Has this been done yet?

I did a little experiment just earlier:

I calculated some ratios for a 7-limit just scale and calculated their
31-tone equal and 53-tone equal values, and came up with something I call a
"justified 53-tone equal temperament". That's a 53-tone ET scale, but the
note assignments aren't based purely on the circle of fifths. The notes are
assigned according to their just intonation values set to the nearest note
in the 53-tET scale, with names decided by their place in the 31-tone scale.

So far I have:

Ratio Name Note #
2/1 C 53.00
3/2 G 31.00
4/3 F 22.00
5/4 E 17.06
5/3 A 39.06
6/5 Eb 13.94
7/6 D# 11.79
7/5 F# 25.73
7/4 A# 42.79
8/7 Ebb/D+ 10.21
8/5 Ab 35.94
9/8 D 9.01
9/7 Fb/E+ 19.22
9/5 Bb 44.94
10/9 D 8.06
10/7 Gb 27.27
12/7 Bbb/A+ 41.21
14/9 G# 33.78
15/14 Db 5.28
15/8 B 48.07
16/15 Db 4.93
16/9 Bb 43.99
21/20 C# 3.73
21/16 E# 20.79
25/24 C# 3.12
25/21 Eb 13.33
25/18 F# 25.12
25/16 G# 34.12
25/14 Bb 44.33
27/25 Db 5.88
27/20 F 22.95
27/16 A 44.01
27/14 Cb/B+ 50.22
28/25 D 8.67
32/27 Eb 12.99
32/25 Fb/E+ 18.88
32/21 Abb/G+ 32.21

I only went 7-limit because I feel an 11- or 13-limit scale is better
applied to a larger scale. Already I have three different D's in this
scale.

Coming soon: an attempt at a 31-tone keyboard -- not Fokker-style, but based
on my three-row "brown key" 17/19-tone keyboard. I place two rows of
buttons, seven to an octave each, for E- and B-sharp, C- and F-flat, the
double sharps and double flats (or Fokker's half-sharps and half-flats).

The tacky little pamphlet in your daddy's bottom drawer,

Danny

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--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:
> I did a little experiment just earlier:
>
> I calculated some ratios for a 7-limit just scale and calculated
their
> 31-tone equal and 53-tone equal values, and came up with something
I call a
> "justified 53-tone equal temperament". That's a 53-tone ET scale,
but the
> note assignments aren't based purely on the circle of fifths. The
notes are
> assigned according to their just intonation values set to the
nearest note
> in the 53-tET scale, with names decided by their place in the 31-
tone scale.
>
> So far I have:
>
> Ratio Name Note #
> 2/1 C 53.00
> 3/2 G 31.00
> 4/3 F 22.00
> 5/4 E 17.06
> 5/3 A 39.06
> 6/5 Eb 13.94
> 7/6 D# 11.79
> 7/5 F# 25.73
> 7/4 A# 42.79
> 8/7 Ebb/D+ 10.21
> 8/5 Ab 35.94
> 9/8 D 9.01
> 9/7 Fb/E+ 19.22
> 9/5 Bb 44.94
> 10/9 D 8.06
> 10/7 Gb 27.27
> 12/7 Bbb/A+ 41.21
> 14/9 G# 33.78
> 15/14 Db 5.28
> 15/8 B 48.07
> 16/15 Db 4.93
> 16/9 Bb 43.99
> 21/20 C# 3.73
> 21/16 E# 20.79
> 25/24 C# 3.12
> 25/21 Eb 13.33
> 25/18 F# 25.12
> 25/16 G# 34.12
> 25/14 Bb 44.33
> 27/25 Db 5.88
> 27/20 F 22.95
> 27/16 A 44.01
> 27/14 Cb/B+ 50.22
> 28/25 D 8.67
> 32/27 Eb 12.99
> 32/25 Fb/E+ 18.88
> 32/21 Abb/G+ 32.21

It sounds like you're describing what we call a "53-tone periodicity
block" around here. Can Gene or Manuel identify a unison vector basis
for it, assuming it really is a periodicity block, before I do?

