All-Interval Scale in 22-tET

Just a tidbit on the All-Interval scale in 22-tET. It is a 10-set,
namely

(0,1,2,4,5,7,11,12,15,18) with Interval Vector <5,5,5,5,5,5,5,5,5,5,5>

It's the only perfect FLID (Flat Interval Distribution) in 22-tET.

Now taking it's inversion and transposing, obtain

(1,3,4,5,9,12,14,15,16,20) which has the same Interval Vector.

This is closely related to a decatonic scale, namely

(1,3,5,7,9,12,14,16,18,20) where I have incremented by one step.

We see that it merely sends 7 to its inverse, 15, and 18 to its
inverse 4.

the Perfect FLID is based on the 5-cycle (1,3,5,9,15) and
the same cycle a half rotation apart (T11): (12,14,16,20,4)
which is also 4x the first cycle. (All semi-FLIDS in 22-tET
are based on:

(1,3,5,9,15), its inverse, the cycle at T11, and that inverse.
This creates a Mosaic pattern that fills out all 20 steps
except (0,11). The 10-cycle generated by powers of the 7-step
is actually just (1,3,5,9,15) and its inverse (21,19,17,15,7)
which in total make those numbers totient to 22, the multiplicative
modulo group or I think the group of units in the ring...

A semi-FLID is either the same value in all slots except 11,
or same values in the odds and same values in the evens.(double-FLID)

Now the decatonic scale mentioned can be based on

81/80
10/9
7/6
5/4
4/3
10/27
7/9
5/3
16/9 (or 7/4)
15/8

So my new "Perfect FLID" scale (the only perfect FLID set in 22-tET
can be found by replacing 5/4 with 8/5 and 7/4 with 8/7 so:

81/80
10/9
8/7 Add
7/6
5/4 Kill
4/3
10/27
14/9
8/5 Add
5/3
16/9 (or 7/4)Kill
15/8

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> Just a tidbit on the All-Interval scale in 22-tET. It is a 10-set,
> namely
>
> (0,1,2,4,5,7,11,12,15,18) with Interval Vector
<5,5,5,5,5,5,5,5,5,5,5>

Sorry it is really:

(0,1,2,4,7,11,12,15,18) with Interval Vector <4,4,4,4,4,4,4,4,4,4,5>

So not Perfect. If you add the 5 you get the first set.

The above set with the Perfect FLID is an 11-set.

The rest still applies:

> Now taking it's inversion and transposing, obtain
>
> (1,3,4,5,9,12,14,15,16,20) which has the same Interval Vector.

> This is closely related to a decatonic scale, namely
>
> (1,3,5,7,9,12,14,16,18,20) where I have incremented by one step.
>
> We see that it merely sends 7 to its inverse, 15, and 18 to its
> inverse 4.
>
> the near-Perfect FLID is based on the 5-cycle (1,3,5,9,15) and
> the same cycle a half rotation apart (T11): (12,14,16,20,4)
> which is also 4x the first cycle. (All semi-FLIDS in 22-tET
> are based on:
>
> (1,3,5,9,15), its inverse, the cycle at T11, and that inverse.
> This creates a Mosaic pattern that fills out all 20 steps
> except (0,11). The 10-cycle generated by powers of the 7-step
> is actually just (1,3,5,9,15) and its inverse (21,19,17,15,7)
> which in total make those numbers totient to 22, the multiplicative
> modulo group or I think the group of units in the ring...
>
> A semi-FLID is either the same value in all slots except 11,
> or same values in the odds and same values in the evens.(double-
FLID)
>
> Now the decatonic scale mentioned can be based on
>
> 81/80
> 10/9
> 7/6
> 5/4
> 4/3
> 10/27
> 7/9
> 5/3
> 16/9 (or 7/4)
> 15/8
>
> So my new "Perfect FLID" scale (the only perfect FLID set in 22-tET
> can be found by replacing 5/4 with 8/5 and 7/4 with 8/7 so:
>
> 81/80
> 10/9
> 8/7 Add
> 7/6
> 5/4 Kill
> 4/3
> 10/27
> 14/9
> 8/5 Add
> 5/3
> 16/9 (or 7/4)Kill
> 15/8
>
> PGH

NOTE: The Perfect FLID set adds 21, so obtain

(1,3,4,5,9,12,14,15,16,20,21) for that with <5,5,5,5,5,5,5,5,5,5,5>

which in a sense just fills in the break with another whole-step
but otherwise doesn't relate to the Decatonic scale much.

PGH

I have no idea what you're talking about, but I have two
problems for you:

1. What is the smallest scale in 22-ET whose dyads cover
all of the available intervals in 22-ET?

2. What is the smallest Rothenberg-proper scale of this type?

-Carl

At 12:54 PM 4/21/2008, you wrote:
>Just a tidbit on the All-Interval scale in 22-tET. It is a 10-set,
>namely
>
>(0,1,2,4,5,7,11,12,15,18) with Interval Vector <5,5,5,5,5,5,5,5,5,5,5>
>
>It's the only perfect FLID (Flat Interval Distribution) in 22-tET.
>
>Now taking it's inversion and transposing, obtain
>
>(1,3,4,5,9,12,14,15,16,20) which has the same Interval Vector.
>
>This is closely related to a decatonic scale, namely
>
>(1,3,5,7,9,12,14,16,18,20) where I have incremented by one step.
>
>We see that it merely sends 7 to its inverse, 15, and 18 to its
>inverse 4.
>
>the Perfect FLID is based on the 5-cycle (1,3,5,9,15) and
>the same cycle a half rotation apart (T11): (12,14,16,20,4)
>which is also 4x the first cycle. (All semi-FLIDS in 22-tET
>are based on:
>
>(1,3,5,9,15), its inverse, the cycle at T11, and that inverse.
>This creates a Mosaic pattern that fills out all 20 steps
>except (0,11). The 10-cycle generated by powers of the 7-step
>is actually just (1,3,5,9,15) and its inverse (21,19,17,15,7)
>which in total make those numbers totient to 22, the multiplicative
>modulo group or I think the group of units in the ring...
>
>A semi-FLID is either the same value in all slots except 11,
>or same values in the odds and same values in the evens.(double-FLID)
>
>Now the decatonic scale mentioned can be based on
>
>81/80
>10/9
>7/6
>5/4
>4/3
>10/27
>7/9
>5/3
>16/9 (or 7/4)
>15/8
>
>So my new "Perfect FLID" scale (the only perfect FLID set in 22-tET
>can be found by replacing 5/4 with 8/5 and 7/4 with 8/7 so:
>
>81/80
>10/9
>8/7 Add
>7/6
>5/4 Kill
>4/3
>10/27
>14/9
>8/5 Add
>5/3
>16/9 (or 7/4)Kill
>15/8
>
>PGH
>

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> I have no idea what you're talking about, but I have two
> problems for you:

Nobody seems to follow my ideas. This post was to compare
a FLID-set in 22/10 with the Decatonic Scale.
>
> 1. What is the smallest scale in 22-ET whose dyads cover
> all of the available intervals in 22-ET?

I will look this up in my database (Excel spreadsheet) of
all the sets in 22-tET. Along those lines, the All-Interval
Set uses all the dyads exactly 5 times each:

>(0,1,2,4,5,7,11,12,15,18) with Interval Vector
<5,5,5,5,5,5,5,5,5,5,5>

Now to use all the dyads at least once each, that would
be 11 intervals or more , so would have to be at least
Binom(6,2)=15 so would have to be at least a hexad.

