standard keyboard tuning based on 3/2 and 7/4 only

Hey all,

I have been doing initial explorations of tunings based on 3/2 and 7/4
only, and with the following two constraints:

1) The tuning should fit on a 12 note per octave keyboard whose tuning
capabilities include going +99 or --99 cents away from the standard 12-tet
note distribution.

2) The 7/4 dimension (let's say 'y axis' of the lattice) should contain 2
vertices at least, in other words, 'three lines of 3/2' spreading out in the
x-axis direction.

It seems that La Monte Young's 'Weel Tuned Piano' tuning, pasted below in
.scl format, and obviously it's transpositions and permutations, is the only
scale that would meet the above criteria, at least where perfect fifths
preserve a keyboard-spelling relationship. The Euler-Fokker genera, (e.g.
33377) certainly do not satisfy them, as the total spread would be too wide
for certain intervals.

Does anyone know if there is a theoretical proof of this?

--
Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

! YOUNG-LM_PIANO.scl
!
LaMonte Young's Well-Tempered Piano
12
!
567/512
9/8
147/128
21/16
1323/1024
189/128
3/2
49/32
7/4
441/256
63/32
2/1

Hey all,

I have been doing initial explorations of tunings based on 3/2 and 7/4
only, and with the following two constraints:

1) The tuning should fit on a 12 note per octave keyboard whose tuning
capabilities include going +99 or --99 cents away from the standard 12-tet
note distribution.

2) The 7/4 dimension (let's say 'y axis' of the lattice) should contain 2
vertices at least, in other words, 'three lines of 3/2' spreading out in the
x-axis direction.

It seems that La Monte Young's 'Weel Tuned Piano' tuning, pasted below in
.scl format, and obviously it's transpositions and permutations, is the only
scale that would meet the above criteria, at least where perfect fifths
preserve a keyboard-spelling relationship. The Euler-Fokker genera, (e.g.
33377) certainly do not satisfy them, as the total spread would be too wide
for certain intervals.

Does anyone know if there is a theoretical proof of this?

--
Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

! YOUNG-LM_PIANO.scl
!
LaMonte Young's Well-Tempered Piano
12
!
567/512
9/8
147/128
21/16
1323/1024
189/128
3/2
49/32
7/4
441/256
63/32
2/1

-------------------------------------------------------

--
Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...> wrote:
>
> Hey all,
>
> I have been doing initial explorations of tunings based on 3/2 and 7/4
> only, and with the following two constraints:
>
> 1) The tuning should fit on a 12 note per octave keyboard whose tuning
> capabilities include going +99 or --99 cents away from the standard
12-tet
> note distribution.
>
> 2) The 7/4 dimension (let's say 'y axis' of the lattice) should
contain 2
> vertices at least, in other words, 'three lines of 3/2' spreading
out in the
> x-axis direction.

One thing you might want to look at are the {3,7} Fokker blocks you
get by taking a pair of 12-et compatible, {3,7}-commas. I've listed
some of these below. The TM basis is {64/63, 729/686}, another
important comma is the Pythagorean comma. The first comma on the list,
33554432/33480783 is not probably really what you are looking for, but
it's an interesting comma (beta 2) related to schismic, where 14
fourths make a 56. Putting 64/63 together with 729/686, 59049/57344 or
6561/6272 seems like a good plan for you.

12-et {3,7} commas

33554432/33480783
|25 -14 0 -1>
rel .000063 epi .646817 cents 3.804081

17793060798303/17592186044416
|-44 26 0 1>
rel .000186 epi .853352 cents 19.655929

531441/524288
|-19 12 0 0>
rel .000514 epi .673912 cents 23.460011

64/63
|6 -2 0 -1>
rel .001897 epi 0.000000 cents 27.264092

2147483648/2109289329
|31 -16 0 -2>
rel .000418 epi .813158 cents 31.068173

72057594037927936/70620658308464607
|56 -30 0 -3>
rel .000260 epi .899608 cents 34.872254

59049/57344
|-13 10 0 -1>
rel .001335 epi .679149 cents 50.724102

137438953472/132885227727
|37 -18 0 -3>
rel .000657 epi .868287 cents 58.332265

31381059609/30064771072
|-32 22 0 -1>
rel .000887 epi .870328 cents 74.184112

6561/6272
|-7 8 0 -2>
rel .002569 epi .648047 cents 77.988194

8796093022208/8371769346801
|43 -20 0 -4>
rel .000830 epi .899780 cents 85.596356

