more on the "Arabian comma"

From "The Music of the Arabs" by Habib Hassan Touma, pp. 22-3:

...

Some Arabian theorists, of course, attempted to define the Arabian tone
system, with reference to the circle of fifths (which was also used by
Turkish theorists), on the basis of the unit of measure known as the comma.
The Syrians, in particular, subdivided the octave into fifty-three
equivalent steps, from which they also selected seven at a time to construct
a heptatonic maqam row. Their calculations were based neither on the
Pythagorean comma (23.46 cents) nor on the syntonic comma (21.306 cents) but
on the so-called Arabian comma, also known as the Holdrian comma, whose
value is 53 *root* 2, or 22.6415 cents. The term Arabian comma is indeed
found only in the European treatises of the Middle Ages, not in the Arabian.
The absence of written documentation in the Arabian world may possibly be
explained by the fact that the comma was handed down from master to student
as a kind of trade secret.

Indeed, as early as 45 B.C., the Chinese Ching-Fang had calculated the value
of this comma. He discovered that the highest tone row of fifty-three
natural fifths built on top of the other is almost identical to the lowest
tone of the row, if the fifty-third fifth is transposed down by thirty-one
octaves. Thus the ratio of the lowest tone of the row to the transposed
higher tone (that is, (3/2)^53 minus 2^31) is 176777/177147, which
corresponds to the value of the Arabian comma.

---------------------------------

First he said the Arabian comma was Holdrian, then he says it is the
Mercator of 3.62 cents.

As for the "trade secret", I personally think it a far-fetched tale. The
first mention of the comma in reference to Maqam Music is by Antoine Murat
(an Europeanized Turk of the late 18th century), who conspicuously implies
55-EDO.

Oz.

Hi Oz,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> From "The Music of the Arabs" by Habib Hassan Touma, pp. 22-3:
>
> ...
>
> Some Arabian theorists, of course, attempted to define
> the Arabian tone system, with reference to the circle of
> fifths (which was also used by Turkish theorists), on the
> basis of the unit of measure known as the comma. The Syrians,
> in particular, subdivided the octave into fifty-three
> equivalent steps, from which they also selected seven
> at a time to construct a heptatonic maqam row. Their
> calculations were based neither on the Pythagorean comma
> (23.46 cents) nor on the syntonic comma (21.306 cents) but
> on the so-called Arabian comma, also known as the Holdrian
> comma, whose value is 53 *root* 2, or 22.6415 cents. The
> term Arabian comma is indeed found only in the European
> treatises of the Middle Ages, not in the Arabian. The absence
> of written documentation in the Arabian world may possibly be
> explained by the fact that the comma was handed down from
> master to student as a kind of trade secret.
>
> Indeed, as early as 45 B.C., the Chinese Ching-Fang had
> calculated the value of this comma. He discovered that the
> highest tone row of fifty-three natural fifths built on top
> of the other is almost identical to the lowest tone of the
> row, if the fifty-third fifth is transposed down by thirty-one
> octaves. Thus the ratio of the lowest tone of the row to the
> transposed higher tone (that is, (3/2)^53 minus 2^31) is
> 176777/177147, which corresponds to the value of the
> Arabian comma.
>
> ---------------------------------
>
> First he said the Arabian comma was Holdrian, then he says
> it is the Mercator of 3.62 cents.

You sure are correct about that.

However, even his ratio is off: (3/2)^53 / 2^31 (which is
what he really means mathematically) has a ratio with
terms which are so huge that my spreadsheet won't
calculate them accurately -- so it's not 176777/177147,
and anyway the numerator should be larger than the denominator,
so those terms should be switched.

> As for the "trade secret", I personally think it a far-fetched
> tale. The first mention of the comma in reference to Maqam
> Music is by Antoine Murat (an Europeanized Turk of the late
> 18th century), who conspicuously implies 55-EDO.

Are you sure about that? 55-edo behaves very differently
from 53-edo.

-monz
http://tonalsoft.com
Tonescape microtonal music software

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

> However, even his ratio is off: (3/2)^53 / 2^31 (which is
> what he really means mathematically) has a ratio with
> terms which are so huge that my spreadsheet won't
> calculate them accurately -- so it's not 176777/177147,
> and anyway the numerator should be larger than the denominator,
> so those terms should be switched.

3^53/2^81 = 19383245667680019896796723/19342813113834066795298816

176777 is a prime, and 177147 = 3^11. I think these numbers have
something to do with Ching-Fang's computations; I seem to recall some
such approximation to the comma (which this is, or is if you invert
it) arising.

> > As for the "trade secret", I personally think it a far-fetched
> > tale. The first mention of the comma in reference to Maqam
> > Music is by Antoine Murat (an Europeanized Turk of the late
> > 18th century), who conspicuously implies 55-EDO.

> Are you sure about that? 55-edo behaves very differently
> from 53-edo.

But it might also work for maqam music.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 26 Mart 2007 Pazartesi 19:18
Subject: [tuning] Re: more on the "Arabian comma"

> Hi Oz,
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
> >
> > From "The Music of the Arabs" by Habib Hassan Touma, pp. 22-3:
> >
> > ...
> >
> > Some Arabian theorists, of course, attempted to define
> > the Arabian tone system, with reference to the circle of
> > fifths (which was also used by Turkish theorists), on the
> > basis of the unit of measure known as the comma. The Syrians,
> > in particular, subdivided the octave into fifty-three
> > equivalent steps, from which they also selected seven
> > at a time to construct a heptatonic maqam row. Their
> > calculations were based neither on the Pythagorean comma
> > (23.46 cents) nor on the syntonic comma (21.306 cents) but
> > on the so-called Arabian comma, also known as the Holdrian
> > comma, whose value is 53 *root* 2, or 22.6415 cents. The
> > term Arabian comma is indeed found only in the European
> > treatises of the Middle Ages, not in the Arabian. The absence
> > of written documentation in the Arabian world may possibly be
> > explained by the fact that the comma was handed down from
> > master to student as a kind of trade secret.
> >
> > Indeed, as early as 45 B.C., the Chinese Ching-Fang had
> > calculated the value of this comma. He discovered that the
> > highest tone row of fifty-three natural fifths built on top
> > of the other is almost identical to the lowest tone of the
> > row, if the fifty-third fifth is transposed down by thirty-one
> > octaves. Thus the ratio of the lowest tone of the row to the
> > transposed higher tone (that is, (3/2)^53 minus 2^31) is
> > 176777/177147, which corresponds to the value of the
> > Arabian comma.
> >
> > ---------------------------------
> >
> > First he said the Arabian comma was Holdrian, then he says
> > it is the Mercator of 3.62 cents.
>
>
> You sure are correct about that.
>
> However, even his ratio is off: (3/2)^53 / 2^31 (which is
> what he really means mathematically) has a ratio with
> terms which are so huge that my spreadsheet won't
> calculate them accurately --

