Decimal conversion chart

This chart gives miracle-based notation in multiple forms. 31-equal gives
the simplest notes using meantone with ^ and v for half sharps and flats
respectively. The 72-equal column uses / and \ for 1 steps shifts, and ^
and v for 3 step shifts. Where convenient, these are relative to the same
notes as the previous column. Finally, the equivalent cents deviations
from 12-equal are given.

31-equal 72-equal 12=+cents

9^^^ B# B^ B+50
9^^ B^ B^\\ B+17
9^ B B\ B-17
9 Bv Bv B-50
9v Bb Bb/ Bb+17
9vv A# A#\ A#-17
9vvv A^ A^ A+50

8^^^ Bv Bv\ Bb+33
8^^ Bb Bb Bb
8^ A# A#\\ A#-33
8 A^ A^\ A+33
8v A A A+0
8vv Av Av/ A-33
8vvv Ab Ab// Ab+33

7^^^ A^ A^\\ A+17
7^^ A A\ A-17
7^ Av Av A-50
7 Ab Ab/ Ab+17
7v G# G#\ G#-17
7vv G^ G^ G+50
7vvv G G/ G+17

6^^^ Ab Ab Ab+0
6^^ G# G#\\ G#-33
6^ G^ G^\ G+33
6 G G G+0
6v Gv Gv/ Gb+33
6vv Gb Gb// Gv-33
6vvv F# F# F#+0

5^^^ G G\ G-17
5^^ Gv Gv G-50
5^ Gb Gb/ Gb+17
5 F# F#\ F#-17
5v F^ F^ F+50
5vv F F/ F+17
5vvv Fv Fv// F-17

4^^^ F# F#\\ F#-33
4^^ F^ F^\ F+33
4^ F F F+0
4 Fv Fv/ F-33
4v E^ E^\ E+33
4vvv E E E+0

3^^^ Fv Fv F-50
3^^ E^ E^\\ E+17
3^ E E\ E-17
3 Ev Ev E-50
3v Eb Eb/ Eb+17
3vvv D# D#\ D#-17

2^^^ Ev Ev\ E+33
2^^ Eb Eb Eb+0
2^ D# D#\\ D#-33
2 D^ D^\ D+33
2v D D D+0
2vv Dv Dv/ D-33
2vvv Db Db// Db+33

1^^^ D^ D^\\ D+17
1^^ D D\ D-17
1^ Dv Dv D+50
1 Db Db/ Db+17
1v C# C#\ C#-17
1vv C^ C^ C+50
1vvv C C/ C+17

0^^^ Db Db Db+0
0^^ C# C#\\ C#-33
0^ C^ C^\ C+33
0 C C C+0
0v Cv Cv/ C-33
0vv Cb Cb// Cb+33
0vvv B# B# B#+0

--- In tuning@y..., graham@m... wrote:

/tuning/topicId_22889.html#22889

> This chart gives miracle-based notation in multiple forms.
>
> ...
>
> 7vv G^ G^ G+50
> 7vvv G G/ G+17
>
> 6^^^ Ab Ab Ab+0
> 6^^ G# G#\\ G#-33
> 6^ G^ G^\ G+33
> 6 G G G+0
> 6v Gv Gv/ Gb+33
> 6vv Gb Gb// Gv-33
> 6vvv F# F# F#+0
>
> 5^^^ G G\ G-17
> 5^^ Gv Gv G-50
> 5^ Gb Gb/ Gb+17
>
> ...
>
> 1vv C^ C^ C+50
> 1vvv C C/ C+17
>
> 0^^^ Db Db Db+0
> 0^^ C# C#\\ C#-33
> 0^ C^ C^\ C+33
> 0 C C C+0
> 0v Cv Cv/ C-33
> 0vv Cb Cb// Cb+33
> 0vvv B# B# B#+0

Graham,

I've begun a table which recapitulates this one, adds my
ASCII 72-EDO notation alongside your 72-tET, and then puts
the scale in order of pitch-height.

The first part I quoted, from 7vv to 6vvv, seems to me like
it has an error somewhere in the 72-tET and/or 12=+cents
(cute name, BTW).

The second part, seems to have an error in that it looks
like the last note should be B not B#.

The first (highest [*]) part of my table looks like this:

(Where Graham's table was ordered by decimal notation,
mine is ordered by the 72- and 12-EDO notations.)

Breed Breed Breed ASCII
decimal 31-EDO 72-EDO 72-EDO 12=+cents

0 C C C C+0
0v Cv Cv/ C< C-33
9^^^ B# B^ Cv|B^ B+50
0vv Cb Cb// B> Cb+33
9^^ B^ B^\\ B+ B+17
0vvv B# B# [?] B B+0 **
9^ B B\ B- B-17
9 Bv Bv Bv|Bb^ B-50
8^^^ Bv Bv\ Bb> Bb+33
9v Bb Bb/ Bb+ Bb+17
8^^ Bb Bb Bb Bb
9vv A# A#\ A#- A#-17
8^ A# A#\\ A#< A#-33
9vvv A^ A^ A#v|A^ A+50
8 A^ A^\ A> A+33
7^^^ A^ A^\\ A+ A+17
8v A A A A+0

[*] to Dave Keenan: Sorry, Dave, I decided a really long
time ago that it simply did not make sense to put the lowest
note in a table of pitches at the top and the highest note
at the bottom.

Thinking about your criticism that Scala can't read it
the way I write it prompted this idea:

Scala should have an option where the user can list a scale
in either direction, and have a remark about which one is
the default, if there was/is a default.

