What to do with a tetrachordal scale (13 limit comma inspired)

Hi Margo~
Since i am starting a new tread off of a tetrachord you mentioned, I direct this to you first. [I probably chould have chosen an 13 limit one]
all papers discussed below are included in this pdf,
http://anaphoria.com/Al-FarabiTetrachord.pdf
which will be added to in the future upon time permitting.
The tetrachord chosen is just for the sake of example and to show how we might depart in a few directions for any single point. I feel it is an important question in that we often present scales here but not how to go from one to the other.

While on the subject of some interesting Tetrachordal scales i thought i would expand it to talk about what one can do with one as far as permutations and modulations. While we might be content to stay in one, even the ancients found great musical use in extending it. Ptolemy associated modulations with "crisis" and we like most ages, flatter ourselves with placing ourselves in what appears to be on the brink of the biggest ones in history. The music of the previous century because of the proliferation of the entire array of the resources of the tuning at hand suppressed modulation. At best when it occured it became more a manner of transposition marking the lack of any real significant change in meaning in the music.
first let us look at the rotation of the tetrachords and one useful way of viewing them.

http://anaphoria.com/Al-FarabiTetrachord.jpg

This shows how the structure of the triadic diamond can be applied to melodic structures as easily as harmonic ones. Considering the Diamond - Lambdoma Equivalence and it found in Greece where there was no harmony on might expect this use of the structure to be even more common than the harmonic.

http://anaphoria.com/Al-FarabiRotations.jpg
Shows as an example of modulating by rotating the positions of two intervals in this case in a cycle of what resembles a cycle of variable "fourths" or applying it every third step of a heptatonic scale at a time. .
The advantage to this is that one is only changing one note at a time and offers the option of these tones to be used just like accidentals are within say a diatonic scale.
This also goes through some mixed tetrachords albeit only three of the possible six but the previous diagram can be used as the basis of such a link if one insist on it.

There is an advantage to knowing the most closely related material first, keeping in mind that it is in no way meant as a rule or restriction. Musically we often need to progress more abruptly, hence we can skip how we want. The two extreme scales omitting the gray one are transposed versions of each other.It is as if the scale as drifted with offers us musical possibilities. for instance one might start and suddenly jump to this new key then slowly modulate back one tone at a time until we find ourselves where we started.

http://anaphoria.com/PurviAl-Farabi.jpg <http://anaphoria.com/PurviAl-Farabi.jpg>
This one uses Wilson's Purvi modulation since it came up recently i thought it worth showing this example.
It is similar to the one before except in between the recurrence of the same scale we find ourselves with some new intervals. These scales are of interest in that they involve as many as 5 diffent ratios in a heptatonic scale. I should point out this is not the only Purvi Permutation of this scale. The 7/5, 45/32, or even one of the 4/3 could be used to act as the closing fourth so there would be potentially 7 different direction we could take off this singular Tetrachordal scale.

There is also the possibilities of the Marwa Permutations, of which the options are even greater but these will have to wait till the future.

--
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--- In tuning@yahoogroups.com, kraiggrady <kraiggrady@...> wrote:
Purvi
> Permutation

Marwa Permutations

Can anyone explicitly and rigorously define these for me, using words only, NO pictures?

Hello Gene~
This is a possible over simple version but i hope will work well.I will have to say it is a challenge in that for any type of permutation, it seem simplier to illustrate the sequence.
but here it goes.

The archetypal model of the Marwa permutations takes a 7 tone tetrachordal scale and then spells it as a chain of `fourths' that very often will vary in size. One of these is chosen to be the closing fourth. Most often a fourth is picked that ois followed by one of the atypical in size
Next one takes the beginning fourth (the one following the closing one) and most often atypical in size. one exchanges it with the following fourth and one continues this until it becomes the one before the closing fourth.

If there are more than one atypical in size which we will call A and B. We still start with A with B next to it and we exchange B with the other one once again moving toward the closing fourth. When we get to the end we exchange a with the next one and once again place B next to it and repeat the sequence again etc.

one takes the second one and exchanges toward the closing one until it reaches the place before.

Sometimes A and B combine will result in an octave or near octave in this case we omit these possibilities.
Examples such as 64/45 and 45/32 cannot occur in a row but are fine with an intermittent 4/3.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, kraiggrady <kraiggrady@> wrote:
> Purvi
> > Permutation
>
> Marwa Permutations
>
> Can anyone explicitly and rigorously define these for me, using words only, NO pictures?
>

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
>
> Hello Gene~
> This is a possible over simple version but i hope will work well.I will have to say it is a challenge in that for any type of permutation, it seem simplier to illustrate the sequence.
> but here it goes.