The ratio is 21519725632224173557990013483376 /
21474836480000000000000000000000 equalling 3.615046 cents.

so it's not 176777/177147,
> and anyway the numerator should be larger than the denominator,
> so those terms should be switched.
>

Yes, because the result is -3.61974227 cents.

>
>
> > As for the "trade secret", I personally think it a far-fetched
> > tale. The first mention of the comma in reference to Maqam
> > Music is by Antoine Murat (an Europeanized Turk of the late
> > 18th century), who conspicuously implies 55-EDO.
>
>
> Are you sure about that? 55-edo behaves very differently
> from 53-edo.
>

From Aksoy, B. "Music of the Ottomans from the Perspective of European
Travellers", 'cf. Antoine Murat', pp. 163-5 (title and contents translated
by me):

...

The compass of Turkish Music spans two octaves. An octave is divided into
two tetrachords, each of whose tonic equals the corresponding tones of
European Music. The tonic of the I. tetrachord is 'SOL', the II. is 'RE'. I.
Tetrachord comprises the tones 'SOL-LA-SI-DO' [rast-dugah-segah-chargah],
and II. 'RE-MI-FA-SOL' [neva-huseyni-evj-gerdaniye]. Each of the tetrachords
possesses three large natural intervals. Moreover, each interval is split
into small intervals of equal value called "commas". The first 9 commas form
the primary basic interval [9:8]. This interval is betwixt the tonic 'sol'
and its major second. The 1. comma between 'SOL-LA' makes a comma interval,
3. comma a "diminished second", 5. comma a "minor second"; at the 9. comma,
the interval of a "major second" is completed. In the 7 comma interval of
'LA-SI', 1. comma yields a "diminished third", 3. comma a "minor second",
and 7. comma (i.e., 16. comma after the tonic) a "major third" [27:22]. The
3. comma of the 'SI-DO' interval compounds the "augmented third", and a
perfect fourth is achieved at the 23. comma [4:3].

The II. tetrachord is divided similar to the I tetrachord. The two
tetrachords are in exact symmetry with each other. The 'DO-RE' major tone
interval that seperates the two tetrachords is divided into smaller
intervals of 1,3,5 commas just like the intervals of 'SOL-LA' and 'RE-MI'.
The tonic of the II. tetrachord forms the fifth of the I. There are 32
commas in this pentachord. In the entirety of the composite tetrachords made
of the union of two disjunct tetrachords, there are 55 commas. As may be
seen, amid each of the 7 intervals forming the octave, there are 3
intermediate tones. Via the division of the primary intervals into three, it
is observed that there are 21 intervals and 22 perdes in the Turkish octave.

...

He also goes on to say that segah is a comma lower than the European E and
evj 3 commas higher than F, and remarks that one might think all this a
cacophony at first, but indispensible nonetheless for Turkish Music - so
much so, that replacing the European SI and FA in place of segah and evj
would result in cacophony. He also says that commatic differences arise from
maqam to maqam and major seconds may be transformed from 9 commas to 8, or
even 6 and that major thirds could frolic about 17, 15 and 14.

>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>

Oz.

Hi Gene,

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

> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > However, even his ratio is off: (3/2)^53 / 2^31 (which is
> > what he really means mathematically) has a ratio with
> > terms which are so huge that my spreadsheet won't
> > calculate them accurately -- so it's not 176777/177147,
> > and anyway the numerator should be larger than the
> > denominator, so those terms should be switched.
>
> 3^53/2^81 = 19383245667680019896796723/19342813113834066795298816

Thanks ... you know how i am -- it was bugging me to not
know exactly what that ratio is.

However, dividing by 2^81 is a typo: that makes it
3 octaves too large. The correct 2-3-monzo is [-84 53> .
Would you please post that ratio?

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Oz,

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> monz wrote:
> >
> > However, even his ratio is off: (3/2)^53 / 2^31 (which is
> > what he really means mathematically) has a ratio with
> > terms which are so huge that my spreadsheet won't
> > calculate them accurately --
>
>
> The ratio is 21519725632224173557990013483376 /
> 21474836480000000000000000000000 equalling 3.615046 cents.

Whatever you used to calculate that ratio, it also
rounded it off, just like my spreadsheet. No power
of 2 is going to have all those zeros at the end.
But your cents value is correct to 6 decimal places.

Gene posted something that looks like the correct ratio,
but either he used the wrong power of 2 in the denominator
in his calculation, or else he got the correct ratio and
just typed the wrong exponent in his description of it.

The correct description of the interval in terms of
prime-factors is 3^53 / 2^84 , or my preference
(2^-84)*(3^53).

The value my spreadsheet gives which i trust the most
is in scientific notation: 1.94E+025 / 5.17E-026 .
That denominator looks suspicious, but the decimal
value i get for the ratio to 13 places is 1.0020903140411,
and the cents value to 14 places is 3.61504586553314 .

I hope to respond to the quote you posted regarding
55-edo, but will need some time to study that in more
depth.

-monz
http://tonalsoft.com
Tonescape microtonal music software

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

> > 3^53/2^84 = 19383245667680019896796723/19342813113834066795298816
>
>
> Thanks ... you know how i am -- it was bugging me to not
> know exactly what that ratio is.
>
> However, dividing by 2^81 is a typo: that makes it
> 3 octaves too large. The correct 2-3-monzo is [-84 53> .
> Would you please post that ratio?

