Yahoo Tuning Groups Ultimate Backup tuning-math diamond

i guess the main point was the observation that u can generalise a
diamond by the product of x+k/x+k-1 fron k=1->x, into 2x cycles
(arrangements where each element is either 1 above or 1 below in the
closed group), half of the cycles are "otonal" and going the other way
is "utonal". the 9lim is as complete as u can get in this respect
with a prime lim of 7.

what is the dekany? i have searched the web and have not been able to
find an example. i did briefly have a look at CPS scales and have
mixed feelings about them. i was wondering if any one has done
drawings similar to the cps strctures for the regular diamonds??

how it maps to the keyboard was not exactly the point just a
demonstration and its obviously not going to be very good. u could
map it to ur grandads ass if u want, it doesnt really matter.

🔗Kraig Grady <kraiggrady@anaphoria.com>

http://anaphoria.com/wilson.html
there is some diamond lattices that can be accessed through this page. beneath that is quite a few papers on CPS structures
one advantages of CPS structures over diamonds is the degree for modulation is far greater. each tone will have basically the same level of interelationshi[p between the other tones whereas in the diamond your 1/1 have a higher degree of dominance than any other tone.
tfllt wrote:
>
> i guess the main point was the observation that u can generalise a
> diamond by the product of x+k/x+k-1 fron k=1->x, into 2x cycles
> (arrangements where each element is either 1 above or 1 below in the
> closed group), half of the cycles are "otonal" and going the other way
> is "utonal". the 9lim is as complete as u can get in this respect
> with a prime lim of 7.
>
> what is the dekany? i have searched the web and have not been able to
> find an example. i did briefly have a look at CPS scales and have
> mixed feelings about them. i was wondering if any one has done
> drawings similar to the cps strctures for the regular diamonds??
>
> how it maps to the keyboard was not exactly the point just a
> demonstration and its obviously not going to be very good. u could
> map it to ur grandads ass if u want, it doesnt really matter.
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗Carl Lumma <ekin@lumma.org>

>what is the dekany? i have searched the web and have not been able to
>find an example. i did briefly have a look at CPS scales and have
>mixed feelings about them.

I suppose the analog of what you're talking about here would be the
2|5 [1 3 5 7 9] dekany. Because the number of factors (5) is odd,
there is no CPS with otonal/utonal symmetry (2|5 favors otonal
chords, 3|5 utonal ones). You can just superimpose the two, and
the result has the same number of otonal/utonal chords...

!
Union of 2|5 and 3|5 [1 3 5 7 9] dekanies.
14
!
21/20
9/8
7/6
5/4
21/16
7/5
35/24
3/2
63/40
5/3
7/4
9/5
15/8
2/1
!

Unlike with the diamond, they aren't complete pentatonics; they
are instead tetrads. As a result, however, you get the tone-center-
free property Kraig mentioned.

>i was wondering if any one has done
>drawings similar to the cps strctures for the regular diamonds??

I've placed one in the photos section.

http://tech.ph.groups.yahoo.com/group/tuning-math/photos/view/9c1f?b=1&m=f&o=0

>how it maps to the keyboard was not exactly the point just a
>demonstration and its obviously not going to be very good. u could
>map it to ur grandads ass if u want, it doesnt really matter.

I think mappings matter.

-Carl

🔗tfllt <nasos.eo@gmail.com>

thanks am looking through some of these documents - i have noticed
none of the diamonds in the diagrams contain the number 2 in the
intervals - why is this? is there any sketch of the 9lim diamond?

i know a cps has a great capacity for modulation but due to the nature
in which pitch is perceived it might be more beneficial to use a
diamond (with more consonant intervals) and use modulation as a
musical event wherein the 1/1 is re adjusted to another member of the
diamond-maybe

--- In tuning-math@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> http://anaphoria.com/wilson.html
> there is some diamond lattices that can be accessed through this page.
> beneath that is quite a few papers on CPS structures
> one advantages of CPS structures over diamonds is the degree for
> modulation is far greater. each tone will have basically the same level
> of interelationshi[p between the other tones whereas in the diamond
your
> 1/1 have a higher degree of dominance than any other tone.
>
>
> tfllt wrote:
> >
> > i guess the main point was the observation that u can generalise a
> > diamond by the product of x+k/x+k-1 fron k=1->x, into 2x cycles
> > (arrangements where each element is either 1 above or 1 below in the
> > closed group), half of the cycles are "otonal" and going the other way
> > is "utonal". the 9lim is as complete as u can get in this respect
> > with a prime lim of 7.
> >
> > what is the dekany? i have searched the web and have not been able to
> > find an example. i did briefly have a look at CPS scales and have
> > mixed feelings about them. i was wondering if any one has done
> > drawings similar to the cps strctures for the regular diamonds??
> >
> > how it maps to the keyboard was not exactly the point just a
> > demonstration and its obviously not going to be very good. u could
> > map it to ur grandads ass if u want, it doesnt really matter.
> >
> >
>
> --
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles
>

🔗Carl Lumma <ekin@lumma.org>

🔗tfllt <nasos.eo@gmail.com>

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗Carl Lumma <ekin@lumma.org>

>> >thanks am looking through some of these documents - i have noticed
>> >none of the diamonds in the diagrams contain the number 2 in the
>> >intervals - why is this?
>>
>> Octave equivalence is assumed, so each point stands for all
>> octaves of that note.
>>
>> >is there any sketch of the 9lim diamond?
>>
>> Wilson hasn't published any that I know of, but he may well
>> have drawn it.
>>
>> -C.
>
>im not talking about octaves, if you look at the picture u posted, or
>in wilsons documents, none of the intervals contain factors of 2...

Octaves *are* factors of 2. If we have 9 and 18, the difference
is one octave. In the diagram, it is assumed a point labeled 9
stands for 9, 18, 9/2, etc.

-Carl

🔗Kraig Grady <kraiggrady@anaphoria.com>

if one is not interested in octave equivalence one can look at the lambdoma material. here tyou will find the number 2.
but what are you going to do with it

tfllt wrote:
>
> --- In tuning-math@yahoogroups.com > <mailto:tuning-math%40yahoogroups.com>, Carl Lumma <ekin@...> wrote:
> >
> > At 09:56 AM 9/18/2006, you wrote:
> > >thanks am looking through some of these documents - i have noticed
> > >none of the diamonds in the diagrams contain the number 2 in the
> > >intervals - why is this?
> >
> > Octave equivalence is assumed, so each point stands for all
> > octaves of that note.
> >
> > >is there any sketch of the 9lim diamond?
> >
> > Wilson hasn't published any that I know of, but he may well
> > have drawn it.
> >
> > -C.
> >
>
> im not talking about octaves, if you look at the picture u posted, or
> in wilsons documents, none of the intervals contain factors of 2...
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗Carl Lumma <ekin@lumma.org>

🔗tfllt <nasos.eo@gmail.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> >thanks am looking through some of these documents - i have noticed
> >> >none of the diamonds in the diagrams contain the number 2 in the
> >> >intervals - why is this?
> >>
> >> Octave equivalence is assumed, so each point stands for all
> >> octaves of that note.
> >>
> >> >is there any sketch of the 9lim diamond?
> >>
> >> Wilson hasn't published any that I know of, but he may well
> >> have drawn it.
> >>
> >> -C.
> >
> >im not talking about octaves, if you look at the picture u posted, or
> >in wilsons documents, none of the intervals contain factors of 2...
>
> Octaves *are* factors of 2. If we have 9 and 18, the difference
> is one octave. In the diagram, it is assumed a point labeled 9
> stands for 9, 18, 9/2, etc.
>
> -Carl
>

o right i thought all the intervals were bound by an octave

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

--- In tuning-math@yahoogroups.com, "tfllt" <nasos.eo@...> wrote:
>
> thanks am looking through some of these documents - i have noticed
> none of the diamonds in the diagrams contain the number 2 in the
> intervals - why is this? is there any sketch of the 9lim diamond?