This is the first one I find in hexads:

[1,1,1,1,1,2,3,1,2,1,1]__22__6 (0,1,7,9,12,16)__22__6

(Dr. Jon Wild created the program to generate this data)

Now the affine action will also send this to 4 more types, with
the interval vector scrambled. How do you define "smallest?"
Lowest canonical set? Or just "hexad" with 6 notes?

> 2. What is the smallest Rothenberg-proper scale of this type?

Don't know what that means

- Paul
>
> -Carl
>
>
> At 12:54 PM 4/21/2008, you wrote:
> >Just a tidbit on the All-Interval scale in 22-tET. It is a 10-set,
> >namely
> >
> >(0,1,2,4,5,7,11,12,15,18) with Interval Vector
<5,5,5,5,5,5,5,5,5,5,5>
> >
> >It's the only perfect FLID (Flat Interval Distribution) in 22-tET.
> >
> >Now taking it's inversion and transposing, obtain
> >
> >(1,3,4,5,9,12,14,15,16,20) which has the same Interval Vector.
> >
> >This is closely related to a decatonic scale, namely
> >
> >(1,3,5,7,9,12,14,16,18,20) where I have incremented by one step.
> >
> >We see that it merely sends 7 to its inverse, 15, and 18 to its
> >inverse 4.
> >
> >the Perfect FLID is based on the 5-cycle (1,3,5,9,15) and
> >the same cycle a half rotation apart (T11): (12,14,16,20,4)
> >which is also 4x the first cycle. (All semi-FLIDS in 22-tET
> >are based on:
> >
> >(1,3,5,9,15), its inverse, the cycle at T11, and that inverse.
> >This creates a Mosaic pattern that fills out all 20 steps
> >except (0,11). The 10-cycle generated by powers of the 7-step
> >is actually just (1,3,5,9,15) and its inverse (21,19,17,15,7)
> >which in total make those numbers totient to 22, the
multiplicative
> >modulo group or I think the group of units in the ring...
> >
> >A semi-FLID is either the same value in all slots except 11,
> >or same values in the odds and same values in the evens.(double-
FLID)
> >
> >Now the decatonic scale mentioned can be based on
> >
> >81/80
> >10/9
> >7/6
> >5/4
> >4/3
> >10/27
> >7/9
> >5/3
> >16/9 (or 7/4)
> >15/8
> >
> >So my new "Perfect FLID" scale (the only perfect FLID set in 22-tET
> >can be found by replacing 5/4 with 8/5 and 7/4 with 8/7 so:
> >
> >81/80
> >10/9
> >8/7 Add
> >7/6
> >5/4 Kill
> >4/3
> >10/27
> >14/9
> >8/5 Add
> >5/3
> >16/9 (or 7/4)Kill
> >15/8
> >
> >PGH
> >
>

Paul H. wrote...
>How do you define "smallest?"
>Lowest canonical set? Or just "hexad" with 6 notes?

Just number of notes. How many such hexads are there?

>> 2. What is the smallest Rothenberg-proper scale of this type?
>
>Don't know what that means

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

But unfortunately,

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

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Paul H. wrote...
> >How do you define "smallest?"
> >Lowest canonical set? Or just "hexad" with 6 notes?
>
> Just number of notes. How many such hexads are there?
>
> >> 2. What is the smallest Rothenberg-proper scale of this type?
> >
> >Don't know what that means
>
> http://en.wikipedia.org/wiki/Rothenberg_propriety
>
> But unfortunately,
>
> http://en.wikipedia.org/wiki/FLID
>
> -Carl

1. Well there are 1782 hexad types. These two types are FLIDs,

[0,3,0,3,0,3,0,3,0,3,0]__22__ 6 (0,2,4,8,10,14)__22__6
[1,2,1,2,1,2,1,2,1,2,0]__22__ 6 (0,1,3,5,9,15)__22__6

Actually, 2-tiered FLIDs, but they have zeros.

Subtracting them and dividing by 5 for the Affine Action we
obtain 1780/5 = 356, but that is not what we are looking for.
What you want is any vector without zeros here. Surprisingly, most
are lumpy, even the Decatonic scales, but I should be able to count
them with simple combinatorics. Consider that they all use
1,2,3 only, and the smoothest ones only 1 and 2. The Affine action
is responsible for some but not all scrambling. Simple combinatorics
tells us that to equal 15 you have these choices:

33111111111
32211111111
22221111111

But only in valid arrangements, of course. Every one will be
related to four other types because of the affine group action.

And that's it. I'll get a full count, I imagine there are not too
many of them. The second class is what I mentioned previously.

Perhaps someone can add an article on Flat Interval Distribution on
Wikipedia. Z-relation (Isomeric relation)is under "Interval Vector"
is on there, that would be a perfect place to mention the FLID.

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <carl@> wrote:
> >
> > Paul H. wrote...
> > >How do you define "smallest?"
> > >Lowest canonical set? Or just "hexad" with 6 notes?
> >
> > Just number of notes. How many such hexads are there?

*** Counted by hand. (That was fun.) 150 types, so 30 * 5 (Sent
by a 5-cycle for the affine group action. Possibly each

33111111111
32211111111
22221111111

is responsible for 50 types, so 3 * 10 * 5 = 150. So the Affine group
action only does 1/3 of the scrambling.

So, the answer is 150 hexads use all the intervals at least once.

> > >> 2. What is the smallest Rothenberg-proper scale of this type?
> > >
> > >Don't know what that means
> >
> > http://en.wikipedia.org/wiki/Rothenberg_propriety

Cool. I will study this and diatonic set theory. Cool that
Rothenberg and Erv Wilson worked entirely outside of academia:)

> > But unfortunately,
> >
> > http://en.wikipedia.org/wiki/FLID
> >
> > -Carl
>
> 1. Well there are 1782 hexad types. These two types are FLIDs,
>
> [0,3,0,3,0,3,0,3,0,3,0]__22__ 6 (0,2,4,8,10,14)__22__6
> [1,2,1,2,1,2,1,2,1,2,0]__22__ 6 (0,1,3,5,9,15)__22__6
>
> Actually, 2-tiered FLIDs, but they have zeros.
>
> Subtracting them and dividing by 5 for the Affine Action we
> obtain 1780/5 = 356, but that is not what we are looking for.
> What you want is any vector without zeros here. Surprisingly, most
> are lumpy, even the Decatonic scales, but I should be able to count
> them with simple combinatorics. Consider that they all use
> 1,2,3 only, and the smoothest ones only 1 and 2. The Affine action
> is responsible for some but not all scrambling. Simple combinatorics
> tells us that to equal 15 you have these choices:
>
> 33111111111
> 32211111111
> 22221111111
>
> But only in valid arrangements, of course. Every one will be
> related to four other types because of the affine group action.
>
> And that's it. I'll get a full count, I imagine there are not too
> many of them. The second class is what I mentioned previously.
>
> Perhaps someone can add an article on Flat Interval Distribution on
> Wikipedia. Z-relation (Isomeric relation)is under "Interval Vector"
> is on there, that would be a perfect place to mention the FLID.
>
> PGH
>

I fail to understand why anyone is spending their time experimenting with 22edo.

Were the users to consider 22edo as a subset of 88edo, they would appreciate that all the harmonic relationships, scales, etc. have already been explored and developed from within 88edo.

To my mind to advocate the exclusive use of 22edo is like trying play an instrument wearing mittens for user will "miss" all the obviously consonant intervals, and limit themselves to generally more

"dissonant" intervals.

