Scales found within 19 eq and 31 eq

Thanks to every one who replied to my request on 22 eq scales. My question
now is, can anyone provide similar information for 19 eq and 31 eq? Thanks.

Rick McGowan:
> Interesting. Have you a pointer to a web page or a ratio list, where I
> could get formula for construction of such a tuning?

Kornerup's temperament with fifth of (15 - sqrt 5) / 22 octaves
1: 73.501 cents
2: 118.928 cents
3: 192.429 cents
4: 265.930 cents
5: 311.357 cents
6: 384.858 cents
7: 458.359 cents
8: 503.786 cents
9: 577.287 cents
10: 622.713 cents
11: 696.215 cents
12: 769.716 cents
13: 815.142 cents
14: 888.643 cents
15: 962.145 cents
16: 1007.571 cents
17: 1081.072 cents
18: 1154.574 cents
19: 2/1

This is a cycle of fifths. You can make it shorter or longer as desired
like Paul described.

Paul Erlich:
> 2 2 2 2 1 2 2 2 2 2 2 2 1 (a mode of the hexachordal dodecatonic scale)
One note too many:
2 2 2 2 1 2 2 2 2 2 2 1

Chris Miller:
> My question now is, can anyone provide similar information for 19 eq
> and 31 eq?
Find it in this list:
ftp://ella.mills.edu/ccm/tuning/papers/modename.txt

Manuel Op de Coul coul@ezh.nl

Thanks. This is extremely helpful. Now I've got some experimenting to do :)

>===== Original Message From tuning@onelist.com =====
>From: <manuel.op.de.coul@ezh.nl>
>
>Find it in this list:
>ftp://ella.mills.edu/ccm/tuning/papers/modename.txt
>
>Manuel Op de Coul coul@ezh.nl
>
>
>
>--------------------------- ONElist Sponsor ----------------------------
>
>ONElist now has T-SHIRTS!
>For details and to order, go to:
><a href=" http://clickme.onelist.com/ad/tshirt1 ">Click Here</a>
>
>------------------------------------------------------------------------
>You do not need web access to participate. You may subscribe through
>email. Send an empty email to one of these addresses:
> tuning-subscribe@onelist.com - subscribe to the tuning list.
> tuning-unsubscribe@onelist.com - unsubscribe from the tuning list.
> tuning-digest@onelist.com - switch your subscription to digest mode.
> tuning-normal@onelist.com - switch your subscription to normal mode.

Chris Miller wrote,

>Thanks to every one who replied to my request on 22 eq scales. My question

>now is, can anyone provide similar information for 19 eq and 31 eq?
Thanks.

19 and 31 are meantone tunings; hence any conventionally notated scales that
do not rely on enharmonic equivalence will work in 19 and 31. That means
that all the modes of the normal diatonic scale, melodic minor, harmonic
minor, harmonic major, gypsy, neapolitan, etc. scales will work in 19 and
31. 31 can also handle the Arabic diatonic scale, which would be 5 4 4 5 5 4
4, and all its modes. 31 is excellent for unambiguously approximating
7-limit harmony; hence any of Erv Wilson's (1.3.5.7) CPS structures would
work in 31, as would the Fokker/Lumma/Keenan 12-tone scale, which would be 3
2 2 3 3 2 3 3 2 2 3 3 (while a traditional meantone chromatic scale would be
2 3 3 2 3 2 3 2 3 3 2 3). I'd be happy to elaborate on any of these ideas if
you're interested.

> [Paul Erlich, TD 294.19]
>
> 19 and 31 are meantone tunings; hence any conventionally
> notated scales that do not rely on enharmonic equivalence
> will work in 19 and 31.

While the rest of Paul's post accurately describes the
implications or approximations of 19- and 31-eq to meantone
tunings, it is important to note that, while these ETs
are close enough that for all intents and purposes they can
be assumed to be identical to their respective meantones,
strictly speaking, by mathematical definition, 19- and 31-eq
are *not* meantone tunings.

Nitpicking, perhaps, but one should be careful about making
statements that 'x and y ARE z', when in fact an approximation
is involved (albeit, in this case, an extremely close one).

