Yahoo Tuning Groups Ultimate Backup tuning The Eikosany's tonal center

For the 2-3-5-7-9-11 3)6 Eikosany and similar combination product sets, much tends to be made of the lack of a strong tonal center inherent in the scale. For instance, a few years back there was this:

http://launch.dir.groups.yahoo.com/group/tuning/message/56325

Carl mentions in a followup that the Eikosany is "symmetrical around
a point at which there is no note." I think it is worth noting that if you add one note, the character of the scale changes to one with a built-in tonal center.

The eikosany can be split into three sections, the central portion and two symmetrically located wings, the seven wing and the eleven wing. In one transposition, the seven wing consists of the notes 35/32, 7/6, 21/16, 35/24, 7/4, 35/18. This is a transposed 5-limit scale. The 11-wing is similar, consisting of 33/32, 55/48, 11/8, 55/36, 55/32, 11/6, again purely 5-limit on transposition. The central portion is a more curious affair: 77/72, 77/64, 5/4, 385/288, 3/2, 77/48, 5/3, 15/8. This might seem like an arbitrary amalgam of the notes with numerators divisible by 77 with the 5-limit notes, unless we allow ourselves to tweak things by 385/384. Then we get 16/15, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 15/8. This is the 3x3 square Euler genus, defined via divisors of 225, minus its central 1/1. Adding that to the Eikosany gives a 21-note scale which suddenly is not tonally ambiguous, as the central 1/1 ties everything together. Tempering by 385/384 is an obvious possibility, but even if we don't temper, 385/384 is small enough that you can't really escape it save by refusing to acknowledge or employ the approximations in question. Going solely by ear, you are stuck with them.

Here is the full 21-note scale:

! eikocenter.scl
The 2-3-5-7-9-11 Eikosany plus a tonal center note
21
!
33/32
77/72
35/32
55/48
7/6
77/64
5/4
21/16
385/288
11/8
35/24
3/2
55/36
77/48
5/3
55/32
7/4
11/6
15/8
35/18
2/1

🔗Kraig Grady <kraiggrady@...>

All of Ervs papers on the eikosany can be found in the Wilson archives http://anaphoria.com/wilson.html
under the subtitle
THE HEXANY , THE EIKOSANY, AND THE OTHER COMBINATION PRODUCT SETS

I probably without doubt have worked more with the eikosany than anyone. i made an entire ensemble of instruments in it and was almost my sole tuning for 15 years. Because of it i got rid of 31 equal which is a good system to try it out in if one in interested before commiting one self.
If functions better in just without a doubt.

The eikosany allows certain structural possibilities and gives to each note a unique function in relationship to the whole. for instance the 1-3-5 will function harmonically as these three factors but function subharmonically as a a 7, a 9 and a 11 in relationship to other tones.
Each tone will have this similar relation to the whole defined by the three factors that make it up.

Yes you can divide it into 2, 3 parts and 6 parts and even get cubes which says allot about its multidimensional space
If one wants it to be tonal one has only to hold a note , any note. half the structure will be related as a reciprocal cross-set. You have only to move beyond this if you wish to move.

Once you reduce the structure to any 1/1 or spell it with reduced ratios i am afraid you would lose track of where you are. It is best to keep the three factors just as they are to easily find by omitting the common factors.

One could add a harmonic series and a subharmonic series at the 1)6 set and the 6)6 set and with these additional 12 if quite well fills out a 31 tone matrix which you have to add an additional row at the top and bottom.This is the full dalessandro layout.

This gives you also the 2)6 set and the 4)6 set each very interesting structures.
If you fill in the two empty places in the keyboard you also get 2 1-3-7-9-11-15 eikosanies a 3/2 apart. these form good 22 tone scales that can stand on their own .
the briefest of outline of patterns can be found here
http://anaphoria.com/eikopapers.html
Even with a single eikosany one can rotate the the factors in the centered pentad lattice and one can have bascially 120 different versions (another 120 mirroring these) of the same pattern.

