centreless 11lim diamond

here is a scl file of the 11lim diamond on a 'centreless' map (each
otonality begins on a utonal element rather than 1/1)
it fits on the chromatic keyboard nicely since 12 tones make up a 2
octave range in this diamond

! D:\TUN\11lim diamond\shiftmap.scl
!

72
!
7/6
4/3
3/2
5/3
11/6
2/1
7/3
8/3
3/1
10/3
11/3
12/11
14/11
16/11
18/11
20/11
2/1
24/11
28/11
32/11
36/11
40/11
4/1
6/5
7/5
8/5
9/5
2/1
11/5
12/5
14/5
16/5
18/5
4/1
22/5
4/3
14/9
16/9
2/1
20/9
22/9
8/3
28/9
32/9
4/1
40/9
44/9
3/2
7/4
2/1
9/4
5/2
11/4
3/1
7/2
4/1
9/2
5/1
11/2
12/7
2/1
16/7
18/7
20/7
22/7
24/7
4/1
32/7
36/7
40/7
44/7
4/1

At 09:45 AM 10/6/2006, you wrote:
>here is a scl file of the 11lim diamond on a 'centreless' map (each
>otonality begins on a utonal element rather than 1/1)
>it fits on the chromatic keyboard nicely since 12 tones make up a 2
>octave range in this diamond

Can you show us a single octave of it in a diamond layout, or
tell us exactly which two scales are being multiplied together?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> At 09:45 AM 10/6/2006, you wrote:
> >here is a scl file of the 11lim diamond on a 'centreless' map (each
> >otonality begins on a utonal element rather than 1/1)
> >it fits on the chromatic keyboard nicely since 12 tones make up a 2
> >octave range in this diamond
>
> Can you show us a single octave of it in a diamond layout, or
> tell us exactly which two scales are being multiplied together?
>
> -Carl
>
not sure what you mean by 'two scales are being multiplied together?

the scale i posted is simply a square block taken from the tonality
diamond (rather than a parallelogram from 1/1 like in the previous
scale i posted)

if you use the tonality diamond on this scale it has holes - all the
intervals/pitches still exist in this map though, they are just not
confined within an octave.

so basically it means all the octaves on your keyboard are different
otonalities however each key is its own utonality if u get my drift
i guess its a bit more logical to do it this way since when u want to
modulate u dont have to think about how much u have to offset your
melodic phrase because

>not sure what you mean by 'two scales are being multiplied together?
>
>the scale i posted is simply a square block taken from the tonality
>diamond (rather than a parallelogram from 1/1 like in the previous
>scale i posted)

Can you show your work?

>so basically it means all the octaves on your keyboard are different
>otonalities however each key is its own utonality if u get my drift

I guess I don't.

>i guess its a bit more logical to do it this way since when u want to
>modulate u dont have to think about how much u have to offset your
>melodic phrase because

...

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >not sure what you mean by 'two scales are being multiplied together?
> >
> >the scale i posted is simply a square block taken from the tonality
> >diamond (rather than a parallelogram from 1/1 like in the previous
> >scale i posted)
>
> Can you show your work?
>
> >so basically it means all the octaves on your keyboard are different
> >otonalities however each key is its own utonality if u get my drift
>
> I guess I don't.
>
> >i guess its a bit more logical to do it this way since when u want to
> >modulate u dont have to think about how much u have to offset your
> >melodic phrase because
>
> ...
>
> -Carl
>

hey carl its very simple if you look at a tonality diamond grid in
this case of the 11limit take a square like this

1/1 7/6 4/3 3/2 5/3 11/6 | 2/1
12/11 14/11 16/11 18/11 20/11 2/1 | 24/11
6/5 7/5 8/5 9/5 2/1 11/5 | 12/5
4/3 14/9 16/9 2/1 20/9 22/9 | 18/3
3/2 7/4 2/1 9/4 5/2 11/4 | 3/1
12/7 2/1 16/7 18/7 20/7 22/7 | 24/7

*1/1 * 7/6 * 8/7 * 9/8 *10/9 *11/10|*12/11

the intervals are the same like above so the utonalities are going to
be the vertical lines

there is the same number of tones in this square as there is the
regular '11 limit diamond' but they are not confined within an octave

>hey carl its very simple if you look at a tonality diamond grid in
>this case of the 11limit take a square like this
>
> 1/1 7/6 4/3 3/2 5/3 11/6 | 2/1
>12/11 14/11 16/11 18/11 20/11 2/1 | 24/11
> 6/5 7/5 8/5 9/5 2/1 11/5 | 12/5
> 4/3 14/9 16/9 2/1 20/9 22/9 | 18/3
> 3/2 7/4 2/1 9/4 5/2 11/4 | 3/1
>12/7 2/1 16/7 18/7 20/7 22/7 | 24/7

Thanks, that's what I needed.