BTW, it's nice (to me) to see someone else using my preferred
abbreviation of 'n-eq' instead of the more usual 'n-tET',
which I've lately even been using myself.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
--------------------------------------------------

___________________________________________________________________
Get the Internet just the way you want it.
Free software, free e-mail, and free Internet access for a month!
Try Juno Web: http://dl.www.juno.com/dynoget/tagj.

In-Reply-To: <935749240.7415@onelist.com>
Chris Miller, digest 294.1, wrote:

> Thanks. This is extremely helpful. Now I've got some experimenting to
> do :)

Does that mean you don't want any more?

On my website at http://www.cix.co.uk/~gbreed/genera.htm are a list of
scales with interesting harmonic properties, all defined in 12- and
19-equal. To get 31-equal, add the two columns together. I haven't tried
them all myself, so for all I know they might be really good.

Also, the two scales right at the bottom of

http://www.cix.co.uk/~gbreed/blues.htm

are in 31-equal.

I've also found this scale in 19-equal:

4 1 4 1 4 1 4

to be surprisingly good with a celesta sound.

Thanks. I'm always looking for something interesting to play with.

>===== Original Message From tuning@onelist.com =====
>From: gbreed@cix.compulink.co.uk (Graham Breed)
>
>
>Does that mean you don't want any more?
>
>On my website at http://www.cix.co.uk/~gbreed/genera.htm are a list of
>scales with interesting harmonic properties, all defined in 12- and
>19-equal. To get 31-equal, add the two columns together. I haven't tried
>them all myself, so for all I know they might be really good.
>

I'll check them out.

>Also, the two scales right at the bottom of
>
>http://www.cix.co.uk/~gbreed/blues.htm
>
>are in 31-equal.
>
>I've also found this scale in 19-equal:
>
>4 1 4 1 4 1 4
>
>to be surprisingly good with a celesta sound.
>

Thanks alot.

>
>--------------------------- ONElist Sponsor ----------------------------
>
>GET WHAT YOU DESERVE! A NextCard Platinum VISA: DOUBLE Rewards points,
> NO annual fee & rates as low as 9.9 percent FIXED APR.
>Apply online today! http://www.onelist.com/ad/nextcard1
>
>------------------------------------------------------------------------
>You do not need web access to participate. You may subscribe through
>email. Send an empty email to one of these addresses:
> tuning-subscribe@onelist.com - subscribe to the tuning list.
> tuning-unsubscribe@onelist.com - unsubscribe from the tuning list.
> tuning-digest@onelist.com - switch your subscription to digest mode.
> tuning-normal@onelist.com - switch your subscription to normal mode.

> From: Chris Miller <vogonpoet@MailAndNews.com>
> 2 2 3 3 2 3 3 2 2 3 3 (while a traditional meantone chromatic scale would be
> 2 3 3 2 3 2 3 2 3 3 2 3). I'd be happy to elaborate on any of these ideas if
> you're interested.
>

Please do, as I've been ponderring 12-out-of-N tunings due to technology
constraints, and 31tet because of its unambiguous 7-goodness.

The chromatic scale I'd been planning on using was

2 3 3 2 3 2 3 3 2 3 2 3

C C# D Eb E F F# G Ab A Bb B

rather than the

C C# D Eb E F F# G G# A Bb B

you showed, but this is just a rotation and doesn't do anything in
particular better.

An alternative that makes less theoretical sense but has a few
nice properties comes from saying "what if 31tet had sharps higher
than flats", and then building the sequence.

3 2 2 3 3 2 2 3 2 3 3

C C#' D Eb' E F F#' G Ab' A Bb' B

0 : =1/1
3 : =14/13
5 : =9/8
7 : =7/6
10 : =5/4
13 : =4/3
16 : =10/7
18 : =3/2
20 : =11/7(?) =25/16
23 : =5/3
25 : =7/4
28 : =15/8

Besides the 'pull' in the leading tones, more of the 31-ness is
presented.

