A septimal 22 tone scale subset of B&C matrix

> Forwarded is a question from Justin White.
>
> He refers to <http://www.anaphoria.com/genus.PDF>. On the bottoms of pages
> 15, 19, 20, and 23, there is a lattice of Wilson's famous 17-tone scale,
> which is clearly a periodicity block with unison vectors schisma and
> chromatic semitone; i.e.,
>
> [8 1]
>
> and
>
> [-1 2].
>
>
> Anyone like to tackle Justin's question below? Justin, if you're reading
> this, you might like to join tuning-math to see what responses this
> generates!
>
>
> -----Original Message-----
> From: Justin White [mailto:justin.white@d...]
> Sent: Monday, May 14, 2001 3:53 AM
> To: Paul H. Erlich
> Subject: Re: adaptive tuning. Can a computer pick a melody from the
> harmony ?
>
>
>
>
>
> Hello Paul, Thanks for your offer of assistance with this one. Have you read
> Erv
> Wilsons paper "Some Basic Patterns Underlying Genus 12 & 17"?
>
> --- In tuning@y..., "Justin White" <justin.white@d...> wrote:
> >>
> >> Yes I was attracted to this scale. I thought of creating a scale in
> the smae
> >> manner using a septimal tetrachord...I haven't found a tetrachord
> that will give
> >> me the tetrad s I want yet.
>
> >Can you explain what you're trying to do? Maybe I can help.
>
>
> What I want to do is use the same methodology to create a [septimal] subset
> of
> the scale I have posted below.
>
>
> 0. 1/1
> 1 25/24
> 2. 135/128
> 3. 35/32
> 4. 9/8
> 5. 7/6
> 6. 75/64
> 7. 1215/1024
> 8. 6/5
> 9. 315/256
> 10. 5/4
> 11. 81/64
> 12. 21/16
> 13. 675/512
> 14. 4/3
> 15. 7/5
> 16. 45/32
> 17. 35/24
> 18. 189/128
> 19. 3/2
> 20. 25/16
> 21. 405/256
> 22. 8/5
> 23. 105/64
> 24. 5/3
> 25. 27/16
> 26. 7/4
> 27. 225/128
> 28. 9/5
> 29. 945/512
> 30. 15/8
> 31. 243/128
> 32. 63/32
> 33. 2/1
>
> Note how Wilsons genus 17 [see below] contains mostly notes from the above
> superset [B&C's blue melodic reference]
>
> 0. 1/1
> 1. 135/128
> 2. 10/9
> 3. 9/8
> 4. 1215
> 5. 5/4
> 6. 81/80
> 7. 4/3
> 8. 45/32.
> 9. 729/512
> 10 .3/2
> 11. 405/256
> 12. 5/3
> 13. 27/16
> 14. 3645/2048
> 15. 15/8
> 16. 243/128
> 17. 2/1
>
> The columns below are to indicate what ratios are more important than
> others.
> The notes in the left hand column should be used before the notes in the
> right
> hand column. [This is to do with th e chain of reference used in that
> scale.]
>
> 1/1
> 9/8 45/32 135/128 405/256
> 1215/1024
> 7/6 35/24
> 6/5
> 5/4 25/16 75/64 225/128
> 675/256
> 4/3
> 7/5 21/16 63/32
> 189/128
> 3/2 15/8
> 8/5 25/24
> 5/3
> 7/4 35/32 105/64 315/256
> 945/512
> 9/5 27/16 81/64
> 243/128
>
>
> I'd be interested to see what you make of it all.
>
>
>
> Best wishes,
>
> Justin White

Hello Mathtuners !

Here is what I came up with as a septimal related to B&C's blue
reference.

Thanks Paul for forwarding this and making me aware of the existence of
this group.

The scale is based on a re-ordering of Al Farabi's chromatic
tetrachord.

When taken as a 7 note scale it looks like this
1/1, 7/6, 5/4, 4/3, 3/2, 7/4, 15/8

The 22 tone scale is this 'blues scale' transposed through six
pythagorean keys [meantone keys ?] : 4/3, 1/1, 3/2, 9/8, 27/16, 81/64.

Wilsons scale is a bit purer in that it is trivalent, the disjunction
interval 8:9 is also actually part of the the diatonic tetrachord. in
my version there are 7 different successive intervals.