I believe that his page illustrates my point.

http://www.lucytune.com/downloads/2288LT.pdf

And shows how the users unnecessarily restrict themselves by only using every fourth interval of 88edo.

Charles Lucy
lucy@lucytune.com

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

At 02:58 PM 4/22/2008, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>>
>> Paul H. wrote...
>> >How do you define "smallest?"
>> >Lowest canonical set? Or just "hexad" with 6 notes?
>>
>> Just number of notes. How many such hexads are there?
>>
>> >> 2. What is the smallest Rothenberg-proper scale of this type?
>> >
>> >Don't know what that means
>>
>> http://en.wikipedia.org/wiki/Rothenberg_propriety
>>
>> But unfortunately,
>>
>> http://en.wikipedia.org/wiki/FLID
>>
>> -Carl
>
>1. Well there are 1782 hexad types. These two types are FLIDs,

Recall that I do not know what a FLID is.

>Perhaps someone can add an article on Flat Interval Distribution on
>Wikipedia. Z-relation (Isomeric relation)is under "Interval Vector"
>is on there, that would be a perfect place to mention the FLID.

Sounds like a job for Paul Hjelmstad!

-Carl

Charles Lucy wrote:
> I fail to understand why anyone is spending their time experimenting > with 22edo.

Yes, it looks like you do.

> Were the users to consider 22edo as a subset of 88edo, they would > appreciate that all the harmonic relationships, scales, etc. have > already been explored and developed from within 88edo.

They have? Maybe Paul can tell us how many harmonic relationships and scales there are in 88-edo to explore.

> To my mind to advocate the exclusive use of 22edo is like trying play > an instrument wearing mittens for user will "miss" all the obviously > consonant intervals, and limit themselves to generally more
> > "dissonant" intervals.
> > I believe that his page illustrates my point.
> > http://www.lucytune.com/downloads/2288LT.pdf
> > > And shows how the users unnecessarily restrict themselves by only > using every fourth interval of 88edo.

I know tuning-math caters to a range of interests and abilities, but "two fours are eight" isn't the kind of insight we need a table to illustrate. Of course you could also produce a table comparing everything to 12-edo to show how out of tune these weird microtonal scales are.

Graham

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> At 02:58 PM 4/22/2008, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <carl@> wrote:
> >>
> >> Paul H. wrote...
> >> >How do you define "smallest?"
> >> >Lowest canonical set? Or just "hexad" with 6 notes?
> >>
> >> Just number of notes. How many such hexads are there?
> >>
> >> >> 2. What is the smallest Rothenberg-proper scale of this type?
> >> >
> >> >Don't know what that means
> >>
> >> http://en.wikipedia.org/wiki/Rothenberg_propriety
> >>
> >> But unfortunately,
> >>
> >> http://en.wikipedia.org/wiki/FLID
> >>
> >> -Carl
> >
> >1. Well there are 1782 hexad types. These two types are FLIDs,
>
> Recall that I do not know what a FLID is.
>
> >Perhaps someone can add an article on Flat Interval Distribution
on
> >Wikipedia. Z-relation (Isomeric relation)is under "Interval
Vector"
> >is on there, that would be a perfect place to mention the FLID.
>
> Sounds like a job for Paul Hjelmstad!
>
> -Carl

FLID: Flat Interval Distribution: When all the entries in an
interval value for a set have the same value, like
<5,5,5,5,5,5,5,5,5,5,5>

Now, it my 22-tET theories, I also consider two-tiered quasi-FLIDs,
and quasi-FLIDs with only 11th place different (it ties
into the Affine action)

<4,4,4,4,4,4,4,4,4,4,5> quasi-FLID
<2,9,2,9,2,9,2,9,2,9,0> two-tiered quasi-FLID

PGH

--- In tuning-math@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Charles Lucy wrote:
> > I fail to understand why anyone is spending their time
experimenting
> > with 22edo.
>
> Yes, it looks like you do.
>
> > Were the users to consider 22edo as a subset of 88edo, they
would
> > appreciate that all the harmonic relationships, scales, etc.
have
> > already been explored and developed from within 88edo.
>
> They have? Maybe Paul can tell us how many harmonic
> relationships and scales there are in 88-edo to explore.
>
> > To my mind to advocate the exclusive use of 22edo is like trying
play
> > an instrument wearing mittens for user will "miss" all the
obviously
> > consonant intervals, and limit themselves to generally more
> >
> > "dissonant" intervals.
> >
> > I believe that his page illustrates my point.
> >
> > http://www.lucytune.com/downloads/2288LT.pdf
> >
> >
> > And shows how the users unnecessarily restrict themselves by
only
> > using every fourth interval of 88edo.
>
> I know tuning-math caters to a range of interests and
> abilities, but "two fours are eight" isn't the kind of
> insight we need a table to illustrate. Of course you could
> also produce a table comparing everything to 12-edo to show
> how out of tune these weird microtonal scales are.
>
>
> Graham

Do you know how big that databases would be for 88-tET? 31-tET
is the biggest Dr. Wild could even go with 32-bit architecture
at this point (interesting coincidence).

In answer to Charle's question, actually 22-tET sounds great
for jazz, take Erlich's "Decatonic Swing". It sounds terrible
for diatonic music, so no Bruce Springsteen or Tom Petty in
22-tET!

I stumbled on 22-tET because my set theory stuff all came
together there, so this is more Pure than Applied Music Theory.

I do want to analyze 31-tET which is a lot messier and doesn't have
the nice Affine action that 22-tET has, with the 5-cycle (1 3 9 5 15)
and evens similar.

I want to prove that FLID's can never be Z-related and that sort
of thing. They never are in 22-tET, because they are all literally
built on the 5-cycle above. They are also responsible for three
rows in the Cycle Grid for Aff(22). It's all in my paper.

I should say I am building much of what I do on the work on
Paul Erlich and Jon Wild, and that both have supplied me with
knowledge and "datasets" even though some ideas are actually my
own.

PGH

Hi Paul,

>> Recall that I do not know what a FLID is.
>
>FLID: Flat Interval Distribution: When all the entries in an
>interval value for a set have the same value, like
><5,5,5,5,5,5,5,5,5,5,5>

What's the reasoning behind this property? I mean, why
is it interesting?

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Hi Paul,
>
> >> Recall that I do not know what a FLID is.
> >
> >FLID: Flat Interval Distribution: When all the entries in an
> >interval value for a set have the same value, like
> ><5,5,5,5,5,5,5,5,5,5,5>
>
> What's the reasoning behind this property? I mean, why
> is it interesting?
>
> -Carl

It has to do with Difference Sets. These tie into other things.

This paper is great:

http://www.mcm2007.info/pdf/sun3-wild.pdf

By the way,

I just got done crunching Aff(31). There are three FLIDs in 31t-ET:

<1,1,1,1,1,1,1,1,1,1,1,1,1,1,1> (31/6)
<3,3,3,3,3,3,3,3,3,3,3,3,3,3,3> (31/10)
<7,7,7,7,7,7,7,7,7,7,7,7,7,7,7> (31/15)

In 31/15, Aff(31) merely divides 300,540,195 by 31 = 9,694,845, which
is nothing special because transpositions do that anyway.

So it looks like every set in 31-tET is a "pole" of Aff(31), it
merely maps to transpositions. Nothing could be more trivial.

But in 22-tET, my quasi-FLIDs (and one FLID) are the only sets for
which the Affine group maps the sets into themselves (poles).