Bob Valentine

[Robert C Valentine:]
> An alternative that makes less theoretical sense but has a few nice
properties comes from saying "what if 31tet had sharps higher than
flats", and then building the sequence.

> 0 : =1/1
> 3 : =14/13
> 5 : =9/8
> 7 : =7/6
> 10 : =5/4
> 13 : =4/3
> 16 : =10/7
> 18 : =3/2
> 20 : =11/7(?) =25/16
> 23 : =5/3
> 25 : =7/4
> 28 : =15/8
>

How about the I-IV-V as +5/4 & +6/5 (+10 & +8) and its inversions as
+9/7 & +7/6 (-11 & -7):

0-10-18
13-23-31
18-28-5

31-20-13
18-7-0
13-2-26

or:

1/1, 5/4, 3/2
4/3, 5/3, 2/1
3/2, 15/8, 9/8

2/1, 14/9, 4/3,
3/2, 7/6, 1/1
4/3, 28/27, 16/9

Where:

C Db D D# E F Gb G G# A A# B C 0 3 5 7 10 13 16 18 20 23 25 28 31
0 2 4 7 10 13 15 17 20 22 25 28 31
0 2 5 8 11 13 15 18 20 23 26 29 31
0 3 6 9 11 13 16 18 21 24 27 29 31
0 3 6 8 10 13 15 18 21 24 26 28 31
0 3 5 7 10 12 15 18 21 23 25 28 31
0 2 4 7 9 12 15 18 20 22 25 28 31
0 2 5 7 10 13 16 18 20 23 26 29 31
0 3 5 8 11 14 16 18 21 24 27 29 31
0 2 5 8 11 13 15 18 21 24 26 28 31
0 3 6 9 11 13 16 19 22 24 26 29 31
0 3 6 8 10 13 16 19 21 23 26 28 31
0 3 5 7 10 13 16 18 20 23 25 28 31

becomes:

C C# D D# E F Gb G G# A Bb B C
0 2 5 7 10 13 16 18 20 23 26 28 31
0 3 5 8 11 14 16 18 21 24 26 29 31
0 2 5 8 11 13 15 18 21 23 26 28 31
0 3 6 9 11 13 16 19 21 24 26 29 31
0 3 6 8 10 13 16 18 21 23 26 28 31
0 3 5 7 10 13 15 18 20 23 25 28 31
0 2 4 7 10 12 15 17 20 22 25 28 31
0 2 5 8 10 13 15 18 20 23 26 29 31
0 3 6 8 11 13 16 18 21 24 27 29 31
0 3 5 8 10 13 15 18 21 24 26 28 31
0 2 5 7 10 12 15 18 21 23 25 28 31
0 3 5 8 10 13 16 19 21 23 26 29 31
0 2 5 7 10 13 16 18 20 23 26 28 31

Dan

Robert!

This is the 13/31MOS, but maybe you already realized this!

Robert C Valentine wrote:

>
>
> The chromatic scale I'd been planning on using was
>
> 2 3 3 2 3 2 3 3 2 3 2 3
>
> C C# D Eb E F F# G Ab A Bb B
>
> rather than the
>
> C C# D Eb E F F# G G# A Bb B
>
> you showed, but this is just a rotation and doesn't do anything in
> particular better.
>
> An alternative that makes less theoretical sense but has a few
> nice properties comes from saying "what if 31tet had sharps higher
> than flats", and then building the sequence.
>
> 3 2 2 3 3 2 2 3 2 3 3
>
> C C#' D Eb' E F F#' G Ab' A Bb' B
>
> 0 : =1/1
> 3 : =14/13
> 5 : =9/8
> 7 : =7/6
> 10 : =5/4
> 13 : =4/3
> 16 : =10/7
> 18 : =3/2
> 20 : =11/7(?) =25/16
> 23 : =5/3
> 25 : =7/4
> 28 : =15/8
>
> Besides the 'pull' in the leading tones, more of the 31-ness is
> presented.
>
> Bob Valentine
>
> .