See above; it's the ratio I did post, albeit incorrectly labeled.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 27 Mart 2007 Sal� 6:41
Subject: [tuning] Re: more on the "Arabian comma"

> Hi Oz,
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> > monz wrote:
> > >
> > > However, even his ratio is off: (3/2)^53 / 2^31 (which is
> > > what he really means mathematically) has a ratio with
> > > terms which are so huge that my spreadsheet won't
> > > calculate them accurately --
> >
> >
> > The ratio is 21519725632224173557990013483376 /
> > 21474836480000000000000000000000 equalling 3.615046 cents.
>
>
> Whatever you used to calculate that ratio, it also
> rounded it off, just like my spreadsheet. No power
> of 2 is going to have all those zeros at the end.
> But your cents value is correct to 6 decimal places.
>

One does what one can do best.

> Gene posted something that looks like the correct ratio,
> but either he used the wrong power of 2 in the denominator
> in his calculation, or else he got the correct ratio and
> just typed the wrong exponent in his description of it.
>
> The correct description of the interval in terms of
> prime-factors is 3^53 / 2^84 , or my preference
> (2^-84)*(3^53).
>
> The value my spreadsheet gives which i trust the most
> is in scientific notation: 1.94E+025 / 5.17E-026 .
> That denominator looks suspicious, but the decimal
> value i get for the ratio to 13 places is 1.0020903140411,
> and the cents value to 14 places is 3.61504586553314 .
>
>
> I hope to respond to the quote you posted regarding
> 55-edo, but will need some time to study that in more
> depth.
>

I'll be anxious to hear what you have to say.

>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>

Oz.

Hi Oz,

I don't know much about Turkish music, and have not
studied any of your posts here enough to really learn
much about it. But basing my reading of Murat on what
i know of Western tuning-theory, here goes ...

I should explain that i wrote my response as i read
each paragraph from Murat. So i didn't see that he
actually said that the octave should be 55 steps until
i had written everything before that.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> > > As for the "trade secret", I personally think it a far-fetched
> > > tale. The first mention of the comma in reference to Maqam
> > > Music is by Antoine Murat (an Europeanized Turk of the late
> > > 18th century), who conspicuously implies 55-EDO.
> >
> >
> > [me, monz]
> >
> > Are you sure about that? 55-edo behaves very differently
> > from 53-edo.
> >
>
>
> From Aksoy, B. "Music of the Ottomans from the Perspective
> of European Travellers", 'cf. Antoine Murat', pp. 163-5
> (title and contents translated by me):
>
> ...
>
> The compass of Turkish Music spans two octaves. An octave
> is divided into two tetrachords, each of whose tonic equals
> the corresponding tones of European Music. The tonic of
> the I. tetrachord is 'SOL', the II. is 'RE'. I. Tetrachord
> comprises the tones 'SOL-LA-SI-DO' [rast-dugah-segah-chargah],
> and II. 'RE-MI-FA-SOL' [neva-huseyni-evj-gerdaniye]. Each of
> the tetrachords possesses three large natural intervals.
> Moreover, each interval is split into small intervals of
> equal value called "commas". The first 9 commas form
> the primary basic interval [9:8]. This interval is betwixt
> the tonic 'sol' and its major second. The 1. comma between
> 'SOL-LA' makes a comma interval, 3. comma a "diminished second",
> 5. comma a "minor second"; at the 9. comma, the interval of
> a "major second" is completed.

OK, well, exactly what he means by the "1.comma" is what
we are trying to determine.

"9.comma" equaling the "major second" is going to be
the same for both 53-edo (~203.77 cents) and 55-edo
(~196.36 cents). What differentiates the two tunings
is how the chromatic and diatonic semitones are arranged.

In 55-edo, "5.comma" (~109.09 cents) does always equal
a "minor second". 55-edo tempers out the syntonic-comma,
so if our reference pitch "SOL" is "G", then "5.comma"
in 55-edo will always be "Ab". All representations of
"G#" in 55-edo will be "4.comma" (~87.27 cents).

However, this does not necessarily rule out 53-edo,
because of the fact that 53-edo gives not only a superb
approximation to pythaogrean tuning, but also a very good
approximation to 5-limit JI. In other words, 53-edo does
*not* temper out the syntonic-comma, but in fact represents
it by 1 step of the tuning.

As a pythagorean approximation, 53-edo uses "4.comma"
(~90.57 cents) as the "minor second" (diatonic semitone)
and "5.comma" (~113.21 cents) as the "augmented prime"
(chromatic semitone).

However, as a 5-limit JI approximation, it's the other
way around: "5.comma" represents the JI "minor second"
of ratio 16:15, and "4.comma" represents one version
of the JI chromatic-semitone, namely the one with ratio
135:128. Note that because the syntonic-comma is not
tempered out, 53-edo also has another version of the
chromatic semitone which represents ratio 25:24 and is
"3.comma".

Now, at first i really had no idea what Murat means by
'3.comma [designates] a "diminished second"'. It doesn't
work that way for either 55-edo or for 53-edo-as-pythagorean.
In fact, in 55-edo, it's nowhere near any kind of "second":
assuming that we're determining intervals by constructing
a chain-of-5ths, the "3.comma" in 55-edo (~65.45 cents)
designates a "doubly-augmented-6th", so that if "SOL" is "G",
then "3.comma" is "Fx#".

The actual "diminished second" in a chain-of-5ths tuning is
the -12 generator 5th, that is, 12 5ths down or 12 4ths up.

(Note that the convention on the tuning list these days
is to use the smaller of two complementary intervals as
the generator, so for meantone the 4th is generally named
as the generator. In that case, reverse the signs ... but
i still like to think of meantone in terms of 5ths.)

Anyway, following this, the "diminished second" of
55-edo is ~21.82 cents, which is in fact "1.comma".
And because the syntonic-comma is tempered out in 55-edo,
the "diminished second" in this tuning will *always*
be "1.comma". So now, unless his interval terminology
is not being used correctly according to Western
procedure, i'm tempted to conclude that Murat is
definitely not talking about 55-edo.

But as for 53-edo, the fact that the syntonic-comma is
not tempered out in 53-edo means that there is no one-to-one
correspondence between letter-notation and number of steps
(commas), which makes things a bit complicated. But let's
explore this further ...

