13 limit comma names

Some 13-limit superparticular commas wandering around without names are 40/39, 65/64, 66/65, 1716/1715 and 6656/6655. Might as well name all of them.

Have we named 169/168?

66/65 mixes 8/5 and 13/8; it prominently appears in flattone and mavila and
negri. It means that both 5/4 and its inverse are rooted intervals. Maybe
the rootsma or something would do.

-Mike

On Dec 23, 2012, at 7:46 AM, genewardsmith <genewardsmith@...>
wrote:

Some 13-limit superparticular commas wandering around without names are
40/39, 65/64, 66/65, 1716/1715 and 6656/6655. Might as well name all of
them.

Oops, meant 65/64 there.

-Mike

On Dec 23, 2012, at 2:33 PM, Mike Battaglia <battaglia01@...> wrote:

Have we named 169/168?

66/65 mixes 8/5 and 13/8; it prominently appears in flattone and mavila and
negri. It means that both 5/4 and its inverse are rooted intervals. Maybe
the rootsma or something would do.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Have we named 169/168?

It's listed as dhanvantarisma, which is an open invitation to change it to something else.

On Dec 23, 2012, at 5:04 PM, genewardsmith <genewardsmith@...>
wrote:

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Have we named 169/168?

It's listed as dhanvantarisma, which is an open invitation to change it to
something else.

169/168 is important because it's a superparticular ratio that's the
quotient of two consecutive superparticular ratios, in this case 14/13 and
13/12. As a result, 7/6 is bisected if this comma is ever tempered out.

It also fits nicely into a sequence of commas having the same properties,
and which is generated by (n^2)/(n^2-1
), yielding 2/1, 9/8, 16/15, 25/24, 36/35, 49/48, 64/63, 81/80, 100/99,
121/120, 144/133, 169/168, 196/195, 225/224... . For each of these cases,
the ratio (n+1)/(n-1) is bisected.

In other words, these are pretty damn important commas, especially when
they're also of a low prime-limit. So excluding the ones which already have
names we don't want to change, like 81/80 and 64/63, it might be nice to
come up with a naming scheme for the whole set at once.

-Mike

Hello, all, and I'd like to comment
on the 169/168, because this is one
of my favorite commas to observe:
Ibn Sina (980-1037) has early dibs,
for one.

But the first and biggest question
is what dhanvantarisma means, and
whether it is based, for example,
on some raga practice or theory
featuring 12:13:14 or the like.
If it is, then dhanvantarisma
should stand.

Emotionally, I am attached to the
169/168 because it's central to my
music and tunings, especially the
12:13:14:16 or 28:26:24:21 division
(the form I originally encountered
back in 2001-2002). Peppermint was
designed to get 12:13:14 virtually
just, and this is a feature of
my favorite maqam tunings.
There's lots of history here, from
Ibn Sina to Kathleen Schlesinger
in "Greek-influenced" traditions;
and, if the name dhanvantarisma has
an historical basis, in raga music
also.

I have heard that 12:13:14 occurs in
certain ragas, and this would have
every bit as much rightful priority
as Ibn Sina -- so there would be no
reason to upset dhanvantarisma. It
might be that dhanvantarisma itself
is indirectly honoring Ibn Sina and
the Persian tradition his writings
reflect, since there is evidence
for a Persian influence on some
Hindustani music.

For example, there's a raga Darbari
Kanada at

1/1 9/8 7/6 4/3 3/2 14/9 16/9 2/1

where "Darbari" may refer to a
Persian Court style. I wonder if
the 12:13:14 pattern may occur in
related ragas with a septimal feeling.

So the question is: is dhanvantarisma
a reference to some traditional raga
music with 12:13:14 or the like where
169/168 is relevant? If so, it should
stand.

If not, maybe "Ibn Sina's tredecimal
seconds comma" would best follow scala,
by analogy to "undecimal seconds comma"
for 121/120 (11:10 vs. 12:11).
He is the first writer of which I'm
aware in the Near Eastern tradition
to introduce ratios of 13, e.g.
16:14:13:12; and he notes that some
musicians, in fretting their instruments,
do not seem to distinguish between
14/13 and 13/12.

In short, the first question is to
clarify what dhanvantarisma means.

Best,

Margo

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Hello, all, and I'd like to comment
> on the 169/168, because this is one
> of my favorite commas to observe:
> Ibn Sina (980-1037) has early dibs,
> for one.

We've already got two commas named after him.

Hello, everyone.

Having written a bit about the 169/168,
I'd like to mention two commas (one
already brought up in a previous article
on the 194481/194480) also connected with
Ibn Sina (980-1037).

The first, which I mentioned, is the 2080/2079
(0.833 cents), the difference between 14/13
and 320/297, the latter the 32/27 complement
of 11/10. Ibn Sina notes that in 9:8-11:10-320:297,
the nonsuperparticular interval is very close
to 14/13.

The second might be called the tredecimal kleisma
or the like -- I'm not sure of the best name,
but it's connected with Ibn Sina -- the 352/351
(4.925 cents).

For Ibn Sina, what he sometimes the nonsuperparticular
"remainder" interval in a tetrachord (analogous to the
256/243 limma in the ditonic diatonic) may sometimes
approximate a superparticular ratio.

Thus in al-Farabi's 9:8-12:11-88:81, Ibn Sina notes
that 88:81 is close to 13:12 -- thus alluding to
the difference, which we know is 352:351.

Likewise, in his own favored 9:8-13:12-128:117, he
notes that 128:117 is close to 12:11 -- the
difference, as we know, again being 352:351.

In modern music, 352:351 also is the difference
between 32/27 and the smaller 13:11. Indeed the
11:12:13 or 13:12:11 division might be seen as
a kind of further development of Ibn Sina's
theory, although he doesn't, at least explicitly,
suggest use of a minor third at 13:11 rather
than the classic 32:27.

So 169/168, 2080/2079, and 352/351 all come up
in Ibn Sina's writings and tunings, as well
as tunings of others (and more specifically
al-Farabi) that he mentions and analyzes.
Here are some relevant tunings:

169/168 16:14:13:12 (14/13, 13/12) 352/351 9:8-13:12-128:117 (12/11, 128/117)
2080/2079 9:8-11:10-320:297 (14/13, 320/297)

While 2080/2079 might be "Ibn Sina's 11-13
schismina" or the like, and 169/168 "Ibn
Sina's tredecimal seconds comma" (maybe a
name to complement dhanvantarisma, as with
Gauss's comma or the xenisma), the 352/351
might have a dual status: either "Ibn Sina's
undecimal-tredecimal seconds kleisma"
(12/11, 128/117); or also the modern
"tredecimal kleisma" (13/11, 32/27) for
situations where 13/11 is in use.

Best,

Margo

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> In other words, these are pretty damn important commas, especially when
> they're also of a low prime-limit. So excluding the ones which already have
> names we don't want to change, like 81/80 and 64/63, it might be nice to
> come up with a naming scheme for the whole set at once.

49/48 square7
64/63 square8
81/80 square9
100/99 square10
121/120 square11
144/143 square12
169/168 square13
196/195 square14
225/224 square15

etc

45/44 triangle9
55/54 triangle10
66/65 triangle11
78/77 triangle12
91/90 triangle13
105/104 triangle14

etc

As I recall, Paul Erlich pointed out that there is only one more superparticular ratio at the 23 limit in toto, not just at the 10,000,000 size limit in my article. I don't recall what it is right now, however.
--John

--- In tuning@yahoogroups.com, "John H. Chalmers" <JHCHALMERS@...> wrote:
>
> As I recall, Paul Erlich pointed out that there is only one more
> superparticular ratio at the 23 limit in toto, not just at the
> 10,000,000 size limit in my article. I don't recall what it is right
> now, however.