Is it a PB ?

JI Calc tuning file
22 tone scale on al farabi tetrachord
,,,,,,,,,,,,
60 440
R22
1/1
135/128
567/512
9/8
7/6
1215/1024
5/4
81/64
21/16
4/3
45/32
189/128
3/2
14/9
405/256
5/3
27/16
7/4
16/9
15/8
243/128
63/32
#
EOF

The scales I am experimenting with are intended for an electric guitar
with a bridge that can switch between 3 notes on every open string.

This obviously makes a JI scale harder as when a string is transposed
to a different note so is the scale !

If someone can suggest a tempered scale that would allow me to play
most [or all] of the the B&C blue reference scale with sufficient
accuracy I will probably go down that route, or else have the entire
scale mapped to one or two strings [1/1=G] which i will not transpose.
I could then play slide guitar on the open strings using the frets on
the g strings for fingered melodies and as a visual guide.

Justin

--- In tuning-math@y..., "Justin White " <JUSTINTONATION@H...> wrote:
>
> Is it a PB ?

No -- 135/128 would be two steps, not one, in a 22-tET PB or CS.

Before I start spouting out 22-tone 7-limit periodicity blocks that
are very similar to this for you, can I ask you, why only 22 tones,
given that you have intervals
as small as a syntonic comma (about 1/56 of an octave) in your scale
(e.g., between 5/4 and 81/64)?

> If someone can suggest a tempered scale that would allow me to play
> most [or all] of the the B&C blue reference scale with sufficient
> accuracy

You'll have to specify what "sufficient accuracy" means for you.
Presumably, each of the consonant intervals in the chains of
reference need to be tuned correctly to within x cents. What is x?
Presumably, you'd also rather have comma differences respected rather
than tempered out/confuted -- yes?

--- In tuning-math@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning-math@y..., "Justin White " <JUSTINTONATION@H...> wrote:
> >
> > Is it a PB ?
>
> No -- 135/128 would be two steps, not one, in a 22-tET PB or CS.
>
> Before I start spouting out 22-tone 7-limit periodicity blocks that
> are very similar to this for you, can I ask you, why only 22 tones,
> given that you have intervals
> as small as a syntonic comma (about 1/56 of an octave) in your scale
> (e.g., between 5/4 and 81/64)?

Hello Paul,

There is no reason for 22 tones per octave [22 tones was just a
consequence of the scale constuction method I used i.e tetrachords.] I
would prefer to accomodate every tone in the B&C matrix, so a 31 or 34
tone PB would be more sensible.

> > If someone can suggest a tempered scale that would allow me to play
> > most [or all] of the the B&C blue reference scale with sufficient
> > accuracy
>
> You'll have to specify what "sufficient accuracy" means for you.
> Presumably, each of the consonant intervals in the chains of
> reference need to be tuned correctly to within x cents. What is x?
> Presumably, you'd also rather have comma differences respected rather
> than tempered out/confuted -- yes?

Sufficient accuracy would be less than 5 cents. Comma differences are
important as they provide the different melodic shades.

If I had a fixed bridge that could not switch open tunings I would
merely tune the matrix to JI, and there would be no problem. But I have
this bridge that can change open string tunings at the flick of a
lever. I would like to play bottleneck chords on the open strings. But
I dont want to change the key of the matrix ! I sing in G and would
like it to be tuned to G whatever the sting is tuned to.

Each string has 3 possibilities so these are what I would find the most
useful notes [for chords] for each string.

String 6 5 4 3 2 1

lever position high E9/5 B5/4 F7/4 A9/8 C4/3 F#15/8


lever position middle D3/2 A#6/5 D3/2 G1/1 B5/4 F7/4

lever position low C4/3 G1/1 C4/3 F9/5 A#6/5 D3/2

I have thought of implementing a fretting that is a subbset of the B&C
matrix that is transposable by each of the open string possibilities.

Anyway there are alot of issues regardin the guitar.

If yourself and others cannot think of aggod solution, I will probably
just make my fretless fingerboard first and really play the scales
proposed. This will give mme a ggood idea of how i will play the guitar
and interact with the bridge in reality.

sincerely,

Justin White