Also strange, with 31/15, the FLID-set is an isomeric quartet, while
in 22tET the FLID <5,5,5,5,5,5,5,5,5,5,5> is not isomeric, but is
a "pole" (my term, please excuse).

Lucky Sevens in 31/15:

ZCount Field2 Field3
4 [777777777777777]__31__15
(0,1,2,4,6,10,11,12,13,15,16,18,19,23,26)__31__15
4 [777777777777777]__31__15
(0,1,2,5,9,11,13,14,15,16,18,19,21,24,25)__31__15
4 [777777777777777]__31__15
(0,1,3,4,7,8,9,10,12,15,17,19,20,21,25)__31__15
4 [777777777777777]__31__15
(0,2,3,4,6,7,9,11,14,15,16,17,21,22,25)__31__15

As I said, in 22-tET, my quasi-FLIDs (and one FLID) are sets for which
the Affine group maps the sets into themselves.

Right now I believe they are key to understanding the Isomeric
Relation (in 22-tET), but I haven't reached that conclusion in my
paper yet. It's a WIP.

I will post the latest version on Files Paul Hj's Stuff. If you
want to glance through it...

PGH

At 04:23 PM 4/23/2008, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>>
>> Hi Paul,
>>
>> >> Recall that I do not know what a FLID is.
>> >
>> >FLID: Flat Interval Distribution: When all the entries in an
>> >interval value for a set have the same value, like
>> ><5,5,5,5,5,5,5,5,5,5,5>
>>
>> What's the reasoning behind this property? I mean, why
>> is it interesting?
>>
>> -Carl
>
>It has to do with Difference Sets. These tie into other things.
>
>This paper is great:
>
>http://www.mcm2007.info/pdf/sun3-wild.pdf
>
>By the way,
>
>I just got done crunching Aff(31). There are three FLIDs in 31t-ET:
>
><1,1,1,1,1,1,1,1,1,1,1,1,1,1,1> (31/6)
><3,3,3,3,3,3,3,3,3,3,3,3,3,3,3> (31/10)
><7,7,7,7,7,7,7,7,7,7,7,7,7,7,7> (31/15)
>
>In 31/15, Aff(31) merely divides 300,540,195 by 31 = 9,694,845, which
>is nothing special because transpositions do that anyway.
>
>So it looks like every set in 31-tET is a "pole" of Aff(31), it
>merely maps to transpositions. Nothing could be more trivial.
>
>But in 22-tET, my quasi-FLIDs (and one FLID) are the only sets for
>which the Affine group maps the sets into themselves (poles).
>
>Also strange, with 31/15, the FLID-set is an isomeric quartet, while
>in 22tET the FLID <5,5,5,5,5,5,5,5,5,5,5> is not isomeric, but is
>a "pole" (my term, please excuse).
>
>Lucky Sevens in 31/15:
>
>ZCount Field2 Field3
>4 [777777777777777]__31__15
> (0,1,2,4,6,10,11,12,13,15,16,18,19,23,26)__31__15
>4 [777777777777777]__31__15
> (0,1,2,5,9,11,13,14,15,16,18,19,21,24,25)__31__15
>4 [777777777777777]__31__15
> (0,1,3,4,7,8,9,10,12,15,17,19,20,21,25)__31__15
>4 [777777777777777]__31__15
> (0,2,3,4,6,7,9,11,14,15,16,17,21,22,25)__31__15
>
>As I said, in 22-tET, my quasi-FLIDs (and one FLID) are sets for which
>the Affine group maps the sets into themselves.
>
>Right now I believe they are key to understanding the Isomeric
>Relation (in 22-tET), but I haven't reached that conclusion in my
>paper yet. It's a WIP.
>
>I will post the latest version on Files Paul Hj's Stuff. If you
>want to glance through it...
>
>PGH

I was still hoping to know the smallest and smallest proper
all-interval scales in 22. Not that I think all-interval
scales (or whatever you want to call them) are particularly
useful (though this has been proposed as an explanation for
the diatonic scale in 12-ET).

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> At 04:23 PM 4/23/2008, you wrote:
> >--- In tuning-math@yahoogroups.com, Carl Lumma <carl@> wrote:
> >>
> >> Hi Paul,
> >>
> >> >> Recall that I do not know what a FLID is.
> >> >
> >> >FLID: Flat Interval Distribution: When all the entries in an
> >> >interval value for a set have the same value, like
> >> ><5,5,5,5,5,5,5,5,5,5,5>
> >>
> >> What's the reasoning behind this property? I mean, why
> >> is it interesting?
> >>
> >> -Carl
> >
> >It has to do with Difference Sets. These tie into other things.
> >
> >This paper is great:
> >
> >http://www.mcm2007.info/pdf/sun3-wild.pdf
> >
> >By the way,
> >
> >I just got done crunching Aff(31). There are three FLIDs in 31t-ET:
> >
> ><1,1,1,1,1,1,1,1,1,1,1,1,1,1,1> (31/6)
> ><3,3,3,3,3,3,3,3,3,3,3,3,3,3,3> (31/10)
> ><7,7,7,7,7,7,7,7,7,7,7,7,7,7,7> (31/15)
> >
> >In 31/15, Aff(31) merely divides 300,540,195 by 31 = 9,694,845,
which
> >is nothing special because transpositions do that anyway.
> >
> >So it looks like every set in 31-tET is a "pole" of Aff(31), it
> >merely maps to transpositions. Nothing could be more trivial.
> >
> >But in 22-tET, my quasi-FLIDs (and one FLID) are the only sets for
> >which the Affine group maps the sets into themselves (poles).
> >
> >Also strange, with 31/15, the FLID-set is an isomeric quartet,
while
> >in 22tET the FLID <5,5,5,5,5,5,5,5,5,5,5> is not isomeric, but is
> >a "pole" (my term, please excuse).
> >
> >Lucky Sevens in 31/15:
> >
> >ZCount Field2 Field3
> >4 [777777777777777]__31__15
> > (0,1,2,4,6,10,11,12,13,15,16,18,19,23,26)__31__15
> >4 [777777777777777]__31__15
> > (0,1,2,5,9,11,13,14,15,16,18,19,21,24,25)__31__15
> >4 [777777777777777]__31__15
> > (0,1,3,4,7,8,9,10,12,15,17,19,20,21,25)__31__15
> >4 [777777777777777]__31__15
> > (0,2,3,4,6,7,9,11,14,15,16,17,21,22,25)__31__15
> >
> >As I said, in 22-tET, my quasi-FLIDs (and one FLID) are sets for
which
> >the Affine group maps the sets into themselves.
> >
> >Right now I believe they are key to understanding the Isomeric
> >Relation (in 22-tET), but I haven't reached that conclusion in my
> >paper yet. It's a WIP.
> >
> >I will post the latest version on Files Paul Hj's Stuff. If you
> >want to glance through it...
> >
> >PGH
>
> I was still hoping to know the smallest and smallest proper
> all-interval scales in 22. Not that I think all-interval
> scales (or whatever you want to call them) are particularly
> useful (though this has been proposed as an explanation for
> the diatonic scale in 12-ET).
>
> -Carl

Well like you said, its all hand-waving. But seriously, I have to
work with what I have at hand. I am working to find the Rothenberg
all-interval scales. Actually, the diatonic scale is opposite from a
FLID in that every interval vector value is different <2,5,4,3,6,1>
and of course <6,5,4,3,2,1> for chromatic scale from 0-6.

There's nothing out there, I don't think anyway, so one has to work
with the patterns that exist. We create reality. haha.