As an approximation to pythagorean tuning, the -12
generator-5th of 53-edo in fact leads to a note which is
*lower* than the tonic. So i am doubtful that this is what
Murat means.

Also according to the pythagorean approximation, 53-edo's
"3.comma" (~67.92 cents) would be a "doubly-diminished-3rd"
at -17 generator-5ths, or "Bbbb". So this also doesn't
correspond to what Murat says.

However, as you can see on my bingo-card lattice page

http://tonalsoft.com/enc/b/bingo.aspx

(go to the "5-limit bingo card viewer" diagram at the
bottom of the page and click on the link to "53")

there are several different JI intervals represented
by "3.comma" in 53-edo. The one closest to 1:1 is the 25:24
chromatic-semitone which i noted above. The next closest is
the ratio 648:625 or 2-3-5-monzo [* 4, -4> (~62.57 cents),
which is in fact one version of the JI "diminished second"
-- it is called "major diesis" in the hexagonal diagram by
Paul Erlich which appears at the top of my bingo-card page.
Note that this interval plays a special role in 1/3-comma
meantone and its close relative 19-edo; see my "diesis" page,
definition #7:

http://tonalsoft.com/enc/d/diesis.aspx

So it's beginning to look like what Murat had in mind
was 53-edo as a representation of 5-limit JI.

> In the 7 comma interval of 'LA-SI', 1. comma yields a
> "diminished third", 3. comma a "minor second", and
> 7. comma (i.e., 16. comma after the tonic) a "major third"
> [27:22]. The 3. comma of the 'SI-DO' interval compounds
> the "augmented third", and a perfect fourth is achieved
> at the 23. comma [4:3].

Oh no ... now things are really getting muddled, unless
i just don't understand Murat's divisions.

If "LA-SI" means the interval from "A" to "B", i can't
figure out why he's dividing it into 7 commas instead
of 8 or 9.

In 53-edo-as-pythagorean, the "major second" or "whole tone"
will always be "9.comma". In 53-edo-as-JI, the interval
can contain varying numbers of commas depending on exactly
which ratio is being considered. But even at that, the
ratios being considered would normally be 9:8 (= "9.comma")
or 10:9 (= "8.comma"). The only "major second" in 5-limit JI
which corresponds to "7.comma" in 53-edo is the ratio
800:729 = 2-3-5-monzo [* -6, 2> (~16.09 cents), which
as you can see from the monzo is quite far from the origin.

The "7.comma" of "LA-SI", "16.comma after the tonic",
indeed does not correspond to the typical 5-limit JI
ratios. The pythagorean "major third" of ratio 81:64
is "18.comma" (~407.55 cents) in 53-edo, and the JI
"major third" of 5:4 ratio is "17.comma" (~384.91 cents).
The "16.comma" (~362.26 cents) certainly is much closer
to the 27:22 ratio which you interpolated into the quote
from Murat

If Murat *does* mean 55-edo, then he seems to be using
it not only as a meantone representation of 5-limit JI,
but rather to represent 11-limit JI, which seems to be
the interpretation you yourself understood him to mean.
The 55-edo "16.comma" (~349.09 cents) does give a very
good representation not only of 27:22 as you note, but
also of the 11:9 "neutral third".

As for "1.comma" of "LA-SI": the JI "diminished third"
represented in 53-edo by "10.comma" (~226.42 cents) is the
closest one to the lattice origin, with ratio 256:225 =
2-3-5-monzo [* -2, -2>. There is another one just a bit
farther at ratio 144:125 which is represented by "11.comma",
but Murat's description of "1.comma" in "LA-SI" shows that
he is referring to "10.comma" after the tonic. So this
interval description seems clear.

If he does mean 55-edo, the "10.comma" (~218.18 cents)
actually does represent the -10 generator-5th "diminished
third" of 55-edo.

His description of 3.comma of "LA-SI", which is 12.comma
after the tonic, as "minor second" makes no sense -- i
believe this must be a typo for "minor third". However,
as with the "7.comma" above, this one also seems to be
one comma too narrow. In 53-edo, the JI "minor third"
of ratio 6:5 is represented by "14.comma" (~316.98 cents),
and the pythagorean "minor third" of ratio 32:27 is
"13.comma" (~294.34 cents). The "12.comma" is only
~271.7 cents.

And again, with the "augmented third", the closest one
in JI is ratio 125:96 = 2-3-5-monzo [* -1, 3>, but this
is represented in 53-edo by "20.comma" (~452.83 cents)
and according to Murat it should be 16+3 = "19.comma"
after the tonic (~430.19 cents).

It looks here as if he either using 53-edo to refer
to some intervals which are from a prime-limit higher
than 5, or else maybe using a multiple-division. If
it's the latter, then he seems to be assuming a 5-limit
JI basis for the tetrachord and dividing the 9:8 into
9 commas, which corresponds to 53-edo, and the 10:9
into 7 commas, which corresponds to 46-edo: (10/9)^(1/7)
is ~26.06 cents, and the EDO which has a step-size close
to this is 46-edo.

> The II. tetrachord is divided similar to the I tetrachord.
> The two tetrachords are in exact symmetry with each other.
> The 'DO-RE' major tone interval that seperates the two
> tetrachords is divided into smaller intervals of 1,3,5 commas
> just like the intervals of 'SOL-LA' and 'RE-MI'.
> The tonic of the II. tetrachord forms the fifth of the I.
> There are 32 commas in this pentachord. In the entirety of
> the composite tetrachords made of the union of two disjunct
> tetrachords, there are 55 commas. As may be seen, amid each
> of the 7 intervals forming the octave, there are 3
> intermediate tones. Via the division of the primary intervals
> into three, it is observed that there are 21 intervals and
> 22 perdes in the Turkish octave.

OK, after spending most of my free time today working
on this, at this point i give up. What Murat says here
clearly refers to 55-edo. So i'm not sure what's going
on at those points where i earlier wrote that 55-edo
doesn't fit his description and he must be referring to
53-edo. I guess i'll leave it to you to work out the
discrepancies.