729/686
|-1 6 0 -3>
rel .004633 epi .575910 cents 105.252286

387420489/359661568
|-20 18 0 -3>
rel .001883 epi .869974 cents 128.712297

2592/2401
|5 4 0 -4>
rel .004893 epi .674784 cents 132.516377

On Sunday 04 April 2004 09:33 pm, Gene Ward Smith wrote:
> --- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...> wrote:
> > Hey all,
> >
> > I have been doing initial explorations of tunings based on 3/2 and 7/4
> > only, and with the following two constraints:
> >
> > 1) The tuning should fit on a 12 note per octave keyboard whose tuning
> > capabilities include going +99 or --99 cents away from the standard
>
> 12-tet
>
> > note distribution.
> >
> > 2) The 7/4 dimension (let's say 'y axis' of the lattice) should
>
> contain 2
>
> > vertices at least, in other words, 'three lines of 3/2' spreading
>
> out in the
>
> > x-axis direction.
>
> One thing you might want to look at are the {3,7} Fokker blocks you
> get by taking a pair of 12-et compatible, {3,7}-commas. I've listed
> some of these below. The TM basis is {64/63, 729/686}, another
> important comma is the Pythagorean comma. The first comma on the list,
> 33554432/33480783 is not probably really what you are looking for, but
> it's an interesting comma (beta 2) related to schismic, where 14
> fourths make a 56. Putting 64/63 together with 729/686, 59049/57344 or
> 6561/6272 seems like a good plan for you.
>
> 12-et {3,7} commas
>
> 33554432/33480783
>
> |25 -14 0 -1>
>
> rel .000063 epi .646817 cents 3.804081
>
> 17793060798303/17592186044416
>
> |-44 26 0 1>
>
> rel .000186 epi .853352 cents 19.655929
>
> 531441/524288
>
> |-19 12 0 0>
>
> rel .000514 epi .673912 cents 23.460011
>
> 64/63
>
> |6 -2 0 -1>
>
> rel .001897 epi 0.000000 cents 27.264092
>
> 2147483648/2109289329
>
> |31 -16 0 -2>
>
> rel .000418 epi .813158 cents 31.068173
>
> 72057594037927936/70620658308464607
>
> |56 -30 0 -3>
>
> rel .000260 epi .899608 cents 34.872254
>
> 59049/57344
>
> |-13 10 0 -1>
>
> rel .001335 epi .679149 cents 50.724102
>
> 137438953472/132885227727
>
> |37 -18 0 -3>
>
> rel .000657 epi .868287 cents 58.332265
>
> 31381059609/30064771072
>
> |-32 22 0 -1>
>
> rel .000887 epi .870328 cents 74.184112
>
> 6561/6272
>
> |-7 8 0 -2>
>
> rel .002569 epi .648047 cents 77.988194
>
> 8796093022208/8371769346801
>
> |43 -20 0 -4>
>
> rel .000830 epi .899780 cents 85.596356
>
> 729/686
>
> |-1 6 0 -3>
>
> rel .004633 epi .575910 cents 105.252286
>
> 387420489/359661568
>
> |-20 18 0 -3>
>
> rel .001883 epi .869974 cents 128.712297
>
> 2592/2401
>
> |5 4 0 -4>
>
> rel .004893 epi .674784 cents 132.516377

Gene,

Excuse my ignorance of all of this math, and terms like 'rel' epi' (I know
what cents are!)

But those matrix-like symbols tell me nothing. How am I to construct a
practical scale with what you just said? Can you walk me through it, as if
you were writing a 'Constructing a 3,7 Fokker block for dummies' book?

Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

Aaron K. Johnson wrote:

>Gene,
>
>Excuse my ignorance of all of this math, and terms like 'rel' epi' (I know >what cents are!)
>
>But those matrix-like symbols tell me nothing. How am I to construct a >practical scale with what you just said? Can you walk me through it, as if >you were writing a 'Constructing a 3,7 Fokker block for dummies' book?
>
No tuning@yahoogroups.com magic decoder ring?

--
* David Beardsley
* microtonal guitar
* http://biink.com/db

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...>
wrote:

> Excuse my ignorance of all of this math, and terms like 'rel' epi'
(I know
> what cents are!)

Sorry, I forgot to explain what those meant. "Rel" is the listing for
log(p/q)/log(p*q) for an interval p/q. "Epi" is log(p-q)/log(q).

> But those matrix-like symbols tell me nothing. How am I to
construct a
> practical scale with what you just said? Can you walk me through
it, as if
> you were writing a 'Constructing a 3,7 Fokker block for dummies'
book?

I have my own way of doing it, if that's OK. I'll post something here
when I log off Windows and log on Linux.

On Monday 05 April 2004 01:25 pm, David Beardsley wrote:
> Aaron K. Johnson wrote:
> >Gene,
> >
> >Excuse my ignorance of all of this math, and terms like 'rel' epi' (I know
> >what cents are!)
> >
> >But those matrix-like symbols tell me nothing. How am I to construct a
> >practical scale with what you just said? Can you walk me through it, as if
> >you were writing a 'Constructing a 3,7 Fokker block for dummies' book?
>
> No tuning@yahoogroups.com magic decoder ring?