There is code that implements Stormer's algorithm at http://www.ics.uci.edu/~eppstein/0xDE/stormer.py

The last 23-limit superparticular ratio is 11859211/11859210.

Keenan

>The last 23-limit superparticular ratio is 11859211/11859210.

Thanks, Keenan!

--John

Hello, Gene and all. And Happy Holidays!

After a bit of thought, Gene, I would urge that even though
Ibn Sina may already have two commas named after him -- the
one I've found so far in Scala and on Xenwiki is a 7-limit
diesis at 525/512 (43.408 cents) -- his 13-limit comma or
kleisma at 169/168 (10.274 cents), the difference beween
13:12 and 14:13, should take precedence. One or both of the
others should, if necessary, be renamed. And 169/168 can be
treated like 9801/9800 (either Gauss's comma or the kalisma),
being named "Ibn Sina's tredecimal seconds kleisma, or
dhanvantarisma."

To cut to the chase, Ibn Sina (980-1037) is the first music
theorist of which I know, at least in the Greek and partly
derivative European and Near Eastern traditions to address
prime 13, and to specify what he terms a "very noble genus"
in the 13-limit with all superparticular steps, and with
remarkably small integers to boot, 16:14:13:12.

He derives this tetrachord by taking the 8:7:6 division of
the 4/3 fourth by Archytas, and then subdividing 7/6 into
14:13:12.

1/1 8/7 4/3
8 7 6
|-----------------------|------------------------|
16 14 13 12
1/1 8/7 16/13 4/3
0 231.2 359.5 498.0
8:7 14:13 13:12
231.2 128.3 138.6

In _Divisions of the Tetrachord_, John Chalmers rightly
lists this 13-limit tetrachord of Ibn Sina as among the
select group with all superparticular ratios. A permutation,
28:26:24:21, is available in the Scala archive as
avicenna_diat.scl.

Making 169/168 the signature comma of Ibn Sina follows the
same pattern we follow with the comma of Didymus at 81/80
(9/8, 10/9; the comma of Archytas at 64/63 (8/7, 9/8); or
the comma of Ptolemy at 100/99 (10/9, 11/10).

Here I can see two reasonable solutions:

(1) Rename one or both of his two other commas, and
give him 169/168 as his signature comma. A
theorist introducing the 13-limit should be known
for a 13-limit comma rather than something like
525/512 (7-limit), if there must be a choice.

(2) Let him have three commas -- maybe a solution
contrary to a very sound policy, but a lesser evil
than not giving him the 169/168.

With best holiday wishes,

Margo Schulter
mschulter@...

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
And 169/168 can be
> treated like 9801/9800 (either Gauss's comma or the kalisma),
> being named "Ibn Sina's tredecimal seconds kleisma, or
> dhanvantarisma."

That is much too verbose for a trivial name. We could call it the ibnsinma, and name 2080/2079 something else as one possibility.

Hello. Gene.

Sorry I wasn't aware of the 2080/2079 being listed as the
ibnsinma, as I now found under "recent changes" when
searching Google. And thank you for getting this in process
so quickly!

My first and foremost concern was that Ibn Sina have at
least one 13-limit comma to his name; which one may not be
that important. While the 169/168 might be more obvious, the
2080/2079 actually may pay him a special honor for his
insightfulness in noting that his 14:13 step at 128.298
cents and al-Farabi's 320:297 at 129.131 cents (in
9:8-11:10-320:297) are very close in size.

As for the 169/168, we might as one alternative name call
that the buzurgisma or buzurg kleisma, since it occurs in a
genus around 1300 called Buzurg, with a lower tetrachord
given by sources of the period as either 14:13-8:7-13:12 or
13:12-8:7-14:13. These are both permutations of Ibn Sina's
16:14:13:12, of course, and the different versions of Buzurg
illustrate the comma (10.274 cents) between 13:12 and 14:13.
Again "dhanvantarisma or buzurgisma" would respect priorities
while noting the Near Eastern (and likely specifically
Persian) history.

Happy Holidays,

Margo

> To cut to the chase, Ibn Sina (980-1037) is the first music
> theorist of which I know, at least in the Greek and partly
> derivative European and Near Eastern traditions to address
> prime 13,

Dear Margo,

It may be relevant to consider the significance of Al-Farabi's Jins 6, which includes 16/13 and 64/39. MM. pp. 647-648, 650-655.

Cris

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Hello, Gene and all. And Happy Holidays!
>
> After a bit of thought, Gene, I would urge that even though
> Ibn Sina may already have two commas named after him -- the
> one I've found so far in Scala and on Xenwiki is a 7-limit
> diesis at 525/512 (43.408 cents) -- his 13-limit comma or
> kleisma at 169/168 (10.274 cents), the difference beween
> 13:12 and 14:13, should take precedence. One or both of the
> others should, if necessary, be renamed. And 169/168 can be
> treated like 9801/9800 (either Gauss's comma or the kalisma),
> being named "Ibn Sina's tredecimal seconds kleisma, or
> dhanvantarisma."
>
> To cut to the chase, Ibn Sina (980-1037) is the first music
> theorist of which I know, at least in the Greek and partly
> derivative European and Near Eastern traditions to address
> prime 13, and to specify what he terms a "very noble genus"
> in the 13-limit with all superparticular steps, and with
> remarkably small integers to boot, 16:14:13:12.
>
> He derives this tetrachord by taking the 8:7:6 division of
> the 4/3 fourth by Archytas, and then subdividing 7/6 into
> 14:13:12.
>
> 1/1 8/7 4/3
> 8 7 6
> |-----------------------|------------------------|
> 16 14 13 12
> 1/1 8/7 16/13 4/3
> 0 231.2 359.5 498.0
> 8:7 14:13 13:12
> 231.2 128.3 138.6
>
> In _Divisions of the Tetrachord_, John Chalmers rightly
> lists this 13-limit tetrachord of Ibn Sina as among the
> select group with all superparticular ratios. A permutation,
> 28:26:24:21, is available in the Scala archive as
> avicenna_diat.scl.
>
> Making 169/168 the signature comma of Ibn Sina follows the
> same pattern we follow with the comma of Didymus at 81/80
> (9/8, 10/9; the comma of Archytas at 64/63 (8/7, 9/8); or
> the comma of Ptolemy at 100/99 (10/9, 11/10).
>
> Here I can see two reasonable solutions:
>
> (1) Rename one or both of his two other commas, and
> give him 169/168 as his signature comma. A
> theorist introducing the 13-limit should be known
> for a 13-limit comma rather than something like
> 525/512 (7-limit), if there must be a choice.
>
> (2) Let him have three commas -- maybe a solution
> contrary to a very sound policy, but a lesser evil
> than not giving him the 169/168.
>
> With best holiday wishes,
>
> Margo Schulter
> mschulter@...
>

In a recent post, I wrote:

>> To cut to the chase, Ibn Sina (980-1037) is the first music
>> theorist of which I know, at least in the Greek and partly
>> derivative European and Near Eastern traditions to address
>> prime 13,

Cris Forster responded:

> Dear Margo,

> It may be relevant to consider the significance of Al-Farabi's Jins 6, which
> includes 16/13 and 64/39. MM. pp. 647-648, 650-655.