PGH

Graham Breed wrote:
> Charles Lucy wrote:
>> I fail to understand why anyone is spending their time experimenting >> with 22edo.
> > Yes, it looks like you do.
> >> Were the users to consider 22edo as a subset of 88edo, they would >> appreciate that all the harmonic relationships, scales, etc. have >> already been explored and developed from within 88edo.
> > They have? Maybe Paul can tell us how many harmonic > relationships and scales there are in 88-edo to explore.

It's meantone if you're looking at 5-limit harmony, cynder/mothra in 7- and 11-limits. Its 3/1 is so close to half a step flat that the second best 3/1 (the 22-ET version) is nearly as good. It has a good 7/4, but it's not the one that's notated as a meantone augmented sixth (which is off about as much as 22-ET's 7/4). So 22-ET wins by being nearly as accurate with only 1/4 of the notes, if you're using 88-ET as a meantone. 22EDO offers a wide range of both harmonic progressions and MOS scales due to all the temperaments it supports (porcupine, doublewide, magic, srutal, superpyth, orwell, among others). On the other hand, if you can accept a scale with as many notes as 88edo, it's hard to come up with one much better than 72.

Of course, everyone has their favorite EDO, and I happen to like 26. A glance at a chart like http://tonalsoft.com/enc/e/equal-temperament.aspx shows that 26 is hardly in a favorable situation compared with 12, 19, or 22; in fact it's way out on the edge of acceptable temperaments, but that doesn't keep me from liking it.

--- In tuning-math@yahoogroups.com, Herman Miller <hmiller@...> wrote:
>
> Graham Breed wrote:
> > Charles Lucy wrote:
> >> I fail to understand why anyone is spending their time
experimenting
> >> with 22edo.

1. 22-tET is great for my set-theory stuff, and it has all those
temperaments, MOS scales, progressions Herman mentions etc.

2. And mystically, 22 is the number of letters in the Hebrew
alphabet, the paths of the Sephiroth, and the highest number in
certain numerological systems (The Master Builder). S(3,6,22)
is a Steiner system and the basis of the Mathieu Group M22.
(Automorphism group of Steiner system).

3. The Hindu scale has 22 notes (of course, far from ET)

Perhaps the answer is a kind of Well-tempered 22-tET.

Has anyone done work on that? It can't be so far off as to
distort the step sizes. I am thinking of an analogy to
Bach's Well-Temperament (which is not Equal Temperament)
but still sounds great in all 12 (Here: 22) keys!

PGH

> > Yes, it looks like you do.
> >
> >> Were the users to consider 22edo as a subset of 88edo, they
would
> >> appreciate that all the harmonic relationships, scales, etc.
have
> >> already been explored and developed from within 88edo.
> >
> > They have? Maybe Paul can tell us how many harmonic
> > relationships and scales there are in 88-edo to explore.
>
> It's meantone if you're looking at 5-limit harmony, cynder/mothra
in 7-
> and 11-limits. Its 3/1 is so close to half a step flat that the
second
> best 3/1 (the 22-ET version) is nearly as good. It has a good 7/4,
but
> it's not the one that's notated as a meantone augmented sixth
(which is
> off about as much as 22-ET's 7/4). So 22-ET wins by being nearly as
> accurate with only 1/4 of the notes, if you're using 88-ET as a
> meantone. 22EDO offers a wide range of both harmonic progressions
and
> MOS scales due to all the temperaments it supports (porcupine,
> doublewide, magic, srutal, superpyth, orwell, among others). On the
> other hand, if you can accept a scale with as many notes as 88edo,
it's
> hard to come up with one much better than 72.
>
> Of course, everyone has their favorite EDO, and I happen to like
26. A
> glance at a chart like http://tonalsoft.com/enc/e/equal-
temperament.aspx
> shows that 26 is hardly in a favorable situation compared with 12,
19,
> or 22; in fact it's way out on the edge of acceptable temperaments,
but
> that doesn't keep me from liking it.
>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad" <phjelmstad@...>
wrote:
>
> Perhaps the answer is a kind of Well-tempered 22-tET.
>
> Has anyone done work on that? It can't be so far off as to
> distort the step sizes. I am thinking of an analogy to
> Bach's Well-Temperament (which is not Equal Temperament)
> but still sounds great in all 12 (Here: 22) keys!

I've tried on several occasions, but I never came up with anything I
would consider satisfactory.

Part of the problem is trying to figure out what you're after. A well-
temperament for porcupine won't necessarily be good for pajara or magic.

--George

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@> wrote:
> >
> > Graham Breed wrote:
> > > Charles Lucy wrote:
> > >> I fail to understand why anyone is spending their time
> experimenting
> > >> with 22edo.
>
> 1. 22-tET is great for my set-theory stuff, and it has all those
> temperaments, MOS scales, progressions Herman mentions etc.
>
> 2. And mystically, 22 is the number of letters in the Hebrew
> alphabet, the paths of the Sephiroth, and the highest number in
> certain numerological systems (The Master Builder). S(3,6,22)
> is a Steiner system and the basis of the Mathieu Group M22.
> (Automorphism group of Steiner system).
>
> 3. The Hindu scale has 22 notes (of course, far from ET)
>
> Perhaps the answer is a kind of Well-tempered 22-tET.
>
> Has anyone done work on that? It can't be so far off as to
> distort the step sizes. I am thinking of an analogy to
> Bach's Well-Temperament (which is not Equal Temperament)
> but still sounds great in all 12 (Here: 22) keys!
>
> PGH
>
>
> > > Yes, it looks like you do.
> > >
> > >> Were the users to consider 22edo as a subset of 88edo, they
> would
> > >> appreciate that all the harmonic relationships, scales, etc.
> have
> > >> already been explored and developed from within 88edo.
> > >
> > > They have? Maybe Paul can tell us how many harmonic
> > > relationships and scales there are in 88-edo to explore.
> >
> > It's meantone if you're looking at 5-limit harmony, cynder/mothra
> in 7-
> > and 11-limits. Its 3/1 is so close to half a step flat that the
> second
> > best 3/1 (the 22-ET version) is nearly as good. It has a good
7/4,
> but
> > it's not the one that's notated as a meantone augmented sixth
> (which is
> > off about as much as 22-ET's 7/4). So 22-ET wins by being nearly
as
> > accurate with only 1/4 of the notes, if you're using 88-ET as a
> > meantone. 22EDO offers a wide range of both harmonic progressions
> and
> > MOS scales due to all the temperaments it supports (porcupine,
> > doublewide, magic, srutal, superpyth, orwell, among others). On
the
> > other hand, if you can accept a scale with as many notes as
88edo,
> it's
> > hard to come up with one much better than 72.
> >
> > Of course, everyone has their favorite EDO, and I happen to like
> 26. A
> > glance at a chart like http://tonalsoft.com/enc/e/equal-
> temperament.aspx
> > shows that 26 is hardly in a favorable situation compared with
12,
> 19,
> > or 22; in fact it's way out on the edge of acceptable
temperaments,
> but
> > that doesn't keep me from liking it.
> >
> While your maths is totally beyond me I think that any exploration
of 22tet is worthwhile. 22tet is my personal favorite and I'm not
surprised to find out lots of other people think so too. From a
practical point of view anything beyond 31tet seems absurd where
singers are concerned. Even the number of theoretically legitimate 3-
note chords seems too many in 31tet.

