For Cameron: Baglama initiative?

Dear Cameron,

Thank you for posting those measurements for the first
tetrachord of your baglama from Istanbul!

I'm having lots of fun imagining some of the possible
tetrachords and divisions available on your baglama --
That great 0-202-370-498 Rast is a great place to start,
of course. And it's fun to imagine what Ozan might say
about some of my "weirder" ideas!

Since you've already done more than enough measuring,
one approach to the rest of the tuning might be simply
to see which fourths or fifths seem pure or close to it,
and which might be obviously otherwise -- if you were so
inclined, and at some point had the time and energy.

The idea that 3/2-2/1 might divided rather like 1/1-4/3
could be a bit overly simple, and I'm wondering if
looking for aurally obvious 3/2 or 4/3 relationships could
give some quick but helpful clues as to the general
nature of the tuning. I recall that there are 15 notes
per octave.

I'd be curious, for example, whether there's a neutral
sixth step about a 3/2 up from 146 cents (say a 44/27,
845 cents); or maybe about a 4/3 up from that great
370-cent Rast third (say around 33/20 at 867 cents)?

Other questions might be 7/4 or 16/9 (especially
given that 7/6!)? And maybe a 13/7 or so at a 3/2
from 370 cents (which I'll join you in calling a
26/21)? I was picturing something for Rast like:

0 202 370 498 702 906 1072 1200

And I'd be curious if between 4/3 and 3/2 there's
maybe a step around 9/8 above 370 cents, or 574
cents -- or maybe a 4/3 above 92 cents, around
588 or 590 cents?

Again, given all the measurements you've
done, I'd want to make this a question of
"Where are the 4/3 or 3/2 relationships that
are easy to hear?" -- as opposed to more
measurements!

Best,

Margo

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Dear Cameron,
>
> Thank you for posting those measurements for the first
> tetrachord of your baglama from Istanbul!
>
> I'm having lots of fun imagining some of the possible
> tetrachords and divisions available on your baglama --
> That great 0-202-370-498 Rast is a great place to start,
> of course. And it's fun to imagine what Ozan might say
> about some of my "weirder" ideas!
>
> Since you've already done more than enough measuring,
> one approach to the rest of the tuning might be simply
> to see which fourths or fifths seem pure or close to it,
> and which might be obviously otherwise -- if you were so
> inclined, and at some point had the time and energy.
>
> The idea that 3/2-2/1 might divided rather like 1/1-4/3
> could be a bit overly simple, and I'm wondering if
> looking for aurally obvious 3/2 or 4/3 relationships could
> give some quick but helpful clues as to the general
> nature of the tuning. I recall that there are 15 notes
> per octave.

The open strings were tuned to fifths, so the possible tetrachords can all be duplicated in an upper disjunct tetrachord. There was no fret at the 3:2! There was a fret at the Pythagorean M6, the scheme seems to be centered on open strings and at the 9:8. I am quite sure of the key position of the 9:8, as it not only has a fourth and fifth above, but I carefully listened and watched guys playing, and noticed that the thumb wrapped over as a kind of capo at the 9:8 seems to be a standard thing to do.

I have the entire original tuning as a Scala file, but I'm loathe to give more than the frets which fall within the region of the first tetrachord, for above that I did not measure with such intense care.

>
> I'd be curious, for example, whether there's a neutral
> sixth step about a 3/2 up from 146 cents (say a 44/27,
> 845 cents); or maybe about a 4/3 up from that great
> 370-cent Rast third (say around 33/20 at 867 cents)?

The neutral sixth is there on the next string a pure 3:2 above, at
845 cents.
>
> Other questions might be 7/4 or 16/9 (especially
> given that 7/6!)? And maybe a 13/7 or so at a 3/2
> from 370 cents (which I'll join you in calling a
> 26/21)? I was picturing something for Rast like:
>
> 0 202 370 498 702 906 1072 1200

0...202....370....498....702....904....1072....1200
is exactly the Rast, disjunct Rast, on the baglama. When it
descends, as is typical, with a "minor 7th" (flatted second step of upper tetrachord), it is:

1:1, 9:8, 26:21, 4:3, 3:2, 27:16, 7:4, 2:1
>
> And I'd be curious if between 4/3 and 3/2 there's
> maybe a step around 9/8 above 370 cents, or 574
> cents -- or maybe a 4/3 above 92 cents, around
> 588 or 590 cents?

Yes, above the 4:3 there is indeed a fret which corresponds to a whole step above the 26:21 and is very close to 7:5. When I get a chance to double-check it, I will post the data.

>
> Again, given all the measurements you've
> done, I'd want to make this a question of
> "Where are the 4/3 or 3/2 relationships that
> are easy to hear?" -- as opposed to more
> measurements!

All the intervals are reduplicated at a 3:2, giving disjunct tetrachords, and by using the thumb as a capo on the 9:8, you can have either conjunct tetrachords, or the new pentachords made of the intervals between the 9:8 and the 27:16, and combinations thereof.

-Cameron Bobro

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

> > Other questions might be 7/4 or 16/9 (especially
> > given that 7/6!)? And maybe a 13/7 or so at a 3/2
> > from 370 cents (which I'll join you in calling a
> > 26/21)? I was picturing something for Rast like:

Oh, Margo, I forgot to mention that of course we should call the "370 cent" fretting here 26:21. This interval is one found in Ibn Sin's soft diatonic tetrachord, a tetrachordal division which is a stellar example of historical writers most certainly NOT playing number games, being perhaps the most physically simple and eminently practical division of the tetrachord of them all. When we have a contemporary example of a traditional instrument fretted to measurements which have been documented for some thousand years at least, and those measurements all ultimately derive from very simple and practical processes, only numerological superstition would prevent use from using traditional names for the intervals. In your case, this is an especially solid approach, as you temper with an intent to create a system with vertical harmonies, and the traditional name here immediately gives you the primes with which to work, very practical!

-Cameron Bobro

Cameron,

Please note carefully that Ibn Sina categorized the
following tetrachord with ascending interval ratios
8/7, 13/12, 14/13, as Strong Genus Diatonic, No. 4,
in "La Musique Arab," Vol. 2, p. 148. The other two
tetrachord categories he gives are Soft Genus
Chromatic and Soft Genus Enharmonic.

Cris

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
>
> > > Other questions might be 7/4 or 16/9 (especially
> > > given that 7/6!)? And maybe a 13/7 or so at a 3/2
> > > from 370 cents (which I'll join you in calling a
> > > 26/21)? I was picturing something for Rast like:
>
> Oh, Margo, I forgot to mention that of course we should call the "370 cent" fretting here 26:21. This interval is one found in Ibn Sin's soft diatonic tetrachord, a tetrachordal division which is a stellar example of historical writers most certainly NOT playing number games, being perhaps the most physically simple and eminently practical division of the tetrachord of them all. When we have a contemporary example of a traditional instrument fretted to measurements which have been documented for some thousand years at least, and those measurements all ultimately derive from very simple and practical processes, only numerological superstition would prevent use from using traditional names for the intervals. In your case, this is an especially solid approach, as you temper with an intent to create a system with vertical harmonies, and the traditional name here immediately gives you the primes with which to work, very practical!
>
> -Cameron Bobro
>

Thanks, Cris- I should have checked rather than go from memory. And I need to get my own copy of your book!

I know the "orthodox" order of the tetrachordal intervals of the ancient Greeks, but it's not clear to me what the equivalent would be for the medieval Islamic musicians- ascending, descending? And in a fair bit of reading on ancient tetrachordal music over the last few years, I haven't noticed anything giving a clear picture as to what extent the intervals of tetrachordal division were freely (or not) re-ordered in practice by the ancients and medievals. This question is of special interest to me as I use the enharmonic of Eratosthenes, which is definitely "micro and macro"tonal, and the ordering of the intervals makes a huge difference in the feeling of it. Since I tend to prefer conjuct tetrachords to disjunct these days, obviously different orderings can make for some very bunched-and-gapped resulting "scales". Which is great, I just wonder if the ancients did such things in actual practice.

At any rate, the 26/21 is right there in the Strong Genus Diatonic,
and the ultimate origin of the tetrachordal division could hardly have been anything other than the extremely simple process of dividing string lengths which is spelled within it. So I think it is the more reasonable as well as richer choice, as opposed to "370 cents". When I measured the song of a particular cuckoo, I called the result "369 cents", but 26/21 is better for describing the fretting of a maqam instrument.

-Cameron Bobro

--- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@...> wrote:
>
> Cameron,
>
> Please note carefully that Ibn Sina categorized the
> following tetrachord with ascending interval ratios
> 8/7, 13/12, 14/13, as Strong Genus Diatonic, No. 4,
> in "La Musique Arab," Vol. 2, p. 148. The other two
> tetrachord categories he gives are Soft Genus
> Chromatic and Soft Genus Enharmonic.
>
> Cris
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> >
> > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> > > > Other questions might be 7/4 or 16/9 (especially
> > > > given that 7/6!)? And maybe a 13/7 or so at a 3/2
> > > > from 370 cents (which I'll join you in calling a
> > > > 26/21)? I was picturing something for Rast like:
> >
> > Oh, Margo, I forgot to mention that of course we should call the "370 cent" fretting here 26:21. This interval is one found in Ibn Sin's soft diatonic tetrachord, a tetrachordal division which is a stellar example of historical writers most certainly NOT playing number games, being perhaps the most physically simple and eminently practical division of the tetrachord of them all. When we have a contemporary example of a traditional instrument fretted to measurements which have been documented for some thousand years at least, and those measurements all ultimately derive from very simple and practical processes, only numerological superstition would prevent use from using traditional names for the intervals. In your case, this is an especially solid approach, as you temper with an intent to create a system with vertical harmonies, and the traditional name here immediately gives you the primes with which to work, very practical!
> >
> > -Cameron Bobro
> >
>

You're welcome, Cameron. Forgive me, but I refer
to the time roughly between the life of Al-Kindi (d. c.
874) and Safi Al-Din (d. 1294) as the Arabian
Renaissance. In the Middle East, this time span is
at least as historically profound as the Italian
Renaissance.