The best way to diagram the 9-limit diamond would be in three
dimensions, but below are some jpg files of projections down to two
dimensions using breed and marvel.

http://www.xenharmony.org/images/diamonds/breed/breed5.jpg
http://www.xenharmony.org/images/diamonds/breed/breed7.jpg
http://www.xenharmony.org/images/diamonds/breed/breed9.jpg

http://www.xenharmony.org/images/diamonds/marvel/marv5.jpg
http://www.xenharmony.org/images/diamonds/marvel/marv7.jpg
http://www.xenharmony.org/images/diamonds/marvel/marv9.jpg

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗Carl Lumma <ekin@lumma.org>

At 02:32 AM 9/19/2006, you wrote:
>--- In tuning-math@yahoogroups.com, "tfllt" <nasos.eo@...> wrote:
>>
>> thanks am looking through some of these documents - i have noticed
>> none of the diamonds in the diagrams contain the number 2 in the
>> intervals - why is this? is there any sketch of the 9lim diamond?
>
>The best way to diagram the 9-limit diamond would be in three
>dimensions, but below are some jpg files of projections down to two
>dimensions using breed and marvel.
>
>http://www.xenharmony.org/images/diamonds/breed/breed5.jpg
>http://www.xenharmony.org/images/diamonds/breed/breed7.jpg
>http://www.xenharmony.org/images/diamonds/breed/breed9.jpg
>
>http://www.xenharmony.org/images/diamonds/marvel/marv5.jpg
>http://www.xenharmony.org/images/diamonds/marvel/marv7.jpg
>http://www.xenharmony.org/images/diamonds/marvel/marv9.jpg

These are meaningless to my eye.

-C.

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗tfllt <nasos.eo@gmail.com>

--- In tuning-math@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:

each tone will have basically the same level
> of interelationshi[p between the other tones whereas in the diamond
your
> 1/1 have a higher degree of dominance than any other tone.
>
>

so in other words t i cant see how this statement is true..?? maybe u
cud explain but like i described in my other post it seems to me that
it doesnt matter where u start at in a diamond - u will always have
the same options interval wise. i cant see how a CPS scale is a
rational alternative to a diamond. but then again i have been smoking
cones constantly for the last 18 hours and i can barely sit up
straight so i will be prepared to eat my words tomorrow if u show me
how that is not ttrue.......:P

🔗tfllt <nasos.eo@gmail.com>

--- In tuning-math@yahoogroups.com, "tfllt" <nasos.eo@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "tfllt" <nasos.eo@> wrote:
> > i know a cps has a great capacity for modulation but due to the nature
> > in which pitch is perceived it might be more beneficial to use a
> > diamond (with more consonant intervals) and use modulation as a
> > musical event wherein the 1/1 is re adjusted to another member of the
> > diamond-maybe
> >
>
> but really, the diamond in itself already has an inherent modulatory
> facility since every melodic passage in a 'mode' is a movement between
> the cycle of the building harmonics, and every mode comes from this
> same cycle, you can transpose any passage from any mode to any other
> having the intervals exactly the same
>

in other words, each mode is mirrored in every other mode. the mirror
scales start at the utonal elements. this is sort of obvious i guess
and it doesnt imply the same sort of modulaton as an equal temperament
scale or even a cps but those types of scales do not have correct
intervals anyway

🔗Carl Lumma <ekin@lumma.org>

🔗tfllt <nasos.eo@gmail.com>

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@...> wrote:
>
> >but really, the diamond in itself already has an inherent modulatory
> >facility since every melodic passage in a 'mode' is a movement between
> >the cycle of the building harmonics, and every mode comes from this
> >same cycle, you can transpose any passage from any mode to any other
> >having the intervals exactly the same
>
> Yes. The only drawback is all these modes will share a single
> pitch, the 1/1. With CPS there is a similar "cycle" of modes
> available, and they have no pitches in common. Your mileage
> may vary as to whether this is important in your music.
>
> -Carl
>

i think i dont understand CPS properly. im looking at the scale u
posted the other day

21/20
9/8
7/6
5/4
21/16
7/5
35/24
3/2
63/40
5/3
7/4
9/5
15/8
2/1

okay so we see here that there is an implied 1/1 by the presence of
2/1. do u mean that u can think of any pitch in the scale as 1/1 and
the relationships are still the same?
well thats clearly not the case in this scale since for instance the
ratio between 9/8 and 3/2 is 4/3, however 4/3 isn't a pitch in this
scale and most of the other pitches do not have a 4/3 complement

so how does this work if u could explain plz.///? :|

🔗Carl Lumma <ekin@lumma.org>

>> >but really, the diamond in itself already has an inherent modulatory
>> >facility since every melodic passage in a 'mode' is a movement between
>> >the cycle of the building harmonics, and every mode comes from this
>> >same cycle, you can transpose any passage from any mode to any other
>> >having the intervals exactly the same
>>
>> Yes. The only drawback is all these modes will share a single
>> pitch, the 1/1. With CPS there is a similar "cycle" of modes
>> available, and they have no pitches in common. Your mileage
>> may vary as to whether this is important in your music.
>
>i think i dont understand CPS properly. im looking at the scale u
>posted the other day
>
>21/20
>9/8
>7/6
>5/4
>21/16
>7/5
>35/24
>3/2
>63/40
>5/3
>7/4
>9/5
>15/8
>2/1
>
>okay so we see here that there is an implied 1/1 by the presence of
>2/1.

That's how Scala files work, yes. You specify the period by the
last entry. It could have been 3/1. Then the scale would still
have a 1/1 below 21/20, but the first note of the period above
would not be 21/10 as it is here but instead 21/20 * 3/1 = 63/20.

>do u mean that u can think of any pitch in the scale as 1/1 and
>the relationships are still the same?

In *any* scale, you can call any note 1/1 if you like, by
multiplying all notes by the reciprocal of the chosen note.
All the intervals will stay the same because you've just
multiplied by a constant factor.

>well thats clearly not the case in this scale since for instance the
>ratio between 9/8 and 3/2 is 4/3, however 4/3 isn't a pitch in this
>scale and most of the other pitches do not have a 4/3 complement

Oh, you mean can you call any note 1/1 and have the same relationships
as you did from the original 1/1. Only equal temperaments do this.
The diamond does it in a limited number of cases.

>so how does this work if u could explain plz.///? :|

The above scale contains two dekany CPSs. For explaining CPSs,
let's start with:

http://tech.ph.groups.yahoo.com/group/tuning-math/photos/view/9c1f?b=1

In the upper-right we see the 7-limit diamond. In the bottom-
right we see the tetrads that are the "modes" of the 7-limit.
You can, as you say, get an otonal mode (major tetrad) on each
note of a utonal one. Or the other way around! Do you see
the cycles of tetrads in the figure in the upper-right?
(Selecting the "large" view might help.)