--- In tuning-math@yahoogroups.com, "robert thomas martin"
<robertthomasmartin@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> <phjelmstad@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@>
wrote:
> > >
> > > Graham Breed wrote:
> > > > Charles Lucy wrote:
> > > >> I fail to understand why anyone is spending their time
> > experimenting
> > > >> with 22edo.
> >
> > 1. 22-tET is great for my set-theory stuff, and it has all those
> > temperaments, MOS scales, progressions Herman mentions etc.
> >
> > 2. And mystically, 22 is the number of letters in the Hebrew
> > alphabet, the paths of the Sephiroth, and the highest number in
> > certain numerological systems (The Master Builder). S(3,6,22)
> > is a Steiner system and the basis of the Mathieu Group M22.
> > (Automorphism group of Steiner system).
> >
> > 3. The Hindu scale has 22 notes (of course, far from ET)
> >
> > Perhaps the answer is a kind of Well-tempered 22-tET.
> >
> > Has anyone done work on that? It can't be so far off as to
> > distort the step sizes. I am thinking of an analogy to
> > Bach's Well-Temperament (which is not Equal Temperament)
> > but still sounds great in all 12 (Here: 22) keys!
> >
> > PGH
> >
> >
> > > > Yes, it looks like you do.
> > > >
> > > >> Were the users to consider 22edo as a subset of 88edo, they
> > would
> > > >> appreciate that all the harmonic relationships, scales, etc.
> > have
> > > >> already been explored and developed from within 88edo.
> > > >
> > > > They have? Maybe Paul can tell us how many harmonic
> > > > relationships and scales there are in 88-edo to explore.
> > >
> > > It's meantone if you're looking at 5-limit harmony,
cynder/mothra
> > in 7-
> > > and 11-limits. Its 3/1 is so close to half a step flat that the
> > second
> > > best 3/1 (the 22-ET version) is nearly as good. It has a good
> 7/4,
> > but
> > > it's not the one that's notated as a meantone augmented sixth
> > (which is
> > > off about as much as 22-ET's 7/4). So 22-ET wins by being
nearly
> as
> > > accurate with only 1/4 of the notes, if you're using 88-ET as a
> > > meantone. 22EDO offers a wide range of both harmonic
progressions
> > and
> > > MOS scales due to all the temperaments it supports (porcupine,
> > > doublewide, magic, srutal, superpyth, orwell, among others). On
> the
> > > other hand, if you can accept a scale with as many notes as
> 88edo,
> > it's
> > > hard to come up with one much better than 72.
> > >
> > > Of course, everyone has their favorite EDO, and I happen to
like
> > 26. A
> > > glance at a chart like http://tonalsoft.com/enc/e/equal-
> > temperament.aspx
> > > shows that 26 is hardly in a favorable situation compared with
> 12,
> > 19,
> > > or 22; in fact it's way out on the edge of acceptable
> temperaments,
> > but
> > > that doesn't keep me from liking it.
> > >
> > While your maths is totally beyond me I think that any
exploration
> of 22tet is worthwhile. 22tet is my personal favorite and I'm not
> surprised to find out lots of other people think so too. From a
> practical point of view anything beyond 31tet seems absurd where
> singers are concerned. Even the number of theoretically legitimate
3-
> note chords seems too many in 31tet.

Yes, that is true. I would put the cutoff at about 31 though myself,
based on the demonstration of his Clavette by Harold Fortuin at
the Minnesota Composers Forum (Now American). Oh it's "Paul G.
Hjelmstad" BTW. Actually it was discovering the use of the logarithm
in my early teens and its relationship to tuning that got me
interested in this field...but that was 1976 or so! Just wondering
why you posted it. No matter.

PGH
>

--- In tuning-math@yahoogroups.com, "George D. Secor" <gdsecor@...>
wrote:
>
> --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@>
> wrote:
> >
> > Perhaps the answer is a kind of Well-tempered 22-tET.
> >
> > Has anyone done work on that? It can't be so far off as to
> > distort the step sizes. I am thinking of an analogy to
> > Bach's Well-Temperament (which is not Equal Temperament)
> > but still sounds great in all 12 (Here: 22) keys!
>
> I've tried on several occasions, but I never came up with anything
I
> would consider satisfactory.
>
> Part of the problem is trying to figure out what you're after. A
well-
> temperament for porcupine won't necessarily be good for pajara or
magic.
>
> --George

Well, of course 22 has superpyth, porcupine, magic and pajara for
starters. What I am after is a 22-tET well-temperament that:

1. Would sound good (duh)
2. Would preserve the basic steps, so nothing as uneven as the
Hindu scale...
3. Would keep maybe 2 of the above 4 commas intact?

It would be fun to play with a sliding scale of those four commas
and see what one could come up with....

PGH
>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, "robert thomas martin"
> <robertthomasmartin@> wrote:
> >
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <phjelmstad@> wrote:
> > >
> > > --- In tuning-math@yahoogroups.com, Herman Miller <hmiller@>
> wrote:
> > > >
> > > > Graham Breed wrote:
> > > > > Charles Lucy wrote:
> > > > >> I fail to understand why anyone is spending their time
> > > experimenting
> > > > >> with 22edo.
> > >
> > > 1. 22-tET is great for my set-theory stuff, and it has all
those
> > > temperaments, MOS scales, progressions Herman mentions etc.
> > >
> > > 2. And mystically, 22 is the number of letters in the Hebrew
> > > alphabet, the paths of the Sephiroth, and the highest number in
> > > certain numerological systems (The Master Builder). S(3,6,22)
> > > is a Steiner system and the basis of the Mathieu Group M22.
> > > (Automorphism group of Steiner system).
> > >
> > > 3. The Hindu scale has 22 notes (of course, far from ET)
> > >
> > > Perhaps the answer is a kind of Well-tempered 22-tET.
> > >
> > > Has anyone done work on that? It can't be so far off as to
> > > distort the step sizes. I am thinking of an analogy to
> > > Bach's Well-Temperament (which is not Equal Temperament)
> > > but still sounds great in all 12 (Here: 22) keys!
> > >
> > > PGH
> > >
> > >
> > > > > Yes, it looks like you do.
> > > > >
> > > > >> Were the users to consider 22edo as a subset of 88edo,
they
> > > would
> > > > >> appreciate that all the harmonic relationships, scales,
etc.
> > > have
> > > > >> already been explored and developed from within 88edo.
> > > > >
> > > > > They have? Maybe Paul can tell us how many harmonic
> > > > > relationships and scales there are in 88-edo to explore.
> > > >
> > > > It's meantone if you're looking at 5-limit harmony,
> cynder/mothra
> > > in 7-
> > > > and 11-limits. Its 3/1 is so close to half a step flat that
the
> > > second
> > > > best 3/1 (the 22-ET version) is nearly as good. It has a good
> > 7/4,
> > > but
> > > > it's not the one that's notated as a meantone augmented sixth
> > > (which is
> > > > off about as much as 22-ET's 7/4). So 22-ET wins by being
> nearly
> > as
> > > > accurate with only 1/4 of the notes, if you're using 88-ET as
a
> > > > meantone. 22EDO offers a wide range of both harmonic
> progressions
> > > and
> > > > MOS scales due to all the temperaments it supports
(porcupine,
> > > > doublewide, magic, srutal, superpyth, orwell, among others).
On
> > the
> > > > other hand, if you can accept a scale with as many notes as
> > 88edo,
> > > it's
> > > > hard to come up with one much better than 72.
> > > >
> > > > Of course, everyone has their favorite EDO, and I happen to
> like
> > > 26. A
> > > > glance at a chart like http://tonalsoft.com/enc/e/equal-
> > > temperament.aspx
> > > > shows that 26 is hardly in a favorable situation compared
with
> > 12,
> > > 19,
> > > > or 22; in fact it's way out on the edge of acceptable
> > temperaments,
> > > but
> > > > that doesn't keep me from liking it.
> > > >
> > > While your maths is totally beyond me I think that any
> exploration
> > of 22tet is worthwhile. 22tet is my personal favorite and I'm not
> > surprised to find out lots of other people think so too. From a
> > practical point of view anything beyond 31tet seems absurd where
> > singers are concerned. Even the number of theoretically
legitimate
> 3-
> > note chords seems too many in 31tet.
>
> Yes, that is true. I would put the cutoff at about 31 though myself,
> based on the demonstration of his Clavette by Harold Fortuin at
> the Minnesota Composers Forum (Now American). Oh it's "Paul G.
> Hjelmstad" BTW. Actually it was discovering the use of the logarithm
> in my early teens and its relationship to tuning that got me
> interested in this field...but that was 1976 or so! Just wondering
> why you posted it. No matter.
>
> PGH
> >
> I just posted a list of 17 chords for 31tet in the tuning group.
It's not as many that I imagined in the previous email. With so many
triads to choose from it seems that I wouldn't even bother with 7ths
if I was working in 31tet.