In general, the Greeks always described their
tetrachords in descending order. The largest
interval descended first. The smallest interval
descended last. Consequently, if one looks at such
a progression of intervals in ascending order, then
the smallest interval ratio is the first interval.

Al-Farabi, in his description of eight ajnas, or eight
separate jins, intentionally inverted the Greek order,
so that when one looks at the progression of
intervals of his tetrachords in ascending order, then
the largest interval ratio is the first interval.

From my text:

******************************

Because Al-Farabi systematically inverted the
intervalic order of all Greek tetrachords, his
interpretations on the fretted `ud profoundly
influenced countless musicians in future
generations. Indeed, the tendency to place the
largest intervals first, and the smallest intervals last,
is very familiar to Western musicians.

******************************

Cris

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> Thanks, Cris- I should have checked rather than go from memory. And I need to get my own copy of your book!
>
> I know the "orthodox" order of the tetrachordal intervals of the ancient Greeks, but it's not clear to me what the equivalent would be for the medieval Islamic musicians- ascending, descending? And in a fair bit of reading on ancient tetrachordal music over the last few years, I haven't noticed anything giving a clear picture as to what extent the intervals of tetrachordal division were freely (or not) re-ordered in practice by the ancients and medievals. This question is of special interest to me as I use the enharmonic of Eratosthenes, which is definitely "micro and macro"tonal, and the ordering of the intervals makes a huge difference in the feeling of it. Since I tend to prefer conjuct tetrachords to disjunct these days, obviously different orderings can make for some very bunched-and-gapped resulting "scales". Which is great, I just wonder if the ancients did such things in actual practice.
>
> At any rate, the 26/21 is right there in the Strong Genus Diatonic,
> and the ultimate origin of the tetrachordal division could hardly have been anything other than the extremely simple process of dividing string lengths which is spelled within it. So I think it is the more reasonable as well as richer choice, as opposed to "370 cents". When I measured the song of a particular cuckoo, I called the result "369 cents", but 26/21 is better for describing the fretting of a maqam instrument.
>
> -Cameron Bobro
>
>
> --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> >
> > Cameron,
> >
> > Please note carefully that Ibn Sina categorized the
> > following tetrachord with ascending interval ratios
> > 8/7, 13/12, 14/13, as Strong Genus Diatonic, No. 4,
> > in "La Musique Arab," Vol. 2, p. 148. The other two
> > tetrachord categories he gives are Soft Genus
> > Chromatic and Soft Genus Enharmonic.
> >
> > Cris
> >
> >
> > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > >
> > >
> > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > >
> > > > > Other questions might be 7/4 or 16/9 (especially
> > > > > given that 7/6!)? And maybe a 13/7 or so at a 3/2
> > > > > from 370 cents (which I'll join you in calling a
> > > > > 26/21)? I was picturing something for Rast like:
> > >
> > > Oh, Margo, I forgot to mention that of course we should call the "370 cent" fretting here 26:21. This interval is one found in Ibn Sin's soft diatonic tetrachord, a tetrachordal division which is a stellar example of historical writers most certainly NOT playing number games, being perhaps the most physically simple and eminently practical division of the tetrachord of them all. When we have a contemporary example of a traditional instrument fretted to measurements which have been documented for some thousand years at least, and those measurements all ultimately derive from very simple and practical processes, only numerological superstition would prevent use from using traditional names for the intervals. In your case, this is an especially solid approach, as you temper with an intent to create a system with vertical harmonies, and the traditional name here immediately gives you the primes with which to work, very practical!
> > >
> > > -Cameron Bobro
> > >
> >
>

It is my understanding of which I apologize for not being able to come up with the source of this, But i understand the Greeks thought of the scales as descending because of it descending from heaven or the heavens.
I think you mentioned at some point years back that this is what caused Boethius confusion over the names of the modes in Plato.
It sounds hear that it is Al-Farabi might have been the first to think of it the opposite, or do you think this is just the West interpreted his permutations this way.

--- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@...> wrote:
>
> You're welcome, Cameron. Forgive me, but I refer
> to the time roughly between the life of Al-Kindi (d. c.
> 874) and Safi Al-Din (d. 1294) as the Arabian
> Renaissance. In the Middle East, this time span is
> at least as historically profound as the Italian
> Renaissance.
>
> In general, the Greeks always described their
> tetrachords in descending order. The largest
> interval descended first. The smallest interval
> descended last. Consequently, if one looks at such
> a progression of intervals in ascending order, then
> the smallest interval ratio is the first interval.
>
> Al-Farabi, in his description of eight ajnas, or eight
> separate jins, intentionally inverted the Greek order,
> so that when one looks at the progression of
> intervals of his tetrachords in ascending order, then
> the largest interval ratio is the first interval.
>
> From my text:
>
> ******************************
>
> Because Al-Farabi systematically inverted the
> intervalic order of all Greek tetrachords, his
> interpretations on the fretted `ud profoundly
> influenced countless musicians in future
> generations. Indeed, the tendency to place the
> largest intervals first, and the smallest intervals last,
> is very familiar to Western musicians.
>
> ******************************
>
> Cris
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> > Thanks, Cris- I should have checked rather than go from memory. And I need to get my own copy of your book!
> >
> > I know the "orthodox" order of the tetrachordal intervals of the ancient Greeks, but it's not clear to me what the equivalent would be for the medieval Islamic musicians- ascending, descending? And in a fair bit of reading on ancient tetrachordal music over the last few years, I haven't noticed anything giving a clear picture as to what extent the intervals of tetrachordal division were freely (or not) re-ordered in practice by the ancients and medievals. This question is of special interest to me as I use the enharmonic of Eratosthenes, which is definitely "micro and macro"tonal, and the ordering of the intervals makes a huge difference in the feeling of it. Since I tend to prefer conjuct tetrachords to disjunct these days, obviously different orderings can make for some very bunched-and-gapped resulting "scales". Which is great, I just wonder if the ancients did such things in actual practice.
> >
> > At any rate, the 26/21 is right there in the Strong Genus Diatonic,
> > and the ultimate origin of the tetrachordal division could hardly have been anything other than the extremely simple process of dividing string lengths which is spelled within it. So I think it is the more reasonable as well as richer choice, as opposed to "370 cents". When I measured the song of a particular cuckoo, I called the result "369 cents", but 26/21 is better for describing the fretting of a maqam instrument.
> >
> > -Cameron Bobro
> >
> >
> > --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> > >
> > > Cameron,
> > >
> > > Please note carefully that Ibn Sina categorized the
> > > following tetrachord with ascending interval ratios
> > > 8/7, 13/12, 14/13, as Strong Genus Diatonic, No. 4,
> > > in "La Musique Arab," Vol. 2, p. 148. The other two
> > > tetrachord categories he gives are Soft Genus
> > > Chromatic and Soft Genus Enharmonic.
> > >
> > > Cris
> > >
> > >
> > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > >
> > > >
> > > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > >
> > > > > > Other questions might be 7/4 or 16/9 (especially
> > > > > > given that 7/6!)? And maybe a 13/7 or so at a 3/2
> > > > > > from 370 cents (which I'll join you in calling a
> > > > > > 26/21)? I was picturing something for Rast like:
> > > >
> > > > Oh, Margo, I forgot to mention that of course we should call the "370 cent" fretting here 26:21. This interval is one found in Ibn Sin's soft diatonic tetrachord, a tetrachordal division which is a stellar example of historical writers most certainly NOT playing number games, being perhaps the most physically simple and eminently practical division of the tetrachord of them all. When we have a contemporary example of a traditional instrument fretted to measurements which have been documented for some thousand years at least, and those measurements all ultimately derive from very simple and practical processes, only numerological superstition would prevent use from using traditional names for the intervals. In your case, this is an especially solid approach, as you temper with an intent to create a system with vertical harmonies, and the traditional name here immediately gives you the primes with which to work, very practical!
> > > >
> > > > -Cameron Bobro
> > > >
> > >
> >
>

Hi Kraig,

I believe there are demonstrable reasons why the
Greeks described their tetrachord divisions in
descending order.

In constructing a tetrachord on a canon with four
strings, the tunings of two strings are
predetermined: one open string sounds the
fundamental, or lowest tone, ratio 1/1; and one
bridged string sounds the highest tone, ratio 4/3.
The question remains where one places bridges
under the two remaining strings. In the case of
Ptolemy's Even Diatonic tetrachord, he first tuned
the largest interval, or the characteristic interval,
down from 4/3 by placing a bridge to locate ratio
6/5. If all the strings have an overall length of
1000.0 mm, then 4/3 would be located at 750.0
mm, and 6/5 at 833.33 mm. Now, if one measures
the short remaining string length of the 6/5 string, or
the distance on the other side of the 6/5 bridge, it
has a length of 166.67 mm. If one then divides this
short length in half one gets 83.33. Now because
833.33 mm + 83.33 mm = 916.66 mm, string length
ratio 1000.00 mm / 916.66 mm simplifies to ratio
12/11 = 150.64 cents. So, now in Greek ascending
order, Ptolemy's Even Diatonic consists of interval
ratios 12/11, 11/10, 10/9; or of frequency ratios 1/1,
12/11, 6/5, 4/3.