Well, the thing in the middle is the hexany, the 2|4 7-limit
CPS. It does not have tetrads but instead triads. Do you see
the cycle of triangles in it? Each of these is an incomplete
o- or utonality. They can be completed by "stellating" the
hexany. The result would look something like this:

http://www.pandragon.com/polyhedra/jpg/st_octo.jpg

As for the assertion about modulation, can you see the point
in the center of the diamond? Every tetrad in the diamond
contains that point. If the listener has good pitch memory,
your modulations can start to sound monotonous. Plenty of
popular tunes can be harmonized with a single note, though, so
it isn't hell and damnation, but... notice the hexany has no
such central point. Pick any point in the hexany and you will
see that it is found in only 4 of its 8 triads.

Make any sense?

-Carl

🔗tfllt <nasos.eo@gmail.com>

> The above scale contains two dekany CPSs. For explaining CPSs,
> let's start with:
>
> http://tech.ph.groups.yahoo.com/group/tuning-math/photos/view/9c1f?b=1
>
> In the upper-right we see the 7-limit diamond. In the bottom-
> right we see the tetrads that are the "modes" of the 7-limit.
> You can, as you say, get an otonal mode (major tetrad) on each
> note of a utonal one. Or the other way around! Do you see
> the cycles of tetrads in the figure in the upper-right?
> (Selecting the "large" view might help.)
>
> Well, the thing in the middle is the hexany, the 2|4 7-limit
> CPS. It does not have tetrads but instead triads. Do you see
> the cycle of triangles in it? Each of these is an incomplete
> o- or utonality. They can be completed by "stellating" the
> hexany. The result would look something like this:
>
> http://www.pandragon.com/polyhedra/jpg/st_octo.jpg
>
> As for the assertion about modulation, can you see the point
> in the center of the diamond? Every tetrad in the diamond
> contains that point. If the listener has good pitch memory,
> your modulations can start to sound monotonous. Plenty of
> popular tunes can be harmonized with a single note, though, so
> it isn't hell and damnation, but... notice the hexany has no
> such central point. Pick any point in the hexany and you will
> see that it is found in only 4 of its 8 triads.
>
> Make any sense?
>
> -Carl
>

okay so u are saying the modes of the 7 limit is in the bottom right
hand corner of the diamond

there r 4 modes in the sense i have been talking about to the 7lim
diamond, how are they modes?

i see here "major tetdrad" and minor and they are both the same except
one is upside down

what pitches are these points supposed to represent and how is that
affected by their orientation?

u said in another post
"It can and does, but the modes I speak of do not all contain
it.
"
what modes r u speaking of?

i just cant see how the double dekany (?) for instance could make
music any more 'interesting' with intervals like 21/16 substituting a
perfect fourth - seriously, listen to how shit it sounds

🔗Carl Lumma <ekin@lumma.org>

>> For explaining CPSs, let's start with:
>>
>> http://tech.ph.groups.yahoo.com/group/tuning-math/photos/view/9c1f?b=1
>>
>> In the upper-right we see the 7-limit diamond. In the bottom-
>> right we see the tetrads that are the "modes" of the 7-limit.
>> You can, as you say, get an otonal mode (major tetrad) on each
>> note of a utonal one. Or the other way around! Do you see
>> the cycles of tetrads in the figure in the upper-right?
>> (Selecting the "large" view might help.)
>>
>> Well, the thing in the middle is the hexany, the 2|4 7-limit
>> CPS. It does not have tetrads but instead triads. Do you see
>> the cycle of triangles in it? Each of these is an incomplete
>> o- or utonality. They can be completed by "stellating" the
>> hexany. The result would look something like this:
>>
>> http://www.pandragon.com/polyhedra/jpg/st_octo.jpg
>>
>> As for the assertion about modulation, can you see the point
>> in the center of the diamond? Every tetrad in the diamond
>> contains that point. If the listener has good pitch memory,
>> your modulations can start to sound monotonous. Plenty of
>> popular tunes can be harmonized with a single note, though, so
>> it isn't hell and damnation, but... notice the hexany has no
>> such central point. Pick any point in the hexany and you will
>> see that it is found in only 4 of its 8 triads.
>>
>> Make any sense?
>>
>> -Carl
>
>okay so u are saying the modes of the 7 limit is in the bottom right
>hand corner of the diamond

Let me try again. Look at figure 6d only (right-hand side of the
image). There are two tetrahedra at the bottom. The one pointing
down represents a minor tetrad (utonality). The one pointing up
a major tetrad (otonality). In the top of the figure is the diamond.
Can you see the eight tetrads in it?

>there r 4 modes in the sense i have been talking about to the 7lim
>diamond, how are they modes?

You called them that. There are four otonalities and four
utonalities.

>i see here "major tetdrad" and minor and they are both the same except
>one is upside down

Exactly.

>what pitches are these points supposed to represent and how is that
>affected by their orientation?

They are points in "tonespace". They represent octave-equivalent
notes in a scale. In tonespace, intervals are lines connecting
these notes. If you see a line parallel to the floor, it is always
a 3:2 (in this example). A line up-and-to-the-right is a 5:4. etc.
Major and minor tetrads contain the same intervals, but their order
is different.

>u said in another post
>"It can and does, but the modes I speak of do not all contain
>it.
>"
>what modes r u speaking of?

Whenever I have said "modes" I have meant o- and utonalities. I
only said that because I thought that's how you were referring to
them.

>i just cant see how the double dekany (?) for instance could make
>music any more 'interesting' with intervals like 21/16 substituting a
>perfect fourth - seriously, listen to how shit it sounds

It's not substituting for a perfect fourth. You have to know where
to find the consonances.

-Carl

🔗tfllt <nasos.eo@gmail.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> For explaining CPSs, let's start with:
> >>
> >>
http://tech.ph.groups.yahoo.com/group/tuning-math/photos/view/9c1f?b=1
> >>
> >> In the upper-right we see the 7-limit diamond. In the bottom-
> >> right we see the tetrads that are the "modes" of the 7-limit.
> >> You can, as you say, get an otonal mode (major tetrad) on each
> >> note of a utonal one. Or the other way around! Do you see
> >> the cycles of tetrads in the figure in the upper-right?
> >> (Selecting the "large" view might help.)
> >>
> >> Well, the thing in the middle is the hexany, the 2|4 7-limit
> >> CPS. It does not have tetrads but instead triads. Do you see
> >> the cycle of triangles in it? Each of these is an incomplete
> >> o- or utonality. They can be completed by "stellating" the
> >> hexany. The result would look something like this:
> >>
> >> http://www.pandragon.com/polyhedra/jpg/st_octo.jpg
> >>
> >> As for the assertion about modulation, can you see the point
> >> in the center of the diamond? Every tetrad in the diamond
> >> contains that point. If the listener has good pitch memory,
> >> your modulations can start to sound monotonous. Plenty of
> >> popular tunes can be harmonized with a single note, though, so
> >> it isn't hell and damnation, but... notice the hexany has no
> >> such central point. Pick any point in the hexany and you will
> >> see that it is found in only 4 of its 8 triads.
> >>
> >> Make any sense?
> >>
> >> -Carl
> >
> >okay so u are saying the modes of the 7 limit is in the bottom right
> >hand corner of the diamond
>
> Let me try again. Look at figure 6d only (right-hand side of the
> image). There are two tetrahedra at the bottom. The one pointing
> down represents a minor tetrad (utonality). The one pointing up
> a major tetrad (otonality). In the top of the figure is the diamond.
> Can you see the eight tetrads in it?
>
> >there r 4 modes in the sense i have been talking about to the 7lim
> >diamond, how are they modes?
>
> You called them that. There are four otonalities and four
> utonalities.
>
> >i see here "major tetdrad" and minor and they are both the same except
> >one is upside down
>
> Exactly.
>
> >what pitches are these points supposed to represent and how is that
> >affected by their orientation?
>
> They are points in "tonespace". They represent octave-equivalent
> notes in a scale. In tonespace, intervals are lines connecting
> these notes. If you see a line parallel to the floor, it is always
> a 3:2 (in this example). A line up-and-to-the-right is a 5:4. etc.
> Major and minor tetrads contain the same intervals, but their order
> is different.
>
> >u said in another post
> >"It can and does, but the modes I speak of do not all contain
> >it.
> >"
> >what modes r u speaking of?
>
> Whenever I have said "modes" I have meant o- and utonalities. I
> only said that because I thought that's how you were referring to
> them.