> Cris

Dear Cris,

Please let me begin by expressing my thanks and admiration, as
always, for your monumental achievement in _Musical Mathematics_.
At the outset, I would also invite any comments from John
Chalmers, since the question of how to interpret al-Farabi's
chromatic genus (Arabic _jins_, plural _ajnas_) we are here
discussing is one also relevant to _Divisions of the
Tetrachord_.

Certainly I agree that it is "relevant to consider the
significance" of al-Farabi's Jins 6, which in one reading -- the
one you may quite correctly consider the best -- would include
16/13 and 64/39. And I further agree that it is appropriate to
mention this possible earlier use of 13 when discussing the
contributions of Ibn Sina, who made ratios of 13 not only an
explicit element of music theory but a central part of his
musical practice, as in his `oud tuning adopting 39/32 as the
wusta Zalzal or neutral third fret.

Revisiting the relevant passages of _Musical Mathematics_, I learn
that al-Farabi's Jins 6 is specified, more or less in the manner
of the Aristoxenian Cleonides, as 44-8-8 parts of a fourth, or
1-5/6, 1/3, 1/3 parts of a tone. (Cleonides divides a tone into
12 parts, or a fourth into 30 parts, but sometimes uses interval
sizes like 4-1/2 parts; so he, like al-Farabi, is effectively
following a 60-part division of the fourth.)[1]

Unlike Ibn Sina, al-Farabi does not explicitly "address prime
13." Rather, he describes a chromatic jins where you make a case
for a rational interpretation of 32:26:25:24 or 16/13-26/25-25/24
(359.5-67.9-70.7 cents).[2]

As you candidly note at p. 650, the calculations leading to this
specific interpretation "are _not_ straightforward" (emphasis in
original). They assume, however reasonably based on al-Farabi's
text and context, that he intends a rational division; that this
division is to be based on the 9/8 tone; and that it is to be
based more specifically on the 9/8 tone found on the `oud between
the frets 32/27 and 4/3.

Honestly, the best interpretation of al-Farabi's 44-8-8 parts in
Jins 6 is a problem I might best leave to you and other scholars
who specialize in this aspect of theory and have the linguistic
as well as mathematical competence needed to make the most just
assessment, in the Near East or elsewhere.

But this is a good occasion to note both the tentative nature of
many interpretations and the virtue of nevertheless daring to
offer them after carefully considering all the evidence and
weighing the likeliest hypotheses from the perspective of the
theorist's text and context, as you have done for al-Farabi.

---------

1. Interestingly, Chrysanthos of Madytos, the famous scholar and
theorist of Byzantine music in the earlier 19th century, mentions
a division of 8-8-44 parts as one of the "shades" known to
Eucleides. See his _Great Theory of Music_, in a translation by
Katy G. Romanou (Master's thesis, Indiana University, 1973),
pp. 109-110. While using a division of the fourth into "60 parts"
for this Classic context, he elsewhere specifies integer ratios
(e.g. al-Farabi's 108:96:88:81 or 9/8-12:11-88:81 as the basis of
the Byzantine Diatonic) or defines tetrachords and modes in terms
of 68-EDO.

2. This is a permutation of Tetrachord #161 in John Chalmers,
_Divisions of the Tetrachord_, p. 174, 26/25 x 25/24 x 16/13.

Most appreciatively, and a Happy New Year,

Margo Schulter

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

> Please let me begin by expressing my thanks and admiration, as
> always, for your monumental achievement in _Musical Mathematics_.
> At the outset, I would also invite any comments from John
> Chalmers, since the question of how to interpret al-Farabi's
> chromatic genus (Arabic _jins_, plural _ajnas_) we are here
> discussing is one also relevant to _Divisions of the
> Tetrachord_.

Speaking of Al-Farabi's Chromatic, I just added this:

http://xenharmonic.wikispaces.com/Graph-theoretic+properties+of+scales#Seven note scales-Gypsy

This is a marvel tempering of, among other things, a scale which claims to be a permutation of Al-Farabi's Chromatic which Scala has listed as "al-farabi_chrom2", and of which I still want to know if it has any connection with Al-Farabi really.

! al-farabi_chrom2.scl
!
Al-Farabi's Chromatic permuted
7
!
16/15
56/45
4/3
3/2
8/5
28/15
2/1

Dear Margo,

Thank you for your kind response. Since Al-
Farabi's entire discussion takes place in the context
of his `ud strings, I interpret his tuning description
as an intentional departure from the non-
mathematical, non-sonorous Aristoxenian model. In
any case, since Al-Farabi's Strong Disjunct (Firm)
tetrachord explicitly includes interval ratio 13/12,
(MM, pp. 662-663) I also interpret his work as an
inspiration to Ibn Sina's monumental achievements.

Thank you again for your thoughts, which I always
enjoy very much.

Sincerely,

Cris

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> In a recent post, I wrote:
>
> >> To cut to the chase, Ibn Sina (980-1037) is the first music
> >> theorist of which I know, at least in the Greek and partly
> >> derivative European and Near Eastern traditions to address
> >> prime 13,
>
> Cris Forster responded:
>
> > Dear Margo,
>
> > It may be relevant to consider the significance of Al-Farabi's Jins 6, which
> > includes 16/13 and 64/39. MM. pp. 647-648, 650-655.
>
> > Cris
>
> Dear Cris,
>
> Please let me begin by expressing my thanks and admiration, as
> always, for your monumental achievement in _Musical Mathematics_.
> At the outset, I would also invite any comments from John
> Chalmers, since the question of how to interpret al-Farabi's
> chromatic genus (Arabic _jins_, plural _ajnas_) we are here
> discussing is one also relevant to _Divisions of the
> Tetrachord_.
>
> Certainly I agree that it is "relevant to consider the
> significance" of al-Farabi's Jins 6, which in one reading -- the
> one you may quite correctly consider the best -- would include
> 16/13 and 64/39. And I further agree that it is appropriate to
> mention this possible earlier use of 13 when discussing the
> contributions of Ibn Sina, who made ratios of 13 not only an
> explicit element of music theory but a central part of his
> musical practice, as in his `oud tuning adopting 39/32 as the
> wusta Zalzal or neutral third fret.
>
> Revisiting the relevant passages of _Musical Mathematics_, I learn
> that al-Farabi's Jins 6 is specified, more or less in the manner
> of the Aristoxenian Cleonides, as 44-8-8 parts of a fourth, or
> 1-5/6, 1/3, 1/3 parts of a tone. (Cleonides divides a tone into
> 12 parts, or a fourth into 30 parts, but sometimes uses interval
> sizes like 4-1/2 parts; so he, like al-Farabi, is effectively
> following a 60-part division of the fourth.)[1]
>
> Unlike Ibn Sina, al-Farabi does not explicitly "address prime
> 13." Rather, he describes a chromatic jins where you make a case
> for a rational interpretation of 32:26:25:24 or 16/13-26/25-25/24
> (359.5-67.9-70.7 cents).[2]
>
> As you candidly note at p. 650, the calculations leading to this
> specific interpretation "are _not_ straightforward" (emphasis in
> original). They assume, however reasonably based on al-Farabi's
> text and context, that he intends a rational division; that this
> division is to be based on the 9/8 tone; and that it is to be
> based more specifically on the 9/8 tone found on the `oud between
> the frets 32/27 and 4/3.
>
> Honestly, the best interpretation of al-Farabi's 44-8-8 parts in
> Jins 6 is a problem I might best leave to you and other scholars
> who specialize in this aspect of theory and have the linguistic
> as well as mathematical competence needed to make the most just
> assessment, in the Near East or elsewhere.
>
> But this is a good occasion to note both the tentative nature of
> many interpretations and the virtue of nevertheless daring to
> offer them after carefully considering all the evidence and
> weighing the likeliest hypotheses from the perspective of the
> theorist's text and context, as you have done for al-Farabi.
>
> ---------
>
> 1. Interestingly, Chrysanthos of Madytos, the famous scholar and
> theorist of Byzantine music in the earlier 19th century, mentions
> a division of 8-8-44 parts as one of the "shades" known to
> Eucleides. See his _Great Theory of Music_, in a translation by
> Katy G. Romanou (Master's thesis, Indiana University, 1973),
> pp. 109-110. While using a division of the fourth into "60 parts"
> for this Classic context, he elsewhere specifies integer ratios
> (e.g. al-Farabi's 108:96:88:81 or 9/8-12:11-88:81 as the basis of
> the Byzantine Diatonic) or defines tetrachords and modes in terms
> of 68-EDO.
>
> 2. This is a permutation of Tetrachord #161 in John Chalmers,
> _Divisions of the Tetrachord_, p. 174, 26/25 x 25/24 x 16/13.
>
>
> Most appreciatively, and a Happy New Year,
>
> Margo Schulter
>