Thanks! Care to give Purvi a shot?

Hello, Kraig and Gene and all.

What I'd emphasize is that actually playing through
some of these Marwa Permutations or Purvi Modulations
for a given tetrachord like 22:21-12:11-7:6, the
Intense Chromatic of Ptolemy, quicky shows what a
creative method this is for finding new modes.

<http://anaphoria.com/xen9mar.PDF>
<http://anaphoria.com/xen10pur.PDF>

The device of treating a heptatonic scale as a chain
of fourths, and doing the kinds of rotations or
permutations shown in these papers, generates some
fascinating tetrachords and modes maybe not so
obvious at first blush. For example, from Ptolemy's
22:12-12:11-7:6, we get 88:81-12:11-9:8 (143-151-204
cents), a permutation of al-Farabi's 9:8-12:11-88:81
which could be a modern Arab Bayyati tetrachord.

The closest analogy I can come up with to this method
is Safi al-Din al-Urmawi in the middle to late 13th
century and his method of trying out various tetrachord
and pentachord combinations. We get all kinds of modes,
some of the usual maqamat, and some which evidently
aren't, at least _yet_, but certainly make for
interesting music.

As the article discuss, these methods of tetrachord
or modal variation were evidently developed in
connection with the raga system. An especially
fascinating instance involves the permutations or
modulations of the Diatonic of Archytas at
28:27-8:7-9:8, including Raga Darbari Kanada,
9:8-28:27-8:7-9:8-28:27-8:7-9:8. An interesting
question is whether this division came by way of
Persia, since I've read that "Darbari Kanada" might
mean "Kanada of the (Persian) Court" or the like.

However, Erv Wilson's examples using ancient Greek
tetrachords and more recent tunings from Kathleen
Schlesinger, for example, show that the method
is relevant to raga and many other musics as well.

In sum, these are _very_ impressive ways of
generating new genera and modes.

Happy New Year,

Margo

Hello Gene~
Purvi is fortunately a little more straight forward.
Once again we have a closing fourth which remain constant and at the end of our cycle of fourths spelling of our tetrachordal scale. The other 6 'fourths' are placed in order and rotated by taking the first one and placing it at the end of the sequence of these 6 and one repeats this till one comes back to the beginning. It is worth noting that both of these patterns are more spiral like than cyclical in that when one returns to the scale one started with one is in a quite different key.
Know your preferences towards ETs . One can still apply such patterns using number of scale steps over ratios, albeit one would need a medium to large enough number of tones for it to be distinctful.
--
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/^_,',',',_ //^/Kraig Grady_^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
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_'''''''_^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

Hello Margo~
I am glad you find these scales useful also.

There is a variation on these i came up by accident. which i will call modified Purvi.

Being stuck in a library at a time i hadn't studied it in detail and trying to recall it from memory I changed what appears on page 2 on the purvi article.<http://anaphoria.com/xen10pur.PDF <http://anaphoria.com/xen10pur.PDF>>
I will express this also in a form Gene requested of the others so he needn't request it.

One takes the entire chain of "fourths" found in our tetrachordal scale and starting at the beginning one exchanges it with the one following. Then one follows this same fourth is like sequence until one gets to the end. At which point one takes the one now found in the first place and repeat the process all over again till we reach the end. If we keep repeating this we will eventually go through all the fourths.
here is a chart version for those like myself prefer it. This is not one of Wilson's Diagram, but later when i which to write it out and compare, i found it easier to cut and paste one of his charts which is a rather unorthodox thing to do.

http://anaphoria.com/pur2.gif

This leads to 42 tone cycle, and also once again differs from the one he mentions on page 2 in that he cycles the the same fourth continually, I had shifted it to the new fourth at the beginning.

One could apply this to this to an ET or even a Constant structure in which case we can think of it being the difference between a real modulation and what is called a tonal modulation. The latter can be awkward at times , but often will give us new patterns in a tuning we might already have and love.

You mention there
"For example, from Ptolemy's 22:12-12:11-7:6, we get 88:81-12:11-9:8 (143-151-204 cents), a permutation of al-Farabi's 9:8-12:11-88:81 which could be a modern Arab Bayyati tetrachord."