> -Carl
>

so what are these 'o and u tonalities of a CPS scale?

🔗tfllt <nasos.eo@gmail.com>

so is the 7 lim made out of 8 tetrads?

🔗Carl Lumma <ekin@lumma.org>

>>>> For explaining CPSs, let's start with:
>>>>
>>>> http://tech.ph.groups.yahoo.com/group/tuning-math/photos/view/9c1f?b=1
>>>>
>>>> In the upper-right we see the 7-limit diamond. In the bottom-
>>>> right we see the tetrads that are the "modes" of the 7-limit.
>>>> You can, as you say, get an otonal mode (major tetrad) on each
>>>> note of a utonal one. Or the other way around! Do you see
>>>> the cycles of tetrads in the figure in the upper-right?
>>>> (Selecting the "large" view might help.)
>>>>
>>>> Well, the thing in the middle is the hexany, the 2|4 7-limit
>>>> CPS. It does not have tetrads but instead triads. Do you see my
>>>> tetrad
>
>so is the 7 lim made out of 8 tetrads?

The 7-limit diamond is, yes. In each limit there is a *family*
of CPS scales, not just one. Here is a family tree of CPSs through
the 11-limit

http://tinyurl.com/zvwbr

In limits with an even number of factors (like 1-3-5-7 or 1-3-5-7-9-11)
there is one CPS usually considered 'best' (the hexany and eikosany in
these examples). The hexany is made of 8 triads.

In limits with an odd number of factors, like the 9-limit you suggest,
there are two 'best' CPSs (the dekanies in this case). The dekanies
contain tetrads (wheras the 9-limit diamond contains pentads). They
are 9-limit o- and utonalities missing one note each. Again, they can
be completed to make a "stellated dekany". Here, according to the
Scala archive, is that scale for you

! steldek1.scl
!
Stellated two out of 1 3 5 7 9 dekany.
30
!
21/20
135/128
15/14
35/32
9/8
7/6
189/160
135/112
315/256
5/4
81/64
21/16
27/20
45/32
35/24
189/128
3/2
49/32
25/16
63/40
45/28
105/64
5/3
27/16
7/4
15/8
27/14
35/18
63/32
2/1
!

-Carl

🔗tfllt <nasos.eo@gmail.com>

🔗Carl Lumma <ekin@lumma.org>

🔗tfllt <nasos.eo@gmail.com>

🔗Carl Lumma <ekin@lumma.org>

>>>i still do not know what u mean by that sentence. what is this cycle
>>>of modes
>>
>> You still haven't answered whether you can see the cycles of
>> tetrahedra in the 7-limit diamond of figure 6d.
>
>i can see eight tetrads in the shape is that what u mean?

Yes. Well now let's talk about your favorite example, the
9-limit diamond. Go to page 6 here

http://anaphoria.com/dia.PDF

And pick out any downward-pointing pentagon. It represents
a utonality. Notice that at each of its points, it is
intersected by an upward-pointing pentagon. That is a "cycle"
of otonalities rooted on the notes of a utonality.

Now go here

http://tech.ph.groups.yahoo.com/group/tuning-math/photos/view/9c1f?b=2&m=f&o=0

and look at the two dekanies (things with the numbers 10 above
them).

There are similar things going on, it's just that the o- and
utonalities are not complete. I thought you got that. Well
there's nothing more to get. Would you like me to explain how
to build CPS scales? You can do it in Scala (just type
"help CPS") for one thing.

-Carl

🔗tfllt <nasos.eo@gmail.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >>>i still do not know what u mean by that sentence. what is this cycle
> >>>of modes
> >>
> >> You still haven't answered whether you can see the cycles of
> >> tetrahedra in the 7-limit diamond of figure 6d.
> >
> >i can see eight tetrads in the shape is that what u mean?
>
> Yes. Well now let's talk about your favorite example, the
> 9-limit diamond. Go to page 6 here
>
> http://anaphoria.com/dia.PDF
>
> And pick out any downward-pointing pentagon. It represents
> a utonality. Notice that at each of its points, it is
> intersected by an upward-pointing pentagon. That is a "cycle"
> of otonalities rooted on the notes of a utonality.
>
> Now go here
>
>
http://tech.ph.groups.yahoo.com/group/tuning-math/photos/view/9c1f?b=2&m=f&o=0
>
> and look at the two dekanies (things with the numbers 10 above
> them).
>
> There are similar things going on, it's just that the o- and
> utonalities are not complete. I thought you got that. Well
> there's nothing more to get. Would you like me to explain how
> to build CPS scales? You can do it in Scala (just type
> "help CPS") for one thing.
>
> -Carl
>

when you used the words "cycle" and linear modes in previous posts i
thought u were saying that CPS scales could be broken up in asimilar
way i did by using circular multiplication on a interval set to make
linear series... i do understand what u r saying but this is not quite
the same as what i am talking about with diamonds i.e. it doesnt
suggest a way to divide all the tones into equal groups with their
harmonic neighbours that share an equal relationship with eachother

this is all very interesting... personally i am only interested in one
solution and i think for me that is going to be the 9lim diamond. and
i do appreciate the time and effort u have expended in explaining and
discussing these topics. thnku ;)

🔗Carl Lumma <ekin@lumma.org>

>when you used the words "cycle" and linear modes in previous posts i
>thought u were saying that CPS scales could be broken up in asimilar
>way i did by using circular multiplication on a interval set to make
>linear series... i do understand what u r saying but this is not quite
>the same as what i am talking about with diamonds i.e. it doesnt
>suggest a way to divide all the tones into equal groups with their
>harmonic neighbours that share an equal relationship with eachother
>
>this is all very interesting... personally i am only interested in one
>solution and i think for me that is going to be the 9lim diamond. and
>i do appreciate the time and effort u have expended in explaining and
>discussing these topics. thnku ;)

No prob. But like I said, I'm pretty sure the stellated dekany
does what you want.

