth
4: 2 7/4 968.826 cents harmonic seventh
4: 1 16/9 996.090 cents Pythagorean minor seventh

At the end of bar 11, all the notes of this pentatonic are sounding.

I use it with Bb as the root instead of C. I.e. successive semitones
from Bb upwards are identical to those of the original scale
from C upwards.

I'm using cross set root control, which keeps the note chosen for the root
fixed, so Bb is 7/4 at this point, as in the original scale.

The C becomes a 9/8 above 7/4, (the 9/8 of the septimal minor scale)
at 63/64.

The Ab then becomes an 8/7 below the Bb, so is at 49/32.

At this point the D is indeed the 35/32, and the F the 21/16,
(or would be if they were played) because of the root control
change to Bb at 7/4.

The key it is playing in at that point is Db, and the 63/64 C is
the leading tone, and I'm using the septimal pentatonic as the minor
pentatonic in Db.

However, this is all with the C at 63/64. My piece doesn't
use the C at that point, and by the time you get to it
(on return to the beginning section) it has returned to 1/1.

The way it gets back to C major is a very fast transition:

(root control Bb)
Db Eb Gb Ab Bb

(root control C)
Bb E G

C E G

This is a much more straightforward return to C from Ab than the earlier
return from G#, with all the chords pure except the 21/16 in the septimal
pentatonic; it's used twice, and ends the piece:

(11)
...
(the septimal pentatonic Db Eb Gb Ab Bb, with all those intervals shown in the
SCALA listing

(12)
Bb E G
7/16 ~20/7~ 5/4 ~24/7, 6/5~ 3/2
C E G
1/2 ~5/2~ 5/4 ~3/1, 6/5~ 3/2

You've got the Bb carrying through to tie the chords together
to ensure that the Bb is a 7/4 if using the 7-limit scale.
(though it is two octaves below the one in the previous chord)

> >If real time adaptive tuning program chose
> >8/5 each time instead, it would end up
> >over a tone sharp at the end of the piece.

> >So, will be a fine challenge for your program I hope.

> As it stands, a bit _too_ much of a challenge, I'm afraid! Could you
> possibly redo it using scale degree 8 for Ab/G#? The bend of -62.35
> cents is still very workable for a MIDI module.

Yes, I've saved the original .midi in 12 tone equal temperament.

7_lim_ap_12_tet.mid

I wonder if your program could automatically add a root control
channel to a midi clip by analysing the j.i. progressions
in some way?

Kind of vague idea at present, but could be useful if a composer
could then edit it and change the root controls as desired.

Also could be useful for transforming a piece into j.i. for
trying out Gene's transformations.

Robert