A diatonic scale

Is anyone familiar with the scale MmMmMss, where M is a major tone, m is a minor tone, and s is a semitone? In both the 46 and 72 ets, it turns out to be quite a good scale; particularly in the 72-et.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

> Is anyone familiar with the scale MmMmMss, where M is a major tone, m is a minor tone, and s is a semitone? In both the 46 and 72 ets, it turns out to be quite a good scale; particularly in the 72-et.

Here's an analysis of this scale. One mode of it in the 5-limit can be considered an RI scale, since 0-7-19-30-42-53-65 approximates

1--16/15--6/5--4/3--3/2--5/3--15/8

If we temper by 225/224~1, we may include a 7/5 interval from
16/15~15/14 to 3/2, and a 7/4 interval from 16/15 to 28/15 or
15/14 to 15/8, depending on how we want to look at it. In the 9-limit, we may add to this a 14/9 from 15/14 to 5/3 and another from
6/5 to 28/15. We therefore have quit a bit of additional harmony, with the considerable tuning accuracy the 225/224 temperament (or the 72-et) affords.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_32461.html#32461

> Is anyone familiar with the scale MmMmMss, where M is a major tone,
m is a minor tone, and s is a semitone? In both the 46 and 72 ets, it
turns out to be quite a good scale; particularly in the 72-et.

Hi Gene!

Would you mind "drawing this out" in 72-tET units for me, just for
clarification...??

Thanks!

Joseph

----- Original Message -----
From: genewardsmith <genewardsmith@juno.com>
To: <tuning@yahoogroups.com>
Sent: Wednesday, January 09, 2002 12:01 AM
Subject: [tuning] Re: A diatonic scale

> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
>
> > Is anyone familiar with the scale MmMmMss, where M is a major tone, m is
a minor tone, and s is a semitone? In both the 46 and 72 ets, it turns out
to be quite a good scale; particularly in the 72-et.
>
> Here's an analysis of this scale. One mode of it in the 5-limit can be
considered an RI scale, since 0-7-19-30-42-53-65 approximates
>
> 1--16/15--6/5--4/3--3/2--5/3--15/8

Here's a triangular lattice of it:

5:3-----()----15:8
/ \ / \ /
/ \ / \ /
4:3----1:1----3:2
/ \ / \ /
/ \ / \ /
16:15---()-----6:5

>
> If we temper by 225/224~1, we may include a 7/5 interval from
> 16/15~15/14 to 3/2, and a 7/4 interval from 16/15 to 28/15 or
> 15/14 to 15/8, depending on how we want to look at it. In the 9-limit, we
may add to this a 14/9 from 15/14 to 5/3 and another from
> 6/5 to 28/15. We therefore have quit a bit of additional harmony, with the
considerable tuning accuracy the 225/224 temperament (or the 72-et) affords.

5:3--------( )-------15:8
/ \ / \ \ / /
/ \ / \15:14/
/ \ / \ | /
4:3-------1:1--------3:2
/ | \ / \ /
/28:15\ / \ /
/ / \ \ / \ /
16:15------( )-------6:5

Pitches are notated in my adaptation of 72-EDO -- legend:

+ - 1/12-tone
< > 1/6-tone
^ v 1/4-tone

72-EDO
degree cents

B- 65 1083&1/3
A- 53 883&1/3
G 42 700
F 30 500
Eb+ 19 316&2/3
C#+ 7 116&2/3
C 0 0

5
/ \
/ 7 \
/ \
4-------6-------9 A-.....().......B-
otonal pentad / \ / \ /
legend / \ / \C#+/
/ \ / \ /
F.......C.......G
utonal pentad / \ / \ /
legend /B- \ / \ /
1:9-----1:6-----1:4 / \ / \ /
\ / C#+......()......Eb+
\1:7/
\ /
1:5

This is a subset of the 19-tone scale posted here by Dave Keenan
on Saturday, April 28, 2001 1:27 AM
Subject: Re: Good 19 of 72-tET (was: what I need to know now...)

love / peace / harmony ...

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

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

> From: monz <joemonz@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, January 09, 2002 8:59 AM
> Subject: Re: [tuning] Re: A diatonic scale
>
>
> > --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> >
> > > Is anyone familiar with the scale MmMmMss, where M is
> > > a major tone, m is a minor tone, and s is a semitone?
> > > <snip>
>
>
> Pitches are notated in my adaptation of 72-EDO -- legend:
>
> + - 1/12-tone
> < > 1/6-tone
> ^ v 1/4-tone
>
>
> 72-EDO
> degree cents
>
> B- 65 1083&1/3
> A- 53 883&1/3
> G 42 700
> F 30 500
> Eb+ 19 316&2/3
> C#+ 7 116&2/3
> C 0 0
>
>
> 5
> / \
> / 7 \
> / \
> 4-------6-------9 A-.....().......B-
> otonal pentad / \ / \ /
> legend / \ / \C#+/
> / \ / \ /
> F.......C.......G
> utonal pentad / \ / \ /
> legend /B- \ / \ /
> 1:9-----1:6-----1:4 / \ / \ /
> \ / C#+......()......Eb+
> \1:7/
> \ /
> 1:5
>

I rushed that off this morning just before leaving the house.
Upon reading it again, I see that there could be some
amibguity over what to call 2^(7/72) here.