BTW, if you're confused, go to:

http://www.ixpres.com/interval/td/erlich/intropblock1.htm

It's called a periodicity block because it tiles with copies of
itself, separated by tiny pitch intervals from one another, to fill
the entire infinite 7-limit lattice.

rom: "Paul Erlich" <paul@stretch-music.com>

> --- In tuning@y..., "Danny Wier" <dawier@y...> wrote:
> > I did a little experiment just earlier:
> >
> > I calculated some ratios for a 7-limit just scale and calculated
> their
> > 31-tone equal and 53-tone equal values, and came up with something
> I call a
> > "justified 53-tone equal temperament". That's a 53-tone ET scale,
> but the
> > note assignments aren't based purely on the circle of fifths. The
> notes are
> > assigned according to their just intonation values set to the
> nearest note
> > in the 53-tET scale, with names decided by their place in the 31-
> tone scale.

blah blah blah blah blah blah blah blah blah blah blah blah blah

> It sounds like you're describing what we call a "53-tone periodicity
> block" around here. Can Gene or Manuel identify a unison vector basis
> for it, assuming it really is a periodicity block, before I do?
>
> BTW, if you're confused, go to:
>
> http://www.ixpres.com/interval/td/erlich/intropblock1.htm
>
> It's called a periodicity block because it tiles with copies of
> itself, separated by tiny pitch intervals from one another, to fill
> the entire infinite 7-limit lattice.

I'm thinking "circular lattice" in this case. I chose 53-tone because of
its historical significance. Also, I decided to switch over to half-sharp
notation when I reached E-double sharp and half-flat notation when I reached
C-double flat, because I did NOT want to get into triple sharps and triple
flats. I use this for 50-tet. Since the juxtaposition of X-sharp and
Y-flat is reversed in 53-tone, the rules obviously change.

So there's a lot of work to be done here. Now I must play my flute badly
out of tune -- I mean, improvise in 97-tone 1/4 comma meantone.

The other white meat,
~DaW~

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--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:
> rom: "Paul Erlich" <paul@s...>
>
> > --- In tuning@y..., "Danny Wier" <dawier@y...> wrote:
> > > I did a little experiment just earlier:
> > >
> > > I calculated some ratios for a 7-limit just scale and calculated
> > their
> > > 31-tone equal and 53-tone equal values, and came up with
something
> > I call a
> > > "justified 53-tone equal temperament". That's a 53-tone ET
scale,
> > but the
> > > note assignments aren't based purely on the circle of fifths.
The
> > notes are
> > > assigned according to their just intonation values set to the
> > nearest note
> > > in the 53-tET scale, with names decided by their place in the
31-
> > tone scale.
>
> blah blah blah blah blah blah blah blah blah blah blah blah blah
>
> > It sounds like you're describing what we call a "53-tone
periodicity
> > block" around here. Can Gene or Manuel identify a unison vector
basis
> > for it, assuming it really is a periodicity block, before I do?
> >
> > BTW, if you're confused, go to:
> >
> > http://www.ixpres.com/interval/td/erlich/intropblock1.htm
> >
> > It's called a periodicity block because it tiles with copies of
> > itself, separated by tiny pitch intervals from one another, to
fill
> > the entire infinite 7-limit lattice.
>
> I'm thinking "circular lattice" in this case.

You can look at it that way, if you temper out one or more of the
unison vectors. Otherwise, there isn't much "circular" about it, as
far as I can see. The fact that it approximates an ET is already
implicit in the tiling property. That's because iteratively
proceeding by any interval 53 times in the lattice, one returns to a
copy (separated by a few of those tiny pitch intervals mentioned
above) of the same note one started with, which is a hallmark of 53-
tET. Is that what you meant by "circular", or did you mean something
else?