As a structure also to have the most compact relationship possible there are numerous tetrachordal scales often one can modulated through a few cycles of 5ths.
Since it is self mirroring the inversions of every thing is possible.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> For the 2-3-5-7-9-11 3)6 Eikosany and similar combination product sets, much tends to be made of the lack of a strong tonal center inherent in the scale. For instance, a few years back there was this:
>
> http://launch.dir.groups.yahoo.com/group/tuning/message/56325
>
> Carl mentions in a followup that the Eikosany is "symmetrical around
> a point at which there is no note." I think it is worth noting that if you add one note, the character of the scale changes to one with a built-in tonal center.
>
> The eikosany can be split into three sections, the central portion and two symmetrically located wings, the seven wing and the eleven wing. In one transposition, the seven wing consists of the notes 35/32, 7/6, 21/16, 35/24, 7/4, 35/18. This is a transposed 5-limit scale. The 11-wing is similar, consisting of 33/32, 55/48, 11/8, 55/36, 55/32, 11/6, again purely 5-limit on transposition. The central portion is a more curious affair: 77/72, 77/64, 5/4, 385/288, 3/2, 77/48, 5/3, 15/8. This might seem like an arbitrary amalgam of the notes with numerators divisible by 77 with the 5-limit notes, unless we allow ourselves to tweak things by 385/384. Then we get 16/15, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 15/8. This is the 3x3 square Euler genus, defined via divisors of 225, minus its central 1/1. Adding that to the Eikosany gives a 21-note scale which suddenly is not tonally ambiguous, as the central 1/1 ties everything together. Tempering by 385/384 is an obvious possibility, but even if we don't temper, 385/384 is small enough that you can't really escape it save by refusing to acknowledge or employ the approximations in question. Going solely by ear, you are stuck with them.
>
> Here is the full 21-note scale:
>
> ! eikocenter.scl
> The 2-3-5-7-9-11 Eikosany plus a tonal center note
> 21
> !
> 33/32
> 77/72
> 35/32
> 55/48
> 7/6
> 77/64
> 5/4
> 21/16
> 385/288
> 11/8
> 35/24
> 3/2
> 55/36
> 77/48
> 5/3
> 55/32
> 7/4
> 11/6
> 15/8
> 35/18
--

/^_,',',',_ //^ /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/>

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

a momentary antenna as i turn to water
this evaporates - an island once again

🔗genewardsmith <genewardsmith@...>

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

> I probably without doubt have worked more with the eikosany than
> anyone. i made an entire ensemble of instruments in it and was
> almost my sole tuning for 15 years. Because of it i got rid of
> 31 equal which is a good system to try it out in if one in
> interested before commiting one self.
> If functions better in just without a doubt.

But even so I doubt it. Did you notice that 385=5*7*11 can function as a 3? Once you allow yourself to think in that way, the eikosany looks very different.

> The eikosany allows certain structural possibilities and gives
> to each note a unique function in relationship to the whole. for
> instance the 1-3-5 will function harmonically as these three
> factors but function subharmonically as a a 7, a 9 and a 11 in
> relationship to other tones.

What does "function subharmonically" mean?

> Yes you can divide it into 2, 3 parts and 6 parts and even
> get cubes which says allot about its multidimensional space.

But which multidimensional space? The 11-limit is rank five, which means the lattice of pitch classes is in a four dimensional space. But if you allow 385~384, so that the product 5*7*11 functions also as a 3, the space is reduced by one dimension, and the eikosany can be seen as a construct of pitch classes in three dimensions. In the middle of it, not part of the eikosany but exactly centrally located and with a dominating connection to the eikosany as a whole, is 9. The scale I gave, which plugs the hole in the middle, simply adds 9 to the eikosany.

> If one wants it to be tonal one has only to hold a note , any
> note.

A drone is hardly the most sophisticated way of introducing tonality. I think putting the 3x3 Euler genus ("Genus chromaticum veterum correctum") in the middle of it all with the rest symmetrically arranged around it changes the whole picture as far as tonality is concerned.

> Once you reduce the structure to any 1/1 or spell it with
> reduced ratios i am afraid you would lose track of where you
> are.

I don't see why, but you can add the 9, and then adopt a rule that 385 can be equated with 3. That's obviously more confusing than pure JI but that's the way the cookie then crumbles, as the approximation in question is sitting there in the notes of the scale in the form of the Euler genus, and etc, whether you acknowledge them or not; your ear will insist that a fifth 4.5 cents flat is a fifth of some kind.

You can look at things in terms of the 7-limit JI lattice, with 11 coming in as 384/35, which is in the 3/35 pitch class, a note on the opposite side of a hexany from 1 and hence not complex at all.

Of course you can also temper everything, for instance by 284edo, where the fifth is now half a cent flat and the whole issue of two kinds of fifth in your scale goes away.

