Yahoo Tuning Groups Ultimate Backup tuning Analysis

Hi Raintree ,

A useful way to analyse a scale is to rotate it to all possible positions like this:

16/15 8/7 128/105 32/25 4/3 64/45 112/75 8/5 128/75 16/9 28/15 2/1
15/14 8/7 60/49 64/49 48/35 10/7 32/21 8/5 12/7 64/35 40/21 2/1
21/20 9/8 6/5 9/7 48/35 36/25 3/2 8/5 42/25 9/5 48/25 2/1
25/24 35/32 75/64 5/4 75/56 10/7 3/2 25/16 5/3 7/4 15/8 2/1
16/15 10/9 7/6 5/4 4/3 10/7 32/21 8/5 5/3 16/9 28/15 2/1
15/14 8/7 25/21 5/4 75/56 10/7 75/49 80/49 12/7 25/14 40/21 2/1
21/20 9/8 6/5 5/4 21/16 45/32 3/2 45/28 12/7 9/5 15/8 2/1
16/15 28/25 6/5 32/25 4/3 7/5 3/2 8/5 12/7 64/35 48/25 2/1
25/24 10/9 7/6 5/4 4/3 25/18 35/24 25/16 5/3 25/14 40/21 2/1
21/20 35/32 7/6 49/40 21/16 7/5 35/24 49/32 105/64 7/4 15/8 2/1
16/15 28/25 7/6 56/45 98/75 7/5 112/75 14/9 49/30 7/4 28/15 2/1
15/14 8/7 6/5 5/4 4/3 7/5 3/2 8/5 5/3 7/4 15/8 2/1

This shows how it will be tuned in all the twelve keys, with your unrotated scale
at the bottom. If your scale was the tuning for C then the line above is the tuning for C#,
then D and so on.

Then the first column shows all the semitone steps in your scale, the second column shows all
the whole tone steps and so forth, so you get a good overview of what it is like.

Robert

http://www.robertinventor.com

🔗Robert Walker <robertwalker@ntlworld.com>

Hi Rainer,
You can view that using Reply and then send to yourself - dont' need to actually send, just preview
But I wonder if sending a pre-formatted html message will lay it out nicely on Yahoo - here is an experiment with it placed within <pre> </pre> tags
16/15 8/7 128/105 32/25 4/3 64/45 112/75 8/5 128/75 16/9 28/15 2/1
15/14 8/7 60/49 64/49 48/35 10/7 32/21 8/5 12/7 64/35 40/21 2/1
21/20 9/8 6/5 9/7 48/35 36/25 3/2 8/5 42/25 9/5 48/25 2/1
25/24 35/32 75/64 5/4 75/56 10/7 3/2 25/16 5/3 7/4 15/8 2/1
16/15 10/9 7/6 5/4 4/3 10/7 32/21 8/5 5/3 16/9 28/15 2/1
15/14 8/7 25/21 5/4 75/56 10/7 75/49 80/49 12/7 25/14 40/21 2/1
21/20 9/8 6/5 5/4 21/16 45/32 3/2 45/28 12/7 9/5 15/8 2/1
16/15 28/25 6/5 32/25 4/3 7/5 3/2 8/5 12/7 64/35 48/25 2/1
25/24 10/9 7/6 5/4 4/3 25/18 35/24 25/16 5/3 25/14 40/21 2/1
21/20 35/32 7/6 49/40 21/16 7/5 35/24 49/32 105/64 7/4 15/8 2/1
16/15 28/25 7/6 56/45 98/75 7/5 112/75 14/9 49/30 7/4 28/15 2/1
15/14 8/7 6/5 5/4 4/3 7/5 3/2 8/5 5/3 7/4 15/8 2/1

Robert

🔗Afmmjr@aol.com

🔗Raintree Goldbach <goldraintree425@hotmail.com>

thanks, shall do.