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Paul H. wrote...
> >How do you define "smallest?"
> >Lowest canonical set? Or just "hexad" with 6 notes?
>
> Just number of notes. How many such hexads are there?
>
> >> 2. What is the smallest Rothenberg-proper scale of this type?
> >
> >Don't know what that means
>
> http://en.wikipedia.org/wiki/Rothenberg_propriety
>
> But unfortunately,
>
> http://en.wikipedia.org/wiki/FLID
>
> -Carl

Working on your question #2. I guess I need to know what
decatonic scale to use as the basis. Perhaps Erlich's Decatonic
Minor: 0 2 4 6 8 11 13 15 17 19

And assign A B C D E F G H I J and hopefully find a few
Rothenberg proper scales. Is that what you meant? They
would be 10-scales.

PGH

Paul Hjelmstad wrote...

>>> 1. What is the smallest scale in 22-ET whose dyads cover
>>> all of the available intervals in 22-ET?
>>>
>>> 2. What is the smallest Rothenberg-proper scale of
>>> this type?
>>>
>>> -Carl
//
>
> Working on your question #2. I guess I need to know what
> decatonic scale to use as the basis. Perhaps Erlich's Decatonic
> Minor: 0 2 4 6 8 11 13 15 17 19
>
> And assign A B C D E F G H I J and hopefully find a few
> Rothenberg proper scales. Is that what you meant? They
> would be 10-scales.
>
> PGH

I didn't ask for a decatonic scale. I asked what is the
smallest scale in 22-ET that contains all the dyads in 22-ET.
I think you answered that there a many 6-tone scales that
do this.

My second question was then, which of these scales is
Rothenberg proper.

-Carl

--- In tuning-math@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Paul Hjelmstad wrote...
>
> >>> 1. What is the smallest scale in 22-ET whose dyads cover
> >>> all of the available intervals in 22-ET?
> >>>
> >>> 2. What is the smallest Rothenberg-proper scale of
> >>> this type?
> >>>
> >>> -Carl
> //
> >
> > Working on your question #2. I guess I need to know what
> > decatonic scale to use as the basis. Perhaps Erlich's Decatonic
> > Minor: 0 2 4 6 8 11 13 15 17 19
> >
> > And assign A B C D E F G H I J and hopefully find a few
> > Rothenberg proper scales. Is that what you meant? They
> > would be 10-scales.
> >
> > PGH
>
> I didn't ask for a decatonic scale. I asked what is the
> smallest scale in 22-ET that contains all the dyads in 22-ET.
> I think you answered that there a many 6-tone scales that
> do this.
>
> My second question was then, which of these scales is
> Rothenberg proper.
>
> -Carl

Yes any hexad with no zero in the IV and no repeats. How
can a 22-hexad be Rothenberg-proper? Don't you need 10 notes?

To help me perhaps give me a Rothenberg proper scale in 12-tET
that is not a heptad (for example a pentad or less)

Thanx

PGH

Paul Hjelmstad wrote...

>> >>> 1. What is the smallest scale in 22-ET whose dyads cover
>> >>> all of the available intervals in 22-ET?
>> >>>
>> >>> 2. What is the smallest Rothenberg-proper scale of
>> >>> this type?
>> >>>
>> >>> -Carl
>> //
>> >
>> > Working on your question #2. I guess I need to know what
>> > decatonic scale to use as the basis. Perhaps Erlich's Decatonic
>> > Minor: 0 2 4 6 8 11 13 15 17 19
>> >
>> > And assign A B C D E F G H I J and hopefully find a few
>> > Rothenberg proper scales. Is that what you meant? They
>> > would be 10-scales.
>> >
>> > PGH
>>
>> I didn't ask for a decatonic scale. I asked what is the
>> smallest scale in 22-ET that contains all the dyads in 22-ET.
>> I think you answered that there a many 6-tone scales that
>> do this.
>>
>> My second question was then, which of these scales is
>> Rothenberg proper.
>>
>> -Carl
>
>Yes any hexad with no zero in the IV and no repeats

What do you means "zero in the IV"?

>How
>can a 22-hexad be Rothenberg-proper? Don't you need 10 notes?

?

>To help me perhaps give me a Rothenberg proper scale in 12-tET
>that is not a heptad (for example a pentad or less)

Sure. The pentatonic scale is proper.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Paul Hjelmstad wrote...
>
> >> >>> 1. What is the smallest scale in 22-ET whose dyads cover
> >> >>> all of the available intervals in 22-ET?
> >> >>>
> >> >>> 2. What is the smallest Rothenberg-proper scale of
> >> >>> this type?
> >> >>>
> >> >>> -Carl
> >> //
> >> >
> >> > Working on your question #2. I guess I need to know what
> >> > decatonic scale to use as the basis. Perhaps Erlich's Decatonic
> >> > Minor: 0 2 4 6 8 11 13 15 17 19
> >> >
> >> > And assign A B C D E F G H I J and hopefully find a few
> >> > Rothenberg proper scales. Is that what you meant? They
> >> > would be 10-scales.
> >> >
> >> > PGH
> >>
> >> I didn't ask for a decatonic scale. I asked what is the
> >> smallest scale in 22-ET that contains all the dyads in 22-ET.
> >> I think you answered that there a many 6-tone scales that
> >> do this.
> >>
> >> My second question was then, which of these scales is
> >> Rothenberg proper.
> >>
> >> -Carl
> >
> >Yes any hexad with no zero in the IV and no repeats
>
> What do you means "zero in the IV"?

IV=Interval Vector, actually you can have repeats

> >How
> >can a 22-hexad be Rothenberg-proper? Don't you need 10 notes?
>
> ?
>
> >To help me perhaps give me a Rothenberg proper scale in 12-tET
> >that is not a heptad (for example a pentad or less)
>
> Sure. The pentatonic scale is proper.

Could you show how this is proper, but not strictly proper?
Do you letter the black keys?

PGH

>

>> Sure. The pentatonic scale is proper.
>
>Could you show how this is proper, but not strictly proper?
>Do you letter the black keys?

It is strictly proper in 12-ET. The diatonic scale in 12-ET
is proper but not strictly so. Scala will tell you all this
in its "show data". Rothenberg propriety is explained on
wikipedia.