> Dear Margo,

> Thank you for your kind response. Since Al- Farabi's
> entire discussion takes place in the context of his `ud
> strings, I interpret his tuning description as an
> intentional departure from the non- mathematical,
> non-sonorous Aristoxenian model. In any case, since
> Al-Farabi's Strong Disjunct (Firm) tetrachord explicitly
> includes interval ratio 13/12, (MM, pp. 662-663) I also
> interpret his work as an inspiration to Ibn Sina's
> monumental achievements. Thank you again for your
> thoughts, which I always enjoy very much.

Dear Cris,

Thank you for this example! I stand corrected, of course.

And I'll make a pre-New Years' resolution to cite this
definitive example of al-Farabi, as well as your
interpretation of his Jins 6, whenever I mention Ibn Sina in
connection with prime 13! Bravo!

The Strong Disjunct (Firm) tetrachord, if I'm right, is then
572:520:480:429 or 11/10-13/12-160/143 (165.0-138.6-194.468
cents).

An interesting aside here is that the interval between the
lowest and third notes of this tetrachord is 143/120, or
303.577 cents. As it happens, this is very close to
al-Farabi's Persian third finger at 81/68 or 302.865 cents,
a difference of 2431/2430 or 0.712 cents.

The topic al-Farabi's Jins 6 also led me to note that his
Jins 7 as you have interpreted it includes a chromatic step
of 128/105 or 342.905 cents. This differs from 39/32
(342.483 cents), an interval which was to become Ibn Sina's
wusta Zalzal but is also found in your octave species based
on Jins 6, by a superparticular ratio of 4096/4095 (0.423
cents), used in the Sagittal notation system, and sometimes
called the Sagittal schismina.

So not only have you enlightened me on al-Farabi's explicit
use of 13/12 and evidence pointing to 16/13 as well, but you
have led me to a 2431/2430 distinction clearly present
between his intervals of 143/120 and 81/68, as well as one
of 4096/4095 between 39/32 and 128/105 in your readings of
Jins 6 and Jins 7.

My warmest thanks for being set straight, and in a most
educational way!

Happy Holidays,

Margo

Gene wrote:

> This is a marvel tempering of, among other things, a
> scale which claims to be a permutation of Al-Farabi's
> Chromatic which Scala has listed as "al-farabi_chrom2",
> and of which I still want to know if it has any
> connection with Al-Farabi really.

> ! al-farabi_chrom2.scl
> !
> Al-Farabi's Chromatic permuted
> 7
> !
> 16/15
> 56/45
> 4/3
> 3/2
> 8/5
> 28/15
> 2/1

Hi, Gene. The short story is that this one is for real: the
original tuning of al-Farabi, that is, of which this is
indeed a permutation. As given by Cris Forster at pp. 663 of
_Musical Mathematics_, #3, al-Farabi's tuning is
7/6-16/15-15/14 or, if I'm correct, 56:48:45:42.

In the original, the large interval of 7/6 is first, then
16:15 (for 56/45), and finally 15:14 -- or 0-267-379-498.
This could be a modern Turkish Sazkar, I might guess, in a
style where the third of the related Makam Rast is placed
near but a tad below 5/4 -- in theory a schisma lower, but
in practice sometimes a bit more (Ozan Yarman suggested that
382 cents might be a sweet spot, and 56/45 about the lower
limit for this type of Rast).

However, a permutation like al-farabi_chrom2.scl placing the
large chromatic step in the middle is most characteristic of
maqam/makam music, and is generally known as a Hijaz
tetrachord. By the 13th century, Safi al-DIn al-Urmawi
advises that it is best to try all six permutations of a
tetrachord; and by 1300 or so, Qutb al-Din al-Urmawi defines
another type of chromatic, 12:11-7:6-22:21, as Hijaz. This
has a higher third step; but for musicians who liked a third
near 5/4, something like 112:105:90:84 or 16:15-7:6-15:14
might be quite attractive. In modern terms, it would be seen
as one fine shading of 5-12-5 commas.

Again, al-Farabi's chromatic is for real, as is the
permutation as something that people might well have been
doing at various points during that general era, and which
Qutb al-Din confirms that people were doing by around 1300.

With best New Year's wishes,

Margo

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

> Hi, Gene. The short story is that this one is for real: the
> original tuning of al-Farabi, that is, of which this is
> indeed a permutation.

Are we talking about a permutation of a tetrachord? Can you give an octave scale for Al-Farabi's Chromatic?

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

> Are we talking about a permutation of a tetrachord? Can you give an octave scale for Al-Farabi's Chromatic?

"al-farabi_chrom2" goes 16/15, 7/6, 15/14, 9/8, 16/15, 7/6, 15/14 in terms of steps. "al-farabi", called "Al-Farabi Syn Chrom" (I hate these truncated names) goes 16/15, 15/14, 7/6, 9/8, 16/15, 15/14, 7/6. Permuting the tetrachord of al-farabi chrom2 as you suggest gives 7/6, 15/14, 16/15, 9/8, 7/6, 15/14, 16/15, which as a scale turns into 7/6-5/4-4/3-3/2-7/4-15/8-2, which is a mode of "al-farabi_g10", called "Al-Farabi's Greek genus chromaticum forte", and is also inverse to "Al-Farabi Syn Chrom". I'm left unclear on the question of whether "al-farabi_chrom2" can reasonably be attributed to Al-Farabi.

> Dear Cris,
>
> Thank you for this example! I stand corrected, of course.
>
> And I'll make a pre-New Years' resolution to cite this
> definitive example of al-Farabi, as well as your
> interpretation of his Jins 6, whenever I mention Ibn Sina in
> connection with prime 13! Bravo!
>
> The Strong Disjunct (Firm) tetrachord, if I'm right, is then
> 572:520:480:429 or 11/10-13/12-160/143 (165.0-138.6-194.468
> cents).

This is correct.

> An interesting aside here is that the interval between the
> lowest and third notes of this tetrachord is 143/120, or
> 303.577 cents. As it happens, this is very close to
> al-Farabi's Persian third finger at 81/68 or 302.865 cents,
> a difference of 2431/2430 or 0.712 cents.

This is not only correct, but highly insightful.

