From the familiar to the exotic.

Here is the familiar:
The major scale in 12tet can be constructed from three similar
chords. Thus:
0-400-700
500-900-1200
700-1100-200
becomes 0-200-400-500-700-900-1100-1200
And it forms part of a continuous cycle of 700cents
500-0-700-200-900-400-1100.

Now for the exotic:
One particular scale in 22tet can be constructed from three similar
chords. Thus:
0-436-655
545-982-1200
655-1091-109
becomes 0-109-436-545-655-982-1091-1200
And it forms part of a continuous cycle of 655cents
982-436-1091-545-0-655-109.

The above is a simple example of how the familiar can be used to
teach the exotic; and to good effect.

Using familiar principles to explore the exotic, that is very much what I am attempting to do with my selection of 12 pitches out of 53edo. Of course one can build a scale out of stacked chords. But what fascinates me these days is the notion of enharmonic equivalence. The regular 7 out of 12edo supports punning - this can be observed by looking at the pattern in the plane of harmonic relationships:

http://i140.photobucket.com/albums/r6/kukulaj/seven.jpg

The little islands of 7 pitches actually link up to make stripes. This means that we can get from one pitch to another by multiple harmonically inequivalent paths.

My choice 12 out of 53 makes a similar striped pattern. Can this vocabulary be used to make poetry?

Jim

how did you arrive at the 12 you did?

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Jim K wrote:
>
>
> Using familiar principles to explore the exotic, that is very much > what I am attempting to do with my selection of 12 pitches out of > 53edo. Of course one can build a scale out of stacked chords. But what > fascinates me these days is the notion of enharmonic equivalence. The > regular 7 out of 12edo supports punning - this can be observed by > looking at the pattern in the plane of harmonic relationships:
>
> http://i140.photobucket.com/albums/r6/kukulaj/seven.jpg > <http://i140.photobucket.com/albums/r6/kukulaj/seven.jpg>
>
> The little islands of 7 pitches actually link up to make stripes. This > means that we can get from one pitch to another by multiple > harmonically inequivalent paths.
>
> My choice 12 out of 53 makes a similar striped pattern. Can this > vocabulary be used to make poetry?
>
> Jim
>
>

--- In MakeMicroMusic@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> how did you arrive at the 12 you did?

I played around with a few possibilities, of course. The basic idea is to plot a path from a pitch to the same pitch at a different spot in the harmonic plane - this path represents one of the commas tempered by the tuning. I then augment the path a bit with local relationships, and try to make the thing somehow regular or logical.

The cool pattern with this set... it is just 11 minor thirds in a row! Turns out that 6 minor thirds is enharmonically equivalent to a fourth in 53edo.

It looks like a fun enough pattern, the regular stripes. So now I will have to keep playing around to see if I can make music out of it!

Jim

Jim K wrote:

> The cool pattern with this set... it is just 11 minor
> thirds in a row! Turns out that 6 minor thirds is
> enharmonically equivalent to a fourth in 53edo.

Hanson/kleismatic

--- In MakeMicroMusic@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Jim K wrote:
>
> > 11 minor thirds in a row!
>
> Hanson/kleismatic
>

Thanks a zillion! It is wonderful to see that I am on charted territory - that I am not totally in the dark!

i think this series of 6/5 is what Hanson came up with. It is not completely clear to me

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Jim K wrote:
>
>
> --- In MakeMicroMusic@yahoogroups.com > <mailto:MakeMicroMusic%40yahoogroups.com>, Kraig Grady > <kraiggrady@...> wrote:
> >
> > how did you arrive at the 12 you did?
>
> I played around with a few possibilities, of course. The basic idea is > to plot a path from a pitch to the same pitch at a different spot in > the harmonic plane - this path represents one of the commas tempered > by the tuning. I then augment the path a bit with local relationships, > and try to make the thing somehow regular or logical.
>
> The cool pattern with this set... it is just 11 minor thirds in a row! > Turns out that 6 minor thirds is enharmonically equivalent to a fourth > in 53edo.
>
> It looks like a fun enough pattern, the regular stripes. So now I will > have to keep playing around to see if I can make music out of it!
>
> Jim
>
>

thanks -missed this before my previous message

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Graham Breed wrote:
>
>
> Jim K wrote:
>
> > The cool pattern with this set... it is just 11 minor
> > thirds in a row! Turns out that 6 minor thirds is
> > enharmonically equivalent to a fourth in 53edo.
>
> Hanson/kleismatic
>
>