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> >> Sure. The pentatonic scale is proper.
> >
> >Could you show how this is proper, but not strictly proper?
> >Do you letter the black keys?
>
> It is strictly proper in 12-ET. The diatonic scale in 12-ET
> is proper but not strictly so. Scala will tell you all this
> in its "show data". Rothenberg propriety is explained on
> wikipedia.
>
> -Carl

Yeah, I read it. So a Class isn't always the ordinary note names
(A through G). In the case of the pentatonic scale you have only
5 classes, and it is strictly proper.

Here is where I get stuck, in how to label the Class steps

Wikipedia has these scales that are proper diatonic scales:

These five scales are:

Diatonic: C D E F G A B

Ascending minor: C D Eó F G A B

Harmonic minor: C D Eó F G Aó B

Harmonic major: C D E F G Aó B

Major locrian: C D E F Gó Aó Bó

It's easy here, just use C,D,E,F,G,A,B,C
For the pentatonic, use B1, B2, B3, B4, B5 for the 5 black keys.

What is my basis for 22-hexads? Perhaps a MOS scale?

0 4 8 11 15 19 works, label these A B C D E F and find
(strictly) proprietary scales from that... That is where I am
stuck, not in the basic definitions. I could play with Scala
and feed in all-interval sets and see what I get.

PGH

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<phjelmstad@...> wrote:
>
> --- In tuning-math@yahoogroups.com, Carl Lumma <carl@> wrote:
> >
> > >> Sure. The pentatonic scale is proper.
> > >
> > >Could you show how this is proper, but not strictly proper?
> > >Do you letter the black keys?
> >
> > It is strictly proper in 12-ET. The diatonic scale in 12-ET
> > is proper but not strictly so. Scala will tell you all this
> > in its "show data". Rothenberg propriety is explained on
> > wikipedia.
> >
> > -Carl
>
> Yeah, I read it. So a Class isn't always the ordinary note names
> (A through G). In the case of the pentatonic scale you have only
> 5 classes, and it is strictly proper.
>
> Here is where I get stuck, in how to label the Class steps
>
> Wikipedia has these scales that are proper diatonic scales:
>
> These five scales are:
>
> Diatonic: C D E F G A B
>
> Ascending minor: C D Eó F G A B
>
> Harmonic minor: C D Eó F G Aó B
>
> Harmonic major: C D E F G Aó B
>
> Major locrian: C D E F Gó Aó Bó
>
> It's easy here, just use C,D,E,F,G,A,B,C
> For the pentatonic, use B1, B2, B3, B4, B5 for the 5 black keys.
>
> What is my basis for 22-hexads? Perhaps a MOS scale?
>
> 0 4 8 11 15 19 works, label these A B C D E F and find
> (strictly) proprietary scales from that... That is where I am
> stuck, not in the basic definitions. I could play with Scala
> and feed in all-interval sets and see what I get.
>
> PGH

Wait, I see, a step is a step is a step, so just number the
notes and go from there. Don't need a baseline. I'll look at the
more even scales first, I don't imagine the weird ones could
possibly be proper much less strictly proper

PGH wrote...

>> >> Sure. The pentatonic scale is proper.
>> >
>> >Could you show how this is proper, but not strictly proper?
>> >Do you letter the black keys?
>>
>> It is strictly proper in 12-ET. The diatonic scale in 12-ET
>> is proper but not strictly so. Scala will tell you all this
>> in its "show data". Rothenberg propriety is explained on
>> wikipedia.
>>
>> -Carl
>
>Yeah, I read it. So a Class isn't always the ordinary note names
>(A through G). In the case of the pentatonic scale you have only
>5 classes, and it is strictly proper.
>
>Here is where I get stuck, in how to label the Class steps

So, it's Interval Class (not just Class). They should be
labeled 2nds, 3rds, 4ths, etc, not that it matters much.

>What is my basis for 22-hexads? Perhaps a MOS scale?

Basis?? For each hexatonic scale meeting the condition I
asked about (containing all dyads in 22-ET), see if any 3rd
is smaller than any 2nd, etc. If so, throw it out. Then
tell us which scale(s) you didn't throw out!

-Carl

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> PGH wrote...
>
> >> >> Sure. The pentatonic scale is proper.
> >> >
> >> >Could you show how this is proper, but not strictly proper?
> >> >Do you letter the black keys?
> >>
> >> It is strictly proper in 12-ET. The diatonic scale in 12-ET
> >> is proper but not strictly so. Scala will tell you all this
> >> in its "show data". Rothenberg propriety is explained on
> >> wikipedia.
> >>
> >> -Carl
> >
> >Yeah, I read it. So a Class isn't always the ordinary note names
> >(A through G). In the case of the pentatonic scale you have only
> >5 classes, and it is strictly proper.
> >
> >Here is where I get stuck, in how to label the Class steps
>
> So, it's Interval Class (not just Class). They should be
> labeled 2nds, 3rds, 4ths, etc, not that it matters much.
>
> >What is my basis for 22-hexads? Perhaps a MOS scale?
>
> Basis?? For each hexatonic scale meeting the condition I
> asked about (containing all dyads in 22-ET), see if any 3rd
> is smaller than any 2nd, etc. If so, throw it out. Then
> tell us which scale(s) you didn't throw out!
>
> -Carl

By Basis, I meant a scale for basing the lettering on (like the
diatonic scale). Of course this really isn't needed.

To be honest, I don't think I will find ANY Rothenberg proper
scales in 22-tET which contain all the dyads. The scales
are just too lumpy if they do, and then they aren't proper.
It's not really worth going through by hand, I will have
to program something. There might be a scarce few. And they
will probably be 10 or 11-chords, 6-chords probably don't have
any. I have found some proper 6-scales, but their IV's are full
of zeros.

PGH

At 08:21 AM 5/14/2008, you wrote:
>--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>>
>> PGH wrote...
>>
>> >> >> Sure. The pentatonic scale is proper.
>> >> >
>> >> >Could you show how this is proper, but not strictly proper?
>> >> >Do you letter the black keys?
>> >>
>> >> It is strictly proper in 12-ET. The diatonic scale in 12-ET
>> >> is proper but not strictly so. Scala will tell you all this
>> >> in its "show data". Rothenberg propriety is explained on
>> >> wikipedia.
>> >>
>> >> -Carl
>> >
>> >Yeah, I read it. So a Class isn't always the ordinary note names
>> >(A through G). In the case of the pentatonic scale you have only
>> >5 classes, and it is strictly proper.
>> >
>> >Here is where I get stuck, in how to label the Class steps
>>
>> So, it's Interval Class (not just Class). They should be
>> labeled 2nds, 3rds, 4ths, etc, not that it matters much.
>>
>> >What is my basis for 22-hexads? Perhaps a MOS scale?
>>
>> Basis?? For each hexatonic scale meeting the condition I
>> asked about (containing all dyads in 22-ET), see if any 3rd
>> is smaller than any 2nd, etc. If so, throw it out. Then
>> tell us which scale(s) you didn't throw out!
>>
>> -Carl
>
>By Basis, I meant a scale for basing the lettering on (like the
>diatonic scale). Of course this really isn't needed.
>
>To be honest, I don't think I will find ANY Rothenberg proper
>scales in 22-tET which contain all the dyads. The scales
>are just too lumpy if they do, and then they aren't proper.
>It's not really worth going through by hand, I will have
>to program something.

I wasn't expecting you to do it by hand!

>There might be a scarce few. And they
>will probably be 10 or 11-chords, 6-chords probably don't have
>any.

Well that's the question, isn't it? :)

-Carl