So, by initially descending from 4/3 with the largest
and therefore the characteristic interval of the
tetrachord, and then by directly dividing the
remaining string length section in half (or to
whatever proportion one likes), this first step and
the second step determine a descending direction
of ratios.

The canon is a great scientific tool, but because of
the rattling bridge problem, it never became a
widely accepted musical instrument. The Arabian
`ud on the other hand, although just as accurate as
a Greek canon, did evolve into a supreme musical
instrument. The practicality of its moveable frets
(like moveable canon bridges) caused musicians
like Al-Farabi to rethink the process of constructing
tetrachords with descending interval ratios. As a
musical instrument, the `ud simply rendered the
Greek method obsolete. Because all the stopped
(or bridged) tones (of a double-octave scale) fell
under the grasp of a single human hand, the `ud
inspired spontaneous explorations of tetrachord
divisions in the more obviously musical direction of
ascending interval ratios.

Cris

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
>
> It is my understanding of which I apologize for not being able to come up with the source of this, But i understand the Greeks thought of the scales as descending because of it descending from heaven or the heavens.
> I think you mentioned at some point years back that this is what caused Boethius confusion over the names of the modes in Plato.
> It sounds hear that it is Al-Farabi might have been the first to think of it the opposite, or do you think this is just the West interpreted his permutations this way.
>
> --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> >
> > You're welcome, Cameron. Forgive me, but I refer
> > to the time roughly between the life of Al-Kindi (d. c.
> > 874) and Safi Al-Din (d. 1294) as the Arabian
> > Renaissance. In the Middle East, this time span is
> > at least as historically profound as the Italian
> > Renaissance.
> >
> > In general, the Greeks always described their
> > tetrachords in descending order. The largest
> > interval descended first. The smallest interval
> > descended last. Consequently, if one looks at such
> > a progression of intervals in ascending order, then
> > the smallest interval ratio is the first interval.
> >
> > Al-Farabi, in his description of eight ajnas, or eight
> > separate jins, intentionally inverted the Greek order,
> > so that when one looks at the progression of
> > intervals of his tetrachords in ascending order, then
> > the largest interval ratio is the first interval.
> >
> > From my text:
> >
> > ******************************
> >
> > Because Al-Farabi systematically inverted the
> > intervalic order of all Greek tetrachords, his
> > interpretations on the fretted `ud profoundly
> > influenced countless musicians in future
> > generations. Indeed, the tendency to place the
> > largest intervals first, and the smallest intervals last,
> > is very familiar to Western musicians.
> >
> > ******************************
> >
> > Cris
> >
> >
> > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > >
> > > Thanks, Cris- I should have checked rather than go from memory. And I need to get my own copy of your book!
> > >
> > > I know the "orthodox" order of the tetrachordal intervals of the ancient Greeks, but it's not clear to me what the equivalent would be for the medieval Islamic musicians- ascending, descending? And in a fair bit of reading on ancient tetrachordal music over the last few years, I haven't noticed anything giving a clear picture as to what extent the intervals of tetrachordal division were freely (or not) re-ordered in practice by the ancients and medievals. This question is of special interest to me as I use the enharmonic of Eratosthenes, which is definitely "micro and macro"tonal, and the ordering of the intervals makes a huge difference in the feeling of it. Since I tend to prefer conjuct tetrachords to disjunct these days, obviously different orderings can make for some very bunched-and-gapped resulting "scales". Which is great, I just wonder if the ancients did such things in actual practice.
> > >
> > > At any rate, the 26/21 is right there in the Strong Genus Diatonic,
> > > and the ultimate origin of the tetrachordal division could hardly have been anything other than the extremely simple process of dividing string lengths which is spelled within it. So I think it is the more reasonable as well as richer choice, as opposed to "370 cents". When I measured the song of a particular cuckoo, I called the result "369 cents", but 26/21 is better for describing the fretting of a maqam instrument.
> > >
> > > -Cameron Bobro
> > >
> > >
> > > --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> > > >
> > > > Cameron,
> > > >
> > > > Please note carefully that Ibn Sina categorized the
> > > > following tetrachord with ascending interval ratios
> > > > 8/7, 13/12, 14/13, as Strong Genus Diatonic, No. 4,
> > > > in "La Musique Arab," Vol. 2, p. 148. The other two
> > > > tetrachord categories he gives are Soft Genus
> > > > Chromatic and Soft Genus Enharmonic.
> > > >
> > > > Cris
> > > >
> > > >
> > > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > > >
> > > > >
> > > > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > > >
> > > > > > > Other questions might be 7/4 or 16/9 (especially
> > > > > > > given that 7/6!)? And maybe a 13/7 or so at a 3/2
> > > > > > > from 370 cents (which I'll join you in calling a
> > > > > > > 26/21)? I was picturing something for Rast like:
> > > > >
> > > > > Oh, Margo, I forgot to mention that of course we should call the "370 cent" fretting here 26:21. This interval is one found in Ibn Sin's soft diatonic tetrachord, a tetrachordal division which is a stellar example of historical writers most certainly NOT playing number games, being perhaps the most physically simple and eminently practical division of the tetrachord of them all. When we have a contemporary example of a traditional instrument fretted to measurements which have been documented for some thousand years at least, and those measurements all ultimately derive from very simple and practical processes, only numerological superstition would prevent use from using traditional names for the intervals. In your case, this is an especially solid approach, as you temper with an intent to create a system with vertical harmonies, and the traditional name here immediately gives you the primes with which to work, very practical!
> > > > >
> > > > > -Cameron Bobro
> > > > >
> > > >
> > >
> >
>

Thanks Cris~
a practical explanation that requires no metaphyscial one:)

--- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@...> wrote:
>
> Hi Kraig,
>
> I believe there are demonstrable reasons why the
> Greeks described their tetrachord divisions in
> descending order.
>
> In constructing a tetrachord on a canon with four
> strings, the tunings of two strings are
> predetermined: one open string sounds the
> fundamental, or lowest tone, ratio 1/1; and one
> bridged string sounds the highest tone, ratio 4/3.
> The question remains where one places bridges
> under the two remaining strings. In the case of
> Ptolemy's Even Diatonic tetrachord, he first tuned
> the largest interval, or the characteristic interval,
> down from 4/3 by placing a bridge to locate ratio
> 6/5. If all the strings have an overall length of
> 1000.0 mm, then 4/3 would be located at 750.0
> mm, and 6/5 at 833.33 mm. Now, if one measures
> the short remaining string length of the 6/5 string, or
> the distance on the other side of the 6/5 bridge, it
> has a length of 166.67 mm. If one then divides this
> short length in half one gets 83.33. Now because
> 833.33 mm + 83.33 mm = 916.66 mm, string length
> ratio 1000.00 mm / 916.66 mm simplifies to ratio
> 12/11 = 150.64 cents. So, now in Greek ascending
> order, Ptolemy's Even Diatonic consists of interval
> ratios 12/11, 11/10, 10/9; or of frequency ratios 1/1,
> 12/11, 6/5, 4/3.
>
> So, by initially descending from 4/3 with the largest
> and therefore the characteristic interval of the
> tetrachord, and then by directly dividing the
> remaining string length section in half (or to
> whatever proportion one likes), this first step and
> the second step determine a descending direction
> of ratios.
>
> The canon is a great scientific tool, but because of
> the rattling bridge problem, it never became a
> widely accepted musical instrument. The Arabian
> `ud on the other hand, although just as accurate as
> a Greek canon, did evolve into a supreme musical
> instrument. The practicality of its moveable frets
> (like moveable canon bridges) caused musicians
> like Al-Farabi to rethink the process of constructing
> tetrachords with descending interval ratios. As a
> musical instrument, the `ud simply rendered the
> Greek method obsolete. Because all the stopped
> (or bridged) tones (of a double-octave scale) fell
> under the grasp of a single human hand, the `ud
> inspired spontaneous explorations of tetrachord
> divisions in the more obviously musical direction of
> ascending interval ratios.
>
> Cris
>
>
> --- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@> wrote:
> >
> > It is my understanding of which I apologize for not being able to come up with the source of this, But i understand the Greeks thought of the scales as descending because of it descending from heaven or the heavens.
> > I think you mentioned at some point years back that this is what caused Boethius confusion over the names of the modes in Plato.
> > It sounds hear that it is Al-Farabi might have been the first to think of it the opposite, or do you think this is just the West interpreted his permutations this way.
> >
> > --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> > >
> > > You're welcome, Cameron. Forgive me, but I refer
> > > to the time roughly between the life of Al-Kindi (d. c.
> > > 874) and Safi Al-Din (d. 1294) as the Arabian
> > > Renaissance. In the Middle East, this time span is
> > > at least as historically profound as the Italian
> > > Renaissance.
> > >
> > > In general, the Greeks always described their
> > > tetrachords in descending order. The largest
> > > interval descended first. The smallest interval
> > > descended last. Consequently, if one looks at such
> > > a progression of intervals in ascending order, then
> > > the smallest interval ratio is the first interval.
> > >
> > > Al-Farabi, in his description of eight ajnas, or eight
> > > separate jins, intentionally inverted the Greek order,
> > > so that when one looks at the progression of
> > > intervals of his tetrachords in ascending order, then
> > > the largest interval ratio is the first interval.
> > >
> > > From my text:
> > >
> > > ******************************
> > >
> > > Because Al-Farabi systematically inverted the
> > > intervalic order of all Greek tetrachords, his
> > > interpretations on the fretted `ud profoundly
> > > influenced countless musicians in future
> > > generations. Indeed, the tendency to place the
> > > largest intervals first, and the smallest intervals last,
> > > is very familiar to Western musicians.
> > >
> > > ******************************
> > >
> > > Cris
> > >
> > >
> > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > >
> > > > Thanks, Cris- I should have checked rather than go from memory. And I need to get my own copy of your book!
> > > >
> > > > I know the "orthodox" order of the tetrachordal intervals of the ancient Greeks, but it's not clear to me what the equivalent would be for the medieval Islamic musicians- ascending, descending? And in a fair bit of reading on ancient tetrachordal music over the last few years, I haven't noticed anything giving a clear picture as to what extent the intervals of tetrachordal division were freely (or not) re-ordered in practice by the ancients and medievals. This question is of special interest to me as I use the enharmonic of Eratosthenes, which is definitely "micro and macro"tonal, and the ordering of the intervals makes a huge difference in the feeling of it. Since I tend to prefer conjuct tetrachords to disjunct these days, obviously different orderings can make for some very bunched-and-gapped resulting "scales". Which is great, I just wonder if the ancients did such things in actual practice.
> > > >
> > > > At any rate, the 26/21 is right there in the Strong Genus Diatonic,
> > > > and the ultimate origin of the tetrachordal division could hardly have been anything other than the extremely simple process of dividing string lengths which is spelled within it. So I think it is the more reasonable as well as richer choice, as opposed to "370 cents". When I measured the song of a particular cuckoo, I called the result "369 cents", but 26/21 is better for describing the fretting of a maqam instrument.
> > > >
> > > > -Cameron Bobro
> > > >
> > > >
> > > > --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> > > > >
> > > > > Cameron,
> > > > >
> > > > > Please note carefully that Ibn Sina categorized the
> > > > > following tetrachord with ascending interval ratios
> > > > > 8/7, 13/12, 14/13, as Strong Genus Diatonic, No. 4,
> > > > > in "La Musique Arab," Vol. 2, p. 148. The other two
> > > > > tetrachord categories he gives are Soft Genus
> > > > > Chromatic and Soft Genus Enharmonic.
> > > > >
> > > > > Cris
> > > > >
> > > > >
> > > > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > > > >
> > > > > >
> > > > > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > > > >
> > > > > > > > Other questions might be 7/4 or 16/9 (especially
> > > > > > > > given that 7/6!)? And maybe a 13/7 or so at a 3/2
> > > > > > > > from 370 cents (which I'll join you in calling a
> > > > > > > > 26/21)? I was picturing something for Rast like:
> > > > > >
> > > > > > Oh, Margo, I forgot to mention that of course we should call the "370 cent" fretting here 26:21. This interval is one found in Ibn Sin's soft diatonic tetrachord, a tetrachordal division which is a stellar example of historical writers most certainly NOT playing number games, being perhaps the most physically simple and eminently practical division of the tetrachord of them all. When we have a contemporary example of a traditional instrument fretted to measurements which have been documented for some thousand years at least, and those measurements all ultimately derive from very simple and practical processes, only numerological superstition would prevent use from using traditional names for the intervals. In your case, this is an especially solid approach, as you temper with an intent to create a system with vertical harmonies, and the traditional name here immediately gives you the primes with which to work, very practical!
> > > > > >
> > > > > > -Cameron Bobro
> > > > > >
> > > > >
> > > >
> > >
> >
>

"cameron" <misterbobro@...> wrote:

> At any rate, the 26/21 is right there in the Strong Genus
> Diatonic, and the ultimate origin of the tetrachordal
> division could hardly have been anything other than the
> extremely simple process of dividing string lengths which
> is spelled within it. So I think it is the more
> reasonable as well as richer choice, as opposed to "370
> cents". When I measured the song of a particular cuckoo,
> I called the result "369 cents", but 26/21 is better for
> describing the fretting of a maqam instrument.

I don't know how cuckoos got into this thread, but Fokker
also mentions them here:

http://www.huygens-fokker.org/docs/fokkerorg.html

"Again, the fifth may be divided into two equal parts. The
result is a pair of 'thirds' intermediate between minor and
major thirds. These intermediate thirds make a very
beautiful rendering of the cuckoo's call. In human
civilization they are used in music of the Arabs according
to the teaching of the famous medieval musician Zalzal."

Graham

Heh, I didn't know Fokker made that observation- I talked out about the "cuckoo thirds" some time ago here, and Andreas Sparschuh pointed out that the descending third of each bird gets lower as the weather gets colder. After he pointed this out, I followed the calls of the two birds who lived near me at the time and found it to be true of them at least- the interval gets darker as the year gets colder.

The recording I was able to measure most precisely (can't remember if it was my recording or from somewhere else, there's a series of bird, insect, amphibian calls of Slovenia from the national museum) had an admirably consistent call of 369 cents. The autumn call outside my house was about an 11/9.

The cuckoo came up because of the natural association with the Zalzalian intervals in question, very groovy that Fokker made the comparison, and because I wanted to point out that calling 370 cents on a maqam instrument 26/21 is reasonable, given the historical background, while it would be misleading to imply any such rational intention in the call of a cuckoo (though if you were intending to incorporate the interval of that cuckoo call into a temperament, it would great to call it a 26/21, as you'd have your primes to temper right there).

-Cameron Bobro

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> "cameron" <misterbobro@...> wrote:
>
> > At any rate, the 26/21 is right there in the Strong Genus
> > Diatonic, and the ultimate origin of the tetrachordal
> > division could hardly have been anything other than the
> > extremely simple process of dividing string lengths which
> > is spelled within it. So I think it is the more
> > reasonable as well as richer choice, as opposed to "370
> > cents". When I measured the song of a particular cuckoo,
> > I called the result "369 cents", but 26/21 is better for
> > describing the fretting of a maqam instrument.
>
> I don't know how cuckoos got into this thread, but Fokker
> also mentions them here:
>
> http://www.huygens-fokker.org/docs/fokkerorg.html
>
> "Again, the fifth may be divided into two equal parts. The
> result is a pair of 'thirds' intermediate between minor and
> major thirds. These intermediate thirds make a very
> beautiful rendering of the cuckoo's call. In human
> civilization they are used in music of the Arabs according
> to the teaching of the famous medieval musician Zalzal."
>
>
> Graham
>

Hi, Cameron, and thanks to you and Cris for all the inforamtion and
ideas in these posts, which I'm just getting caught up with now!

> The open strings were tuned to fifths, so the possible
> tetrachords can all be duplicated in an upper disjunct
> tetrachord. There was no fret at the 3:2! There was a fret at
> the Pythagorean M6, the scheme seems to be centered on open
> strings and at the 9:8. I am quite sure of the key position of
> the 9:8, as it not only has a fourth and fifth above, but I
> carefully listened and watched guys playing, and noticed that
> the thumb wrapped over as a kind of capo at the 9:8 seems to be
> a standard thing to do.

Thanks for giving me exactly what I was asking: the "upper
disjunct tetrachord" idea is a very helpful answer, and gives me
some ideas about Persian as well as Turkish tetrachords and modes
that might be available.

> I have the entire original tuning as a Scala file, but I'm
> loathe to give more than the frets which fall within the region
> of the first tetrachord, for above that I did not measure with
> such intense care.

Yes, I understand, when giving concrete figures in a Scala file,
you want be as accurate as possible.

However, for my purposes, the information you give is perfect.
I'll respond to some points in this post, and then write another
article to respond to you and Cris on Ibn Sina's diatonic with
septimal ratios (steps of 8:7-13:12-14:13).

>
>> I'd be curious, for example, whether there's a neutral
>> sixth step about a 3/2 up from 146 cents (say a 44/27,
>> 845 cents); or maybe about a 4/3 up from that great
>> 370-cent Rast third (say around 33/20 at 867 cents)?

> The neutral sixth is there on the next string a pure 3:2 above,
> at 845 cents.

Ah, that reminds me of the one step out of the 15 per octave I
might not have asked about: is there a step at around a 4:3 above
146 cents, which would also be a 9:8 or so below 845 cents, say
around 352/243 or 642 cents?