Hanson didn't do any music so the territory is yours unless i have missed others actually using it.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Jim K wrote:
>
>
> --- In MakeMicroMusic@yahoogroups.com > <mailto:MakeMicroMusic%40yahoogroups.com>, Graham Breed <gbreed@...> > wrote:
> >
> > Jim K wrote:
> >
> > > 11 minor thirds in a row!
> >
> > Hanson/kleismatic
> >
>
> Thanks a zillion! It is wonderful to see that I am on charted > territory - that I am not totally in the dark!
>
>

At 08:17 PM 5/16/2009, you wrote:
>--- In MakeMicroMusic@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>>
>> Jim K wrote:
>>
>> > 11 minor thirds in a row!
>>
>> Hanson/kleismatic
>>
>
>Thanks a zillion! It is wonderful to see that I am on charted
>territory - that I am not totally in the dark!

In fact you can add yourself to the list of folks who have
independently discovered periodicity blocks -- still quite an
elite group. They have been seriously studied for the first
time in recent years by some of the members here.

As far as hanson/kleismic, see also:
http://en.wikipedia.org/wiki/Shohe_Tanaka

-Carl

--- In MakeMicroMusic@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> At 08:17 PM 5/16/2009, you wrote:
> >--- In MakeMicroMusic@yahoogroups.com, Graham Breed <gbreed@> wrote:
>
> As far as hanson/kleismic, see also:
> http://en.wikipedia.org/wiki/Shohe_Tanaka

Wow! This feels a bit like what happened to me in grad school... I was supposed to be studying physics, but spent an awful lot of time on tuning theory! I never did get my PhD, no surprise! In those days my favorite scale had equal steps of size 10**(1/91). My bedroom walls were covered with charts of those harmonic hexagonal grids, which I had never seen anywhere else. Then I found McClain's _Myth of Invariance_ and just about fell over.

I didn't know until just now about Tanaka. What a blast! Thanks!

Jim

periodicity blocks being the same as constant structures

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
>
> At 08:17 PM 5/16/2009, you wrote:
> >--- In MakeMicroMusic@yahoogroups.com > <mailto:MakeMicroMusic%40yahoogroups.com>, Graham Breed <gbreed@...> > wrote:
> >>
> >> Jim K wrote:
> >>
> >> > 11 minor thirds in a row!
> >>
> >> Hanson/kleismatic
> >>
> >
> >Thanks a zillion! It is wonderful to see that I am on charted
> >territory - that I am not totally in the dark!
>
> In fact you can add yourself to the list of folks who have
> independently discovered periodicity blocks -- still quite an
> elite group. They have been seriously studied for the first
> time in recent years by some of the members here.
>
> As far as hanson/kleismic, see also:
> http://en.wikipedia.org/wiki/Shohe_Tanaka > <http://en.wikipedia.org/wiki/Shohe_Tanaka>
>
> -Carl
>
>

Perhaps someone could add to this. something that might be useful to those beyond a mathematical explanation which severely limits it audience. I for one use them daily and for years and have no need for any of this math so obviously there might be other ways of defining it

Their are advantages to 'constant structures' i now realize over periodicity blocks. the latter is included but also all those JI scales not limit based and offers a better bridge to mapping recurrent sequences where any one particular interval might never be repeated 'exactly'.
How one could add this possibility to the page of formula is beyond me.

both regardless are fundamental in understanding in one way to use microtones.

That it has been 'rediscovered' by many both shows that it probably something more to it for more than a few.
On the other hand we could point to the 'Johnston' and/or 'Tenney' schools which seem to have never stumbled into this way of thinking, either directly or indirectly. possibly in their interest in 'atonal' landscapes. How the CPS escapes their notice though is bewildering not to mention that it too can be fit into a constant structure. here again the approach of a periodicity block would miss this possibility.

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Related, but not the same.

-Carl

At 08:36 PM 5/16/2009, you wrote:
>periodicity blocks being the same as constant structures
>
>
>/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
>Mesotonal Music from:
>_'''''''_ ^North/Western Hemisphere:
>North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
>_'''''''_ ^South/Eastern Hemisphere:
>Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
>
>',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>

yes my latter post pointed out that out.
but all periodicity blocks are constant structures. or do you know of an exception?