I would just like to comment that some of these scales will result in as many as 5 different size steps in our scale which is quite unusual and hard to have a good theoretical way to explore.
I will say that if i had only one 7 tone scale i was going to tune up, such a scale would interest me in that it would give me the greatest variety with the least amount of notes. I would see this in the same fascination certain people like Walter O'Connell did with those 4 note sequences in 12 that give one all interval sets, but of which Dave Keenan pointed out work better in 13. Wilson too mapped out such 6 note all interval sets in 31 ET. But perhaps i am placing more importance on a technical aspect of like scales, still it is nice to have a tool to access them and be what will.

The subject of Raga Darbari Kanada came up in a private conversation with Cris Forster and as we found out is quite an abnormality in Rage scales, being a much later additions as we discovered.
http://en.wikipedia.org/wiki/Darbari_Kanada

I will say despite the many years when i heard a 28/27 in like scale, i still succumb to it beauty. It will be interesting to see how much further this and like scales will develop in their music. And surely too it cannot be overvalued with it being a wonderful stand alone scale.

--
signature file

/^_,',',',_ //^/Kraig Grady_^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
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_'''''''_^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

> Hello Margo~

> I am glad you find these scales useful also.

Dear Kraig,

These scales are very useful indeed! And one of the interesting
aspects, as I discuss below, is exploring some of the things that
are getting compromised in a "near-just" system. It's a bit like
those tests for different forms of color blindness, where we find
that a comma has been tempered out, although others may be
observed so that we still have a lot of step sizes in a
heptatonic mode.

> One could apply this to this to an ET or even a Constant structure
> in which case we can think of it being the difference between a
> real modulation and what is called a tonal modulation. The latter
> can be awkward at times , but often will give us new patterns in a
> tuning we might already have and love.

Please remind me on the distinction between real and tonal
modulation. I recall something like this in counterpoint theory
distinguishing between "real" and "tonal" imitation -- maybe it
was "real answer" and "tonal answer." But I'd like to understand
your meaning as it relates to modulation.

>> "For example, from Ptolemy's 22:12-12:11-7:6, [my
>> mistake, 22:21 not 22:12 -- M.S.] we get 88:81-12:11-9:8
>> (143-151-204 cents), a permutation of al-Farabi's
>> 9:8-12:11-88:81 which could be a modern Arab Bayyati
>> tetrachord."

> I would just like to comment that some of these scales will result
> in as many as 5 different size steps in our scale which is quite
> unusual and hard to have a good theoretical way to explore.

The curious thing is that 5 step sizes may not be so unusual in
some of the maqam tunings which often use diverse ajnas
(genera). To take a relatively "garden variety" example, here's a
version of Makam Suznak, using some of Ptolemy's ratios, see
Erv's Purvi modulations, p. 7, Fig. 8a, mode 3. Here I'll give
the just values and the tempered values in MET-24.

207.4 150.0 139.5 207.4 150.0 264.8 80.9
203.9 150.6 143.5 203.9 150.6 266.9 80.5
9:8 12:11 88:81 9:8 12:11 7:6 22:21
1/1 9/8 27/22 4/3 3/2 18/11 21/11 2/1
0 203.9 354.5 498.0 702.0 852.6 1119.5 1200
0 207.4 357.4 496.9 704.3 854.3 1119.1 1200
|--------------------|.......|----------------------|
Rast tone Hijaz

In either JI or MET-24 we get 5 different step sizes. But another
of Erv's modulations, Fig. 8a, mode 4, will show that MET-24 is
indeed not quite JI when it comes to observing these
distinctions:

207.4 207.4 82.0 207.4 150.0 264.8 80.9
203.9 213.6 80.5 203.9 150.6 266.9 80.5
9:8 112:99 22:21 9:8 12:11 7:6 22:21
1/1 9/8 14/11 4/3 3/2 18/11 21/11 2/1
0 203.9 417.5 498.0 702.0 852.6 1119.5 1200
0 207.4 414.8 496.9 704.3 854.3 1119.1 1200
|--------------------|.......|----------------------|
`Ajam tone Hijaz

Again, the JI version has 5 step sizes. In MET-24, however, we
only get 4, because 9:8 and 112:99 are both represented as 207.4
cents. This is actually a tad closer to a just 9/8 than 12n-EDO,
but not very close to 112:99 at 896:891 larger, one of the commas
(an undecimal kleisma) tempered out in MET-24. Curiously, we
could say that the two tempered steps of 207.4 cents might most
accurately represent 44:39 (208.835 cents) and 273:242 (208.673
cents), thus tempering out the 10648/10647! But it's not quite a
representation of the full variety of Erv's permutation.