From: "Robert Walker" <robertwalker@ntlworld.com>
Reply-To: tuning@yahoogroups.com
To: <tuning@yahoogroups.com>
Subject: [tuning] Analysis
Date: Fri, 12 Dec 2003 19:24:50 -0000

Hi Raintree ,

A useful way to analyse a scale is to rotate it to all possible positions like this:

This shows how it will be tuned in all the twelve keys, with your unrotated scale
at the bottom. If your scale was the tuning for C then the line above is the tuning for C#,
then D and so on.

Then the first column shows all the semitone steps in your scale, the second column shows all
the whole tone steps and so forth, so you get a good overview of what it is like.

Robert

http://www.robertinventor.com

_________________________________________________________________
Our best dial-up offer is back. Get MSN Dial-up Internet Service for 6 months @ $9.95/month now! http://join.msn.com/?page=dept/dialup

🔗Raintree Goldbach <goldraintree425@hotmail.com>

🔗Carl Lumma <ekin@lumma.org>

🔗Raintree Goldbach <goldraintree425@hotmail.com>

agreed, it is a scale.

From: Carl Lumma <ekin@lumma.org>
Reply-To: tuning@yahoogroups.com
To: tuning@yahoogroups.com
Subject: RE: [tuning] Re: Analysis
Date: Fri, 12 Dec 2003 12:09:17 -0800

>Why is it not a tuning, since it is designed to work in all keys?

http://sonic-arts.org/dict/scale.htm

I notice there's no entry for "tuning". We usually us "tuning"
more to mean a way of approximating a scale. If you were aiming
for 12-tET then you might consider this a tuning, I suppose.
Maybe others have something to add here.

Please note that these words are sometimes used interchangeably.
And sometimes instead of correcting it, as Gene has done, I simply
go with the flow. There may be something to gain by enforcing
the preferred terminology, though.

-Carl

_________________________________________________________________
Tired of slow downloads and busy signals? Get a high-speed Internet connection! Comparison-shop your local high-speed providers here. https://broadband.msn.com

🔗Carl Lumma <ekin@lumma.org>

***NOTE To see this post correctly on the web, you must hit
"Reply" and view it in the subsequent text box!***

Hi Raintree,

The first thing you should have is a Scala file for your scale, so
you can share it with others, have it in a searchable archive, and
perform all kinds of neat tricks with it. Here's one...

!
Raintree Goldbach 1
12
!
15/14
8/7
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8
2/1
!

The Scala .scl format is defined at:
http://www.xs4all.nl/~huygensf/scala/scl_format.html

Now you can create an *interval matrix* as Robert did...

15/14 8/7 6/5 5/4 4/3 7/5 3/2 8/5 5/3 7/4 15/8
16/15 28/25 7/6 56/45 98/75 7/5 112/75 14/9 49/30 7/4 28/15
21/20 35/32 7/6 49/40 21/16 7/5 35/24 49/32 105/64 7/4 15/8
25/24 10/9 7/6 5/4 4/3 25/18 35/24 25/16 5/3 25/14 40/21
16/15 28/25 6/5 32/25 4/3 7/5 3/2 8/5 12/7 64/35 48/25
21/20 9/8 6/5 5/4 21/16 45/32 3/2 45/28 12/7 9/5 15/8
15/14 8/7 25/21 5/4 75/56 10/7 75/49 80/49 12/7 25/14 40/21
16/15 10/9 7/6 5/4 4/3 10/7 32/21 8/5 5/3 16/9 28/15
25/24 35/32 75/64 5/4 75/56 10/7 3/2 25/16 5/3 7/4 15/8
21/20 9/8 6/5 9/7 48/35 36/25 3/2 8/5 42/25 9/5 48/25
15/14 8/7 60/49 64/49 48/35 10/7 32/21 8/5 12/7 64/35 40/21
16/15 8/7 128/105 32/25 4/3 64/45 112/75 8/5 128/75 16/9 28/15

And you can also convert these to cents, to give a different sort of
judgement on consonance (how bad of a wolf is 112/75? how bad 35/24?).
This is done in Scala with "quantize 1200". Then "view" the
"inverval matrix" again...