>there is the same number of tones in this square as there is the
>regular '11 limit diamond' but they are not confined within an octave

This is the regular 11-limit diamond, just transposed up a 4/3.

-Carl

> Thanks, that's what I needed.
>
> >there is the same number of tones in this square as there is the
> >regular '11 limit diamond' but they are not confined within an octave
>
> This is the regular 11-limit diamond, just transposed up a 4/3.
>
> -Carl
>

what do you mean it is transposed by a 4/3?

>> Thanks, that's what I needed.
>>
>> >there is the same number of tones in this square as there is the
>> >regular '11 limit diamond' but they are not confined within an octave
>>
>> This is the regular 11-limit diamond, just transposed up a 4/3.
>
>what do you mean it is transposed by a 4/3?

If you take the 11-limit diamond and multiply everything by 4/3,
you get the diamond you just posted. Maybe I misinterpreted
what you meant by "centerless".

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> Thanks, that's what I needed.
> >>
> >> >there is the same number of tones in this square as there is the
> >> >regular '11 limit diamond' but they are not confined within an
octave
> >>
> >> This is the regular 11-limit diamond, just transposed up a 4/3.
> >
> >what do you mean it is transposed by a 4/3?
>
> If you take the 11-limit diamond and multiply everything by 4/3,
> you get the diamond you just posted. Maybe I misinterpreted
> what you meant by "centerless".
>
> -Carl
>

hmm i dont see how u get that result

the 9th otonality is multiplied by 4/3 since that is the utonal
position from 6 however all the other otonalities are multiplied by
their own respective utonalities so the next one 10

1/1 11/10 6/5 7/5 8/5 9/5

by 4/3

4/3 22/15 8/5 28/15 32/15 12/5

as u can see that is not in the scale i posted. so what is it u r
doing to get that result?

>> >what do you mean it is transposed by a 4/3?
>>
>> If you take the 11-limit diamond and multiply everything by 4/3,
>> you get the diamond you just posted. Maybe I misinterpreted
>> what you meant by "centerless".
>
>hmm i dont see how u get that result
>
>the 9th otonality is multiplied by 4/3 since that is the utonal
>position from 6 however all the other otonalities are multiplied by
>their own respective utonalities so the next one 10
>
>1/1 11/10 6/5 7/5 8/5 9/5
>
>by 4/3
>
>4/3 22/15 8/5 28/15 32/15 12/5
>
>as u can see that is not in the scale i posted. so what is it u r
>doing to get that result?

Here is the 11-limit tonality diamond

1/1 9/8 5/4 11/8 3/2 7/4
8/7 9/7 10/7 11/7 12/7 1/1
4/3 3/2 5/3 11/6 1/1 7/6
16/11 18/11 20/11 1/1 12/11 14/11
8/5 9/5 1/1 11/10 6/5 7/5
16/9 1/1 10/9 11/9 4/3 14/9

It is the result of multiplying this scale

1/1 9/8 5/4 11/8 3/2 7/4

with its inverse. Let's multiply this
scale by 4/3

4/3 3/2 5/3 11/6 1/1 7/6

Reordering by pitch height

1/1 7/6 4/3 3/2 5/3 11/6

The scale you posted

1/1 7/6 4/3 3/2 5/3 11/6
12/11 14/11 16/11 18/11 20/11 2/1
6/5 7/5 8/5 9/5 2/1 11/5
4/3 14/9 16/9 2/1 20/9 22/9
3/2 7/4 2/1 9/4 5/2 11/4
12/7 2/1 16/7 18/7 20/7 22/7

is the diamond of that scale. I misspoke when I said "trasposed"
and "everything by 4/3". Your scale is more like a 'mode' of
the 11-limit diamond -- the relationships are the same, the ratios
are just spelled as if 3 (otonality and utonality) were 1.

-Carl

> is the diamond of that scale. I misspoke when I said "trasposed"
> and "everything by 4/3". Your scale is more like a 'mode' of
> the 11-limit diamond -- the relationships are the same, the ratios
> are just spelled as if 3 (otonality and utonality) were 1.
>
> -Carl
>

thats sort of what i was trying to say - each otonal element is
multiplied by the utonal of the first otononal in the dimond, in this
case 6. the consequence is obvious when you use the tuning file, as
any chord in any octave is the same in every other octave just
modulated to a different utonalitiy.