> The topic al-Farabi's Jins 6 also led me to note that his
> Jins 7 as you have interpreted it includes a chromatic step
> of 128/105 or 342.905 cents. This differs from 39/32
> (342.483 cents), an interval which was to become Ibn Sina's
> wusta Zalzal but is also found in your octave species based
> on Jins 6, by a superparticular ratio of 4096/4095 (0.423
> cents), used in the Sagittal notation system, and sometimes
> called the Sagittal schismina.

Yes, the inversion of 64/39 leads directly to Ibn Sina's wusta al Zalzal 39/32. Another brilliant observation. Thank you.

> So not only have you enlightened me on al-Farabi's explicit
> use of 13/12 and evidence pointing to 16/13 as well, but you
> have led me to a 2431/2430 distinction clearly present
> between his intervals of 143/120 and 81/68, as well as one
> of 4096/4095 between 39/32 and 128/105 in your readings of
> Jins 6 and Jins 7.

A wonderful summation. And thank you especially also for your implicit endorsement (I don't want to put words in your mouth) that Al-Farabi's "parts" or "step" may have a physical presence on his sound-producing `ud strings.

> My warmest thanks for being set straight, and in a most
> educational way!
>
> Happy Holidays,
>
> Margo

Dear Margo,

I wish you a wonderful New Year and many moments of great insight
for all of us to value and appreciate.

Sincerely,

Cris

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

> I'm left unclear on the question of whether "al-farabi_chrom2" can reasonably be attributed to Al-Farabi.

Margo tells us that Safi al-Din al-Urmawi tells us that you should always try all six permutations of a given tetrachord, and I think this is the justification for attributing this scale to Al Farabi. If we marvel temper, the six permutations become three: Tss, sTs and ssT, where T is a subminor third and s is a secor. If we project to the 5-limit, T becomes 75/64, the classic augmented second, and s becomes 16/15. The first permutation, Tss, leads to a 5-limit version of 75/64-5/4-4/3-3/2-225/128-15/8-2. Adjusting by 225/224, this is equivalent to 7/6-5/4-4/3-3/2-7/4-15/8-2, which is inverse to al-farabi and is a mode of al-farabi_g10, and is one of al-Urmawi's permutations: 7/6, 15/14, 16/15, 9/8, 7/6, 15/14, 16/15. The second permutation sTs, leads to Gypsy. The third permutation, ssT, has a 5-limit version 16/15-256/225-4/3-3/2-8/5-128/75-2, which can be adjusted to 16/15-8/7-4/3-3/2-8/5-12/7-2, which is al-farabi.

Adding 32/25 to the 5-limit projection of a-farabi leads to a scale which is nicely symmetrical in the 5-limit lattice of pitch classes and which does not seem to correspond to any known scale. Another mode of the same scale can be obtained by adding 25/16 to the 5-limit version of the first permutation, hence both scales can be played within this 8-note scale.

> Are we talking about a permutation of a tetrachord? Can you give an
> octave scale for Al-Farabi's Chromatic?

Dear Gene,

Indeed we are! While you canvass some of the possibilities
and intonational variations or temperings in your subsequent
posts, I'll go through all six permutations of the
tetrachord we're discussing, and also the conjunct and
disjunct octave species from each. So we'll have 12 octave
species in all.

Whether all permutations of a given jins (Arab for "genus")
may rightly be attributed to the person describing it is
another question, with Safi al-Din al-Urmawi's advice to
consider all the permutations as one possible basis for
taking this view. In some cases, however, we may have either
practical knowledge or contextual clues suggesting that a
given theorist or style would _not_ equally value all six
permutations or accept them all as idiomatic. That may be
another discussion, and here I just want to acknowledge that
while all six permutations will be interesting, that doesn't
necessarily mean that all six will be used in practice -- at
least at one given time and place.

Now to our tetrachord of al-Farabi, 7:6-16:15-15:14 or
56:48:45:42 (1/1-7/6-56/45-4/3), and its permutations and
octave species.

Original tetrachord (Permutation 0) -- Conjunct

lower upper 9/8
|----------------|-------------------|.....|
1/1 7/6 56/45 4/3 14/9 224/135 16/9 2/1
7:6 16:15 15:14 7:6 16:15 15:14 9:8

Original tetrachord (Permutation 0) -- Disjunct

lower 9/8 upper
|----------------|.....|----------------|
1/1 7/6 56/45 4/3 3/2 7/4 28/15 2/1
7:6 16:15 15:14 9:8 7:6 16:15 15:14

Note that we get this tetrachord by taking 7:6 as the
chromatic interval greater than (4/3)^(1/2), and having a
remainder or fourth's complement of 8:7, which al-Farabi
divides into two superparticular intervals, 16:15:14. This
form reflects, among other things, both a general preference
for superparticular steps where they are available, and a
tendency to favor arithmetic divisions like 16:15:14 here.
But as we go through the permutations, we'll encounter
different orderings of these steps.

I should add that while the _lower_ tetrachord might be a
Turkish Sazkar, in that makam the upper tetrachord would be
Rast, if I am correct. Often in maqam/makam, there's a
tendency to think of the lower tetrachord as the "root" one,
and mainly definitive of a given "family" -- with the upper
tetrachord more subject to variation without changing the
basic identity of the family. So if I say, "Yes, that's
al-Farabi," or "Yes, that's a permutation of one his
tunings," I'll often be focusing on the lower tetrachord.
Clarifying questions like the ones you asked about an octave
species are thus very helpful.

Permutation 1 -- Conjunct

lower upper 9/8
|----------------|-------------------|.....|
1/1 16/15 8/7 4/3 64/45 32/21 16/9 2/1
16:15 15:14 7:6 16:15 15:14 7:6 9:8

Permutation 1 -- Disjunct

lower 9/8 upper
|-----------------|.....|----------------|
1/1 16/15 8/7 4/3 3/2 8/5 12/7 2/1
16:15 15:14 7:6 9:8 16:15 15:14 7:6

This permutation is like a typical ancient Greek chromatic,
with smallest step first, then middle step, and the large
chromatic step last going up to the 4/3 step of the
tetrachord. Interestingly, the positions with the large
chromatic step as the highest seem least characteristic of
maqam music today, as systematists such as Amine Beyhom have
concluded. Having the large step in the middle is most
characteristic (Arab or Turkish Hijaz, Persian Chahargah),
but Sazkar has it in the lowest position, as in our original
tetrachord.

Permutation 2 -- Conjunct

lower upper 9/8
|-----------------|-------------------|.....|
1/1 15/14 5/4 4/3 10/7 5/3 16/9 2/1
15:14 7:6 16:15 15:14 7:6 16/15 9:8

Permutation 2 -- Disjunct

lower 9/8 upper
|-----------------|.....|----------------|
1/1 16/15 8/7 4/3 3/2 45/28 15/8 2/1
15:14 7:6 16:15 9:8 15:14 7:6 16:15

This could be an excellent modern Turkish Hijaz, following
the often preferred but not invariable pattern having the
lower step larger than the upper (15:14, 16:15), with the
large chromatic step in the middle. More specifically, the
disjunct form would be Makam Hijazkar. The form with two
conjunct Hijaz tetrachords is sometimes known in the Arab
world as Hijazayn or the like -- a twin or double Hijaz.
While octave species with conjunct tetrachords are often
preferred in medieval theory -- a general comment, not one
relating especially to Hijaz -- both conjunct and disjunct
forms are described (in keeping with Greek theory), and in
modern theory and practice disjunct forms are sometimes the
most common (e.g. modern Rast, Hijazkar).