-Carl

🔗tfllt <nasos.eo@gmail.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >when you used the words "cycle" and linear modes in previous posts i
> >thought u were saying that CPS scales could be broken up in asimilar
> >way i did by using circular multiplication on a interval set to make
> >linear series... i do understand what u r saying but this is not quite
> >the same as what i am talking about with diamonds i.e. it doesnt
> >suggest a way to divide all the tones into equal groups with their
> >harmonic neighbours that share an equal relationship with eachother
> >
> >this is all very interesting... personally i am only interested in one
> >solution and i think for me that is going to be the 9lim diamond. and
> >i do appreciate the time and effort u have expended in explaining and
> >discussing these topics. thnku ;)
>
> No prob. But like I said, I'm pretty sure the stellated dekany
> does what you want.
>
> -Carl
>

really? is the stellated dekany the 30 tone one? can u plz
demonstrate cause i still dont get it then.

for instance the 7 lim diamon broken into the otonal tetrads bound by
an octave

5/4 * 6/5 * 7/6 * 8/7 -> 5/4 3/2 7/4 2
6/5 * 7/6 * 8/7 * 6/5 -> 6/5 7/5 8/5 2
7/6 * 8/7 * 9/8 * 6/5 -> 7/6 4/3 5/3 2
8/7 * 9/8 * 5/4 * 6/5 -> 8/7 10/7 12/7 2

or for any diamond:

cycle of superparticulars:

G[x] = {(x+1)/x,,2x/(2x-1)}
(G[z] = G[z mod x], if z < 0, G[z]=G[x - |z| mod x], G[x]! = 2 )

otonal linear sets:
M0 = {1, (x+1)/x, (x+2)/x, , 2, }

Mz = {1,G[z],,(PRODUCT[G(z+k)] 1<=k<=x-1), 2, }

Mx = {1, 2x/(2x-1), (2x+2)/(2x-1), , 2, }

how is it done with a cps?

🔗tfllt <nasos.eo@gmail.com>

🔗Carl Lumma <ekin@lumma.org>

>> >when you used the words "cycle" and linear modes in previous posts i
>> >thought u were saying that CPS scales could be broken up in asimilar
>> >way i did by using circular multiplication on a interval set to make
>> >linear series... i do understand what u r saying but this is not quite
>> >the same as what i am talking about with diamonds i.e. it doesnt
>> >suggest a way to divide all the tones into equal groups with their
>> >harmonic neighbours that share an equal relationship with eachother
>> >
>> >this is all very interesting... personally i am only interested in one
>> >solution and i think for me that is going to be the 9lim diamond. and
>> >i do appreciate the time and effort u have expended in explaining and
>> >discussing these topics. thnku ;)
>>
>> No prob. But like I said, I'm pretty sure the stellated dekany
>> does what you want.
>
>really? is the stellated dekany the 30 tone one?

Yes.

>can u plz demonstrate cause i still dont get it then.
>
>for instance the 7 lim diamon broken into the otonal tetrads bound by
>an octave
>
>5/4 * 6/5 * 7/6 * 8/7 -> 5/4 3/2 7/4 2
>6/5 * 7/6 * 8/7 * 6/5 -> 6/5 7/5 8/5 2
>7/6 * 8/7 * 9/8 * 6/5 -> 7/6 4/3 5/3 2
>8/7 * 9/8 * 5/4 * 6/5 -> 8/7 10/7 12/7 2

Here's a hexany with only the otonal chords stellated

!
! 5/4
! /|\
! / | \
! / | \
! / 7/4 \
! /.-'/|\'-.\
! 1/1--/-|-\--3/2
! /|\ / | \ /|\
! / | / 49/20 \ | \
! / |/.-' '-.\| \
! / 7/5--------21/20 \
! /.-' '-.\ /.-' '-. \
! 8/5---------6/5---------9/5

It's a big tetrahedron in tonespace. It has 10 notes instead
of the diamond's 13, but the same number (four) of otonal tetrads.

>or for any diamond:
>
>cycle of superparticulars:
>G[x] = {(x+1)/x, ..., 2x/(2x-1)}
>(G[z] = G[z mod x], if z < 0, G[z]=G[x - |z| mod x], G[x]! = 2 )

I dunno about superparticulars in CPSs. Why are they important?

By the way, does []! really mean the product of the []'s members?

>otonal linear sets:
>M0 = {1, (x+1)/x, (x+2)/x, ..., 2, ...}

Yes, this is an otonality. Stellated CPSs contain them.

>Mz = {1, G[z], ..., (PRODUCT[G(z+k)] 1<=k<=x-1), 2, ...}
>...
>Mx = {1, 2x/(2x-1), (2x+2)/(2x-1), ..., 2, ...}
>
>how is it done with a cps?

You mean, how does one find the otonalities? I'm sure there's a
simple formula, but I've just always looked at the diagrams.

-Carl

🔗tfllt <nasos.eo@gmail.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> >when you used the words "cycle" and linear modes in previous posts i
> >> >thought u were saying that CPS scales could be broken up in
asimilar
> >> >way i did by using circular multiplication on a interval set to make
> >> >linear series... i do understand what u r saying but this is not
quite
> >> >the same as what i am talking about with diamonds i.e. it doesnt
> >> >suggest a way to divide all the tones into equal groups with their
> >> >harmonic neighbours that share an equal relationship with eachother
> >> >
> >> >this is all very interesting... personally i am only interested
in one
> >> >solution and i think for me that is going to be the 9lim
diamond. and
> >> >i do appreciate the time and effort u have expended in
explaining and
> >> >discussing these topics. thnku ;)
> >>
> >> No prob. But like I said, I'm pretty sure the stellated dekany
> >> does what you want.
> >
> >really? is the stellated dekany the 30 tone one?
>
> Yes.
>
> >can u plz demonstrate cause i still dont get it then.
> >
> >for instance the 7 lim diamon broken into the otonal tetrads bound by
> >an octave
> >
> >5/4 * 6/5 * 7/6 * 8/7 -> 5/4 3/2 7/4 2
> >6/5 * 7/6 * 8/7 * 6/5 -> 6/5 7/5 8/5 2
> >7/6 * 8/7 * 9/8 * 6/5 -> 7/6 4/3 5/3 2
> >8/7 * 9/8 * 5/4 * 6/5 -> 8/7 10/7 12/7 2
>
> Here's a hexany with only the otonal chords stellated
>
> !
> ! 5/4
> ! /|\
> ! / | \
> ! / | \
> ! / 7/4 \
> ! /.-'/|\'-.\
> ! 1/1--/-|-\--3/2
> ! /|\ / | \ /|\
> ! / | / 49/20 \ | \
> ! / |/.-' '-.\| \
> ! / 7/5--------21/20 \
> ! /.-' '-.\ /.-' '-. \
> ! 8/5---------6/5---------9/5
>
> It's a big tetrahedron in tonespace. It has 10 notes instead
> of the diamond's 13, but the same number (four) of otonal tetrads.
>
> >or for any diamond:
> >
> >cycle of superparticulars:
> >G[x] = {(x+1)/x, ..., 2x/(2x-1)}
> >(G[z] = G[z mod x], if z < 0, G[z]=G[x - |z| mod x], G[x]! = 2 )
>
> I dunno about superparticulars in CPSs. Why are they important?
>
> By the way, does []! really mean the product of the []'s members?
>
> >otonal linear sets:
> >M0 = {1, (x+1)/x, (x+2)/x, ..., 2, ...}
>
> Yes, this is an otonality. Stellated CPSs contain them.
>
> >Mz = {1, G[z], ..., (PRODUCT[G(z+k)] 1<=k<=x-1), 2, ...}
> >...
> >Mx = {1, 2x/(2x-1), (2x+2)/(2x-1), ..., 2, ...}
> >
> >how is it done with a cps?
>
> You mean, how does one find the otonalities? I'm sure there's a
> simple formula, but I've just always looked at the diagrams.
>
> -Carl
>

hi carl that looks interesting - but i dont understand why 49/20 is in
the scale? i think im going to have to read up about cps' cause to be
honest all i know is what u have said in the past week or so so i am
really in no position to make a criticism. could u idrect me to a
good source for finding out exactly the process for forming a cps..
stellated means filling out the otonal elements right? so how would i
use this 30 tone stellated dekany? does it divide evenly into
different harmonic areas? can u show me how it is made (with numbers
not pictures)? ill have to search through the messages to get another
copy of the .scl.i am not sure whether what i wrote in the previous
post has any connection to CPS - but if like u say the cps is taken
fro mthe same structure as the diamond then there should be a
relation. superparticular multiplications define the otonalities so
they should be there if there are complete otonalities in the cps.