🔗Kraig Grady <kraiggrady@...>

Re: The Eikosany's tonal center

Gene~
In order to avoid a point by point argument
This should at least confirm that what you are talking about was considered
page 31of http://anaphoria.com/dal.PDF shows the small interval and how they conflict on a 31 tone mapping

If you then look at the http://anaphoria.com/xen11.PDF [ The article was written in a haste at a time in which i was working long hours.] you will see on the marimba i indeed did temper out one of these small intervals in the middle of the marimba bar 23 [ for fans of numerology]. Elsewhere i made a choice between one and the other.
Erv strategy for keyboard mapping here is quite interesting where certain intervals can occur in some octaves and alternates in others.

You could temper the entire structure if you wish with minute tempering stretching the over 31 fifths.
This possibility was discussed at the time and I decided against it. I would understand how this would appeal to some. The goal was to have instruments where other traditional instruments could be added to play along with it and it seemed all the help one could get would be needed.

So I decided for going for the pure version to give as many acoustical cues as possible .also no one had heard it and i was the first to tune up the full dallesandro CPS with James Wyness being the only other person i know who has.

It is conjecture at this point what would happen to such a microtemperment of this structure on either of our parts.
It is like the "humanize' function on many software programs. It shifts everything slightly, but what happens if we try to humanize the humanize, and what is the limit before it causes problems.

We can all witness the Midi standard where tuning of 768[?] seemed like it would solve all.
surely a difference of no more than a cent and a half could make any difference.
It has been one of the greatest mistakes this field has endured and maybe set it back 30 years.

Certain aspects of working with this material are more apparent when one works with it on a day to day level.
I remember a discussion of putting a 22 tone scale a conventional keyboard and the list decided that it was better to skip the two notes at the end of the second octave in order to have the appearance look the same.

That i had and used two such keyboards daily for years were ignored and the fact that many thing would no longer be possible, such as perfect fifths in one hand matter little.

There were many assumption i made and for all practical purposes seemed correct about the eikosany.
It was only by working with the material did certain things come up and kick me in the face.
What i put up i did in order to having others not to fall into the same traps as myself.

I placed the 1-3-7-9-11-15 22 tone eikosany in three different keyboard patterns over the years and the final one was thought to be the worse by the list and would still appear so. It was the best for what i was doing.
and would not go back to any of the previous versions.

/^_,',',',_ //^ /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/>

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

a momentary antenna as i turn to water
this evaporates - an island once again

🔗genewardsmith <genewardsmith@...>

Nothing I said had anything very much to do with 31et, and I'm sorry if I left that impression. Erv has a lot of nifty diagrams of the eikosany and other scales as regular graphs, but I don't see anything relevant to my comments. I suppose two regular graph diagrams, one showing the JI relationships you get when adding 9 to the eikosany and the other showing the full complement of 385/384 tempered intervals might be illuminating.

385/384 tempering is quite a lot more accurate than 31et is able to demonstrate; you can get the error of the 11-limit to within two cents. But I wouldn't call it microtempering. My point however is that even if you do no tempering at all, a fifth flat by 385/384 is going to sound like a fifth. Hence after adding 9 you have a 21-note just intonation scale, with a clearly audible 9-note Euler genus in the middle of it dominating things so far as tonal centering goes, even though some of the fifths are flat by 4.5 cents (about 1/5 comma) and others are just. From that point of view the eikosany is a scale where the natural choice for 1/1 was omitted, rendering it more fragmented and tonally ambiguous.

🔗sevishmusic <sevish@...>

Kraig,

you mentioned that Eikosanies should not be reduced to an octave, because it makes the patterns harder to understand. The 1-3-5-7-9-11 Eikosany scale in the Scala database looks like this:

3)6 1.3.5.7.9.11 Eikosany (1.3.5 tonic)
|
0: 1/1 0.000 unison, perfect prime
1: 33/32 53.273 undecimal comma, al-Farabi's 1/4-tone
2: 21/20 84.467 minor semitone
3: 11/10 165.004 4/5-tone, Ptolemy's second
4: 9/8 203.910 major whole tone
5: 7/6 266.871 septimal minor third
6: 99/80 368.914
7: 77/60 431.875
8: 21/16 470.781 narrow fourth
9: 11/8 551.318 undecimal semi-augmented fourth
10: 7/5 582.512 septimal or Huygens' tritone, BP fourth
11: 231/160 635.785
12: 3/2 701.955 perfect fifth
13: 63/40 786.422 narrow minor sixth
14: 77/48 818.189
15: 33/20 866.959
16: 7/4 968.826 harmonic seventh
17: 9/5 1017.596 just minor seventh, BP seventh
18: 11/6 1049.363 21/4-tone, undecimal neutral seventh
19: 77/40 1133.830
20: 2/1 1200.000 octave

As a beginner, should I avoid using this Eikosany?