-monz
http://tonalsoft.com
Tonescape microtonal music software

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 28 Mart 2007 �ar�amba 12:27
Subject: [tuning] Re: more on the "Arabian comma"

> Hi Oz,
>
>
> I don't know much about Turkish music, and have not
> studied any of your posts here enough to really learn
> much about it. But basing my reading of Murat on what
> i know of Western tuning-theory, here goes ...
>
> I should explain that i wrote my response as i read
> each paragraph from Murat. So i didn't see that he
> actually said that the octave should be 55 steps until
> i had written everything before that.
>
>

Great, just great.

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> > > > As for the "trade secret", I personally think it a far-fetched
> > > > tale. The first mention of the comma in reference to Maqam
> > > > Music is by Antoine Murat (an Europeanized Turk of the late
> > > > 18th century), who conspicuously implies 55-EDO.
> > >
> > >
> > > [me, monz]
> > >
> > > Are you sure about that? 55-edo behaves very differently
> > > from 53-edo.
> > >
> >

What else is there to conclude?

> >
> > From Aksoy, B. "Music of the Ottomans from the Perspective
> > of European Travellers", 'cf. Antoine Murat', pp. 163-5
> > (title and contents translated by me):
> >
> > ...
> >
> > The compass of Turkish Music spans two octaves. An octave
> > is divided into two tetrachords, each of whose tonic equals
> > the corresponding tones of European Music. The tonic of
> > the I. tetrachord is 'SOL', the II. is 'RE'. I. Tetrachord
> > comprises the tones 'SOL-LA-SI-DO' [rast-dugah-segah-chargah],
> > and II. 'RE-MI-FA-SOL' [neva-huseyni-evj-gerdaniye]. Each of
> > the tetrachords possesses three large natural intervals.
> > Moreover, each interval is split into small intervals of
> > equal value called "commas". The first 9 commas form
> > the primary basic interval [9:8]. This interval is betwixt
> > the tonic 'sol' and its major second. The 1. comma between
> > 'SOL-LA' makes a comma interval, 3. comma a "diminished second",
> > 5. comma a "minor second"; at the 9. comma, the interval of
> > a "major second" is completed.
>
>
> OK, well, exactly what he means by the "1.comma" is what
> we are trying to determine.
>
> "9.comma" equaling the "major second" is going to be
> the same for both 53-edo (~203.77 cents) and 55-edo
> (~196.36 cents). What differentiates the two tunings
> is how the chromatic and diatonic semitones are arranged.
>

In no other temperament besides 55 and 53 do we have the possibility of
defining "9 commas" per whole tone in the traditionalist sense.

> In 55-edo, "5.comma" (~109.09 cents) does always equal
> a "minor second". 55-edo tempers out the syntonic-comma,
> so if our reference pitch "SOL" is "G", then "5.comma"
> in 55-edo will always be "Ab". All representations of
> "G#" in 55-edo will be "4.comma" (~87.27 cents).
>

Good enough.

>
> However, this does not necessarily rule out 53-edo,
> because of the fact that 53-edo gives not only a superb
> approximation to pythaogrean tuning, but also a very good
> approximation to 5-limit JI. In other words, 53-edo does
> *not* temper out the syntonic-comma, but in fact represents
> it by 1 step of the tuning.
>

How 53-edo is ruled out is admittedly clarified below.

> As a pythagorean approximation, 53-edo uses "4.comma"
> (~90.57 cents) as the "minor second" (diatonic semitone)
> and "5.comma" (~113.21 cents) as the "augmented prime"
> (chromatic semitone).
>
> However, as a 5-limit JI approximation, it's the other
> way around: "5.comma" represents the JI "minor second"
> of ratio 16:15, and "4.comma" represents one version
> of the JI chromatic-semitone, namely the one with ratio
> 135:128.

5. comma also represents the JI major second at 15:14.

Note that because the syntonic-comma is not
> tempered out, 53-edo also has another version of the
> chromatic semitone which represents ratio 25:24 and is
> "3.comma".

This 3 comma stuff clearly refers to Archytas' 1/3 tone expressed as 28/27.

>
>
> Now, at first i really had no idea what Murat means by
> '3.comma [designates] a "diminished second"'. It doesn't
> work that way for either 55-edo or for 53-edo-as-pythagorean.
> In fact, in 55-edo, it's nowhere near any kind of "second":
> assuming that we're determining intervals by constructing
> a chain-of-5ths, the "3.comma" in 55-edo (~65.45 cents)
> designates a "doubly-augmented-6th", so that if "SOL" is "G",
> then "3.comma" is "Fx#".
>

You are thinking in terms foreign to Ottoman Music. Diminished second here
surely means a second much smaller than minor. Saying "doubly-augmented-6th"
has little musical meaning for the genre.

> The actual "diminished second" in a chain-of-5ths tuning is
> the -12 generator 5th, that is, 12 5ths down or 12 4ths up.
>
> (Note that the convention on the tuning list these days
> is to use the smaller of two complementary intervals as
> the generator, so for meantone the 4th is generally named
> as the generator. In that case, reverse the signs ... but
> i still like to think of meantone in terms of 5ths.)
>

As do I. Employing the complementary fourth as the generator interval has
been adopted by Arel-Ezgi-Uzdilek in describing the 24-tone Pythagorean
model, thus does not appeal to me.

> Anyway, following this, the "diminished second" of
> 55-edo is ~21.82 cents, which is in fact "1.comma".
> And because the syntonic-comma is tempered out in 55-edo,
> the "diminished second" in this tuning will *always*
> be "1.comma". So now, unless his interval terminology
> is not being used correctly according to Western
> procedure, i'm tempted to conclude that Murat is
> definitely not talking about 55-edo.
>

He is definitely talking about 55-edo despite his usage of 3 commas as the
diminished second.

>
> But as for 53-edo, the fact that the syntonic-comma is
> not tempered out in 53-edo means that there is no one-to-one
> correspondence between letter-notation and number of steps
> (commas), which makes things a bit complicated. But let's
> explore this further ...
>
>
> As an approximation to pythagorean tuning, the -12
> generator-5th of 53-edo in fact leads to a note which is
> *lower* than the tonic. So i am doubtful that this is what
> Murat means.
>
> Also according to the pythagorean approximation, 53-edo's
> "3.comma" (~67.92 cents) would be a "doubly-diminished-3rd"
> at -17 generator-5ths, or "Bbbb". So this also doesn't
> correspond to what Murat says.
>

Confer above. The best way to define a 3 comma interval is by direct
reference to the 5 comma interval. It is a second, but not as large as the 5
comma version, thus Murat's usage of the term "diminished" no matter how
discordant this may seem with his general terminology. Hence, saying
"doubly-diminished-third" does not hold much musical meaning for Maqam
Music.