While "C#+" is technically a more correct way to represent
15:14 *harmonically*, it should of course be "Db+" for the
case of 16:15. And as far as pseudo-diatonic scale nomenclature,
"Db+" is more correct for both cases.

Gene is calling this version of the scale a mode, with
"C" as the "root". So removing the modal rotation and giving
"Db+" as the "root", it becomes:

Db+ Eb+ F G A- B- C ( Db+ )
ratios: 16/15 -- 6/5 -- 4/3 -- 3/2 -- 5/3 -- 15/8 -- 1/1 (-- 16/15)
72-EDO: 7 19 30 42 53 65 0 ( 7 )
72-EDO degrees \/ \/ \/ \/ \/ \/ \/
between: 12 11 12 11 12 7 7

So:
cents

"major tone" M = 2^(12/72) = 2^(2/12) 200
"minor tone" m = 2^(11/72) 183 1/3
"semitone" s = 2^(7/72) 116 2/3

It should be easy to see from the step-sizes why this scale
is so good at emulating JI: the "major" and "minor tones"
are excellent approximations to 9:8 and 10:9, respectively.

Note that the "semitone" is a "secor".

Also note that the full 9-limit 4:5:6:7:9 otonal and
1/(4:5:6:7:9) utonal pentads are not possible, because
the scale does not contain the "3-identity" (= "6") for
either of the chord structures which follow that paradigm.

So allowing for an enharmonic change in "spelling",
those two chords can have at most:

note 72-EDO chord identities

Eb+ 19 9
Cb- 65 7
F 30 5
Db+ 7 1
----
Db+

B-
---
B- 65 1
G 42 5
C#+ 7 7
A- 53 9

The only other "complete" chords with low-integer
proportions are the otonal F:A-:C and utonal
C:Eb+:G triads.

So there aren't many "traditional" chord-progressions
that you could do with this scale.

love / peace / harmony ...

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

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

>Here's an analysis of this scale. One mode of it in the 5-limit
>can be considered an RI scale, since 0-7-19-30-42-53-65
>approximates
>
> 1--16/15--6/5--4/3--3/2--5/3--15/8

This would be the most stable mode, judging from the pattern of
2nds alone. I'll fire it up tonight.

!
"diatonic" by Gene Ward Smith, Tuning, Yahoo 32463.
7
!
116.667 !7
316.667 !19
500.000 !30
700.000 !42
883.333 !53
1083.333 !65
1200.000 !72
!

() Pitch set under Miller limit of 7-9.

Yep.

() Variety

Mean variety 3.33

() Tetrachordality

((62 $ 53 % rms) (53 $ 45 % mad))

() Propriety

Rothenberg stability: 1
Lumma stability: 0.36

() At least one interval class gives the same consonance in
a majority of modes.

Fails.

() At least one int. class gives different consonances in
a majority of modes.

yep.
3rds: 5:4, 6:5, 8:7
6ths: 8:5, 5:3, 7:4

This may be considered a version of the normal diatonic scale
into which 7-limit harmony has been strangely warped.

-Carl

I find that Monz's (or anyone else's) ASCII lattices
are FUBARed when normally viewed.

Indeed, if one views them in a "Reply" or "Forward"
window, they are readable. If I email myself the
message link with "?expand=1" appended to the message
URL, they are also visible.

The real mystery *is*, why is it that when Monz
tried appending "?expand=1" to message URLs -
in THAT situation, Yahoo does NOT properly
display the ASCII lattices. Downright wierd!

J Gill

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> --- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
>
> /tuning/topicId_32461.html#32461
>
> > Is anyone familiar with the scale MmMmMss, where M is a major
tone,
> m is a minor tone, and s is a semitone? In both the 46 and 72 ets,
it
> turns out to be quite a good scale; particularly in the 72-et.
>
> Hi Gene!
>
> Would you mind "drawing this out" in 72-tET units for me, just for
> clarification...??
>
> Thanks!
>
> Joseph

In case Gene hasn't done this yet, it would be

12 + 11 + 12 + 11 + 12 + 7 + 7.

This is, I believe, a subset of the Lumma/Fokker 12-tone scale.

>>Here's an analysis of this scale. One mode of it in the 5-limit
>>can be considered an RI scale, since 0-7-19-30-42-53-65
>>approximates
>>
>> 1--16/15--6/5--4/3--3/2--5/3--15/8
>
> This would be the most stable mode, judging from the pattern of
> 2nds alone. I'll fire it up tonight.

See, it's _tonight_, like I said!

I've got to work the kinks out of my MIDI setup. I'm getting
latency, that randomly varies from about normal to .3 seconds,
from the time MIDI events are generated on my Kawai keyboard,
sent over a USB/MIDI converter to my laptop, retuned by Midi
Relay, sounded by my soft synth, and recorded as wave audio in
cool edit.

Anywho, files section. Four mp3s, about 4megs total, for a
limited time only.

Ack! The Yahoo files section sucks (And it looks like we've
only got two megs left on our 20meg limit?). See:

http://lumma.org/fourmp3s/one.mp3
http://lumma.org/fourmp3s/two.mp3
http://lumma.org/fourmp3s/three.mp3
http://lumma.org/fourmp3s/four.mp3

-Carl