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
>
> Related, but not the same.
>
> -Carl
>
> At 08:36 PM 5/16/2009, you wrote:
> >periodicity blocks being the same as constant structures
> >
> >
> >/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> >Mesotonal Music from:
> >_'''''''_ ^North/Western Hemisphere:
> >North American Embassy of Anaphoria Island <http://anaphoria.com/ > <http://anaphoria.com/>>
> >
> >_'''''''_ ^South/Eastern Hemisphere:
> >Austronesian Outpost of Anaphoria > <http://anaphoriasouth.blogspot.com/ > <http://anaphoriasouth.blogspot.com/>>
> >
> >',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> >
>
>

Kraig wrote:

>yes my latter post pointed out that out.
>but all periodicity blocks are constant structures. or do you know of an
>exception?

"Epimorphic" periodicity blocks are, I think, constant structures.
Non-epimorphic PBs are sometimes considered pathological.

Way back when we had you ask Erv about the definition of CS, the
definition he gave would include scales that aren't even JI.

-Carl

yes that would be correct in that they don't have to be JI.
i am not sure why the distinction between epimorphic and non epimorphic?
especially why they might might not be pathological. i guess i missed this discussion.
an example of description of what happened there.
/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
>
> Kraig wrote:
>
> >yes my latter post pointed out that out.
> >but all periodicity blocks are constant structures. or do you know of an
> >exception?
>
> "Epimorphic" periodicity blocks are, I think, constant structures.
> Non-epimorphic PBs are sometimes considered pathological.
>
> Way back when we had you ask Erv about the definition of CS, the
> definition he gave would include scales that aren't even JI.
>
> -Carl
>
>

Kraig wrote:

>yes that would be correct in that they don't have to be JI.

PBs do have to be JI, so there are many constant structures
that are not PBs.

>i am not sure why the distinction between epimorphic and non epimorphic?

Basically it means "this PB is also a constant structure". It
seems most or all of the blocks Fokker considered were epimorphic.
If the commas defining the PB are all smaller than the smallest
scale step in the PB then that also assures you have an
epimorphic block.

The formal definition of epimorphic says that there must be a
val that, if you multiply it by some ratio found in the block,
the result is the scale degree on which that ratio is found.
For example, the 5-limit val of the diatonic scale in JI is
< 7 11 16 |. That means 7 steps to the 2/1, 11 to the 3/1,
and 16 to the 5/1. The monzo for 3/2 is | -1 1 0 >. That means
2^-1 * 3^1 * 5^0 = 3/2. Their product is written

< 7 11 16 | -1 1 0 > = 4

or

(7 * -1) + (11 * 1) + (16 * 0) = 4

And sure enough, 3/2 is the 4th step in the diatonic scale
(not counting the starting note). We can try it with 10/9

< 7 11 16 | 1 -2 1 > = 1

9/8 is also always a 2nd so the answer will again be 1

< 7 11 16 | -3 2 0 > = 1

45/32 is a kind of 4th so the answer will be 3

< 7 11 16 | -5 2 1 > = 3

And so on.

-Carl

>
> The formal definition of epimorphic says that there must be a
> val that, if you multiply it by some ratio found in the block,
> the result is the scale degree on which that ratio is found.
> For example, the 5-limit val of the diatonic scale in JI is
> < 7 11 16 |. That means 7 steps to the 2/1, 11 to the 3/1,
> and 16 to the 5/1. The monzo for 3/2 is | -1 1 0 >. That means
> 2^-1 * 3^1 * 5^0 = 3/2. Their product is written
>
> < 7 11 16 | -1 1 0 > = 4
>
> or
>
> (7 * -1) + (11 * 1) + (16 * 0) = 4
>
> And sure enough, 3/2 is the 4th step in the diatonic scale
> (not counting the starting note). We can try it with 10/9
>
> < 7 11 16 | 1 -2 1 > = 1
>
> 9/8 is also always a 2nd so the answer will again be 1
>
> < 7 11 16 | -3 2 0 > = 1
>
> 45/32 is a kind of 4th so the answer will be 3
>
> < 7 11 16 | -5 2 1 > = 3
>
> And so on.
>

Very intersting looking, never saw it before.
Thanks for the clear explanation!

But is there supposed to be a higher meaning to it or is it just a nice
trick?
And does it stops working at some point for more complex ratios?

[Non-text portions of this message have been removed]

I agree it is fascinating
and likewise i am curious what this supplies beyond just taking the log2 of an interval and multiplying it the number of units in the scale.

i am guess the impulse toward epimoric intervals being more abundant is good scales ( one of those subjective things no one has really nailed down why, although their are many good answers)

an example of a pathological PB?