--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:

> So far I have:
>
> Ratio Name Note #
> 2/1 C 53.00
> 3/2 G 31.00
> 4/3 F 22.00
> 5/4 E 17.06
> 5/3 A 39.06
> 6/5 Eb 13.94
> 7/6 D# 11.79
> 7/5 F# 25.73
> 7/4 A# 42.79
> 8/7 Ebb/D+ 10.21
> 8/5 Ab 35.94
> 9/8 D 9.01
> 9/7 Fb/E+ 19.22
> 9/5 Bb 44.94
> 10/9 D 8.06
> 10/7 Gb 27.27
> 12/7 Bbb/A+ 41.21
> 14/9 G# 33.78
> 15/14 Db 5.28
> 15/8 B 48.07
> 16/15 Db 4.93
> 16/9 Bb 43.99
> 21/20 C# 3.73
> 21/16 E# 20.79
> 25/24 C# 3.12
> 25/21 Eb 13.33
> 25/18 F# 25.12
> 25/16 G# 34.12
> 25/14 Bb 44.33
> 27/25 Db 5.88
> 27/20 F 22.95
> 27/16 A 44.01
> 27/14 Cb/B+ 50.22
> 28/25 D 8.67
> 32/27 Eb 12.99
> 32/25 Fb/E+ 18.88
> 32/21 Abb/G+ 32.21

Oops -- I didn't realize you didn't have all 53 notes yet! And I also
didn't realize that you have two different ratios for note #19: 9/7
and 32/25. Having both of these ratios in your scale means it can't
be a 53-tone periodicity block. Perhaps you want to eliminate one of
them.

Also note #5 is being assigned to both 15/14 and 16/15, note #44 is
being assigned to both 25/14 and 16/9, and note #13 is being assigned
to both 25/21 and 32/27. Same deal.

Here's what a lattice of this scale so far looks like:

25/18-----25/24-----25/16
/ \`. ,'/ \`. ,'/ \
/ \25/21/---\25/14/ \
/ \ |\/ \/|\/ \
/ \|/\ /\|/\ \
10/9-------5/3-------5/4------15/8
/|\ /|\`.\ /,'/|\`.\ ,'/|\
/ | \ / | \10/7-/ | \15/14/ | \
/14/9-------7/6-------7/4------21/16\
/,' `.\ /,' \`.\|/,'/ \`.\|/,'/ `.\
32/27-----16/9-------4/3-----\-1/1-/---\-3/2-/-----9/8------27/16
\`. ,'/ \`. / \/|\/ \ ,\/|\/. ,'/ \`. ,'/
\32/21/---\-8/7-/\|/\12/7-/\|/\-9/7-/---\27/14/
\ | / \ | / 7/5------21/20\ | / \ | /
\|/ \|/,'/ `.\|/,' `.\|/ \|/
16/15------8/5-/-----6/5-------9/5------27/20
/| / \ /
/ |/ \ /
/28/25 \ /
/,' \ /
32/25 27/25

This shows you the consonant just intervals and chords you'll have in
your scale. Do you see that?

Don't you think that, if you did include all these ratios, you'd want
to also include ratios such as 48/25 and 36/25? These seem to involve
more complex ratios, but don't really -- if you were to look at the
octave range _below_ 1/1 instead of that _above_ 1/1, these ratios
would be written at 24/25 and 18/25, while your current 25/24 and
25/18 would become 25/48 and 25/36.

From: "Paul Erlich" <paul@stretch-music.com>

> Oops -- I didn't realize you didn't have all 53 notes yet! And I also
> didn't realize that you have two different ratios for note #19: 9/7
> and 32/25. Having both of these ratios in your scale means it can't
> be a 53-tone periodicity block. Perhaps you want to eliminate one of
> them.

Both ratios are approximations of the 19th note in the 53-equal scale, and
the scale I had in mind is 53-tet tuning. So both apply. Or you could
match it with the smaller interval (9/7).

> Also note #5 is being assigned to both 15/14 and 16/15, note #44 is
> being assigned to both 25/14 and 16/9, and note #13 is being assigned
> to both 25/21 and 32/27. Same deal.