>> Other questions might be 7/4 or 16/9 (especially given that
>> 7/6!)? And maybe a 13/7 or so at a 3/2 > from 370 cents (which
>> I'll join you in calling a > 26/21)? I was picturing something
>> for Rast like: > > 0 202 370 498 702 906 1072 1200

> 0...202....370....498....702....904....1072....1200
> is exactly the Rast, disjunct Rast, on the baglama. When it
> descends, as is typical, with a "minor 7th" (flatted second step of upper
> tetrachord), it is:

> 1:1, 9:8, 26:21, 4:3, 3:2, 27:16, 7:4, 2:1

This is really interesting, and helps me emulate your instrument
(more or less) in O3. For the ascending disjunct Rast, we have:

baglama: 0 202 370 498 702 906 1072 1200
O3: 0 207 369 496 703 912 1073 1200
rast dugah segah chargah neva huseyni evdj gerdaniye

For a typical descending from, which Ozan calls an "acemli rast"
because of the minor seventh step acem, and using your suggested
ratios on which the baglama has very small variations:

1/1 9/8 26/21 4/3 3/2 27/16 7/4 2/1
JI: 0 204 370 498 702 906 969 1200
O3: 0 207 369 496 703 912 969 1200
rast dugah segah chargah neva huseyni acem gerdaniue

A fine point which may tie in with some of Ozan's comments is
that we have evdj, the high neutral seventh at 13/7, and acem,
the minor seventh at 7/4, at around 103 or 104 cents apart --
notably more than a Pythagorean limma at 90 cents, let alone the
usual limma in O3 at 22/21 or 81 cents!

Similarly, and going directly to Ozan's point, we can consider
the 7/6 step as kurdi, or a minor third step; and 26/21 as segah,
the middle third, again with a distance of a bit over 100
cents. One JI ratio might be 52/49 (103 cents), the difference
between 7/6 and 26/21; in O3, it's about 104 cents.

Ozan's point was that if segah is in its high Turkish position a
few cents below 5/4 (say 380 cents), then kurdi might be around
13/11 (289 cents), as in one arrangement found in his 79-MOS; but
if segah is lower, around 26/21, then kurdi would normally be
lower as well, here 7/6. Your baglama is a good illustration!

Another fine point is that in the baglama tuning, that small
semitone step in the version of Rast with a minor seventh,
27/16-7/4, is very close to a just 28/27 or 63 cents. In O3,
because the major sixth is a bit wider at around 22/13, we have a
step of around 22/13-7/4, yet narrower at around 57 cents, which
could be called a 91/88. This is, of course, one effect of the
temperament, with the baglama's step very near the classical
28:27 of Archytas probably just about optimal.

>> And I'd be curious if between 4/3 and 3/2 there's maybe a step
>> around 9/8 above 370 cents, or 574 cents -- or maybe a 4/3
>> above 92 cents, around 588 or 590 cents?

> Yes, above the 4:3 there is indeed a fret which corresponds to
> a whole step above the 26:21 and is very close to 7:5. When I
> get a chance to double-check it, I will post the data.

It sounds like you have a great Maqam Penchgah, then -- basically
a form of Rast which often uses that raised fourth, which Ozan
ideally would tune at 7:5 or so. In O3 it's around 577 cents, and
indeed not too far from 7:5.

>> Again, given all the measurements you've done, I'd want to
>> make this a question of "Where are the 4/3 or 3/2
>> relationships that > are easy to hear?" -- as opposed to more
>> > measurements!

> All the intervals are reduplicated at a 3:2, giving disjunct
> tetrachords, and by using the thumb as a capo on the 9:8, you
> can have either conjunct tetrachords, or the new pentachords
> made of the intervals between the 9:8 and the 27:16, and
> combinations thereof.

A few quick ideas occur to me, now that the only step I might
want to confirm is that possible 642 cents or so.

The first is an interesting form of Hijazkar on rast which I
often use in O3, although I'm not sure what Ozan would say:

baglama: 0 92 370 498 702 ~792 1072 1200
O3: 0 81 369 496 703 784 1073 1200

The second is that your baglama should have a great Maqam Huseyni
that I think Ozan might really appreciate, starting on the 9/8:

baglama: 0 168 296 ~498 ~702 ~870 ~998 1200
O3: 0 162 288 496 704 866 993 1200
steps: 9/8 26/21 4/3 3/2 27/16 13/7 2/1 9/4

Here the sixth degree is sometimes lowered, for example in
descending, we have an alternative form of:

baglama: 0 168 296 ~498 ~702 ~792 ~998 1200
O3: 0 162 288 496 704 785 993 1200
steps: 9/8 26/21 4/3 3/2 27/16 16/9 2/1 9/4

Also, there's a Maqam Kurdi, or actually Hijazkar Kurdi, on rast
(Kurdi would be on the 9/8, dugah) with the low minor third and
seventh:

baglama: 0 92 269 498 702 ~792 ~969 1200
O3: 0 81 265 496 703 784 969 1200

Stating the baglama tuning in 53-commas, we have an interesting
pattern: 4-8-10-9-4-8-10.

> Oh, Margo, I forgot to mention that of course we should call
> the "370 cent" fretting here 26:21. This interval is one found
> in Ibn Sin's soft diatonic tetrachord, a tetrachordal division
> which is a stellar example of historical writers most certainly
> NOT playing number games, being perhaps the most physically
> simple and eminently practical division of the tetrachord of
> them all.

An interesting point that Cris clarifies, and I'll address more
in another post, is that Ibn Sina's tuning often referred to as
his "soft diatonic," as in the Scala archive (avicenna_diat.scl),
is actually 1/1-8/7-26/21-4/3, making the 26/21 all the more
important.

> When we have a contemporary example of a traditional
> instrument fretted to measurements which have been documented
> for some thousand years at least, and those measurements all
> ultimately derive from very simple and practical processes,
> only numerological superstition would prevent use from using
> traditional names for the intervals.

Yes, sometimes taking the theorists seriously and drawing the
obvious connections is the most parsimonious way to slice with
Occam's Razor. The devaluation of medieval Near Eastern and
European theory can interfere with the empirical tests which
anti-medievalists are so often holding up as an ideal.

> In your case, this is an especially solid approach, as you
> temper with an intent to create a system with vertical
> harmonies, and the traditional name here immediately gives you
> the primes with which to work, very practical!

That's true, with the obvious caution that something like Maqam
Penchgah is beautiful and melodically compelling quite apart from
any polyphonic context. For example, in O3, we have a two-voice
progression like this, which might be available on your baglama
if there were some way of double-stopping or whatever so to
combine the two relevant steps:

577 703 ~574? 702
207 0 or on the baglama(?) 202 0

Here we have a 26/21 expanding to a 3/2, which can sound like an
interesting variation on a two-voice setting by the 14th-century
European composer Guillaume de Machaut. It's very effective
because a 26/21 does sound like a small or Zalzalian ditone, not
exactly the same as an 81/64 or the like, but somehow "in a
similar category."

And in three voices:

1073 1200 1072 1200
577 703 ~574? 702
207 0 or on the baglama(?) 202 0

> -Cameron Bobro

Best,

Margo

Hello, Cris and Cameron.

Cris wrote:

> Cameron,

> Please note carefully that Ibn Sina categorized the following
> tetrachord with ascending interval ratios 8/7, 13/12, 14/13, as
> Strong Genus Diatonic, No. 4, in "La Musique Arab," Vol. 2,
> p. 148. The other two tetrachord categories he gives are Soft Genus
> Chromatic and Soft Genus Enharmonic.

Thank you for this critical information! In fact, for the last nine
years almost, I had supposed that it was 14:13, 13:12, 8:7 -- or
possibly 13:12, 14:13, 8:7 -- a form of what might now be called a
septimal Persian shur or a very low Turkish Ushshaq or maybe a very
low Arab Bayyati.

In fact, 8:7, 13:12, 14:13 is what I would call a bright Rast, if that
term may apply to a form with the first step at 8:7 rather than a
usual tone at 9:8. This is 1/1-8/7-26/21-4/3 or 0-231-370-498 cents.

In the O3 temperament, a form of Rast using this tetrachord (an
asterisk shows a note in the higher chain of fifths) would be:

|----------------| |-----------------|
F#* A Bb* B* C#* E F* F#*
JI 1/1 8/7 26/21 4/3 3/2 12/7 13/7 2/1
0 231 370 498 702 933 1072 1200 O3 0 232 370 497 703 935 1073 1200

Of course, learning Ibn Sina's original form (compare
avicenna_diat.scl in the Scala archives, which I've been quoting for
almost a decade) doesn't prevent me from also using various
permutations including 28:26:24:21 (14:13-13:12-8:7) or 12:13:14:16
(13:12-14:13-8:7), either of which I might use for Shur. But it does
clarify the nature of his original, of which these are more like
inversions (precisely so with 28:26:24:21).

[Cameron]

> I know the "orthodox" order of the tetrachordal intervals of the
> ancient Greeks, but it's not clear to me what the equivalent would
> be for the medieval Islamic musicians- ascending, descending? And in
> a fair bit of reading on ancient tetrachordal music over the last
> few years, I haven't noticed anything giving a clear picture as to
> what extent the intervals of tetrachordal division were freely (or
> not) re-ordered in practice by the ancients and medievals.

I've wondered about this also. We do know that medieval writers
sometimes used permutations of ancient Greek tetrachords. A famous
example:

81 151 267
Ptolemy Intense Chromatic: 22:21 12:11 7:6
1/1 22/21 8/7 4/3
0 81 231 498

151 267 81
12:11 7:6 22:21
Qutb al-Din al-Shirazi Hijaz: 1/1 12/11 14/11 4/3
0 151 418 498

Now an example of rather subtle reordering does occur to me, now that
Cris has given us the original form of Ibn Sina's diatonic. This is
especially interesting to me because it's a distinction I make between
two shades of Rast. However, since I know this is likely to be a
detailed enough discussion just sticking to the ancient and medieval
sources, why don't I put that digression aside for the moment.