Besides this I am having many troubles writing my first piece of music using any Eikosany. My first problem is which one to use? My second problem is how do I know which notes should be used with others?

Many thanks if you can help me to understand!

Sean

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
>
> Re: The Eikosany's tonal center
>
> Gene~
> In order to avoid a point by point argument
> This should at least confirm that what you are talking about
> was considered
> page 31of http://anaphoria.com/dal.PDF shows the small
> interval and how they conflict on a 31 tone mapping
>
> If you then look at the http://anaphoria.com/xen11.PDF [ The
> article was written in a haste at a time in which i was working
> long hours.] you will see on the marimba i indeed did temper out
> one of these small intervals in the middle of the marimba bar 23
> [ for fans of numerology]. Elsewhere i made a choice between one
> and the other.
> Erv strategy for keyboard mapping here is quite interesting
> where certain intervals can occur in some octaves and alternates
> in others.
>
> You could temper the entire structure if you wish with minute
> tempering stretching the over 31 fifths.
> This possibility was discussed at the time and I decided
> against it. I would understand how this would appeal to some.
> The goal was to have instruments where other traditional
> instruments could be added to play along with it and it seemed
> all the help one could get would be needed.
>
> So I decided for going for the pure version to give as many
> acoustical cues as possible .also no one had heard it and i was
> the first to tune up the full dallesandro CPS with James Wyness
> being the only other person i know who has.
>
> It is conjecture at this point what would happen to such a
> microtemperment of this structure on either of our parts.
> It is like the "humanize' function on many software programs.
> It shifts everything slightly, but what happens if we try to
> humanize the humanize, and what is the limit before it causes
> problems.
>
> We can all witness the Midi standard where tuning of 768[?]
> seemed like it would solve all.
> surely a difference of no more than a cent and a half could
> make any difference.
> It has been one of the greatest mistakes this field has
> endured and maybe set it back 30 years.
>
> Certain aspects of working with this material are more apparent
> when one works with it on a day to day level.
> I remember a discussion of putting a 22 tone scale a
> conventional keyboard and the list decided that it was better to
> skip the two notes at the end of the second octave in order to
> have the appearance look the same.
>
> That i had and used two such keyboards daily for years were
> ignored and the fact that many thing would no longer be
> possible, such as perfect fifths in one hand matter little.
>
> There were many assumption i made and for all practical purposes
> seemed correct about the eikosany.
> It was only by working with the material did certain things
> come up and kick me in the face.
> What i put up i did in order to having others not to fall into
> the same traps as myself.
>
> I placed the 1-3-7-9-11-15 22 tone eikosany in three different
> keyboard patterns over the years and the final one was thought
> to be the worse by the list and would still appear so. It was
> the best for what i was doing.
> and would not go back to any of the previous versions.
>
>
>
>
>
> --
>
>
> /^_,',',',_ //^ /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/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> a momentary antenna as i turn to water
> this evaporates - an island once again
>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "sevishmusic" <sevish@...> wrote:
>
> Kraig,
>
> you mentioned that Eikosanies should not be reduced to an octave, because it makes the patterns harder to understand. The 1-3-5-7-9-11 Eikosany scale in the Scala database looks like this:
>
> 3)6 1.3.5.7.9.11 Eikosany (1.3.5 tonic)

And if you add a 9 to the Eikosany, and divide by the
1.3.5 tonic, you will add 6/5 to the list of scale elements below. Since the notes of the Eikosany are more closely related to this 6/5 than to anything else, I think it's quite likely it would become easier to understand after adding it. You could try working your way up from the bottom: first find the 5-limit triads, then the 7-limit tetrads, and then add the 9 and 11 limit stuff. It would depend, I suppose, on how you work, but if you feed this scale, with or without the added 6/5, into Scala and do "show location" on various sorts of chords you will get listings for the chords in question, which might help. As I remarked, you can break it apart in the 5-limit into two "wings" and a central region, and you could try to find these, though the central region doesn't make much sense unless you add the 6/5.