Hmmm... there should be an option where the user could show
a window with historical and theoretical commentary about the
scale, with links to audio files, score graphics, other
webpages, etc. The user could edit this to add more links
or data if he wishes.

Sound good, Manuel? Seems like it would be pretty easy
to add these capabilities - at least the bi-directional
listing one.

-monz
http://www.monz.org
"All roads lead to n^0"

Thanks for that Graham. I assume we'd need two more notes (0vvvv and
9^^^^) to cover all of 72-EDO?

Is there any good reason to align 0 with C? Wouldn't it be better to
align the points of symmetry of both the 7 nominal and the 10 nominal
notations. Of course we have a problem, because while D is the point
of symmetry in the 7-nominal notations, 10 is even, so there is no
such note. However the 21, 31 and 41 tone miracle chains all have a
point of symmetry. What note is that? By my thinking it it should be
either 4 or 5. Is there a good reason to choose one of these over the
other? When we go from M-10 to M-21 should we add 6 generators up and
5 down, or the other way?

-- Dave Keenan

I have a preference for aligning 4 with D. This allows folks who just
can't handle using digits for their nominals (think of the potential
for confusion when octave numbers are added), to use the following
letters for the 10 nominals (a modification of Manuel's suggestion).

Decml 31 72 12-ET + cents
-----------------------------
0 A A^ A^\ A#-33 or A -67
1 B Bv Bv B -50 or Bb+50
2 C Cv Cv/ C -33
3 K C# C#\ C#-17
4 D D D D
5 L Eb Eb/ Eb+17
6 E E E^\ E +33
7 F F^ F^ F +50 or F#-50
8 G Gv Gv/ Gb+33 or G-67
9 H G# G#\ G#-17

There is also the meantone precedent for favouring G# over Ab.

Note that the 31-tone naming scheme can be also used for miracle
scales tuned to a 7/72 oct generator (not only the horrible 3/31),
just so long as they are subsets of Canasta (M-31) (or are Canasta
itself).

-- Dave Keenan

Oops. Some mistakes in the cents. It should have been:

Decml 31 72 12-ET + cents
-----------------------------
0 A A^ A^\ A#-67 or A +33
1 B Bv Bv B -50 or Bb+50
2 C Cv Cv/ C -33
3 K C# C#\ C#-17
4 D D D D
5 L Eb Eb/ Eb+17
6 E E E^\ E +33
7 F F^ F^ F +50 or F#-50
8 G Gv Gv/ Gb+67 or G -33
9 H G# G#\ G#-17

Graham, I think some of those cents errors came from your table.

-- Dave Keenan

Dave Keenan wrote:

> Thanks for that Graham. I assume we'd need two more notes (0vvvv and
> 9^^^^) to cover all of 72-EDO?

Yes, I hadn't noticed that. But for converting from 72-EDO you'd want
the table the other way round, and also include decimal enharmonies.

If you're finding errors, you could try posting a revised table. It
was all done by hand, but I think I could write a program to do the
job.

I also meant to reverse the order before I posted it.

> Is there any good reason to align 0 with C? Wouldn't it be better to
> align the points of symmetry of both the 7 nominal and the 10
nominal
> notations. Of course we have a problem, because while D is the point
> of symmetry in the 7-nominal notations, 10 is even, so there is no
> such note. However the 21, 31 and 41 tone miracle chains all have a
> point of symmetry. What note is that? By my thinking it it should be
> either 4 or 5. Is there a good reason to choose one of these over
the
> other? When we go from M-10 to M-21 should we add 6 generators up
and
> 5 down, or the other way?

The reason is that the logarithm of the frequency in Hz is an
approximation to the decimal pitch. If you can get it to line up with
meantone names, that might be a better reason for a different
standard.

The symmetry will be between 4 and 5. I chose 5 for the Partch
diagrams, partly because vw is easier to write than ^^^, but it
doesn't really matter.

Graham

monz wrote:
> I've begun a table which recapitulates this one, adds my
> ASCII 72-EDO notation alongside your 72-tET, and then puts
> the scale in order of pitch-height.
>
>
> The first part I quoted, from 7vv to 6vvv, seems to me like
> it has an error somewhere in the 72-tET and/or 12=+cents
> (cute name, BTW).

Gb and Gv are the wrong way round in the last column. I think the /
and \ are corredct.

> The second part, seems to have an error in that it looks
> like the last note should be B not B#.

Yes, swap B# for B everywhere, so the row above would be simpler as

0vv B^ B^\ B+33

> Scala should have an option where the user can list a scale
> in either direction, and have a remark about which one is
> the default, if there was/is a default.

It'd be a fairly easy Python script for reversing the order. Like

text = open('decimal.conversion').readlines()
text.reverse()
open('decimal.conversion.reordered','w').write('\n'.join(text))

Graham

Joe,

>Thinking about your criticism that Scala can't read it
>the way I write it prompted this idea:

Scales can be read with the notes in any order. Typing
"sort" after reading will put them right.

>Scala should have an option where the user can list a scale
>in either direction, and have a remark about which one is
>the default, if there was/is a default.

It's the first time I get this request so I think there's not
a high demand for a feature like this but I'll think about it.
It would involve a change to several routines.

>Hmmm... there should be an option where the user could show
>a window with historical and theoretical commentary about the
>scale, with links to audio files, score graphics, other
>webpages, etc. The user could edit this to add more links
>or data if he wishes.

This could go along with the suggestion to add text files with
commentary to the archive. Until now I haven't received any.
Links to webpages change too often for a scale archive that
should last a long time, at least I wouldn't want to maintain
them.

Manuel