First, let's look at Ibn Sina's Strong Diatonic, and then at a
division given by Safi al-Din al-Urmawi maybe about 250 years later:

231 139 128
8:7 13:12 14:13
Ibn Sina Strong Diatonic: 1/1 8/7 26/21 4/3
0 231 370 498

231 128 139
Safi al-Din (Arslan): 8:7 14:13 13:12
"Soft" Diatonic 1/1 8/7 16/13 4/3
0 231 359 498

According to Dr. Fazli Arslan, Safi al-Din calls his tetrachord _qawi_
or "strong" in the sense of diatonic; but also _daif_ or "feeble, Ozan
suggests that "soft," taking the concept of the Greek _malakon_, might
be a better Englishing.

While d'Erlanger's Volume III might be good to check, Arslan does give
string ratios of 64-56-52-48, which would simplify to 16-14-13-12, so
if he's right, we do have a permutation of Ibn Sina.

While this could be seen as a small variation between two similar
tetrachords, with Safi al-Din and Qutb al-Din we also meet another
permutation of Ibn Sina's original tetrachord which we'd clearly
consider a different type of genus: Buzurg, also using steps of 8:7,
14:13, and 13:12 in the lower tetrachord of a pentachordal genus:

14:13 8:7 13:12 14:13 117:112
128 231 139 128 76
Buzurg: 1/1 14/13 16/13 4/3 56/39 3/2
0 128 359 498 626 702

Here Ibn Sina's 8:7-13:12-14:13 or Safi al-Din's 8:7-14:13-13:12
becomes 14:13-8:7-13:12.

Another form of Buzurg discussed by Safi al-Din has the lower
tetrachord as a permutation of this last example:

13:12 8:7 14:13 13:12 27:26
128 231 128 139 65
Buzurg: 1/1 13/12 26/21 4/3 13/9 3/2
0 139 370 498 637 702

This time we have 13:12-8:7-14:13.

Aside from illustrating some medieval permutations of ancient or
medieval tetrachords, these examples have a number of 26/21 steps.

[Cameron]

> This question is of special interest to me as I use the enharmonic
> of Eratosthenes, which is definitely "micro and macro"tonal, and the
> ordering of the intervals makes a huge difference in the feeling of
> it. Since I tend to prefer conjuct tetrachords to disjunct these
> days, obviously different orderings can make for some very
> bunched-and-gapped resulting "scales". Which is great, I just wonder
> if the ancients did such things in actual practice.

From Scala, I get that this tetrachord of Eratosthenes is
40/39-20/19-4/3 (43-45-409 cents or so).

What the ancients may have done, I'm not sure. In the medieval Islamic
sources I'd say there are some interesting examples of permutations of
the kind we're discussing. And in modern Near Eastern theory and
practice there are many examples we could discuss; here I'm going in
good part by writers such as Amine Beyhom, and I suspect that Ozan
would take a similar view.

> At any rate, the 26/21 is right there in the Strong Genus Diatonic,
> and the ultimate origin of the tetrachordal division could hardly
> have been anything other than the extremely simple process of
> dividing string lengths which is spelled within it. So I think it is
> the more reasonable as well as richer choice, as opposed to "370
> cents". When I measured the song of a particular cuckoo, I called
> the result "369 cents", but 26/21 is better for describing the
> fretting of a maqam instrument.

A curious thing is that while Safi al-Din's 8:7-14:13-13:12, maybe
because it divides the upper 7:6 arithematically (14-13-12), has a
very simple string ratio, 16-14-13-12, Ibn Sina's 8:7-13:12-14:13 can
look rather more complicated on paper: 104-91-84-78, if I'm right.

However, in practice: "Start at the 1/1, and remove an eighth of the
string length (i.e. 8/7). Now, starting at the 4/3, add a thirteenth
of the string lenth (i.e. 26/21)." And voila, there's our tetrachord.

A fine point is that while 370 cents very much makes sense as 26/21,
of which 369 cents might be an approximation, Safi al-Din shows how
369 cents could also be a basic ratio.

If we take 9:8 plus 11:10, we get 99/80 or 369 cents, almost identical
to 26/21, and differing only by 2080:2079. So subtracting 14:13 from
a 4:3 fourth and adding 11:10 to a 9:8 tone give almost the same
interval.

Again, it's easy to envision a tuning instruction: "Start at 9/8, and
remove an eleventh of the remaining length." And there we have it.

All things being equal, I join you in finding 26/21 simpler and more
intuitive.

Permutations in a modern maqam setting are fascinating to explore, and
that could be another thread.

Again, thanks to both Cameron and Cris for your invaluable information
about tetrachords and the acoustical instruments on which they may
appear.

Best,

Margo

Isn't the age of which you speak also referred to as the Islamic Golden Age?

--- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@...> wrote:
>
> You're welcome, Cameron. Forgive me, but I refer
> to the time roughly between the life of Al-Kindi (d. c.
> 874) and Safi Al-Din (d. 1294) as the Arabian
> Renaissance. In the Middle East, this time span is
> at least as historically profound as the Italian
> Renaissance.
>
> In general, the Greeks always described their
> tetrachords in descending order. The largest
> interval descended first. The smallest interval
> descended last. Consequently, if one looks at such
> a progression of intervals in ascending order, then
> the smallest interval ratio is the first interval.
>
> Al-Farabi, in his description of eight ajnas, or eight
> separate jins, intentionally inverted the Greek order,
> so that when one looks at the progression of
> intervals of his tetrachords in ascending order, then
> the largest interval ratio is the first interval.
>
> From my text:
>
> ******************************
>
> Because Al-Farabi systematically inverted the
> intervalic order of all Greek tetrachords, his
> interpretations on the fretted `ud profoundly
> influenced countless musicians in future
> generations. Indeed, the tendency to place the
> largest intervals first, and the smallest intervals last,
> is very familiar to Western musicians.
>
> ******************************
>
> Cris
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> > Thanks, Cris- I should have checked rather than go from memory. And I need to get my own copy of your book!
> >
> > I know the "orthodox" order of the tetrachordal intervals of the ancient Greeks, but it's not clear to me what the equivalent would be for the medieval Islamic musicians- ascending, descending? And in a fair bit of reading on ancient tetrachordal music over the last few years, I haven't noticed anything giving a clear picture as to what extent the intervals of tetrachordal division were freely (or not) re-ordered in practice by the ancients and medievals. This question is of special interest to me as I use the enharmonic of Eratosthenes, which is definitely "micro and macro"tonal, and the ordering of the intervals makes a huge difference in the feeling of it. Since I tend to prefer conjuct tetrachords to disjunct these days, obviously different orderings can make for some very bunched-and-gapped resulting "scales". Which is great, I just wonder if the ancients did such things in actual practice.
> >
> > At any rate, the 26/21 is right there in the Strong Genus Diatonic,
> > and the ultimate origin of the tetrachordal division could hardly have been anything other than the extremely simple process of dividing string lengths which is spelled within it. So I think it is the more reasonable as well as richer choice, as opposed to "370 cents". When I measured the song of a particular cuckoo, I called the result "369 cents", but 26/21 is better for describing the fretting of a maqam instrument.
> >
> > -Cameron Bobro
> >
> >
> > --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> > >
> > > Cameron,
> > >
> > > Please note carefully that Ibn Sina categorized the
> > > following tetrachord with ascending interval ratios
> > > 8/7, 13/12, 14/13, as Strong Genus Diatonic, No. 4,
> > > in "La Musique Arab," Vol. 2, p. 148. The other two
> > > tetrachord categories he gives are Soft Genus
> > > Chromatic and Soft Genus Enharmonic.
> > >
> > > Cris
> > >
> > >
> > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > >
> > > >
> > > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > >
> > > > > > Other questions might be 7/4 or 16/9 (especially
> > > > > > given that 7/6!)? And maybe a 13/7 or so at a 3/2
> > > > > > from 370 cents (which I'll join you in calling a
> > > > > > 26/21)? I was picturing something for Rast like:
> > > >
> > > > Oh, Margo, I forgot to mention that of course we should call the "370 cent" fretting here 26:21. This interval is one found in Ibn Sin's soft diatonic tetrachord, a tetrachordal division which is a stellar example of historical writers most certainly NOT playing number games, being perhaps the most physically simple and eminently practical division of the tetrachord of them all. When we have a contemporary example of a traditional instrument fretted to measurements which have been documented for some thousand years at least, and those measurements all ultimately derive from very simple and practical processes, only numerological superstition would prevent use from using traditional names for the intervals. In your case, this is an especially solid approach, as you temper with an intent to create a system with vertical harmonies, and the traditional name here immediately gives you the primes with which to work, very practical!
> > > >
> > > > -Cameron Bobro
> > > >
> > >
> >
>

I am not familiar with the term "Islamic Golden Age" nor
do I know the timeframe it encompasses. A source
where I often encountered the term "Muslim
renaissance" is in the monumental 5-volume work
(~6000 pages) "Introduction to the History of Science,"
by George Sarton. This work covers the period from
Homer to the second half of the fourteenth century in all
scientifically oriented civilizations around the world.
However, I prefer "Arabian Renaissance" because it
comes close to avoiding nationalistic and religious
overtones: Arabian was the _lingua franca_ of the
Arabian Empire.