> 0: 1/1 0.000 unison, perfect prime
> 1: 33/32 53.273 undecimal comma, al-Farabi's 1/4-tone
> 2: 21/20 84.467 minor semitone
> 3: 11/10 165.004 4/5-tone, Ptolemy's second
> 4: 9/8 203.910 major whole tone
> 5: 7/6 266.871 septimal minor third
6/5 "center note"
> 6: 99/80 368.914
> 7: 77/60 431.875
> 8: 21/16 470.781 narrow fourth
> 9: 11/8 551.318 undecimal semi-augmented fourth
> 10: 7/5 582.512 septimal or Huygens' tritone, BP fourth
> 11: 231/160 635.785
> 12: 3/2 701.955 perfect fifth
> 13: 63/40 786.422 narrow minor sixth
> 14: 77/48 818.189
> 15: 33/20 866.959
> 16: 7/4 968.826 harmonic seventh
> 17: 9/5 1017.596 just minor seventh, BP seventh
> 18: 11/6 1049.363 21/4-tone, undecimal neutral seventh
> 19: 77/40 1133.830
> 20: 2/1 1200.000 octave
>
> As a beginner, should I avoid using this Eikosany?

Why?

> Besides this I am having many troubles writing my first piece of music using any Eikosany. My first problem is which one to use? My second problem is how do I know which notes should be used with others?

For any scale, you can find the chords using Scala.

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> Carl mentions in a followup that the Eikosany is "symmetrical around
> a point at which there is no note."

We find not only 385/384 approximations in the Eikosany, but also 441/440 and 540/539. Putting these together yields miracle temperament, which suggests looking at the Eikosany through miracle spectacles. That gives us this:

[-21, -15, -13, -9, -8, -7, -6, -4, -2, -1,
1, 2, 4, 6, 7, 8, 9, 13, 15, 21]

This is symmetrical around 0, a point at which there is no note. Adding the 0 gives the miracle tempering of the Eikosany+9 scale I've been promoting:

[-21, -15, -13, -9, -8, -7, -6, -4, -2, -1,
0, 1, 2, 4, 6, 7, 8, 9, 13, 15, 21]

Miracle tempering is accurate enough that it seems to me there is a lot to be said in favor of using it on the Eikosany. It smooths out the irregularity of having some intervals pure and others flat or sharp by small commas, which some people will wish to do and others will not. In any case, it's a aid towards analyzing relationships.

"The eikosany can be split into three sections, the central portion and two symmetrically located wings, the seven wing and the eleven wing. In one transposition, the seven wing consists of the notes 35/32, 7/6, 21/16, 35/24, 7/4, 35/18. This is a transposed 5-limit scale."

In miracle, [-21, -15, -9, -8, -2, 4].

"The 11-wing is similar, consisting of 33/32, 55/48, 11/8, 55/36, 55/32, 11/6, again purely 5-limit on transposition."

In miracle, [-4, 2, 8, 9, 15, 21]

"The central portion is a more curious affair: 77/72, 77/64, 5/4, 385/288, 3/2, 77/48, 5/3, 15/8."

In miracle, [-13, -7, -6, -1, 1, 6, 7, 13]

"This might seem like an arbitrary amalgam of the notes with numerators divisible by 77 with the 5-limit notes, unless we allow ourselves to tweak things by 385/384. Then we get 16/15, 6/5, 5/4, 4/3, 3/2, 8/5, 5/3, 15/8. This is the 3x3 square Euler genus, defined via divisors of 225, minus its central 1/1. Adding that to the Eikosany gives a 21-note scale which suddenly is not tonally ambiguous, as the central 1/1 ties everything together."

1-16/15-6/5-5/4-4/3-3/2-8/5-5/3-15/8 in miracle is:

[-13, -7, -6, -1, 0, 1, 6, 7, 13]

🔗Kraig Grady <kraiggrady@...>

Does the addition of just this one note make a constant structure? is that what i am missing?
which Euler genus
The reason i brought up the 31 tone context is because the 1-3-5-7-9-11 eikosany is not a constant structure (per.block) and to build an instrument or map it to a keyboard you are probably want the rest of these notes much in the same way if one has a diamond one is going to be inclined to add the tones Partch added to bring it as close to a constant structure one can.
the interesting thing with the 1-3-7-9-11-15 eikosany one has to only add two tones to make it a constant structure.

--
Re: The Eikosany's tonal center

--- In tuning@yahoogroups.com </tuning/post?postID=o_jPoNtGUvD3k1NdVCaHrRd8qRJXmyVqOtKAsX4bInOdgOXVlQyl-O1JgXcP04K-IvJHL5-QlUM-tfvY>, Nothing I said had anything very much to do with 31et, and I'm sorry if I left
that impression. Erv has a lot of nifty diagrams of the eikosany and other
scales as regular graphs, but I don't see anything relevant to my comments. I
suppose two regular graph diagrams, one showing the JI relationships you get
when adding 9 to the eikosany and the other showing the full complement of
385/384 tempered intervals might be illuminating.