119.0 231.0 316.0 386.0 498.0 583.0 702.0 814.0 884.0 969.0 1088.0
112.0 197.0 267.0 379.0 464.0 583.0 695.0 765.0 850.0 969.0 1081.0
85.0 155.0 267.0 352.0 471.0 583.0 653.0 738.0 857.0 969.0 1088.0
70.0 182.0 267.0 386.0 498.0 568.0 653.0 772.0 884.0 1003.0 1115.0
112.0 197.0 316.0 428.0 498.0 583.0 702.0 814.0 933.0 1045.0 1130.0
85.0 204.0 316.0 386.0 471.0 590.0 702.0 821.0 933.0 1018.0 1088.0
119.0 231.0 301.0 386.0 505.0 617.0 736.0 848.0 933.0 1003.0 1115.0
112.0 182.0 267.0 386.0 498.0 617.0 729.0 814.0 884.0 996.0 1081.0
70.0 155.0 274.0 386.0 505.0 617.0 702.0 772.0 884.0 969.0 1088.0
85.0 204.0 316.0 435.0 547.0 632.0 702.0 814.0 899.0 1018.0 1130.0
119.0 231.0 350.0 462.0 547.0 617.0 729.0 814.0 933.0 1045.0 1115.0
112.0 231.0 343.0 428.0 498.0 610.0 695.0 814.0 926.0 996.0 1081.0

In addition to this, you should most definitely *PLAY* the scale.
Using Scala, you can retune your MIDI keyboard and try it out!
Or you can retune any MIDI file in it!!

Finally, a good tool is the lattice. Here's a 7-limit lattice for
your scale (some things must still be done by hand, alas; the
"lattice" function in Scala seems rather useless to me)...

5/3-------5/4------15/8
/ \ /|\`. ,'/
/ \ / | \15/14/
/ \ / 7/4 \ | /
/ \ /,'/ `.\|/
4/3-------1/1-/-----3/2
\`. ,'/|\/ /
\ 8/7 / |/\ /
\ | / 7/5 \ /
\|/,' `.\ /
8/5-------6/5

This shows all the pure-JI dyads that occur in your scale, when
we are limited to ratios built from prime numbers of 7 or less.
This is a 3-D structure. Legend:

------- = 3:2

/ \
/ = 5:4 \ = 6:5
/ \
/ \

|
| = 7:5 ,' = 7/4 `. = 7/6

Our general theory on this is that all other things being equal,
the more compact your scale looks on a lattice like this, the
better it is. Here your scale is fairly compact. It contains
the 5-limit Partch Tonality Diamond (the 7-note central hexagon),
which is maximally compact. Your scale could be made more compact
by replacing 15/8 and 15/14 with pitches from the 7-limit Tonality
Diamond, such as 7/6, 10/7 and 12/7, though not without coming
within a comma of 6/5, 7/5 and 5/3. So it looks pretty good.

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗Raintree Goldbach <goldraintree425@hotmail.com>

amazing, thanks for the help, it'll take a little time to process.

From: Carl Lumma <ekin@lumma.org>
Reply-To: tuning@yahoogroups.com
To: tuning@yahoogroups.com
Subject: RE: [tuning] Analysis
Date: Fri, 12 Dec 2003 12:55:02 -0800

***NOTE To see this post correctly on the web, you must hit
"Reply" and view it in the subsequent text box!***

Hi Raintree,

The first thing you should have is a Scala file for your scale, so
you can share it with others, have it in a searchable archive, and
perform all kinds of neat tricks with it. Here's one...

!
Raintree Goldbach 1
12
!
15/14
8/7
6/5
5/4
4/3
7/5
3/2
8/5
5/3
7/4
15/8
2/1
!

The Scala .scl format is defined at:
http://www.xs4all.nl/~huygensf/scala/scl_format.html

Now you can create an *interval matrix* as Robert did...

In addition to this, you should most definitely *PLAY* the scale.
Using Scala, you can retune your MIDI keyboard and try it out!
Or you can retune any MIDI file in it!!

Finally, a good tool is the lattice. Here's a 7-limit lattice for
your scale (some things must still be done by hand, alas; the
"lattice" function in Scala seems rather useless to me)...