So, far, we've been doing simple rotation, leaving the
sequence of intervals in 7/6, 16/15, 15/14 alone and simply
changing where we start in the sequence. For the remaining
three permutations, we'll reverse the order of 15/14 and
16/15.

Permutation 3 -- Conjunct

lower upper 9/8
|----------------|-------------------|.....|
1/1 7/6 5/4 4/3 14/9 5/3 16/9 2/1
7:6 15:14 16:15 7:6 15:14 16:15 9:8

Permutation 3 -- Disjunct

lower 9/8 upper
|----------------|.....|----------------|
1/1 7/6 5/4 4/3 3/2 7/4 15/8 2/1
7:6 15:14 16:15 9:8 7:6 15:14 16:15

Like the original, this could be a shading of Turkish
Sazkar, I would guess. Some Near Eastern theory recognizes
5/4 itself as a correct interval to use in a high Rast, for
example (e.g. Qutb al-Din al-Shirazi), although there is a
school of Turkish theory, as I mentioned, favoring a
position somewhere around 2-7 cents lower for a Rast in this
region -- and the same might apply to Sazkar, which would
have an upper tetrachord, if the Arab model applies to
Turkish music, of 3/2 27/16 15/8 2/1.

Permutation 4 -- Conjunct

lower upper 9/8
|----------------|-------------------|.....|
1/1 15/14 8/7 4/3 0/7 32/21 16/9 2/1
15:14 16:15 7:6 15:14 16:15 7:6 9:8

Permutation 4 -- Disjunct

lower 9/8 upper
|----------------|.....|-----------------|
1/1 15/14 8/7 4/3 3/2 45/28 12/7 2/1
15:14 16:15 7:6 9:8 15:14 15:14 7:6

This is another Greek type of chromatic with the upper
interval as the large step, here 7:6, and a harmonic
division of the remaining 8:7 interval -- in modern
frequency ratios, 14:15:16. A string division for this
tetrachord would be 120:112:105:90. Note that in traditional
theory, the concept of a "harmonic division" is defined in
terms of string lengths, with the first three terms the
relevant ones. In 120:112:105, the difference of the first
two terms is 8, while that of the last two is 7 -- forming a
ratio identical to that between the extreme terms, 120:105,
or 8:7. In some medieval theory (e.g. in Europe), the
harmonic division is seen as especially pleasing; but
arithmetic divisions are very common, and often easier to
find on a monochord or as fret positions.

Permutation 5 -- Conjunct

lower upper 9/8
|-----------------|----------------------|.....|
1/1 16/15 56/45 4/3 64/45 224/135 16/9 2/1
16:15 7:6 15:14 16:15 7:6 16:15 9:8

Permutation 5 -- Disjunct

lower 9/8 upper
|-----------------|.....|------------------|
1/1 16/15 56/45 4/3 3/2 8/5 28/15 2/1
16:15 7:6 15:14 9:8 16:15 7:6 15:14

This is the final permutation, and the one shown in
al-farabi_chrom2.scl. In modern Turkish terms, it would be
another shading of Hijaz. Either this or Permutation 2
follows the general pattern of 5-12-5 commas, with the
middle interval around 7:6 or slightly larger, and the two
outer intervals each around a 5-limit semitone -- or a
Pythagorean apotome at 2187/2048 (113.7 cents). Literally
interpreted, 5-12-5 commas could be read in terms either of
53-EDO steps, or of a Pythagorean apotome, a small minor
third equal to a "trilimma" or 15 fourths up less the
requisite number of octaves (270.7 cents), and another
apotome. In practice, the subtly unequal steps of 16:15 and
15:14 -- in either Permutation 2 or this one -- could be
seen as a pleasant refinement.

So that's a rundown of the six permutations and their octave
species, conjunct and disjunct. Again, Cris or John Chalmers
might have further comments or corrections.

Happy New Year,

Margo

One question I have i regard to this subject is the history of the word 'limit' in regard to tuning.

I have heard it said that Partch was the first to use it, which seems probably not right, but it might point to viewpoint that the ancients might not have thought about in that way. For instance the Greeks did not appear very shy in regard to using ratios without regard to them and possibly if they limited themselves to 11 it might been out of practice and convenience and not as being out of bounds. Did Euler use the term?

--
signature file

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

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

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

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

Margo Schulter <mschulter@...> wrote:

> Permutation 5 -- Disjunct
>
> lower 9/8 upper
> |-----------------|.....|------------------|
> 1/1 16/15 56/45 4/3 3/2 8/5 28/15 2/1
> 16:15 7:6 15:14 9:8 16:15 7:6 15:14
>
> This is the final permutation, and the one shown in
> al-farabi_chrom2.scl. In modern Turkish terms, it would be
> another shading of Hijaz. Either this or Permutation 2
> follows the general pattern of 5-12-5 commas, with the
> middle interval around 7:6 or slightly larger, and the two
> outer intervals each around a 5-limit semitone -- or a
> Pythagorean apotome at 2187/2048 (113.7 cents). Literally
> interpreted, 5-12-5 commas could be read in terms either
> of 53-EDO steps, or of a Pythagorean apotome, a small
> minor third equal to a "trilimma" or 15 fourths up less
> the requisite number of octaves (270.7 cents), and another
> apotome. In practice, the subtly unequal steps of 16:15
> and 15:14 -- in either Permutation 2 or this one -- could
> be seen as a pleasant refinement.

Ah, right, this. That 5-12-5 comma pattern is consistent with
225:224 being tempered out, giving marvel temperament. So this
is where Gene started. I'll copy it with meantone names on E:

lower 9/8 upper
|-----------------|.....|------------------|
E F G# A B C D# E
1/1 15/14 5/4 4/3 3/2 8/5 15/8 2/1
s 7:6 s 9:8 s 7:6 s

The "s" could be either 16:15 or 15:14. It doesn't require
any tempering other than marvel.

A link to the wiki with the right target:

http://xenharmonic.wikispaces.com/Graph-theoretic+properties+of+scales#Seven%20note%20scales-Gypsy

The same scale can be explained harmonically as a combination
of the E, B7/5, and F#6 chords. That is the tonic, E major;
the dominant 7th, B7, with an extended pythagorean tuning of
the seventh and the fifth removed (I think this is acceptable
for jazz harmony); and the marvel tempering of 4:5:6:7 that
shares the tritone of B7.

The 5-limit lattice shows that we have four triads even
without the septimal implications:

~~~~~~~~~~~~G#----D#
~~~~~~~~~~~/~\~~~/
~~~~~~~~~~/~~~\~/
~~~A-----E-----B
~~/~\~~~/
~/~~~\~/
F-----C

It looks promising as a scale for use with marvel
tritone substitutions.

Graham

Hi Kraig,

Ptolemy's "Catalog of Scales," (MM, pp. 330-332),
which consists of scales by Archytas, Aristoxenus
(as rational interpretations by Ptolemy),
Eratosthenes, Didymus, Ptolemy, and Philolaus,
includes the first twelve prime numbers: 2, 3, 5, 7,
11, 13, 17, 19, 23, 29, 31, 37.

On p. 330, I give a possible derivation of
Eratosthenes' Enharmonic Tetrachord, which
includes two 13-limit ratios: 40/39 and 39/38. To my
knowledge, this marks the first appearance of such
ratios.