>
> However, as you can see on my bingo-card lattice page
>
> http://tonalsoft.com/enc/b/bingo.aspx
>
> (go to the "5-limit bingo card viewer" diagram at the
> bottom of the page and click on the link to "53")
>

> there are several different JI intervals represented
> by "3.comma" in 53-edo. The one closest to 1:1 is the 25:24
> chromatic-semitone which i noted above. The next closest is
> the ratio 648:625 or 2-3-5-monzo [* 4, -4> (~62.57 cents),
> which is in fact one version of the JI "diminished second"
> -- it is called "major diesis" in the hexagonal diagram by
> Paul Erlich which appears at the top of my bingo-card page.
> Note that this interval plays a special role in 1/3-comma
> meantone and its close relative 19-edo; see my "diesis" page,
> definition #7:
>
> http://tonalsoft.com/enc/d/diesis.aspx
>
>
> So it's beginning to look like what Murat had in mind
> was 53-edo as a representation of 5-limit JI.
>

Quite the contrary. He is defining 55-edo as a representation of 11-limit
JI.

>
>
> > In the 7 comma interval of 'LA-SI', 1. comma yields a
> > "diminished third", 3. comma a "minor second", and
> > 7. comma (i.e., 16. comma after the tonic) a "major third"
> > [27:22]. The 3. comma of the 'SI-DO' interval compounds
> > the "augmented third", and a perfect fourth is achieved
> > at the 23. comma [4:3].
>
>
> Oh no ... now things are really getting muddled, unless
> i just don't understand Murat's divisions.
>
> If "LA-SI" means the interval from "A" to "B", i can't
> figure out why he's dividing it into 7 commas instead
> of 8 or 9.
>

For good reason. He is attaining a neutral third between G-B. This is in
accordance with the Arabic rendition of Maqam Rast, which we Turks also
execute at times, but fail to acknowledge.

> In 53-edo-as-pythagorean, the "major second" or "whole tone"
> will always be "9.comma". In 53-edo-as-JI, the interval
> can contain varying numbers of commas depending on exactly
> which ratio is being considered. But even at that, the
> ratios being considered would normally be 9:8 (= "9.comma")
> or 10:9 (= "8.comma"). The only "major second" in 5-limit JI
> which corresponds to "7.comma" in 53-edo is the ratio
> 800:729 = 2-3-5-monzo [* -6, 2> (~16.09 cents), which
> as you can see from the monzo is quite far from the origin.
>

You are consigning JI to 5-limit. This is not the case for Maqam Music.
Remember Murat's remarks on cacophony.

> The "7.comma" of "LA-SI", "16.comma after the tonic",
> indeed does not correspond to the typical 5-limit JI
> ratios. The pythagorean "major third" of ratio 81:64
> is "18.comma" (~407.55 cents) in 53-edo, and the JI
> "major third" of 5:4 ratio is "17.comma" (~384.91 cents).
> The "16.comma" (~362.26 cents) certainly is much closer
> to the 27:22 ratio which you interpolated into the quote
> from Murat

Exactly.

>
> If Murat *does* mean 55-edo, then he seems to be using
> it not only as a meantone representation of 5-limit JI,
> but rather to represent 11-limit JI, which seems to be
> the interpretation you yourself understood him to mean.
> The 55-edo "16.comma" (~349.09 cents) does give a very
> good representation not only of 27:22 as you note, but
> also of the 11:9 "neutral third".
>

I hadn't thought of that one. Good for you.

>
> As for "1.comma" of "LA-SI": the JI "diminished third"
> represented in 53-edo by "10.comma" (~226.42 cents) is the
> closest one to the lattice origin, with ratio 256:225 =
> 2-3-5-monzo [* -2, -2>. There is another one just a bit
> farther at ratio 144:125 which is represented by "11.comma",
> but Murat's description of "1.comma" in "LA-SI" shows that
> he is referring to "10.comma" after the tonic. So this
> interval description seems clear.
>
> If he does mean 55-edo, the "10.comma" (~218.18 cents)
> actually does represent the -10 generator-5th "diminished
> third" of 55-edo.
>

Agreeably so. He is a Europeanized Turk after all.

>
> His description of 3.comma of "LA-SI", which is 12.comma
> after the tonic, as "minor second" makes no sense -- i
> believe this must be a typo for "minor third".

I concur.

However,
> as with the "7.comma" above, this one also seems to be
> one comma too narrow. In 53-edo, the JI "minor third"
> of ratio 6:5 is represented by "14.comma" (~316.98 cents),
> and the pythagorean "minor third" of ratio 32:27 is
> "13.comma" (~294.34 cents). The "12.comma" is only
> ~271.7 cents.
>

The septimal minor third, yes. He definitely means 7:6.

>
> And again, with the "augmented third", the closest one
> in JI is ratio 125:96 = 2-3-5-monzo [* -1, 3>, but this
> is represented in 53-edo by "20.comma" (~452.83 cents)
> and according to Murat it should be 16+3 = "19.comma"
> after the tonic (~430.19 cents).
>

He implies the septimal major third at 9:7. Deviation from the original
nomenclature is again understandable, since any other definition would have
little musical meaning. 9:7 is a third, and larger than 11:9, hence the term
"augmented".

> It looks here as if he either using 53-edo to refer
> to some intervals which are from a prime-limit higher
> than 5, or else maybe using a multiple-division. If
> it's the latter, then he seems to be assuming a 5-limit
> JI basis for the tetrachord and dividing the 9:8 into
> 9 commas, which corresponds to 53-edo, and the 10:9
> into 7 commas, which corresponds to 46-edo: (10/9)^(1/7)
> is ~26.06 cents, and the EDO which has a step-size close
> to this is 46-edo.
>

Such complicated procedures are unheard of in Maqam Music.