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> Isn't the age of which you speak also referred to as the Islamic Golden Age?
>
> --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> >
> > You're welcome, Cameron. Forgive me, but I refer
> > to the time roughly between the life of Al-Kindi (d. c.
> > 874) and Safi Al-Din (d. 1294) as the Arabian
> > Renaissance. In the Middle East, this time span is
> > at least as historically profound as the Italian
> > Renaissance.
> >
> > In general, the Greeks always described their
> > tetrachords in descending order. The largest
> > interval descended first. The smallest interval
> > descended last. Consequently, if one looks at such
> > a progression of intervals in ascending order, then
> > the smallest interval ratio is the first interval.
> >
> > Al-Farabi, in his description of eight ajnas, or eight
> > separate jins, intentionally inverted the Greek order,
> > so that when one looks at the progression of
> > intervals of his tetrachords in ascending order, then
> > the largest interval ratio is the first interval.
> >
> > From my text:
> >
> > ******************************
> >
> > Because Al-Farabi systematically inverted the
> > intervalic order of all Greek tetrachords, his
> > interpretations on the fretted `ud profoundly
> > influenced countless musicians in future
> > generations. Indeed, the tendency to place the
> > largest intervals first, and the smallest intervals last,
> > is very familiar to Western musicians.
> >
> > ******************************
> >
> > Cris
> >
> >
> > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > >
> > > Thanks, Cris- I should have checked rather than go from memory. And I need to get my own copy of your book!
> > >
> > > I know the "orthodox" order of the tetrachordal intervals of the ancient Greeks, but it's not clear to me what the equivalent would be for the medieval Islamic musicians- ascending, descending? And in a fair bit of reading on ancient tetrachordal music over the last few years, I haven't noticed anything giving a clear picture as to what extent the intervals of tetrachordal division were freely (or not) re-ordered in practice by the ancients and medievals. This question is of special interest to me as I use the enharmonic of Eratosthenes, which is definitely "micro and macro"tonal, and the ordering of the intervals makes a huge difference in the feeling of it. Since I tend to prefer conjuct tetrachords to disjunct these days, obviously different orderings can make for some very bunched-and-gapped resulting "scales". Which is great, I just wonder if the ancients did such things in actual practice.
> > >
> > > At any rate, the 26/21 is right there in the Strong Genus Diatonic,
> > > and the ultimate origin of the tetrachordal division could hardly have been anything other than the extremely simple process of dividing string lengths which is spelled within it. So I think it is the more reasonable as well as richer choice, as opposed to "370 cents". When I measured the song of a particular cuckoo, I called the result "369 cents", but 26/21 is better for describing the fretting of a maqam instrument.
> > >
> > > -Cameron Bobro
> > >
> > >
> > > --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> > > >
> > > > Cameron,
> > > >
> > > > Please note carefully that Ibn Sina categorized the
> > > > following tetrachord with ascending interval ratios
> > > > 8/7, 13/12, 14/13, as Strong Genus Diatonic, No. 4,
> > > > in "La Musique Arab," Vol. 2, p. 148. The other two
> > > > tetrachord categories he gives are Soft Genus
> > > > Chromatic and Soft Genus Enharmonic.
> > > >
> > > > Cris
> > > >
> > > >
> > > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > > >
> > > > >
> > > > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > > >
> > > > > > > Other questions might be 7/4 or 16/9 (especially
> > > > > > > given that 7/6!)? And maybe a 13/7 or so at a 3/2
> > > > > > > from 370 cents (which I'll join you in calling a
> > > > > > > 26/21)? I was picturing something for Rast like:
> > > > >
> > > > > Oh, Margo, I forgot to mention that of course we should call the "370 cent" fretting here 26:21. This interval is one found in Ibn Sin's soft diatonic tetrachord, a tetrachordal division which is a stellar example of historical writers most certainly NOT playing number games, being perhaps the most physically simple and eminently practical division of the tetrachord of them all. When we have a contemporary example of a traditional instrument fretted to measurements which have been documented for some thousand years at least, and those measurements all ultimately derive from very simple and practical processes, only numerological superstition would prevent use from using traditional names for the intervals. In your case, this is an especially solid approach, as you temper with an intent to create a system with vertical harmonies, and the traditional name here immediately gives you the primes with which to work, very practical!
> > > > >
> > > > > -Cameron Bobro
> > > > >
> > > >
> > >
> >
>

I see the term has a page on Wikipedia:

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

humorously enough I first became aware of the term a couple of years ago when coming across essays and rants on the Internet arguing against the validity or very concept of the term.

I've been aware of the idea since as long as I can remember, the shock of the Crusaders encountering a flowering civilization is famous, and of course most educated people in the West would know about the Aquinas-Avicenna connection. Whether Islamic Golden Age or Muslim or Arabian Renaissance, it's obviously a very valid and important era and concept.

-Cameron Bobro

--- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@...> wrote:
>
> I am not familiar with the term "Islamic Golden Age" nor
> do I know the timeframe it encompasses. A source
> where I often encountered the term "Muslim
> renaissance" is in the monumental 5-volume work
> (~6000 pages) "Introduction to the History of Science,"
> by George Sarton. This work covers the period from
> Homer to the second half of the fourteenth century in all
> scientifically oriented civilizations around the world.
> However, I prefer "Arabian Renaissance" because it
> comes close to avoiding nationalistic and religious
> overtones: Arabian was the _lingua franca_ of the
> Arabian Empire.
>
>
> --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> >
> > Isn't the age of which you speak also referred to as the Islamic Golden Age?
> >
> > --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> > >
> > > You're welcome, Cameron. Forgive me, but I refer
> > > to the time roughly between the life of Al-Kindi (d. c.
> > > 874) and Safi Al-Din (d. 1294) as the Arabian
> > > Renaissance. In the Middle East, this time span is
> > > at least as historically profound as the Italian
> > > Renaissance.
> > >
> > > In general, the Greeks always described their
> > > tetrachords in descending order. The largest
> > > interval descended first. The smallest interval
> > > descended last. Consequently, if one looks at such
> > > a progression of intervals in ascending order, then
> > > the smallest interval ratio is the first interval.
> > >
> > > Al-Farabi, in his description of eight ajnas, or eight
> > > separate jins, intentionally inverted the Greek order,
> > > so that when one looks at the progression of
> > > intervals of his tetrachords in ascending order, then
> > > the largest interval ratio is the first interval.
> > >
> > > From my text:
> > >
> > > ******************************
> > >
> > > Because Al-Farabi systematically inverted the
> > > intervalic order of all Greek tetrachords, his
> > > interpretations on the fretted `ud profoundly
> > > influenced countless musicians in future
> > > generations. Indeed, the tendency to place the
> > > largest intervals first, and the smallest intervals last,
> > > is very familiar to Western musicians.
> > >
> > > ******************************
> > >
> > > Cris
> > >
> > >
> > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > >
> > > > Thanks, Cris- I should have checked rather than go from memory. And I need to get my own copy of your book!
> > > >
> > > > I know the "orthodox" order of the tetrachordal intervals of the ancient Greeks, but it's not clear to me what the equivalent would be for the medieval Islamic musicians- ascending, descending? And in a fair bit of reading on ancient tetrachordal music over the last few years, I haven't noticed anything giving a clear picture as to what extent the intervals of tetrachordal division were freely (or not) re-ordered in practice by the ancients and medievals. This question is of special interest to me as I use the enharmonic of Eratosthenes, which is definitely "micro and macro"tonal, and the ordering of the intervals makes a huge difference in the feeling of it. Since I tend to prefer conjuct tetrachords to disjunct these days, obviously different orderings can make for some very bunched-and-gapped resulting "scales". Which is great, I just wonder if the ancients did such things in actual practice.
> > > >
> > > > At any rate, the 26/21 is right there in the Strong Genus Diatonic,
> > > > and the ultimate origin of the tetrachordal division could hardly have been anything other than the extremely simple process of dividing string lengths which is spelled within it. So I think it is the more reasonable as well as richer choice, as opposed to "370 cents". When I measured the song of a particular cuckoo, I called the result "369 cents", but 26/21 is better for describing the fretting of a maqam instrument.
> > > >
> > > > -Cameron Bobro
> > > >
> > > >
> > > > --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> > > > >
> > > > > Cameron,
> > > > >
> > > > > Please note carefully that Ibn Sina categorized the
> > > > > following tetrachord with ascending interval ratios
> > > > > 8/7, 13/12, 14/13, as Strong Genus Diatonic, No. 4,
> > > > > in "La Musique Arab," Vol. 2, p. 148. The other two
> > > > > tetrachord categories he gives are Soft Genus
> > > > > Chromatic and Soft Genus Enharmonic.
> > > > >
> > > > > Cris
> > > > >
> > > > >
> > > > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > > > >
> > > > > >
> > > > > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > > > >
> > > > > > > > Other questions might be 7/4 or 16/9 (especially
> > > > > > > > given that 7/6!)? And maybe a 13/7 or so at a 3/2
> > > > > > > > from 370 cents (which I'll join you in calling a
> > > > > > > > 26/21)? I was picturing something for Rast like:
> > > > > >
> > > > > > Oh, Margo, I forgot to mention that of course we should call the "370 cent" fretting here 26:21. This interval is one found in Ibn Sin's soft diatonic tetrachord, a tetrachordal division which is a stellar example of historical writers most certainly NOT playing number games, being perhaps the most physically simple and eminently practical division of the tetrachord of them all. When we have a contemporary example of a traditional instrument fretted to measurements which have been documented for some thousand years at least, and those measurements all ultimately derive from very simple and practical processes, only numerological superstition would prevent use from using traditional names for the intervals. In your case, this is an especially solid approach, as you temper with an intent to create a system with vertical harmonies, and the traditional name here immediately gives you the primes with which to work, very practical!
> > > > > >
> > > > > > -Cameron Bobro
> > > > > >
> > > > >
> > > >
> > >
> >
>

Regarding the Arabian Renaissance, Ibn Sina's
most important and famous contributions are his
medical texts.