Now it happens that this last example could be a form of Maqam
Shawq Afza, with a lower `Ajam tetrachord (tone-tone-semitone),
and one type of upper Hijaz (in Arab 24-step notation this would
be 3-5-2, meaning a neutral second, a large step somewhere
between 8:7 and 7:6 or so, and a semitone). And I've learned
something: `Ajam and disjunct Hijaz _is_ a recognized Maqam, as
well as a great permutation even if it weren't!

So one theoretical way to explore might be to sing or play
through the permutations or modulations, and see if they line up
with recognized maqamat, say -- or how they fill in some of the
gaps in the recognized forms. But I know that Erv and colleagues
including yourself have developed some very sophisticated methods
of exploration, so it might be some synthesis of traditional
modal categories, new ones, and techniques for deriving more
modes.

With best New Year's wishes,

Margo

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

>
> Please remind me on the distinction between real and tonal
> modulation. I recall something like this in counterpoint theory
> distinguishing between "real" and "tonal" imitation -- maybe it
> was "real answer" and "tonal answer." But I'd like to understand
> your meaning as it relates to modulation.
Yes this is the same distinction. In an ET it will make not difference unless we we are working with a subset. With say a large constant structure, lets say treating the Partch fabric as a 41 tone structure with two addition, the transposition of the same number of steps throughout the tuning will result in different ratios but all within a certain range of variability. This is would call a tonal transposition as opposed to a real one. Perhaps a better term might be commatic transposition for instead of tempering out commas the constant structure chooses one or the other.
>
> >> "For example, from Ptolemy's 22:12-12:11-7:6, [my
> >> mistake, 22:21 not 22:12 -- M.S.] we get 88:81-12:11-9:8
> >> (143-151-204 cents), a permutation of al-Farabi's
> >> 9:8-12:11-88:81 which could be a modern Arab Bayyati
> >> tetrachord."
>

>
> The curious thing is that 5 step sizes may not be so unusual in
> some of the maqam tunings which often use diverse ajnas
> (genera). To take a relatively "garden variety" example, here's a
> version of Makam Suznak, using some of Ptolemy's ratios, see
> Erv's Purvi modulations, p. 7, Fig. 8a, mode 3.

Thank you for pointing out that a heptatonic with 5 different sizes is not as uncommon as i was implying. I was referring to that many Purvi modulations can produce such scales, but the traditional Persian sources might point that they too found a more varied scale to their liking. There is much to be said about mixed tetrachords in regard to repeating ones. While the latter will have more simple ratios of 4/3 and 3/2 ,It is the lack of them in most places outside the tonic or tonal center of our scale in the former that helps keep the scale centered in the mode being played.

> So one theoretical way to explore might be to sing or play
> through the permutations or modulations, and see if they line up
> with recognized maqamat, say -- or how they fill in some of the
> gaps in the recognized forms. But I know that Erv and colleagues
> including yourself have developed some very sophisticated methods
> of exploration, so it might be some synthesis of traditional
> modal categories, new ones, and techniques for deriving more
> modes.

Erv was one to always draw upon traditional sources for the the development of new material, usually but simple steps not out of place in possibility from the source. It is not much different how the Persians did with the Greek scales themselves. We might say our approach is more of a going deeper into what already exist as opposed to outside of it. Also i would say we are less interested in creating "schism" with the past or with other cultures. On the other hand much has happened around the world, in the west and elsewhere that has broaden our musical expression as I will not surrender how these have enriched our disposition with all due respect to the past. They each stand with an ability to speak and alternate in their influence in a way we might follow forward, knowing we cannot and should not go back.

>
> With best New Year's wishes,
>
> Margo
>

> Yes this is the same distinction. In an ET it will make not
> difference unless we we are working with a subset. With say a
> large constant structure, lets say treating the Partch fabric
> as a 41 tone structure with two addition, the transposition of
> the same number of steps throughout the tuning will result in
> different ratios but all within a certain range of variability.
> This is would call a tonal transposition as opposed to a real
> one. Perhaps a better term might be commatic transposition for
> instead of tempering out commas the constant structure chooses
> one or the other.