>
> > The II. tetrachord is divided similar to the I tetrachord.
> > The two tetrachords are in exact symmetry with each other.
> > The 'DO-RE' major tone interval that seperates the two
> > tetrachords is divided into smaller intervals of 1,3,5 commas
> > just like the intervals of 'SOL-LA' and 'RE-MI'.
> > The tonic of the II. tetrachord forms the fifth of the I.
> > There are 32 commas in this pentachord. In the entirety of
> > the composite tetrachords made of the union of two disjunct
> > tetrachords, there are 55 commas. As may be seen, amid each
> > of the 7 intervals forming the octave, there are 3
> > intermediate tones. Via the division of the primary intervals
> > into three, it is observed that there are 21 intervals and
> > 22 perdes in the Turkish octave.
>
>
> OK, after spending most of my free time today working
> on this, at this point i give up. What Murat says here
> clearly refers to 55-edo. So i'm not sure what's going
> on at those points where i earlier wrote that 55-edo
> doesn't fit his description and he must be referring to
> 53-edo. I guess i'll leave it to you to work out the
> discrepancies.
>

Hopefully, I have clarified for you the situation. Surely if 79/80 MOS
159-tET is unbearable, the next best thing for Maqam Music is 55-edo. And
the nonsense behind the "Arabian comma" becomes moot. Why insist on
Pythagoreanism when 11-limit JI approximation by a cyclic meantone is
clearly implied?

>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>

Oz.

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

> If Murat *does* mean 55-edo, then he seems to be using
> it not only as a meantone representation of 5-limit JI,
> but rather to represent 11-limit JI, which seems to be
> the interpretation you yourself understood him to mean.
> The 55-edo "16.comma" (~349.09 cents) does give a very
> good representation not only of 27:22 as you note, but
> also of the 11:9 "neutral third".

The 16deg55 neutral third is complex if you use chains of fifths, being
16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
alternative would be to use it as a generator; you are then in the 11-
limit version of semififths temperament, the 11-limit 31&55 temperament.
I wonder if this is a good enough excuse to call this
temperament "murat"? I like it, and Paul isn't here to object.

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

> If Murat *does* mean 55-edo, then he seems to be using
> it not only as a meantone representation of 5-limit JI,
> but rather to represent 11-limit JI, which seems to be
> the interpretation you yourself understood him to mean.
> The 55-edo "16.comma" (~349.09 cents) does give a very
> good representation not only of 27:22 as you note, but
> also of the 11:9 "neutral third".

The 16deg55 neutral third is complex if you use chains of fifths, being
16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
alternative would be to use it as a generator; you are then in the 11-
limit version of semififths temperament, the 11-limit 31&55 temperament.
I wonder if this is a good enough excuse to call this
temperament "murat"? I like it, and Paul isn't here to object.

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

> If Murat *does* mean 55-edo, then he seems to be using
> it not only as a meantone representation of 5-limit JI,
> but rather to represent 11-limit JI, which seems to be
> the interpretation you yourself understood him to mean.
> The 55-edo "16.comma" (~349.09 cents) does give a very
> good representation not only of 27:22 as you note, but
> also of the 11:9 "neutral third".

The 16deg55 neutral third is complex if you use chains of fifths, being
16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
alternative would be to use it as a generator; you are then in the 11-
limit version of semififths temperament, the 11-limit 31&55 temperament.
I wonder if this is a good enough excuse to call this
temperament "murat"? I like it, and Paul isn't here to object.

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains
> of fifths, being 16/32 = 1/2 = -27 fifths, as complex as
> is possible. A logical alternative would be to use it as
> a generator; you are then in the 11-limit version of
> semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to
> object.

Exactly. I created a Tonescape 3-5-11 JI Tonespace with
a 55-tone periodicity-block using TM-basis, to get the
ratios i used in my response to Ozan. From there, it's
a simple matter to temper out the commas and get 55-edo.

... and i have no objection to "murat".

-monz
http://tonalsoft.com
Tonescape microtonal music software

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

> If Murat *does* mean 55-edo, then he seems to be using
> it not only as a meantone representation of 5-limit JI,
> but rather to represent 11-limit JI, which seems to be
> the interpretation you yourself understood him to mean.
> The 55-edo "16.comma" (~349.09 cents) does give a very
> good representation not only of 27:22 as you note, but
> also of the 11:9 "neutral third".

The 16deg55 neutral third is complex if you use chains of fifths, being
16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
alternative would be to use it as a generator; you are then in the 11-
limit version of semififths temperament, the 11-limit 31&55 temperament.
I wonder if this is a good enough excuse to call this
temperament "murat"? I like it, and Paul isn't here to object.

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains
> of fifths, being 16/32 = 1/2 = -27 fifths, as complex as
> is possible. A logical alternative would be to use it as
> a generator; you are then in the 11-limit version of
> semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to
> object.

Exactly. I created a Tonescape 3-5-11 JI Tonespace with
a 55-tone periodicity-block using TM-basis, to get the
ratios i used in my response to Ozan. From there, it's
a simple matter to temper out the commas and get 55-edo.

... and i have no objection to "murat".

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains
> of fifths, being 16/32 = 1/2 = -27 fifths, as complex as
> is possible. A logical alternative would be to use it as
> a generator; you are then in the 11-limit version of
> semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to
> object.

Exactly. I created a Tonescape 3-5-11 JI Tonespace with
a 55-tone periodicity-block using TM-basis, to get the
ratios i used in my response to Ozan. From there, it's
a simple matter to temper out the commas and get 55-edo.

... and i have no objection to "murat".

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains
> of fifths, being 16/32 = 1/2 = -27 fifths, as complex as
> is possible. A logical alternative would be to use it as
> a generator; you are then in the 11-limit version of
> semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to
> object.

Exactly. I created a Tonescape 3-5-11 JI Tonespace with
a 55-tone periodicity-block using TM-basis, to get the
ratios i used in my response to Ozan. From there, it's
a simple matter to temper out the commas and get 55-edo.

... and i have no objection to "murat".

-monz
http://tonalsoft.com
Tonescape microtonal music software

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@sbcglobal.net>
To: <tuning@yahoogroups.com>
Sent: 29 Mart 2007 Per�embe 0:28
Subject: [tuning] Re: more on the "Arabian comma"

> --- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains of fifths, being
> 16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
> alternative would be to use it as a generator; you are then in the 11-
> limit version of semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to object.
>
>

Where is this temperament?