in a way a tonality diamond i ssort of a combination product set in
the broadest sense of the word but with strict parameters on the
multiplication so it always breaks into equal membered linear sets

cool cool i will have to do some reading when i get some more free
time thx

naso

🔗Carl Lumma <ekin@lumma.org>

>> Here's a hexany with only the otonal chords stellated
>>
>> !
>> ! 5/4
>> ! /|\
>> ! / | \
>> ! / | \
>> ! / 7/4 \
>> ! /.-'/|\'-.\
>> ! 1/1--/-|-\--3/2
>> ! /|\ / | \ /|\
>> ! / | / 49/20 \ | \
>> ! / |/.-' '-.\| \
>> ! / 7/5--------21/20 \
>> ! /.-' '-.\ /.-' '-. \
>> ! 8/5---------6/5---------9/5
>>
>> It's a big tetrahedron in tonespace. It has 10 notes instead
>> of the diamond's 13, but the same number (four) of otonal tetrads.
>>
>> >or for any diamond:
>> >
>> >cycle of superparticulars:
>> >G[x] = {(x+1)/x, ..., 2x/(2x-1)}
>> >(G[z] = G[z mod x], if z < 0, G[z]=G[x - |z| mod x], G[x]! = 2 )
>>
>> I dunno about superparticulars in CPSs. Why are they important?
>>
>> By the way, does []! really mean the product of the []'s members?
>>
>> >otonal linear sets:
>> >M0 = {1, (x+1)/x, (x+2)/x, ..., 2, ...}
>>
>> Yes, this is an otonality. Stellated CPSs contain them.
>>
>> >Mz = {1, G[z], ..., (PRODUCT[G(z+k)] 1<=k<=x-1), 2, ...}
>> >...
>> >Mx = {1, 2x/(2x-1), (2x+2)/(2x-1), ..., 2, ...}
>> >
>> >how is it done with a cps?
>>
>> You mean, how does one find the otonalities? I'm sure there's a
>> simple formula, but I've just always looked at the diagrams.
>
>hi carl that looks interesting - but i dont understand why 49/20 is in
>the scale?

Because it's part of a complete otonality rooted on 7/5.

>i think im going to have to read up about cps' cause to be
>honest all i know is what u have said in the past week or so so i am
>really in no position to make a criticism. could u idrect me to a
>good source for finding out exactly the process for forming a cps.

Last I checked Wikipedia doesn't have an article, but

http://www.google.com/search?q=combination+product+sets

turns up

http://www.tonalsoft.com/enc/c/combination-product-set.aspx

>stellated means filling out the otonal elements right?

Usually it means filling in both otonal and utonal elements.
I was just doing the otonal ones above because that's what
you seemed interested in.

>so how would i use this 30 tone stellated dekany?

How do you *want* to use it?

>does it divide evenly into different harmonic areas?

Yes, as I've tried to demonstrate.

>can u show me how it is made (with numbers not pictures)?

Yes, one takes all the k-combinations of whatever factors
one wants. To get the 2|5 dekany, take all pairs of the
factors [1 3 5 7 9] (actually they could be any factors
you want) and multiply them

1 * 3 3 * 5 5 * 7 7 * 9
1 * 5 3 * 7 5 * 9
1 * 7 ...
...

These are the tones of the scale. If you want a 1/1 (for
convenience) just pick any tone and multiply all tones by
its reciprocal. To stellate, simply complete the chords.

All of this can be done with Scala rather instantly. As
I already mentioned, use "help cps".

>ill have to search through the messages to get another copy
>of the .scl.i am not sure whether what i wrote in the previous
>post has any connection to CPS - but if like u say the cps is taken
>fro mthe same structure as the diamond then there should be a
>relation. superparticular multiplications define the otonalities so
>they should be there if there are complete otonalities in the cps.

Yup.

-Carl

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

* Guys,

I get 5/4 * 6/5 * 7/6 * 8/7
6/5 * 7/6 * 8/7 * 5/4
7/6 * 8/7 * 5/4 * 6/5
8/7 * 5/4 * 6/5 * 7/6

(- Paul Hj), otherwise nice!

>
> Here's a hexany with only the otonal chords stellated
>
> !
> ! 5/4
> ! /|\
> ! / | \
> ! / | \
> ! / 7/4 \
> ! /.-'/|\'-.\
> ! 1/1--/-|-\--3/2
> ! /|\ / | \ /|\
> ! / | / 49/20 \ | \
> ! / |/.-' '-.\| \
> ! / 7/5--------21/20 \
> ! /.-' '-.\ /.-' '-. \
> ! 8/5---------6/5---------9/5
>
> It's a big tetrahedron in tonespace. It has 10 notes instead
> of the diamond's 13, but the same number (four) of otonal tetrads.
>
> >or for any diamond:
> >
> >cycle of superparticulars:
> >G[x] = {(x+1)/x, ..., 2x/(2x-1)}
> >(G[z] = G[z mod x], if z < 0, G[z]=G[x - |z| mod x], G[x]! = 2 )
>
> I dunno about superparticulars in CPSs. Why are they important?
>
> By the way, does []! really mean the product of the []'s members?
>
> >otonal linear sets:
> >M0 = {1, (x+1)/x, (x+2)/x, ..., 2, ...}
>
> Yes, this is an otonality. Stellated CPSs contain them.
>
> >Mz = {1, G[z], ..., (PRODUCT[G(z+k)] 1<=k<=x-1), 2, ...}
> >...
> >Mx = {1, 2x/(2x-1), (2x+2)/(2x-1), ..., 2, ...}
> >
> >how is it done with a cps?
>
> You mean, how does one find the otonalities? I'm sure there's a
> simple formula, but I've just always looked at the diagrams.
>
> -Carl
>

🔗Carl Lumma <ekin@lumma.org>

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> >for instance the 7 lim diamon broken into the otonal tetrads
> >> >bound by an octave
> >> >
> >> >5/4 * 6/5 * 7/6 * 8/7 -> 5/4 3/2 7/4 2
> >> >6/5 * 7/6 * 8/7 * 6/5 -> 6/5 7/5 8/5 2
> >> >7/6 * 8/7 * 9/8 * 6/5 -> 7/6 4/3 5/3 2
> >> >8/7 * 9/8 * 5/4 * 6/5 -> 8/7 10/7 12/7 2
> >
> >* Guys,
> >
> >I get 5/4 * 6/5 * 7/6 * 8/7
> > 6/5 * 7/6 * 8/7 * 5/4
> > 7/6 * 8/7 * 5/4 * 6/5
> > 8/7 * 5/4 * 6/5 * 7/6
> >
> >(- Paul Hj), otherwise nice!
>
> Thanks for spotting that Paul. I didn't even notice. I
> guess the fact that intervals between adjacent harmonics are
> superparticular isn't particularly exciting to my eye.
>
> -Carl
>
I have been interested in superparticularity these days. For example,
(5*3/7)->15/14, (3*7/5)->21/20, BUT (5*7/3)->35/24, not
superparticular.