"the relationships are the same, the ratios
> are just spelled as if 3 (otonality and utonality) were 1."

that is not entirely accurate but i think u get what i mean anyway.

of course it is not really 'centreless; because the element 2/1
appears in every otonality, however it is no longer the base of the
otonality, so when playing/listening it gives the feeling of
centrelessness. this sort of arrangement is already implied by the
diamond grid its nice to have in a tuning file if u enjoy playing the
diamond though

> Here is the 11-limit tonality diamond
>...
> It is the result of multiplying this scale
>
> 1/1 9/8 5/4 11/8 3/2 7/4
>
> with its inverse.

what do you mean by that?

>> Here is the 11-limit tonality diamond
>>...
>> It is the result of multiplying this scale
>>
>> 1/1 9/8 5/4 11/8 3/2 7/4
>>
>> with its inverse.
>
>what do you mean by that?

"Inverse" in the same sense as the transformation that is sometimes
applied to themes in canons, fugues, and serial music -- a set of
intervals is applied in the opposite direction from the starting
point. So the E D C D E E E of Mary Had A Little Lamb would become
E F# G# F# E E E. Actually there is both literal inversion like
this, and diatonic inversion where diatonic intervals (2nds, 3rds,
etc.) are used instead (producing E F G F E E E).

Multiplying, as in the "cross set" operation as Wilson and
Scala call it, producing

1/1 9/8 5/4 11/8 3/2 7/4
8/7 9/7 10/7 11/7 12/7 1/1
4/3 3/2 5/3 11/6 1/1 7/6
16/11 18/11 20/11 1/1 12/11 14/11
8/5 9/5 1/1 11/10 6/5 7/5
16/9 1/1 10/9 11/9 4/3 14/9

-Carl

>of course it is not really 'centreless; because the element 2/1
>appears in every otonality, however it is no longer the base of the
>otonality, so when playing/listening it gives the feeling of
>centrelessness.

Now I think I understand your intent... you wanted to pick a less
stable mode for the tonalities. However I'm don't think you
succeeded. This mode is if anything more stable because it has
both 4/3 and 3/2, and a sharper leading tone (11/6 vs. 7/4).
Meanwhile, the *pitch* 1/1 occupies a different role in each of
the tonalities in either diamond.

-Carl

Now I think I understand your intent... you wanted to pick a less
> stable mode for the tonalities. However I'm don't think you
> succeeded. This mode is if anything more stable because it has
> both 4/3 and 3/2, and a sharper leading tone (11/6 vs. 7/4).
> Meanwhile, the *pitch* 1/1 occupies a different role in each of
> the tonalities in either diamond.
>
> -Carl

not sure what u r talking about with 'stability' or leading tones

in a physical implementation of the diamond, u would probably want ur
pitches set out in layers of otonalities, each otonality offset by its
distance from the 'root' otonality (in this case 6). this is just
like it appears in a diamond grid. it makes a parralelogram shape.
but because of the layout of the chromatic keyboard, it makes more
sense to start each otonality at the utonal element instead of 1/1,
because *every chord/phrase in any octave has a direct transposition
in every other octave* (otonality). its nothing fancy.

the only reason i posted the scale was because i went to the trouble
of typing it up for my own use and i thought i may aswell share it
incase another diamond enthusiast found it useful.

At 08:32 PM 10/8/2006, you wrote:
>Now I think I understand your intent... you wanted to pick a less
>> stable mode for the tonalities. However I'm don't think you
>> succeeded. This mode is if anything more stable because it has
>> both 4/3 and 3/2, and a sharper leading tone (11/6 vs. 7/4).
>> Meanwhile, the *pitch* 1/1 occupies a different role in each of
>> the tonalities in either diamond.
>>
>> -Carl
>
>not sure what u r talking about with 'stability' or leading tones

The stability of a mode (like the ionion vs. lydian) is often
attributed to the presence of perfect fifths above and below
its tonic, and a narrow leading tone below the tonic. Stability
here means how likely the tonic is to be *heard* as the root
of the scale without the composer having to sound it every other
beat.

>the only reason i posted the scale was because i went to the trouble
>of typing it up for my own use and i thought i may aswell share it
>incase another diamond enthusiast found it useful.

Yes, I'm glad you did. I'll play around with it soon.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:

> The stability of a mode (like the ionion vs. lydian) is often
> attributed to the presence of perfect fifths above and below
> its tonic, and a narrow leading tone below the tonic. Stability
> here means how likely the tonic is to be *heard* as the root
> of the scale without the composer having to sound it every other
> beat.

Which makes the diatonic, melodic minor, harmonic minor, and harmonic
major scales all stable as well as strictly proper.