385/384 tempering is quite a lot more accurate than 31et is able to demonstrate;
you can get the error of the 11-limit to within two cents. But I wouldn't call
it microtempering. My point however is that even if you do no tempering at all,
a fifth flat by 385/384 is going to sound like a fifth. Hence after adding 9 you
have a 21-note just intonation scale, with a clearly audible 9-note Euler genus
in the middle of it dominating things so far as tonal centering goes, even
though some of the fifths are flat by 4.5 cents (about 1/5 comma) and others are
just. From that point of view the eikosany is a scale where the natural choice
for 1/1 was omitted, rendering it more fragmented and tonally ambiguous.

/^_,',',',_ //^ /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/>

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

a momentary antenna as i turn to water
this evaporates - an island once again

🔗Kraig Grady <kraiggrady@...>

Sean~
It is not a matter of not octave reducing you are wanting to do that. It is a matter of not reducing the factors into a fraction. so we are talking more about notation when working with it
For instance I cannot even tell looking at the diagram below what factor they are using for 1/1.
for instance 1*3*11 is a C a 6/5 from A=440 which is 1*5*11 which you will notice is 55 a sub octave of a 440.
the factors work because if you say have 1*9*11 ,3*9*11, 5*9*11,7*9*11 you know you have a 1-3-5-7 because all the 9*11s cancel each other out. you can also quickly find a 3*5*7*9*11 dekany because you know they are all going to have I and the complement will not have 1. I don't think you can find those type of structures if you are referring to the set of ratios below. can you find the 1-3-5-7 tetrad within a minute even?
I can't see it in there.

It is a hard tuning to work with unless you have someway to map it out where you can see it.
I recently added this paper http://anaphoria.com/dekanyconstantstructures.pdf
to this page http://anaphoria.com/journal.html
as what i think is a good starting point. maybe offlist i can show you some mapping to how to
Some of the added tones are outside the eikosany.
Allot can be done with dekanies. this early experiment of a piece was done entirely on the 1-3-7-11-15
http://anaphoria.com/boehme.mp3
some of the players were not even musicians and did not read music which placed great limitations, but gave great energy to things i was trying at the time.
those who were players had to work with unusual keyboard layouts of the bars and due to the size of the instruments practice on them was only possible at group
info on it can found on this page
http://anaphoria.com/samples.html

Kraig,

you mentioned that Eikosanies should not be reduced to an octave, because it
makes the patterns harder to understand. The 1-3-5-7-9-11 Eikosany scale in the
Scala database looks like this:

As a beginner, should I avoid using this Eikosany?

Besides this I am having many troubles writing my first piece of music using any
Eikosany. My first problem is which one to use? My second problem is how do I
know which notes should be used with others?

Many thanks if you can help me to understand!

Sean
--

/^_,',',',_ //^ /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/>

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

a momentary antenna as i turn to water
this evaporates - an island once again

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Does the addition of just this one note make a constant
> structure? is that what i am missing?

No, but the Eikosany+9 scale comes closer to CS than the Eikosany in the sense that Scala lists a bunch of nonmonotonic mappings for it but not for the Eikosany.

> which Euler genus

The 3x3 Euler genus consisting of the divisors of 225, namely {1, 3, 5, 9, 15, 25, 45, 75, 225}. To see how this works, transpose the genus by multiplying through by 3/5, giving {3/5, 9/5, 3, 27/5, 9, 15, 27, 45, 135}. Of these, 15, 27, 45, and 135 belong to the Eikosany. If we multiply 3/5, 9/5, 3, 27/5 by 385/3, we get 77, 231, 385 and 693 which also belong to the Eikosany. The point of this proceedure, of course, is that (385/3)/128 = 385/384 is small (4.5 cents.) Hence in this approximate sense the 225 Euler genus is in the Eikosany save for the central note, 9.

This kind of approximate hole is not unique to the Eikosany. Another and to me very interesting example is the 9-limit tonality diamond, which has two such holes which we can patch by adding 16/15 and 15/8, which makes it into two fused approximate 3x4 Euler genuses, that is, Euler genuses defined by the divisors of 1125.

> the interesting thing with the 1-3-7-9-11-15 eikosany one has to
> only add two tones to make it a constant structure.