🔗Carl Lumma <ekin@lumma.org>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>

🔗tfllt <nasos.eo@gmail.com>

yeah of course sorry about that like i said below the cycle is always
ascending superparticular starting from x to 2x that was the point of
the message i am always very stoned and tend to miss details ull have
to be patient with me =D

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >I have been interested in superparticularity these days. For
example,
> >(5*3/7)->15/14, (3*7/5)->21/20, BUT (5*7/3)->35/24, not
> >superparticular.
>
> What's the significance of that?
>
> -Carl

It relates to parsimonious voice leading, where parts move in
semitones. Like Chd7->F7. I made an error in my other post - 21/20
is actually right in a hexany, not the completion of a tetrad. Can
15/14 occur in a hexany? Yes; if you divide out by 7. So can 35/24,
but it doesn't occur in voice-leading.

So what's in a hexany?

I get

35, 5, 21, 3, 7, 15 -> (8/7, 21/20, 8/7, 7/6, 15/14, 7/6 steps)
also get (6/5, 6/5, 4/3, 5/4, 5/4, 4/3)
and (48/35, 7/5, 10/7, 35/24, 10/7 and 7/5)

All of these can occur in a original hexany, by setting each one of
the original list to 1/1.

The first two rows are superparticular but not the third. I'll have
to look at the stellated hexany diagram and see which axes go with
each ratio. Of course, only superparticular ratios between 15/14 and
28/27 (plus 49/48) in the 7-limit are semitones.

So I guess I am just trying to study the 7-limit Tonality Diamond
and the Stellated Hexany for configurations that relate to voice-
leading. For example, G7-> C7 uses both 15/14 and 21/20, once again,
I will need to check to see if these are two tetrads in the
stellated hexany

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul_hjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <ekin@> wrote:
> >
> > >I have been interested in superparticularity these days. For
> example,
> > >(5*3/7)->15/14, (3*7/5)->21/20, BUT (5*7/3)->35/24, not
> > >superparticular.
> >
> > What's the significance of that?
> >
> > -Carl
>
> It relates to parsimonious voice leading, where parts move in
> semitones. Like Chd7->F7. I made an error in my other post - 21/20
> is actually right in a hexany, not the completion of a tetrad. Can
> 15/14 occur in a hexany? Yes; if you divide out by 7. So can
35/24,
> but it doesn't occur in voice-leading.
>
> So what's in a hexany?
>
> I get
>
> 35, 5, 21, 3, 7, 15 -> (8/7, 21/20, 8/7, 7/6, 15/14, 7/6 steps)
> also get (6/5, 6/5, 4/3, 5/4, 5/4, 4/3)
> and (48/35, 7/5, 10/7, 35/24, 10/7 and 7/5)
>
> All of these can occur in a original hexany, by setting each one
of
> the original list to 1/1.
>
> The first two rows are superparticular but not the third. I'll
have
> to look at the stellated hexany diagram and see which axes go with
> each ratio. Of course, only superparticular ratios between 15/14
and
> 28/27 (plus 49/48) in the 7-limit are semitones.
>
> So I guess I am just trying to study the 7-limit Tonality Diamond
> and the Stellated Hexany for configurations that relate to voice-
> leading. For example, G7-> C7 uses both 15/14 and 21/20, once
again,
> I will need to check to see if these are two tetrads in the
> stellated hexany

Here's what I get -

Two lower otonal tetrads in the stellated hexany (actually Ab7 and
Eb7)

Ab7: 8/5, 1/1, 6/5, 7/5
Eb7: 6/5, 3/2, 9/5, 21/20

Moving from Eb7 to Ab7 gives 21/20 in the soprano, 9/8 in the alto,
15/14 in the tenor and 1/1 in the bass (Eb,G,Bb,Db->Eb,Gb,Ab,C)

So there it is. I guess it doesn't give the significance of
superparticulars, just their usage. Another use of course, is
in unison vectors, like 126/125, 50/49, 36/35, and the like.

🔗Kraig Grady <kraiggrady@anaphoria.com>

see http://anaphoria.com/CPStoC-pt1.PDF

Paul G Hjelmstad wrote:
>
> --- In tuning-math@yahoogroups.com > <mailto:tuning-math%40yahoogroups.com>, Carl Lumma <ekin@...> wrote:
> >
> > >I have been interested in superparticularity these days. For
> example,
> > >(5*3/7)->15/14, (3*7/5)->21/20, BUT (5*7/3)->35/24, not
> > >superparticular.
> >
> > What's the significance of that?
> >
> > -Carl
>
> It relates to parsimonious voice leading, where parts move in
> semitones. Like Chd7->F7. I made an error in my other post - 21/20
> is actually right in a hexany, not the completion of a tetrad. Can
> 15/14 occur in a hexany? Yes; if you divide out by 7. So can 35/24,
> but it doesn't occur in voice-leading.
>
> So what's in a hexany?
>
> I get
>
> 35, 5, 21, 3, 7, 15 -> (8/7, 21/20, 8/7, 7/6, 15/14, 7/6 steps)
> also get (6/5, 6/5, 4/3, 5/4, 5/4, 4/3)
> and (48/35, 7/5, 10/7, 35/24, 10/7 and 7/5)
>
> All of these can occur in a original hexany, by setting each one of
> the original list to 1/1.
>
> The first two rows are superparticular but not the third. I'll have
> to look at the stellated hexany diagram and see which axes go with
> each ratio. Of course, only superparticular ratios between 15/14 and
> 28/27 (plus 49/48) in the 7-limit are semitones.
>
> So I guess I am just trying to study the 7-limit Tonality Diamond
> and the Stellated Hexany for configurations that relate to voice-
> leading. For example, G7-> C7 uses both 15/14 and 21/20, once again,
> I will need to check to see if these are two tetrads in the
> stellated hexany
>
> >
>
> -- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

🔗Carl Lumma <ekin@lumma.org>

> > >I have been interested in superparticularity these days.
> > >For example, (5*3/7)->15/14, (3*7/5)->21/20, BUT
> > >(5*7/3)->35/24, not superparticular.
> >
> > What's the significance of that?
>
> It relates to parsimonious voice leading, where parts move in
> semitones. Like Chd7->F7.

What chord is Chd7?

> I made an error in my other post - 21/20 is actually right in
> a hexany, not the completion of a tetrad. Can 15/14 occur in
> a hexany?

Sure.

> Yes; if you divide out by 7. So can 35/24,

Any ratio can occur in a hexany. Or maybe I'm not getting
your line of thinking.

> but it doesn't occur in voice-leading.

> So what's in a hexany?
>
> I get
>
> 35, 5, 21, 3, 7, 15 -> (8/7, 21/20, 8/7, 7/6, 15/14, 7/6 steps)

Right...