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

From the latter source:

******************************

"Ibn Sina studied medicine under a physician
named Koushyar. He wrote almost 450 treatises on
a wide range of subjects, of which around 240 have
survived. In particular, 150 of his surviving treatises
concentrate on philosophy and 40 of them
concentrate on medicine.[6][13] His most famous
works are The Book of Healing, a vast
philosophical and scientific encyclopaedia, and The
Canon of Medicine,[14] which was a standard
medical text at many medieval universities.[15] The
Canon of Medicine was used as a text-book in the
universities of Montpellier and Louvain as late as
1650.[16]"

******************************

Cris

--- In tuning@yahoogroups.com, "cameron" <misterbobro@...> wrote:
>
> I see the term has a page on Wikipedia:
>
> http://en.wikipedia.org/wiki/Islamic_golden_age
>
> humorously enough I first became aware of the term a couple of years ago when coming across essays and rants on the Internet arguing against the validity or very concept of the term.
>
> I've been aware of the idea since as long as I can remember, the shock of the Crusaders encountering a flowering civilization is famous, and of course most educated people in the West would know about the Aquinas-Avicenna connection. Whether Islamic Golden Age or Muslim or Arabian Renaissance, it's obviously a very valid and important era and concept.
>
> -Cameron Bobro
>
> --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> >
> > I am not familiar with the term "Islamic Golden Age" nor
> > do I know the timeframe it encompasses. A source
> > where I often encountered the term "Muslim
> > renaissance" is in the monumental 5-volume work
> > (~6000 pages) "Introduction to the History of Science,"
> > by George Sarton. This work covers the period from
> > Homer to the second half of the fourteenth century in all
> > scientifically oriented civilizations around the world.
> > However, I prefer "Arabian Renaissance" because it
> > comes close to avoiding nationalistic and religious
> > overtones: Arabian was the _lingua franca_ of the
> > Arabian Empire.
> >
> >
> > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > >
> > > Isn't the age of which you speak also referred to as the Islamic Golden Age?
> > >
> > > --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> > > >
> > > > You're welcome, Cameron. Forgive me, but I refer
> > > > to the time roughly between the life of Al-Kindi (d. c.
> > > > 874) and Safi Al-Din (d. 1294) as the Arabian
> > > > Renaissance. In the Middle East, this time span is
> > > > at least as historically profound as the Italian
> > > > Renaissance.
> > > >
> > > > In general, the Greeks always described their
> > > > tetrachords in descending order. The largest
> > > > interval descended first. The smallest interval
> > > > descended last. Consequently, if one looks at such
> > > > a progression of intervals in ascending order, then
> > > > the smallest interval ratio is the first interval.
> > > >
> > > > Al-Farabi, in his description of eight ajnas, or eight
> > > > separate jins, intentionally inverted the Greek order,
> > > > so that when one looks at the progression of
> > > > intervals of his tetrachords in ascending order, then
> > > > the largest interval ratio is the first interval.
> > > >
> > > > From my text:
> > > >
> > > > ******************************
> > > >
> > > > Because Al-Farabi systematically inverted the
> > > > intervalic order of all Greek tetrachords, his
> > > > interpretations on the fretted `ud profoundly
> > > > influenced countless musicians in future
> > > > generations. Indeed, the tendency to place the
> > > > largest intervals first, and the smallest intervals last,
> > > > is very familiar to Western musicians.
> > > >
> > > > ******************************
> > > >
> > > > Cris
> > > >
> > > >
> > > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > > >
> > > > > Thanks, Cris- I should have checked rather than go from memory. And I need to get my own copy of your book!
> > > > >
> > > > > I know the "orthodox" order of the tetrachordal intervals of the ancient Greeks, but it's not clear to me what the equivalent would be for the medieval Islamic musicians- ascending, descending? And in a fair bit of reading on ancient tetrachordal music over the last few years, I haven't noticed anything giving a clear picture as to what extent the intervals of tetrachordal division were freely (or not) re-ordered in practice by the ancients and medievals. This question is of special interest to me as I use the enharmonic of Eratosthenes, which is definitely "micro and macro"tonal, and the ordering of the intervals makes a huge difference in the feeling of it. Since I tend to prefer conjuct tetrachords to disjunct these days, obviously different orderings can make for some very bunched-and-gapped resulting "scales". Which is great, I just wonder if the ancients did such things in actual practice.
> > > > >
> > > > > At any rate, the 26/21 is right there in the Strong Genus Diatonic,
> > > > > and the ultimate origin of the tetrachordal division could hardly have been anything other than the extremely simple process of dividing string lengths which is spelled within it. So I think it is the more reasonable as well as richer choice, as opposed to "370 cents". When I measured the song of a particular cuckoo, I called the result "369 cents", but 26/21 is better for describing the fretting of a maqam instrument.
> > > > >
> > > > > -Cameron Bobro
> > > > >
> > > > >
> > > > > --- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@> wrote:
> > > > > >
> > > > > > Cameron,
> > > > > >
> > > > > > Please note carefully that Ibn Sina categorized the
> > > > > > following tetrachord with ascending interval ratios
> > > > > > 8/7, 13/12, 14/13, as Strong Genus Diatonic, No. 4,
> > > > > > in "La Musique Arab," Vol. 2, p. 148. The other two
> > > > > > tetrachord categories he gives are Soft Genus
> > > > > > Chromatic and Soft Genus Enharmonic.
> > > > > >
> > > > > > Cris
> > > > > >
> > > > > >
> > > > > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > > > > >
> > > > > > >
> > > > > > > --- In tuning@yahoogroups.com, "cameron" <misterbobro@> wrote:
> > > > > > >
> > > > > > > > > Other questions might be 7/4 or 16/9 (especially
> > > > > > > > > given that 7/6!)? And maybe a 13/7 or so at a 3/2
> > > > > > > > > from 370 cents (which I'll join you in calling a
> > > > > > > > > 26/21)? I was picturing something for Rast like:
> > > > > > >
> > > > > > > Oh, Margo, I forgot to mention that of course we should call the "370 cent" fretting here 26:21. This interval is one found in Ibn Sin's soft diatonic tetrachord, a tetrachordal division which is a stellar example of historical writers most certainly NOT playing number games, being perhaps the most physically simple and eminently practical division of the tetrachord of them all. When we have a contemporary example of a traditional instrument fretted to measurements which have been documented for some thousand years at least, and those measurements all ultimately derive from very simple and practical processes, only numerological superstition would prevent use from using traditional names for the intervals. In your case, this is an especially solid approach, as you temper with an intent to create a system with vertical harmonies, and the traditional name here immediately gives you the primes with which to work, very practical!
> > > > > > >
> > > > > > > -Cameron Bobro
> > > > > > >
> > > > > >
> > > > >
> > > >
> > >
> >
>

Avicenna and the Visionary Recital by Henri Corbin
is an excellent book on his works depth and influence that reached and defined many features of western alchemical and metaphysical thought that still is unrecognized

--- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@...> wrote:
>
> Regarding the Arabian Renaissance, Ibn Sina's
> most important and famous contributions are his
> medical texts.
>
> http://en.wikipedia.org/wiki/Avicenna
>
> From the latter source:
>
> ******************************
>
> "Ibn Sina studied medicine under a physician
> named Koushyar. He wrote almost 450 treatises on
> a wide range of subjects, of which around 240 have
> survived. In particular, 150 of his surviving treatises
> concentrate on philosophy and 40 of them
> concentrate on medicine.[6][13] His most famous
> works are The Book of Healing, a vast
> philosophical and scientific encyclopaedia, and The
> Canon of Medicine,[14] which was a standard
> medical text at many medieval universities.[15] The
> Canon of Medicine was used as a text-book in the
> universities of Montpellier and Louvain as late as
> 1650.[16]"
>
> ******************************
>
> Cris
>

> >

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

> A fine point is that while 370 cents very much makes sense as 26/21,
> of which 369 cents might be an approximation, Safi al-Din shows how
> 369 cents could also be a basic ratio.
>
> If we take 9:8 plus 11:10, we get 99/80 or 369 cents, almost >identical
> to 26/21, and differing only by 2080:2079. So subtracting 14:13 from
> a 4:3 fourth and adding 11:10 to a 9:8 tone give almost the same
> interval.
>
> Again, it's easy to envision a tuning instruction: "Start at 9/8, >and
> remove an eleventh of the remaining length." And there we have it.
>

As I mentioned before, I suspect that there is temperament-in-effect of tiny commas implied in historical Islamic tuning. The fretting plans given in the ratios 26/21 and 99/80 both give superparticular intervals, using different primes, but wind up in effectively the same place (.8 cents difference).

> All things being equal, I join you in finding 26/21 simpler and more
> intuitive.

And of course, if you're going to temper, there you have your primes, neat as can be. And striving for the description of things in terms of superparticulars is also good if you're going to temper, as your example above demonstrates.

>
> Permutations in a modern maqam setting are fascinating to explore, >and that could be another thread.

Personally I'm much more interested in the ancient tetrachords and space-age "reboots" of such, but of course they share quite a bit with current tetrachords, linked via the "Golden Age" musicians.

-Cameron Bobro