Interesting indeed! I'll take a look. Do you know off-hand the two tones in question?

🔗Carl Lumma <carl@...>

🔗Kraig Grady <kraiggrady@...>

I think i understand now . you are adding this pitch to fill out a different structure that one can then interact with the eikosany.
the 22 Eikosany the 1-3-7-9-11-15 that was made out of two jenco vibes ( which i have since added a third to fillout the whole 1)6-6)6CPS [dallesandro] added 3*9**33 and the complement 1*5*7
since there are two of these found in this overall structure Erv and Paul Beaver tuned to the one that lowers this whole structure down a 3/2. This because it give a just major scale on C.
a very weird byproduct of the 1-3-5-7-9-11 if you add the 11.

--- In tuning@yahoogroups.com </tuning/post?postID=RaKYq4Wxmivo0wmucMtGUuiwYj4hv5O8yRR1SjC4ARmFLLVhqKY4JUOlkY8i39KS6LGs-9fb5JjcbCPPcSgJTg>, Kraig Grady <kraiggrady@...> wrote:
>
> Does the addition of just this one note make a constant
> structure? is that what i am missing?

No, but the Eikosany+9 scale comes closer to CS than the Eikosany in the sense
that Scala lists a bunch of nonmonotonic mappings for it but not for the
Eikosany.

> which Euler genus

The 3x3 Euler genus consisting of the divisors of 225, namely {1, 3, 5, 9, 15,
25, 45, 75, 225}. To see how this works, transpose the genus by multiplying
through by 3/5, giving {3/5, 9/5, 3, 27/5, 9, 15, 27, 45, 135}. Of these, 15,
27, 45, and 135 belong to the Eikosany. If we multiply 3/5, 9/5, 3, 27/5 by
385/3, we get 77, 231, 385 and 693 which also belong to the Eikosany. The point
of this proceedure, of course, is that (385/3)/128 = 385/384 is small (4.5
cents.) Hence in this approximate sense the 225 Euler genus is in the Eikosany
save for the central note, 9.

This kind of approximate hole is not unique to the Eikosany. Another and to me
very interesting example is the 9-limit tonality diamond, which has two such
holes which we can patch by adding 16/15 and 15/8, which makes it into two fused
approximate 3x4 Euler genuses, that is, Euler genuses defined by the divisors of
1125.

> the interesting thing with the 1-3-7-9-11-15 eikosany one has to
> only add two tones to make it a constant structure.

Interesting indeed! I'll take a look. Do you know off-hand the two tones in
question?
--

/^_,',',',_ //^ /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/>

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

a momentary antenna as i turn to water
this evaporates - an island once again

🔗genewardsmith <genewardsmith@...>

🔗w_orld_s <warrensummers@...>

Two cents...

Hi Sean, I had this same problem initially...

When we talk amongst each other I agree with Kraig, that demonstrating the factors help other people enter into the scale on hand...

At the same time though, in terms of my use of these scales (which I should say is in the context of meditation, change ringing and personal improvisations)... I have found that after a time, I have drifted away from the technical origins of the pitches.

Once I build the house, I like to live in it, without all those markings left on the walls by builders. If it is a well built house and if I have a sense of the engineering, I can usually find my way to the bathroom in the dark of night. And if I trip on the way? Stub my toe on a step that I did not expect? I will laugh HA HA! At my discovery.

Give the notes your own names... base them on friends, whims and absurd rationalisations. If you give them enough space they all sound good and if you spend enough time with them they do show their secrets.

Happy travels in CPS land.