> also get (6/5, 6/5, 4/3, 5/4, 5/4, 4/3)
> and (48/35, 7/5, 10/7, 35/24, 10/7 and 7/5)

What are these?

> All of these can occur in a original hexany, by setting each one of
> the original list to 1/1.

Can you explain?

> So I guess I am just trying to study the 7-limit Tonality Diamond
> and the Stellated Hexany for configurations that relate to voice-
> leading. For example, G7-> C7 uses both 15/14 and 21/20,

Is that bad?

-Carl

🔗Carl Lumma <ekin@lumma.org>

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@...> wrote:
>
> > > >I have been interested in superparticularity these days.
> > > >For example, (5*3/7)->15/14, (3*7/5)->21/20, BUT
> > > >(5*7/3)->35/24, not superparticular.
> > >
> > > What's the significance of that?
> >
> > It relates to parsimonious voice leading, where parts move in
> > semitones. Like Chd7->F7.
>
> What chord is Chd7?

C half-diminished-7 (C, Eb, Gb, Bb)
>
> > I made an error in my other post - 21/20 is actually right in
> > a hexany, not the completion of a tetrad. Can 15/14 occur in
> > a hexany?
>
> Sure.
>
> > Yes; if you divide out by 7. So can 35/24,
>
> Any ratio can occur in a hexany. Or maybe I'm not getting
> your line of thinking.
>
> > but it doesn't occur in voice-leading.
>
> ??
>
> > So what's in a hexany?
> >
> > I get
> >
> > 35, 5, 21, 3, 7, 15 -> (8/7, 21/20, 8/7, 7/6, 15/14, 7/6 steps)
>
> Right...
>
> > also get (6/5, 6/5, 4/3, 5/4, 5/4, 4/3)
> > and (48/35, 7/5, 10/7, 35/24, 10/7 and 7/5)
>
> What are these?

Other ratios, by skipping one for the second line and two
for the third line
>
> > All of these can occur in a original hexany, by setting each one
of
> > the original list to 1/1.
>
> Can you explain?

By setting any value of the original hexany to 1/1, these values can
all appear.
>
> > So I guess I am just trying to study the 7-limit Tonality
Diamond
> > and the Stellated Hexany for configurations that relate to voice-
> > leading. For example, G7-> C7 uses both 15/14 and 21/20,
>
> Is that bad?

No not bad! It's also cool that 15/14 * 21/20 equals 9/8. So for
example (in blues harmony) Going from the seventh of F7 (Eb) to
E natural is 15/14, and then going to the seventh of G7 (F) is
21/20, the product is 9/8, which makes sense, because the seventh
chords are 9:8 apart from each other. But can this be proven using
cognitive science? Does the ear hear seventh chords this way, and
furthermore, calculate ratios like 21/20? That I don't know.
>
> -Carl
>

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

Paul Hj said:
> > > So I guess I am just trying to study the 7-limit Tonality
> Diamond
> > > and the Stellated Hexany for configurations that relate to
voice-
> > > leading. For example, G7-> C7 uses both 15/14 and 21/20,
> >
Carl said:
> > Is that bad?
Me:
> No not bad! It's also cool that 15/14 * 21/20 equals 9/8. So for
> example (in blues harmony) Going from the seventh of F7 (Eb) to
> E natural is 15/14, and then going to the seventh of G7 (F) is
> 21/20, the product is 9/8, which makes sense, because the seventh
> chords are 9:8 apart from each other. But can this be proven using
> cognitive science? Does the ear hear seventh chords this way, and
> furthermore, calculate ratios like 21/20? That I don't know.

Also, a tempered semitone in 12-tET is about 18/17. Notice that
15/14, 18/17, 21/20, the numerators and denominators are each three
apart. So a tempered semitone with a little decoherence can take
on the guise of 15/14 or 21/20, depending on context. Also,
36/35, 50/49, 64/63, the numerators and denominators are all 14
apart, but this is really pushing things since they are commatic
unison vectors.

Paul Hj.

> >
> >
>

🔗Carl Lumma <ekin@lumma.org>

>> > It relates to parsimonious voice leading, where parts move in
>> > semitones. Like Chd7->F7.
>>
>> What chord is Chd7?
>
>C half-diminished-7 (C, Eb, Gb, Bb)

Ah yes.

>> > So I guess I am just trying to study the 7-limit Tonality
>> > Diamond and the Stellated Hexany for configurations that
>> > relate to voice-leading. For example, G7-> C7 uses both
>> > 15/14 and 21/20,
>>
>> Is that bad?
>
>No not bad! It's also cool that 15/14 * 21/20 equals 9/8. So for
>example (in blues harmony) Going from the seventh of F7 (Eb) to
>E natural is 15/14, and then going to the seventh of G7 (F) is
>21/20, the product is 9/8, which makes sense, because the seventh
>chords are 9:8 apart from each other. But can this be proven using
>cognitive science? Does the ear hear seventh chords this way, and
>furthermore, calculate ratios like 21/20? That I don't know.

I guess I don't know what about superparticularity makes for
parsimonious voice leading. Wouldn't any small intervals do?

-Carl

🔗Paul G Hjelmstad <paul_hjelmstad@allianzlife.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@...> wrote:
>
> >> > It relates to parsimonious voice leading, where parts move in
> >> > semitones. Like Chd7->F7.
> >>
> >> What chord is Chd7?
> >
> >C half-diminished-7 (C, Eb, Gb, Bb)
>
> Ah yes.
>
> >> > So I guess I am just trying to study the 7-limit Tonality
> >> > Diamond and the Stellated Hexany for configurations that
> >> > relate to voice-leading. For example, G7-> C7 uses both
> >> > 15/14 and 21/20,
> >>
> >> Is that bad?
> >
> >No not bad! It's also cool that 15/14 * 21/20 equals 9/8. So for
> >example (in blues harmony) Going from the seventh of F7 (Eb) to
> >E natural is 15/14, and then going to the seventh of G7 (F) is
> >21/20, the product is 9/8, which makes sense, because the seventh
> >chords are 9:8 apart from each other. But can this be proven using
> >cognitive science? Does the ear hear seventh chords this way, and
> >furthermore, calculate ratios like 21/20? That I don't know.
>
> I guess I don't know what about superparticularity makes for
> parsimonious voice leading. Wouldn't any small intervals do?
>
> -Carl

Yes, you are correct, for example there is 27/25. However, you
can (for example) multiply this out by the syntonic comma, 27/25 *
80/81 to obtain 16/15. I am just saying the most basic ratios
used in (semitone) voice-leading are: 15/14, 16/15, 21/20, 25/24 and
49/48. I really need to check the stellated hexany for all
possibilities and see if this is true.

Paul Hj
>

🔗Carl Lumma <ekin@lumma.org>

>Yes, you are correct, for example there is 27/25.

And many irrational intervals...

>However, you
>can (for example) multiply this out by the syntonic comma, 27/25 *
>80/81 to obtain 16/15. I am just saying the most basic ratios
>used in (semitone) voice-leading are: 15/14, 16/15, 21/20, 25/24 and
>49/48. I really need to check the stellated hexany for all
>possibilities and see if this is true.

It will tend to be true, because you're restricting yourself
to simple ratios (the diamond and CPSs live in compact regions
of the lattic) and you're hunting for small intervals.

-Carl