5/3-------5/4------15/8
/ \ /|\`. ,'/
/ \ / | \15/14/
/ \ / 7/4 \ | /
/ \ /,'/ `.\|/
4/3-------1/1-/-----3/2
\`. ,'/|\/ /
\ 8/7 / |/\ /
\ | / 7/5 \ /
\|/,' `.\ /
8/5-------6/5

This shows all the pure-JI dyads that occur in your scale, when
we are limited to ratios built from prime numbers of 7 or less.
This is a 3-D structure. Legend:

------- = 3:2

/ \
/ = 5:4 \ = 6:5
/ \
/ \

|
| = 7:5 ,' = 7/4 `. = 7/6

-Carl

_________________________________________________________________
Wonder if the latest virus has gotten to your computer? Find out. Run the FREE McAfee online computer scan! http://clinic.mcafee.com/clinic/ibuy/campaign.asp?cid=3963

🔗monz <monz@attglobal.net>

wonderful, Robert!

... now let's see if it works when quoting for a reply ...

-monz

--- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
wrote:
> Hi Rainer,
> You can view that using Reply and then send to yourself
> - dont' need to actually send, just preview. But I wonder
> if sending a pre-formatted html message will lay it out
> nicely on Yahoo - here is an experiment with it placed
> within <pre> </pre> tags

<PRE>
> 16/15 8/7 128/105 32/25 4/3 64/45 112/75 8/5
128/75 16/9 28/15 2/1
> 15/14 8/7 60/49 64/49 48/35 10/7 32/21 8/5
12/7 64/35 40/21 2/1
> 21/20 9/8 6/5 9/7 48/35 36/25 3/2 8/5
42/25 9/5 48/25 2/1
> 25/24 35/32 75/64 5/4 75/56 10/7 3/2 25/16
5/3 7/4 15/8 2/1
> 16/15 10/9 7/6 5/4 4/3 10/7 32/21 8/5
5/3 16/9 28/15 2/1
> 15/14 8/7 25/21 5/4 75/56 10/7 75/49 80/49
12/7 25/14 40/21 2/1
> 21/20 9/8 6/5 5/4 21/16 45/32 3/2 45/28
12/7 9/5 15/8 2/1
> 16/15 28/25 6/5 32/25 4/3 7/5 3/2 8/5
12/7 64/35 48/25 2/1
> 25/24 10/9 7/6 5/4 4/3 25/18 35/24 25/16
5/3 25/14 40/21 2/1
> 21/20 35/32 7/6 49/40 21/16 7/5 35/24 49/32
105/64 7/4 15/8 2/1
> 16/15 28/25 7/6 56/45 98/75 7/5 112/75 14/9
49/30 7/4 28/15 2/1
> 15/14 8/7 6/5 5/4 4/3 7/5 3/2 8/5
5/3 7/4 15/8 2/1
</PRE>
>
> Robert

🔗monz <monz@attglobal.net>

well, i just used the Yahoo web interface "Reply" button,
but it didn't format correctly.