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Marcel de Velde wrote:
>
>
> >
> > The formal definition of epimorphic says that there must be a
> > val that, if you multiply it by some ratio found in the block,
> > the result is the scale degree on which that ratio is found.
> > For example, the 5-limit val of the diatonic scale in JI is
> > < 7 11 16 |. That means 7 steps to the 2/1, 11 to the 3/1,
> > and 16 to the 5/1. The monzo for 3/2 is | -1 1 0 >. That means
> > 2^-1 * 3^1 * 5^0 = 3/2. Their product is written
> >
> > < 7 11 16 | -1 1 0 > = 4
> >
> > or
> >
> > (7 * -1) + (11 * 1) + (16 * 0) = 4
> >
> > And sure enough, 3/2 is the 4th step in the diatonic scale
> > (not counting the starting note). We can try it with 10/9
> >
> > < 7 11 16 | 1 -2 1 > = 1
> >
> > 9/8 is also always a 2nd so the answer will again be 1
> >
> > < 7 11 16 | -3 2 0 > = 1
> >
> > 45/32 is a kind of 4th so the answer will be 3
> >
> > < 7 11 16 | -5 2 1 > = 3
> >
> > And so on.
> >
>
> Very intersting looking, never saw it before.
> Thanks for the clear explanation!
>
> But is there supposed to be a higher meaning to it or is it just a nice
> trick?
> And does it stops working at some point for more complex ratios?
>
> [Non-text portions of this message have been removed]
>
>

Kraig wrote:

>I agree it is fascinating
>and likewise i am curious what this supplies beyond just taking the log2
>of an interval and multiplying it the number of units in the scale.

It means the scale is a constant structure, for one thing.

>i am guess the impulse toward epimoric intervals being more abundant is
>good scales ( one of those subjective things no one has really nailed
>down why, although their are many good answers)

Don't get confused between epimoric and epimorphic!
The thing I just described is an epimorphism between periodicity
blocks and just intonation:

http://en.wikipedia.org/wiki/Epimorphism

> an example of a pathological PB?

Here's one:

http://www.lumma.org/music/theory/tctmo/135-128.gif

Notice that one of the unison vectors (32/25) is larger
than one of the 2nds (9/8).

-Carl

At 02:18 PM 5/17/2009, you wrote:

>Very intersting looking, never saw it before.
>Thanks for the clear explanation!
>
>But is there supposed to be a higher meaning to it or is it just
>a nice trick?
>And does it stops working at some point for more complex ratios?

It works for any ratio. It's connected to several deep
things about scales and just intonation, which have been
independently discovered again and again over the years in
different contexts. We can't go into details on this list,
but you are invited to read Paul's paper on the subject

http://eceserv0.ece.wisc.edu/~sethares/paperspdf/Erlich-MiddlePath.pdf

-Carl

yes confusion with the words.
the pathological example would not be a constant structure because from 27/16 to 135/128 is 4 units and all the other 5/4s are 3.
I don't actually understand how this is a PB, [at least how Fokker used the term]

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

Carl Lumma wrote:
>
>
> Kraig wrote:
>
> >I agree it is fascinating
> >and likewise i am curious what this supplies beyond just taking the log2
> >of an interval and multiplying it the number of units in the scale.
>
> It means the scale is a constant structure, for one thing.
>
> >i am guess the impulse toward epimoric intervals being more abundant is
> >good scales ( one of those subjective things no one has really nailed
> >down why, although their are many good answers)
>
> Don't get confused between epimoric and epimorphic!
> The thing I just described is an epimorphism between periodicity
> blocks and just intonation:
>
> http://en.wikipedia.org/wiki/Epimorphism > <http://en.wikipedia.org/wiki/Epimorphism>
>
> > an example of a pathological PB?
>
> Here's one:
>
> http://www.lumma.org/music/theory/tctmo/135-128.gif > <http://www.lumma.org/music/theory/tctmo/135-128.gif>
>
> Notice that one of the unison vectors (32/25) is larger
> than one of the 2nds (9/8).
>
> -Carl
>
>

Kraig wrote:

>> http://www.lumma.org/music/theory/tctmo/135-128.gif
>
>yes confusion with the words.
> the pathological example would not be a constant structure because from
>27/16 to 135/128 is 4 units and all the other 5/4s are 3.
>I don't actually understand how this is a PB, [at least how Fokker used
>the term]

It's a PB just in the sense that it is a convex section of the
lattice defined by two unison vectors (32/16 and 81/80). It
is a 'block' and it does tile the lattice periodically.

I can't recall if Fokker addressed the issue, but his examples
were always epimorphic I think. There's no doubt the above is
a poor PB if it is one at all.

-Carl