Thank you, Kraig, for an explanation that makes things very clear!
Interestingly, a Constant Structure "near-just" temperament like
MET-24 is similar. Here it is possible to divide a 2/1 octave into
17 unequal thirdtones with sizes of 57.4; 68.0 or 69.1; and 80.9
or 82.0 cents. And with the Purvi modulations or a similar system,
we can express a given mode in terms of thirdtone steps, which
could be transposed over the system. Say we start with this mode
from a previous post, and count thirdtone steps:

3 2 2 3 2 4 1
207.4 150.0 139.5 207.4 150.0 264.8 80.9
203.9 150.6 143.5 203.9 150.6 266.9 80.5
9:8 12:11 88:81 9:8 12:11 7:6 22:21
1/1 9/8 27/22 4/3 3/2 18/11 21/11 2/1
0 203.9 354.5 498.0 702.0 852.6 1119.5 1200
0 207.4 357.4 496.9 704.3 854.3 1119.1 1200
|--------------------|.......|----------------------|
Rast tone Hijaz

In terms of thirdtone steps, this becomes:

3 2 2 3 2 4 1
0 3 5 7 10 12 16 17
|--------------------|.......|----------------------|
Rast tone Hijaz

And this pattern of "commatic transposition" could be applied from
other locations "all within a certain range of variability." For
example, with tempered values and one possible JI interpretation:

3 2 2 3 2 4 1
207.4 139.5 150.0 207.4 138.3 276.6 80.9
208.8 138.6 150.6 203.9 138.6 278.9 80.5
44:39 13:12 12:11 9:8 13:12 168:143 22:21
1/1 44/39 11/9 4/3 3/2 13/8 21/11 2/1
0 208.8 346.9 498.0 702.0 840.5 1119.5 1200
0 207.4 347.4 496.9 704.3 842.6 1119.1 1200
|--------------------|.......|----------------------|
Rast tone Hijaz

Either tuning is "3-2-2-3-2-4-1" thirdtones. Now the fact that the
thirdtones are always within a range of a rounded 57.4-82.0 cents
somewhat constrains your "range of variability," as does the fact
that in this tuning set we do not find two adjacent steps of 57.4
cents. Thus a "2-step" interval is categorically a neutral second
in the range of 125.4-162.9 cents, or in JI terms from around
14/13 to 11/10.

> Thank you for pointing out that a heptatonic with 5 different
> sizes is not as uncommon as i was implying. I was referring to
> that many Purvi modulations can produce such scales, but the
> traditional Persian sources might point that they too found a
> more varied scale to their liking. There is much to be said
> about mixed tetrachords in regard to repeating ones. While the
> latter will have more simple ratios of 4/3 and 3/2 ,It is the
> lack of them in most places outside the tonic or tonal center
> of our scale in the former that helps keep the scale centered
> in the mode being played.

This is a kind of creative tension in the "Systematist" theory of
around 1250-1300 developed by Safi al-Din al-Urmawi, Qutb al-Din
al-Shirazi, and others. One criterion for a good mode, at least in
theory, was the degree of "consonance" -- basically, the number of
3/2 and 4/3 intervals you could find. However, asymmetrical
tetrachords are clearly also highly prized.

> Erv was one to always draw upon traditional sources for the the
> development of new material, usually but simple steps not out
> of place in possibility from the source. It is not much
> different how the Persians did with the Greek scales
> themselves. We might say our approach is more of a going deeper
> into what already exist as opposed to outside of it.

The way that the Marwa permutations or Purvi modulations with a
Greek genus like 22:21-12:11-7:6 lead to divisions like the
9:8-12:11-88:81 of al-Farabi is really intriguing -- with the
Persian theorists you discuss of course relevant also.

What Safi al-Din does, for example, is, at a level of tetrachords,
to urge that all six permutations be studied. At a level of
"cycles" or octave modes, he lists the possible pairings of seven
lower tetrachords with 12 upper pentachords, or 84 modes at all,
noting those most important or favored. So the kinds of techniques
we're discussing could be considered indeed "going deeper" into
historical approaches.

> Also i would say we are less interested in creating "schism"
> with the past or with other cultures. On the other hand much has
> happened around the world, in the west and elsewhere that has
> broaden our musical expression as I will not surrender how these
> have enriched our disposition with all due respect to the past.

Actually the Anaphorian approach, if I may call it that, could be
quite in keeping with aspects of the Maqam or Dastgah tradition
(the latter really a specific Persian development in fairly recent
times of the former). In Ottoman circles, for example, I have
heard that there were awards offered to a musician who could
arrive at a pleasing new maqam, with various maqamat attributed to
great composers who thus set precedents for later usage.

What sometimes happens with the Marwa permutations or Purvi
modulations is that, among various easily recognizable maqamat, I
see something "new" -- which, in at least one instance, turned out
to be in fact a recognized maqam!

> They each stand with an ability to speak and alternate in their
> influence in a way we might follow forward, knowing we cannot
> and should not go back.

Of course it is impossible to "go back" by uninventing
developments in practice and theory, but happily possible to seek
some reconciliation of the new and old.

With best New Year's wishes,

Margo