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> wonderful, Robert!
>
> ... now let's see if it works when quoting for a reply ...
>
>
> -monz
>
>
>
> --- In tuning@yahoogroups.com, "Robert Walker" <robertwalker@n...>
> wrote:
> > Hi Rainer,
> > You can view that using Reply and then send to yourself
> > - dont' need to actually send, just preview. But I wonder
> > if sending a pre-formatted html message will lay it out
> > nicely on Yahoo - here is an experiment with it placed
> > within <pre> </pre> tags
>
> <PRE>
> > 16/15 8/7 128/105 32/25 4/3 64/45 112/75 8/5
> 128/75 16/9 28/15 2/1
> > 15/14 8/7 60/49 64/49 48/35 10/7 32/21 8/5
> 12/7 64/35 40/21 2/1
> > 21/20 9/8 6/5 9/7 48/35 36/25 3/2 8/5
> 42/25 9/5 48/25 2/1
> > 25/24 35/32 75/64 5/4 75/56 10/7 3/2
25/16
> 5/3 7/4 15/8 2/1
> > 16/15 10/9 7/6 5/4 4/3 10/7 32/21
8/5
> 5/3 16/9 28/15 2/1
> > 15/14 8/7 25/21 5/4 75/56 10/7 75/49 80/49
> 12/7 25/14 40/21 2/1
> > 21/20 9/8 6/5 5/4 21/16 45/32 3/2 45/28
> 12/7 9/5 15/8 2/1
> > 16/15 28/25 6/5 32/25 4/3 7/5 3/2 8/5
> 12/7 64/35 48/25 2/1
> > 25/24 10/9 7/6 5/4 4/3 25/18 35/24
25/16
> 5/3 25/14 40/21 2/1
> > 21/20 35/32 7/6 49/40 21/16 7/5 35/24 49/32
> 105/64 7/4 15/8 2/1
> > 16/15 28/25 7/6 56/45 98/75 7/5 112/75 14/9
> 49/30 7/4 28/15 2/1
> > 15/14 8/7 6/5 5/4 4/3 7/5 3/2
8/5
> 5/3 7/4 15/8 2/1
> </PRE>
> >
> > Robert

🔗kraig grady <kraiggrady@anaphoria.com>

🔗Peter Wakefield Sault <sault@cyberware.co.uk>

--- In tuning@yahoogroups.com, "Raintree Goldbach"
<goldraintree425@h...> wrote:
> are there any faults, theoretically or acoustically, with the
following
> tuning?
>
> thanks
>
> 1/1 15/14 8/7 6/5 4/3 7/5 3/2 8/5 5/3 7/4 15/8
>

Doesn't it really rather depend upon what you do with it?

Since you are defining a dodekaphonic scale, you may like to take a
look at my dodekaphonic comb filter. This shows a way of defining
upper and lower limits for each interval. The width of each passband
is a pythagorean comma. If each of your ratios resolves to a
coefficient which falls within its respective passband then you have
a true dodekaphonic scale, otherwise you have something else. You can
find this at:-
http://www.odeion.org/atlantis/chapter-1.html#table1-4

Although passbands are shown for intervals 0 and 12, the values of
the coefficients are of course always precisely 1 and 2 respectively.

Peter S.

🔗Peter Wakefield Sault <sault@cyberware.co.uk>

🔗Werner Mohrlok <wmohrlok@hermode.com>

-----Ursprï¿½ngliche Nachricht-----
Von: Raintree Goldbach [mailto:goldraintree425@hotmail.com]
Gesendet: Freitag, 12. Dezember 2003 19:36
An: tuning@yahoogroups.com
Betreff: [tuning] Analysis

are there any faults, theoretically or acoustically, with the following
tuning?

thanks

1/1 15/14 8/7 6/5 4/3 7/5 3/2 8/5 5/3 7/4 15/8

At least a simple fault by carelessness. One value is missing:

1/1, 15/14, 8/7, 6/5, 5/4(!), 4/3, 7/5, 3/2, 8/5, 5/3, 7/4, 15/8

Theoretically???
This depends what you want to do with this scale.

Werner Mohrlok

_________________________________________________________________
Winterize your home with tips from MSN House & Home.
http://special.msn.com/home/warmhome.armx

Yahoo! Groups Sponsor
ADVERTISEMENT

You do not need web access to participate. You may subscribe through
email. Send an empty email to one of these addresses:
tuning-subscribe@yahoogroups.com - join the tuning group.
tuning-unsubscribe@yahoogroups.com - unsubscribe from the tuning group.
tuning-nomail@yahoogroups.com - put your email message delivery on hold
for the tuning group.
tuning-digest@yahoogroups.com - change your subscription to daily digest
mode.
tuning-normal@yahoogroups.com - change your subscription to individual
emails.
tuning-help@yahoogroups.com - receive general help information.

Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

🔗Werner Mohrlok <wmohrlok@hermode.com>

-----Ursprï¿½ngliche Nachricht-----
Von: Raintree Goldbach [mailto:goldraintree425@hotmail.com]
Gesendet: Freitag, 12. Dezember 2003 21:03
An: tuning@yahoogroups.com
Betreff: Re: [tuning] Analysis

thanks, i have been using scala to test every possible chord combination
and, so far, everything sounds ok, and yet it seems more likely, i would
think, that some flaw exists in the system either harmonically or
melodically.

From: Afmmjr@aol.com
Reply-To: tuning@yahoogroups.com
To: tuning@yahoogroups.com
Subject: Re: [tuning] Analysis
Date: Fri, 12 Dec 2003 14:52:15 EST

It looks good to me. Johnny Reinhard

I wouldn't like so much detuned fifth. Not for traditional music. But as
you like it, I don't want to oppose...
As to my opinion it is a scale for experimental music and for this aim as
good as
many other scales of "natural ratios"

Werner Mohrlok

_________________________________________________________________
Tired of slow downloads and busy signals? Get a high-speed Internet
connection! Comparison-shop your local high-speed providers here.
https://broadband.msn.com

Yahoo! Groups Sponsor

Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

🔗Werner Mohrlok <wmohrlok@hermode.com>

-----Ursprï¿½ngliche Nachricht-----
Von: Raintree Goldbach [mailto:goldraintree425@hotmail.com]
Gesendet: Freitag, 12. Dezember 2003 20:54
An: tuning@yahoogroups.com
Betreff: RE: [tuning] Analysis

thanks, shall do.

From: "Robert Walker" <robertwalker@ntlworld.com>
Reply-To: tuning@yahoogroups.com
To: <tuning@yahoogroups.com>
Subject: [tuning] Analysis
Date: Fri, 12 Dec 2003 19:24:50 -0000

Hi Raintree ,

A useful way to analyse a scale is to rotate it to all possible positions
like this:

16/15 8/7 128/105 32/25 4/3 64/45 112/75 8/5 128/75
16/9 28/15 2/1
15/14 8/7 60/49 64/49 48/35 10/7 32/21 8/5 12/7
64/35 40/21 2/1
21/20 9/8 6/5 9/7 48/35 36/25 3/2 8/5 42/25
9/5 48/25 2/1
25/24 35/32 75/64 5/4 75/56 10/7 3/2 25/16 5/3
7/4 15/8 2/1
16/15 10/9 7/6 5/4 4/3 10/7 32/21 8/5 5/3
16/9 28/15 2/1
15/14 8/7 25/21 5/4 75/56 10/7 75/49 80/49 12/7
25/14 40/21 2/1
21/20 9/8 6/5 5/4 21/16 45/32 3/2 45/28 12/7
9/5 15/8 2/1
16/15 28/25 6/5 32/25 4/3 7/5 3/2 8/5 12/7
64/35 48/25 2/1
25/24 10/9 7/6 5/4 4/3 25/18 35/24 25/16 5/3
25/14 40/21 2/1
21/20 35/32 7/6 49/40 21/16 7/5 35/24 49/32 105/64
7/4 15/8 2/1
16/15 28/25 7/6 56/45 98/75 7/5 112/75 14/9 49/30
7/4 28/15 2/1
15/14 8/7 6/5 5/4 4/3 7/5 3/2 8/5 5/3
7/4 15/8 2/1

This shows how it will be tuned in all the twelve keys, with your
unrotated
scale
at the bottom. If your scale was the tuning for C then the line above is
the
tuning for C#,
then D and so on.

Then the first column shows all the semitone steps in your scale, the
second
column shows all
the whole tone steps and so forth, so you get a good overview of what it
is
like.

Robert

http://www.robertinventor.com

A "tuning for C" with a poor fifth D - A.
But this problem is known by most of us.