Cris

--- In tuning@yahoogroups.com, kraiggrady <kraiggrady@...> wrote:
>
> One question I have i regard to this subject is the history of the word
> 'limit' in regard to tuning.
>
> I have heard it said that Partch was the first to use it, which seems
> probably not right, but it might point to viewpoint that the ancients
> might not have thought about in that way. For instance the Greeks did
> not appear very shy in regard to using ratios without regard to them and
> possibly if they limited themselves to 11 it might been out of practice
> and convenience and not as being out of bounds. Did Euler use the term?
>
> --
> signature file
>
> /^_,',',',_ //^/Kraig Grady_^_,',',',_
> Mesotonal Music from:
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> a momentary antenna as i turn to water
> this evaporates - an island once again
>

Thank you Cris~
This reminded me that the Greeks hence did not think in Limits at all, but as we know got to these often through the process of mediants. If we look at http://anaphoria.com/4-4to4-3scaletree.pdf we see how quickly these primes come up.

--- In tuning@yahoogroups.com, "c_ml_forster" <cris.forster@...> wrote:
>
> Hi Kraig,
>
> Ptolemy's "Catalog of Scales," (MM, pp. 330-332),
> which consists of scales by Archytas, Aristoxenus
> (as rational interpretations by Ptolemy),
> Eratosthenes, Didymus, Ptolemy, and Philolaus,
> includes the first twelve prime numbers: 2, 3, 5, 7,
> 11, 13, 17, 19, 23, 29, 31, 37.
>
> On p. 330, I give a possible derivation of
> Eratosthenes' Enharmonic Tetrachord, which
> includes two 13-limit ratios: 40/39 and 39/38. To my
> knowledge, this marks the first appearance of such
> ratios.
>
> Cris
>
>
> --- In tuning@yahoogroups.com, kraiggrady <kraiggrady@> wrote:
> >
> > One question I have i regard to this subject is the history of the word
> > 'limit' in regard to tuning.
> >
> > I have heard it said that Partch was the first to use it, which seems
> > probably not right, but it might point to viewpoint that the ancients
> > might not have thought about in that way. For instance the Greeks did
> > not appear very shy in regard to using ratios without regard to them and
> > possibly if they limited themselves to 11 it might been out of practice
> > and convenience and not as being out of bounds. Did Euler use the term?
> >
> > --
> > signature file
> >
> > /^_,',',',_ //^/Kraig Grady_^_,',',',_
> > Mesotonal Music from:
> > _'''''''_ ^North/Western Hemisphere:
> > North American Embassy of Anaphoria Island <http://anaphoria.com/>
> >
> > _'''''''_^South/Eastern Hemisphere:
> > Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
> >
> > ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
> >
> > a momentary antenna as i turn to water
> > this evaporates - an island once again
> >
>

Dear Gene,

The short and sweet answer to your question
is that al-Farabi's 7:6-16:15:15:14 tetrachord
or 56:48:45:42 is definitely given in rational
terms. Thus al-farabi_chrom2.scl is a permutation
of his actual tuning as defined in integer ratios.

As Cris discusses in his book, the interpretation
of al-Farabi's tunings given in "parts" of a tone
or fourth after the general manner of Cleonides
is more controversial; but a rational
interpretation may be the best alternative.

Clearly al-Farabi is oriented to a Ptolemaic
perspective favoring integer ratios and
especially superparticular ones as the main
basis for tunings, although, as Cris observes,
he may not be a bit more flexible on using
superpartient (neither multiplex like 2/1 nor
superparticular like 3/2) ratios than Ptolemy.

Assuming that al-Farabi's tunings in "parts"
are meant to be implemented, how would one
implement them on a fretted instrument, for
example? Some kind of rational division seems
a likely answer, and arithmetic divisions are
both the easiest to implement, as a general
rule, and the most characteristically used
in explicitly rational tunings, something
that Cris documents in great and admirable
detail.

But the 7:6-16:15-15:14 tuning is defined
as exactly that, so no interpretatin of
the step sizes is necessary, although we
might speculate as to which permutations
were used when or how often.

Best,

Margo

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
> A link to the wiki with the right target:
>
> http://xenharmonic.wikispaces.com/Graph-theoretic+properties+of+scales#Seven%20note%20scales-Gypsy
>
> The same scale can be explained harmonically as a combination
> of the E, B7/5, and F#6 chords. That is the tonic, E major;
> the dominant 7th, B7, with an extended pythagorean tuning of
> the seventh and the fifth removed (I think this is acceptable
> for jazz harmony); and the marvel tempering of 4:5:6:7 that
> shares the tritone of B7.
>
> The 5-limit lattice shows that we have four triads even
> without the septimal implications:
>
> ~~~~~~~~~~~~G#----D#
> ~~~~~~~~~~~/~\~~~/
> ~~~~~~~~~~/~~~\~/
> ~~~A-----E-----B
> ~~/~\~~~/
> ~/~~~\~/
> F-----C
>
> It looks promising as a scale for use with marvel
> tritone substitutions.

Yes, indeed! Other ways to look at Gypsy are as two stacked maj7 chords, or as the marvel scale that results when putting a 4:5:6:7 and a (1/7):(1/6):(1/5):(1/4) chord into the same 7:4, then filling the "hole" in the 5-limit lattice: An otonal F7, a utonal Fm7b5, and the note E to fill the gap in the 5-limit.

Another thing that I love about this scale is that it is a nice way to modulate between the major scale and orwell[9]:
(works at least in 31- and 53-edo)

1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1 (major scale)
1:1 16:15 5:4 4:3 3:2 8:5 15:8 2:1 (Gypsy)
1:1 16:15 7:6 5:4 4:3 3:2 8:5 12:7 15:8 2:1 (orwell[9] MODMOS)
1:1 16:15 7:6 5:4 11:8 16:11 8:5 12:7 15:8 2:1 (orwell[9])

--- In tuning@yahoogroups.com, "gedankenwelt94" <gedankenwelt94@...> wrote:

> Yes, indeed! Other ways to look at Gypsy are as two stacked maj7 chords, or as the marvel scale that results when putting a 4:5:6:7 and a (1/7):(1/6):(1/5):(1/4) chord into the same 7:4, then filling the "hole" in the 5-limit lattice: An otonal F7, a utonal Fm7b5, and the note E to fill the gap in the 5-limit.

You can have even more fun if you allow essentially tempered chords. The 5-limit version ("transversal") of the scale is 1-16/15-5/4-4/3-3/2-8/5-15/8-2, from note 0 to note 7. The dyads 5/4-3/2, 5/4-8/5, 4/3-3/2, and 4/3-8/5 when combined with 1 make for triads, and with 16/15 and 15/8, tetrads. To get the tetrads, add to one of the dyads 16/15 and 15/8, divide through by 16/15, and marvel-reduce to the least complex result; in this way obtaining 1-7/6-7/5-7/4, 1-7/6-3/2-7/4, 1-5/4-7/5-7/4 and 1-5/4-3/2-7/4. Similarly, add 1 to one of the dyads and obtain four triads which are not found as a part of an ambient tetrad, 1-5/4-3/2, 1-5/4-8/5, 1-4/3-3/2 and 1-4/3-8/5.

> Another thing that I love about this scale is that it is a nice way to modulate between the major scale and orwell[9]:
> (works at least in 31- and 53-edo)

Nice indeed!