Again, two approximations each of the 5th, 13th and 44th degress on the
53-equal scale. In the future I'll do this to an even larger
equal-temperament scale, perhaps with hundreds of tones.

> Don't you think that, if you did include all these ratios, you'd want
> to also include ratios such as 48/25 and 36/25? These seem to involve
> more complex ratios, but don't really -- if you were to look at the
> octave range _below_ 1/1 instead of that _above_ 1/1, these ratios
> would be written at 24/25 and 18/25, while your current 25/24 and
> 25/18 would become 25/48 and 25/36.

Oh of course I would! I have to keep going until I fill all 53 slots and
name all 53 notes. When I finish it, my "53-tone justified equal
temperament" scale will be formally presented and included in the Files
section. I'll get to it soon....

~DaW~

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--- In tuning@y..., "Danny Wier" <dawier@y...> wrote:
> From: "Paul Erlich" <paul@s...>
>
> > Oops -- I didn't realize you didn't have all 53 notes yet! And I
also
> > didn't realize that you have two different ratios for note #19:
9/7
> > and 32/25. Having both of these ratios in your scale means it
can't
> > be a 53-tone periodicity block. Perhaps you want to eliminate one
of
> > them.
>
> Both ratios are approximations of the 19th note in the 53-equal
scale, and
> the scale I had in mind is 53-tet tuning. So both apply. Or you
could
> match it with the smaller interval (9/7).
>
> > Also note #5 is being assigned to both 15/14 and 16/15, note #44
is
> > being assigned to both 25/14 and 16/9, and note #13 is being
assigned
> > to both 25/21 and 32/27. Same deal.
>
> Again, two approximations each of the 5th, 13th and 44th degress on
the
> 53-equal scale. In the future I'll do this to an even larger
> equal-temperament scale, perhaps with hundreds of tones.

171-tET would work extremely well for this particular problem.

> > Don't you think that, if you did include all these ratios, you'd
want
> > to also include ratios such as 48/25 and 36/25? These seem to
involve
> > more complex ratios, but don't really -- if you were to look at
the
> > octave range _below_ 1/1 instead of that _above_ 1/1, these ratios
> > would be written at 24/25 and 18/25, while your current 25/24 and
> > 25/18 would become 25/48 and 25/36.
>
> Oh of course I would! I have to keep going until I fill all 53
slots and
> name all 53 notes. When I finish it, my "53-tone justified equal
> temperament" scale will be formally presented and included in the
Files
> section. I'll get to it soon....

Oh, so that's what you mean. Well, then, this may help. Here is a 53-
tone 7-limit Fokker periodicity block, which approximates the full 53-
tET rather well, without redundancy, and with the smallest ratios of
any of the 62 versions I tried:

cents numerator denominator
0 1 1
34.976 50 49
48.77 36 35
70.672 25 24
84.467 21 20
119.44 15 14
146.71 160 147
155.14 35 32
182.4 10 9
203.91 9 8
231.17 8 7
252.68 81 70
266.87 7 6
301.85 25 21
315.64 6 5
350.62 60 49
351.34 49 40
386.31 5 4
413.58 80 63
435.08 9 7
449.27 35 27
470.78 21 16
498.04 4 3
519.55 27 20
546.82 48 35
568.72 25 18
582.51 7 5
617.49 10 7
631.28 36 25
653.18 35 24
680.45 40 27
701.96 3 2
729.22 32 21
750.73 54 35
764.92 14 9
786.42 63 40
813.69 8 5
848.66 80 49
849.38 49 30
884.36 5 3
898.15 42 25
933.13 12 7
947.32 140 81
968.83 7 4
996.09 16 9
1017.6 9 5
1044.9 64 35
1053.3 147 80
1080.6 28 15
1115.5 40 21
1129.3 48 25
1151.2 35 18
1165 49 25

I'd draw a lattice if I had time. For those interested, the unison
vectors defining this FPB are 225:224, 1728:1715, and 4000:3969 (if I
counted right).