Werner Mohrlok

_________________________________________________________________
Our best dial-up offer is back. Get MSN Dial-up Internet Service for 6
months @ $9.95/month now! http://join.msn.com/?page=dept/dialup

Yahoo! Groups Sponsor
ADVERTISEMENT

Your use of Yahoo! Groups is subject to the Yahoo! Terms of Service.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> ***NOTE To see this post correctly on the web, you must hit
> "Reply" and view it in the subsequent text box!***
>
>
> Hi Raintree,
>
> Finally, a good tool is the lattice. Here's a 7-limit lattice for
> your scale (some things must still be done by hand, alas; the
> "lattice" function in Scala seems rather useless to me)...
>
> 5/3-------5/4------15/8
> / \ /|\`. ,'/
> / \ / | \15/14/
> / \ / 7/4 \ | /
> / \ /,'/ `.\|/
> 4/3-------1/1-/-----3/2
> \`. ,'/|\/ /
> \ 8/7 / |/\ /
> \ | / 7/5 \ /
> \|/,' `.\ /
> 8/5-------6/5
>
> This shows all the pure-JI dyads that occur in your scale, when
> we are limited to ratios built from prime numbers of 7 or less.
> This is a 3-D structure. Legend:
>
> ------- = 3:2
>
> / \
> / = 5:4 \ = 6:5
> / \
> / \
>
> |
> | = 7:5 ,' = 7/4 `. = 7/6

Carl -- thanks for doing what I would have done if I had been
available. But . . . don't you mean "odd" and not "prime" above?

🔗Carl Lumma <ekin@lumma.org>

>> Hi Raintree,
>>
>> Finally, a good tool is the lattice. Here's a 7-limit lattice for
>> your scale (some things must still be done by hand, alas; the
>> "lattice" function in Scala seems rather useless to me)...
>>
>> 5/3-------5/4------15/8
>> / \ /|\`. ,'/
>> / \ / | \15/14/
>> / \ / 7/4 \ | /
>> / \ /,'/ `.\|/
>> 4/3-------1/1-/-----3/2
>> \`. ,'/|\/ /
>> \ 8/7 / |/\ /
>> \ | / 7/5 \ /
>> \|/,' `.\ /
>> 8/5-------6/5
>>
>> This shows all the pure-JI dyads that occur in your scale, when
>> we are limited to ratios built from prime numbers of 7 or less.
>> This is a 3-D structure. Legend:
>>
>> ------- = 3:2
>>
>> / \
>> / = 5:4 \ = 6:5
>> / \
>> / \
>>
>> |
>> | = 7:5 ,' = 7/4 `. = 7/6
>
>Carl -- thanks for doing what I would have done if I had been
>available. But . . . don't you mean "odd" and not "prime" above?

Big shoes to fill. :) Yes "odd" would have been better since it
excludes 2.

-Carl

🔗wallyesterpaulrus <paul@stretch-music.com>

--- In tuning@yahoogroups.com, Carl Lumma <ekin@l...> wrote:
> >> Hi Raintree,
> >>
> >> Finally, a good tool is the lattice. Here's a 7-limit lattice
for
> >> your scale (some things must still be done by hand, alas; the
> >> "lattice" function in Scala seems rather useless to me)...
> >>
> >> 5/3-------5/4------15/8
> >> / \ /|\`. ,'/
> >> / \ / | \15/14/
> >> / \ / 7/4 \ | /
> >> / \ /,'/ `.\|/
> >> 4/3-------1/1-/-----3/2
> >> \`. ,'/|\/ /
> >> \ 8/7 / |/\ /
> >> \ | / 7/5 \ /
> >> \|/,' `.\ /
> >> 8/5-------6/5
> >>
> >> This shows all the pure-JI dyads that occur in your scale, when
> >> we are limited to ratios built from prime numbers of 7 or less.
> >> This is a 3-D structure. Legend:
> >>
> >> ------- = 3:2
> >>
> >> / \
> >> / = 5:4 \ = 6:5
> >> / \
> >> / \
> >>
> >> |
> >> | = 7:5 ,' = 7/4 `. = 7/6
> >
> >Carl -- thanks for doing what I would have done if I had been
> >available. But . . . don't you mean "odd" and not "prime" above?
>
> Big shoes to fill. :) Yes "odd" would have been better since it
> excludes 2.

It also excludes things like 25, a more to-the-point example in
case "prime" was read as implying "prime factors" which it does to
many people . . .