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
>
> Sean~
> It is not a matter of not octave reducing you are wanting to
> do that. It is a matter of not reducing the factors into a
> fraction. so we are talking more about notation when working with it
> For instance I cannot even tell looking at the diagram below
> what factor they are using for 1/1.
> for instance 1*3*11 is a C a 6/5 from A=440 which is 1*5*11
> which you will notice is 55 a sub octave of a 440.
> the factors work because if you say have 1*9*11 ,3*9*11,
> 5*9*11,7*9*11 you know you have a 1-3-5-7 because all the 9*11s
> cancel each other out. you can also quickly find a 3*5*7*9*11
> dekany because you know they are all going to have I and the
> complement will not have 1. I don't think you can find those
> type of structures if you are referring to the set of ratios
> below. can you find the 1-3-5-7 tetrad within a minute even?
> I can't see it in there.
>
> It is a hard tuning to work with unless you have someway to map
> it out where you can see it.
> I recently added this paper
> http://anaphoria.com/dekanyconstantstructures.pdf
> to this page http://anaphoria.com/journal.html
> as what i think is a good starting point. maybe offlist i can
> show you some mapping to how to
> Some of the added tones are outside the eikosany.
> Allot can be done with dekanies. this early experiment of a
> piece was done entirely on the 1-3-7-11-15
> http://anaphoria.com/boehme.mp3
> some of the players were not even musicians and did not read
> music which placed great limitations, but gave great energy to
> things i was trying at the time.
> those who were players had to work with unusual keyboard
> layouts of the bars and due to the size of the instruments
> practice on them was only possible at group
> info on it can found on this page
> http://anaphoria.com/samples.html
>
>
> Kraig,
>
> you mentioned that Eikosanies should not be reduced to an
> octave, because it
> makes the patterns harder to understand. The 1-3-5-7-9-11
> Eikosany scale in the
> Scala database looks like this:
>
> 3)6 1.3.5.7.9.11 Eikosany (1.3.5 tonic)
> |
> 0: 1/1 0.000 unison, perfect prime
> 1: 33/32 53.273 undecimal comma, al-Farabi's 1/4-tone
> 2: 21/20 84.467 minor semitone
> 3: 11/10 165.004 4/5-tone, Ptolemy's second
> 4: 9/8 203.910 major whole tone
> 5: 7/6 266.871 septimal minor third
> 6: 99/80 368.914
> 7: 77/60 431.875
> 8: 21/16 470.781 narrow fourth
> 9: 11/8 551.318 undecimal semi-augmented fourth
> 10: 7/5 582.512 septimal or Huygens' tritone, BP fourth
> 11: 231/160 635.785
> 12: 3/2 701.955 perfect fifth
> 13: 63/40 786.422 narrow minor sixth
> 14: 77/48 818.189
> 15: 33/20 866.959
> 16: 7/4 968.826 harmonic seventh
> 17: 9/5 1017.596 just minor seventh, BP seventh
> 18: 11/6 1049.363 21/4-tone, undecimal neutral seventh
> 19: 77/40 1133.830
> 20: 2/1 1200.000 octave
>
> As a beginner, should I avoid using this Eikosany?
>
> Besides this I am having many troubles writing my first piece of
> music using any
> Eikosany. My first problem is which one to use? My second
> problem is how do I
> know which notes should be used with others?
>
> Many thanks if you can help me to understand!
>
> Sean
> --
>
>
> /^_,',',',_ //^ /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/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> a momentary antenna as i turn to water
> this evaporates - an island once again
>

🔗Mike Battaglia <battaglia01@...>

🔗martinsj013 <martinsj@...>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
>
> I think i understand now . you are adding this pitch to fill out
> a different structure that one can then interact with the eikosany.
> the 22 Eikosany the 1-3-7-9-11-15 that was made out of two jenco
> vibes ( which i have since added a third to fillout the whole
> 1)6-6)6CPS [dallesandro] added 3*9**33 and the complement 1*5*7
> since there are two of these found in this overall structure Erv
> and Paul Beaver tuned to the one that lowers this whole
> structure down a 3/2. This because it give a just major scale on C.
> a very weird byproduct of the 1-3-5-7-9-11 if you add the 11.

It's a very interesting scale, in the Scala directory as wilson_16, with the notation "Wilson 1 3 7 9 11 15 eikosany plus 9/8 and tritone. Used Stearns: Jewel". It has a major scale on one end, and a complementary minor scale at the other, with a juicy bunch of 11-limit harmony in the sandwich in between, all of it epimorphic and constant structure.

However, it isn't really too much like the Eikosany+9 scale, as the two added notes are on the periphery of the harmonic relationships, whereas the added 9 is right in the middle of them. If you look at what is complementary to 9 with respect to 10395 = 3*5*7*9*11*15, you get 10395/9 = 1155. Reducing both to the octave gives 9/8 and 1155/1024, which are 385/384 apart. In an octave equivalent sense, 9 is approximately the "square root" of 10395 in the sense that 81 is complementary to 385/3 (10395/81 = 385/3) and 385/3 is approximately 128, which is equivalent to 1.

Wilson_16 is another scale which it would make a lot of sense to temper via miracle. In fact, since there are a lot of 225/224 approximate relationships around, and since those don't work nearly as well as 385/384, 441/440 or 540/539 relationships if not tempered, you might claim it could use it more.

Did Stearns come up with this idea originally and call it "Jewel"?