> 1:1 9:8 5:4 4:3 3:2 5:3 15:8 2:1 (major scale)
> 1:1 16:15 5:4 4:3 3:2 8:5 15:8 2:1 (Gypsy)
> 1:1 16:15 7:6 5:4 4:3 3:2 8:5 12:7 15:8 2:1 (orwell[9] MODMOS)
> 1:1 16:15 7:6 5:4 11:8 16:11 8:5 12:7 15:8 2:1 (orwell[9])

In terms of orwell generators this is:

-10, -7, -3, 0, 4, 7, 14
-7, -4, -3, 0, 3, 4, 7
-7, -4, -3, -1, 0, 1, 3, 4, 7
-4, -3, -2, -1, 0, 1, 2, 3, 4

Are you known to any of us under an actual name, by the way?

"gedankenwelt94" <gedankenwelt94@...> wrote:

> Another thing that I love about this scale is that it is
> a nice way to modulate between the major scale and
> orwell[9]: (works at least in 31- and 53-edo)

Oh, yes, and this Gypsy is also a subset of the tripod
scale, which is how I originally found it. That may not be
of general interest, but what you have here is the tripod
scale:

> 1:1 16:15 7:6 5:4 4:3 3:2 8:5 12:7 15:8 2:1 (orwell[9]
> MODMOS)

Orwell[9] is indeed 2 pitches from the tripod scale in
any orwell tuning. That means it can be written in tripod
notation with two accidentals.

Graham

--- In tuning@yahoogroups.com, "genewardsmith" wrote:
> Are you known to any of us under an actual name, by the way?

Thanks for the question, but for now I've decided to use a pseudonym as
a microtonal musician. Not that it matters, knowing my real name
wouldn't give you much useful information, anyway.

Btw, we had a short discussion in the XA freeforums, if you remember
<http://www.abload.de/img/tsbtreerep5h.png> . ;)

--- In tuning@yahoogroups.com, "gedankenwelt94" wrote:
>
> --- In tuning@yahoogroups.com, "genewardsmith" wrote:
> > Are you known to any of us under an actual name, by the way?
>
> Thanks for the question, but for now I've decided to use
> a pseudonym as a microtonal musician. Not that it matters,
> knowing my real name wouldn't give you much useful
> information, anyway.

You're missing the point. All of the serious contributors
here use our real names. We're a community.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" wrote:
>
> --- In tuning@yahoogroups.com, "gedankenwelt94" wrote:
> >
> > --- In tuning@yahoogroups.com, "genewardsmith" wrote:
> > > Are you known to any of us under an actual name, by the way?
> >
> > Thanks for the question, but for now I've decided to use
> > a pseudonym as a microtonal musician. Not that it matters,
> > knowing my real name wouldn't give you much useful
> > information, anyway.
>
> You're missing the point. All of the serious contributors
> here use our real names. We're a community.
>
> -Carl

...and I still don't get your point. How is it important for the community to know my "real name"? Isn't a name I chose by myself more personal, anyway?

However, there are two attributes I expect from a microtonal community: To be undogmatic, and tolerant. If this community expects me to use my "real name" here, it is neither.

-Gedankenwelt

P.S.: It's funny this discussion came up after I mentioned Orwell. :)

--- In tuning@yahoogroups.com, "gedankenwelt94" wrote:

> > > Thanks for the question, but for now I've decided to use
> > > a pseudonym as a microtonal musician. Not that it matters,
> > > knowing my real name wouldn't give you much useful
> > > information, anyway.
> >
> > You're missing the point. All of the serious contributors
> > here use our real names. We're a community.
> >
> > -Carl
>
> ...and I still don't get your point. How is it important for
> the community to know my "real name"? Isn't a name I chose
> by myself more personal, anyway?
>
> However, there are two attributes I expect from a microtonal
> community: To be undogmatic, and tolerant. If this community
> expects me to use my "real name" here, it is neither.

Using real names is a big benefit to our community, whether
it's to do with meeting up in real life, publishing/citing
work, or just bringing us closer together.

Would you show up to work in a ski mask? If everyone wore a
ski mask at your office it would be no issue. The issue is
that everybody else is offering something and you're not.
It's nothing to do with "tolerance" or some other dogmatic
idea that *you're* invoking. People have used pseudonyms
here before and of course we're always grateful to have their
contributions -- I'm already grateful for yours! But doing
so naturally produces distance and is borderline disrespectful,
in a way that only a moron could fail to understand.

You said you want to use a pseudonym as a microtonal musician.
Stage names are more common in the performing arts. Have you
made any music?

-Carl

On Sat, Jan 19, 2013 at 1:24 PM, Carl Lumma <carl@...> wrote:
>
> But doing
> so naturally produces distance and is borderline disrespectful,
> in a way that only a moron could fail to understand.

As you can see, we pride ourselves on tolerance around here.

"Gedankenwelt94," I suggest checking out the Xenharmonic Alliance
Facebook group, which is here:
http://www.facebook.com/groups/xenharmonic2/

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia wrote:
>
> "Gedankenwelt94," I suggest checking out the Xenharmonic Alliance
> Facebook group, which is here:
> http://www.facebook.com/groups/xenharmonic2/

http://www.facebook.com/help/112146705538576/

-Carl

Gedankenwelt, the real names of a musician are the musics they make. All the serious contributors here upload music they make to illustrate- emody, realize- tuning ideas.

--- In tuning@yahoogroups.com, "gedankenwelt94" wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" wrote:
> >
> > --- In tuning@yahoogroups.com, "gedankenwelt94" wrote:
> > >
> > > --- In tuning@yahoogroups.com, "genewardsmith" wrote:
> > > > Are you known to any of us under an actual name, by the way?
> > >
> > > Thanks for the question, but for now I've decided to use
> > > a pseudonym as a microtonal musician. Not that it matters,
> > > knowing my real name wouldn't give you much useful
> > > information, anyway.
> >
> > You're missing the point. All of the serious contributors
> > here use our real names. We're a community.
> >
> > -Carl
>
> ...and I still don't get your point. How is it important for the community to know my "real name"? Isn't a name I chose by myself more personal, anyway?
>
> However, there are two attributes I expect from a microtonal community: To be undogmatic, and tolerant. If this community expects me to use my "real name" here, it is neither.
>
> -Gedankenwelt
>
> P.S.: It's funny this discussion came up after I mentioned Orwell. :)
>

How do we know what is a real name or not? maybe there is not such person as Carl Lumma and you are trying to cover that up by insisting on others real names:)
,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

--- In tuning@yahoogroups.com, kraiggrady@... wrote:
>
>
> How do we know what is a real name or not? maybe there is not such person as Carl Lumma and you are trying to cover that up by insisting on others real names:)

I've met Carl Lumma if that helps any.

I don't actually care, I'm just pointing out what I consider to
be a common but obviously wrongheaded approach to such matters.

The Carl Lumma who visited your house in LA and met you backstage
in SF a decade later sends his regards.

-Carl L.

--- In tuning@yahoogroups.com, kraiggrady@... wrote:
>
> How do we know what is a real name or not? maybe there is not
> such person as Carl Lumma and you are trying to cover that up
> by insisting on others real names:)
> ,',',',Kraig Grady,',',',
> '''''''North/Western Hemisphere:
> North American Embassy of Anaphoria island
> '''''''South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>

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

> I've met Carl Lumma if that helps any.

But have you encountered "Beefman" on the citizens' band,
Wikipedia, or Hacker News? If so, he owes me some money.

-Carl

It is hard to argue with corporeal presence as proof:)

,',',',Kraig Grady,',',',
'''''''North/Western Hemisphere:
North American Embassy of Anaphoria island
'''''''South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria
',',',',',',',',',',',',',',',',',',',',',',',',',',',',',