David-

Yeah, I guess so!!!! How much do they cost and where do I get one? ;)

--
Aaron Krister Johnson
http://www.akjmusic.com
http://www.dividebypi.com

OCEAN, n. A body of water occupying about two-thirds of a world made
for man -- who has no gills. -Ambrose Bierce 'The Devils Dictionary'

--- In tuning@yahoogroups.com, "Aaron K. Johnson" <akjmicro@c...> wrote:

> But those matrix-like symbols tell me nothing. How am I to construct a
> practical scale with what you just said? Can you walk me through it,
as if
> you were writing a 'Constructing a 3,7 Fokker block for dummies' book?

Suppose I take 64/63 and 729/686 for my commas. I now choose (it
doesn't matter how) a {3,7} interval giving a single step of 12-et;
this could be 28/27. I now find the 3x3 matrix of exponents (exluding
5), for [28/27, 64/63, 729/686]. This is

[2, -3 1]
[6 -2 -1]
[-1 6 -3]

Inverting this matrix gives me

[12 -3 5]
[19 -5 8]
[34 -9 14]

It is the first row of this which is important, since we are computing
an octave-repeating scale. We choose offsets 0<=a, b<1 and then for
every i from 0 to 11, compute

scale[i] = (28/27)^i (64/63)^floor(-3i/12+a) (729/686)^floor(5/12+b)

This gives a Fokker block. We can get all the Fokker blocks possible
from our pair of commas by choosing all pairs of integers 0<= u, v <
24 and then setting a = u/24, b = v/24. This gives 48 possible scales,
which up to transposition reduce to the following four:

[1, 28/27, 9/8, 7/6, 9/7, 4/3, 81/56, 3/2, 14/9, 12/7, 7/4, 27/14]
[1, 28/27, 9/8, 7/6, 9/7, 4/3, 49/36, 3/2, 14/9, 12/7, 7/4, 49/27]
[1, 28/27, 9/8, 7/6, 9/7, 4/3, 49/36, 3/2, 14/9, 12/7, 7/4, 27/14]
[1, 49/48, 9/8, 7/6, 9/7, 21/16, 49/36, 3/2, 14/9, 27/16, 7/4, 49/27]

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> [1, 28/27, 9/8, 7/6, 9/7, 4/3, 49/36, 3/2, 14/9, 12/7, 7/4, 49/27]

If we take the above "aaron" scale and reduce it via the
superpythagorean temperament, which has commas of 245/243 and
1728/1715, we can add in a good bit of 5-limit harmony. The above
scale tempers to -7 to 4 in terms of a chain of superpythagorean sharp
fifth generators, which could be 13/22, 16/27 or 29/49. If we just
take the comma 245/243, we get a planar temperament, but the tuning is
pretty much the same as octacot, which has a TM basis 245/243 and
2401/2400, and which is the linear temperament which has a "nonoctave"
88 cent generator, plus octaves; we can take the generator to be 5/68.
This gives a great deal more tuning accuracy. Reduced to octacot, the
aaron scale becomes -13,-8,-5,-3,-2,0,3,5,6,8,11,16.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> This gives a Fokker block. We can get all the Fokker blocks possible
> from our pair of commas by choosing all pairs of integers 0<= u, v <
> 24 and then setting a = u/24, b = v/24. This gives 48 possible
scales,
> which up to transposition reduce to the following four:
>
> [1, 28/27, 9/8, 7/6, 9/7, 4/3, 81/56, 3/2, 14/9, 12/7, 7/4, 27/14]
> [1, 28/27, 9/8, 7/6, 9/7, 4/3, 49/36, 3/2, 14/9, 12/7, 7/4, 49/27]
> [1, 28/27, 9/8, 7/6, 9/7, 4/3, 49/36, 3/2, 14/9, 12/7, 7/4, 27/14]
> [1, 49/48, 9/8, 7/6, 9/7, 21/16, 49/36, 3/2, 14/9, 27/16, 7/4,
49/27]

Each of these turns out to have six supermajor triads (1--9/7--3/2
chords) and six subminor triads (1--7/6--3/2 chords.) They differ in
the number of 1--3/2--7/4 and 1--7/6--7/4 chords they possess, the
best being the second listed, with five each. A scl file for this is

! aaron.scl
akj 64/63 729/686 Fokker block
12
!
28/27
9/8
7/6
9/7
4/3
49/36
3/2
14/9
12/7
7/4
49/27
2

Like the rest of the scales, it is as expected a constant structure
and JI-epimorphic, but none of them is proper. The 3's exponents
range from -3 to 2, and the 7's from -1 to 2. The step sizes range
from 49/48 to 54/49, giving a high-low size ratio of 4.712. There are
a total of eight fifths. This scale seems suited to actual use if
anyone wants to tune it up.