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

Bob Valentine wrote:

> An alternative that makes less theoretical sense but has a few
> nice properties comes from saying "what if 31tet had sharps higher
> than flats", and then building the sequence.
>
> 3 2 2 3 3 2 2 3 2 3 3
>
> C C#' D Eb' E F F#' G Ab' A Bb' B
>
> 0 : =1/1
> 3 : =14/13

I would never use ratios of 13 to describe 31-tET, since 31-tET is not
consistent in the 13-limit.

> 5 : =9/8
> 7 : =7/6
> 10 : =5/4
> 13 : =4/3
> 16 : =10/7
> 18 : =3/2
> 20 : =11/7(?) =25/16
> 23 : =5/3
> 25 : =7/4
> 28 : =15/8
>
> Besides the 'pull' in the leading tones, more of the 31-ness is
> presented.

Kraig Grady wrote,

>This is the 13/31MOS, but maybe you already realized this!

How can it be the 13/31 MOS if it only has 12 notes? Can the 13/31 MOS have
the C-major scale in it? I don't think so!

[this is where I mentioned the Fokker/Lumma/Keenan tuning; an initial "3"
was snipped from the following]:
>> 2 2 3 3 2 3 3 2 2 3 3 (while a traditional meantone chromatic scale would
be
>> 2 3 3 2 3 2 3 2 3 3 2 3). I'd be happy to elaborate on any of these ideas
if
>> you're interested.
>

>Please do, as I've been ponderring 12-out-of-N tunings due to technology
>constraints, and 31tet because of its unambiguous 7-goodness.

Although the original idea dates back to Fokker, and was rediscovered by
Carl Lumma, Dave Keenan described both the 31-tET and the "wafso-just"
versions of the scale with 12 notes and 6 consonant 7-limit tetrads. Keenan
has diagrammed the scale as:

f---------c
/ \ /
/ \ /
a---------e---------b---------f#
/|\ /|\`. /,'/ \`.\ /,'/
/ | \ / | \ db-/---\--ab /
/ d#--------a# \ | / \ | /
/,'/ \`.\ /,'/ `.\|/ \|/
f--/---\--c--/------g---------d
/ \ | /
/ \|/
b---------f#

The six consonant tetrads are shown as tetrahedra

The 31-tET version of this scale is clear. In the "wafso-just" version, all
six consonant tetrads have all intervals within 2 cents of JI. These are the
offsets from 12-tET in the wafso-just version.

C 0.0
Db 15.6
D 0.1
D# -31.2
E -15.6
F 0.0
F# -15.6
G 0.0
Ab 15.6
A -15.6
A# -31.2
B -15.6

Unfortunately, the fifth D-A is 18 cents flat in this version, which may
lead one to prefer the 31-tET version.

You can get more consonant 7-limit tetrads by using 12 notes out of
22-equal. But first, how do you like the 7-limit tetrads in 22-equal?

"Paul H. Erlich" wrote:

> How can it be the 13/31 MOS if it only has 12 notes? Can the 13/31 MOS have
> the C-major scale in it? I don't think so!

The 13/31 generator produces MOS scales that are 1,2,3,5,7,12,19 tones

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com

I wrote,

>> How can it be the 13/31 MOS if it only has 12 notes? Can the 13/31 MOS
have
>> the C-major scale in it? I don't think so!

Kraig Grady wrote,

>The 13/31 generator produces MOS scales that are 1,2,3,5,7,12,19 tones

I'm sorry, I didn't understand your terminology. You were referring to 13/31
octave as a generator, while I thought you meant a 13-note MOS in 31-tone.

Kraig, my misunderstanding came about because it seemed you were talking
about Bob Valentine's scale, which is not an MOS. You were in fact talking
about the meantone scale mentioned earlier. Right?

"Paul H. Erlich" wrote:

> From: "Paul H. Erlich" <PErlich@Acadian-Asset.com>
>
> Kraig, my misunderstanding came about because it seemed you were talking
> about Bob Valentine's scale, which is not an MOS. You were in fact talking
> about the meantone scale mentioned earlier. Right?

It was the 12 tone subset that goes (I don't remember of hand where in the
sequence he started it)
\3-3-2-3-2-3-3-2-3-2-3-2

-- Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com