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

> If Murat *does* mean 55-edo, then he seems to be using
> it not only as a meantone representation of 5-limit JI,
> but rather to represent 11-limit JI, which seems to be
> the interpretation you yourself understood him to mean.
> The 55-edo "16.comma" (~349.09 cents) does give a very
> good representation not only of 27:22 as you note, but
> also of the 11:9 "neutral third".

The 16deg55 neutral third is complex if you use chains of fifths, being
16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
alternative would be to use it as a generator; you are then in the 11-
limit version of semififths temperament, the 11-limit 31&55 temperament.
I wonder if this is a good enough excuse to call this
temperament "murat"? I like it, and Paul isn't here to object.

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains
> of fifths, being 16/32 = 1/2 = -27 fifths, as complex as
> is possible. A logical alternative would be to use it as
> a generator; you are then in the 11-limit version of
> semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to
> object.

Exactly. I created a Tonescape 3-5-11 JI Tonespace with
a 55-tone periodicity-block using TM-basis, to get the
ratios i used in my response to Ozan. From there, it's
a simple matter to temper out the commas and get 55-edo.

... and i have no objection to "murat".

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains
> of fifths, being 16/32 = 1/2 = -27 fifths, as complex as
> is possible. A logical alternative would be to use it as
> a generator; you are then in the 11-limit version of
> semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to
> object.

Exactly. I created a Tonescape 3-5-11 JI Tonespace with
a 55-tone periodicity-block using TM-basis, to get the
ratios i used in my response to Ozan. From there, it's
a simple matter to temper out the commas and get 55-edo.

... and i have no objection to "murat".

-monz
http://tonalsoft.com
Tonescape microtonal music software

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

> If Murat *does* mean 55-edo, then he seems to be using
> it not only as a meantone representation of 5-limit JI,
> but rather to represent 11-limit JI, which seems to be
> the interpretation you yourself understood him to mean.
> The 55-edo "16.comma" (~349.09 cents) does give a very
> good representation not only of 27:22 as you note, but
> also of the 11:9 "neutral third".

The 16deg55 neutral third is complex if you use chains of fifths, being
16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
alternative would be to use it as a generator; you are then in the 11-
limit version of semififths temperament, the 11-limit 31&55 temperament.
I wonder if this is a good enough excuse to call this
temperament "murat"? I like it, and Paul isn't here to object.

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains
> of fifths, being 16/32 = 1/2 = -27 fifths, as complex as
> is possible. A logical alternative would be to use it as
> a generator; you are then in the 11-limit version of
> semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to
> object.

Exactly. I created a Tonescape 3-5-11 JI Tonespace with
a 55-tone periodicity-block using TM-basis, to get the
ratios i used in my response to Ozan. From there, it's
a simple matter to temper out the commas and get 55-edo.

... and i have no objection to "murat".

-monz
http://tonalsoft.com
Tonescape microtonal music software

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

> If Murat *does* mean 55-edo, then he seems to be using
> it not only as a meantone representation of 5-limit JI,
> but rather to represent 11-limit JI, which seems to be
> the interpretation you yourself understood him to mean.
> The 55-edo "16.comma" (~349.09 cents) does give a very
> good representation not only of 27:22 as you note, but
> also of the 11:9 "neutral third".

The 16deg55 neutral third is complex if you use chains of fifths, being
16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
alternative would be to use it as a generator; you are then in the 11-
limit version of semififths temperament, the 11-limit 31&55 temperament.
I wonder if this is a good enough excuse to call this
temperament "murat"? I like it, and Paul isn't here to object.

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains
> of fifths, being 16/32 = 1/2 = -27 fifths, as complex as
> is possible. A logical alternative would be to use it as
> a generator; you are then in the 11-limit version of
> semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to
> object.

Exactly. I created a Tonescape 3-5-11 JI Tonespace with
a 55-tone periodicity-block using TM-basis, to get the
ratios i used in my response to Ozan. From there, it's
a simple matter to temper out the commas and get 55-edo.

... and i have no objection to "murat".

-monz
http://tonalsoft.com
Tonescape microtonal music software

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@sbcglobal.net>
To: <tuning@yahoogroups.com>
Sent: 29 Mart 2007 Per�embe 0:28
Subject: [tuning] Re: more on the "Arabian comma"

> --- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains of fifths, being
> 16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
> alternative would be to use it as a generator; you are then in the 11-
> limit version of semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to object.
>
>

Where is this temperament?

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

> If Murat *does* mean 55-edo, then he seems to be using
> it not only as a meantone representation of 5-limit JI,
> but rather to represent 11-limit JI, which seems to be
> the interpretation you yourself understood him to mean.
> The 55-edo "16.comma" (~349.09 cents) does give a very
> good representation not only of 27:22 as you note, but
> also of the 11:9 "neutral third".

The 16deg55 neutral third is complex if you use chains of fifths, being
16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
alternative would be to use it as a generator; you are then in the 11-
limit version of semififths temperament, the 11-limit 31&55 temperament.
I wonder if this is a good enough excuse to call this
temperament "murat"? I like it, and Paul isn't here to object.

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains
> of fifths, being 16/32 = 1/2 = -27 fifths, as complex as
> is possible. A logical alternative would be to use it as
> a generator; you are then in the 11-limit version of
> semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to
> object.

Exactly. I created a Tonescape 3-5-11 JI Tonespace with
a 55-tone periodicity-block using TM-basis, to get the
ratios i used in my response to Ozan. From there, it's
a simple matter to temper out the commas and get 55-edo.

... and i have no objection to "murat".

-monz
http://tonalsoft.com
Tonescape microtonal music software

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@sbcglobal.net>
To: <tuning@yahoogroups.com>
Sent: 29 Mart 2007 Per�embe 0:28
Subject: [tuning] Re: more on the "Arabian comma"

> --- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains of fifths, being
> 16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
> alternative would be to use it as a generator; you are then in the 11-
> limit version of semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to object.
>
>

Where is this temperament?

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@sbcglobal.net>
To: <tuning@yahoogroups.com>
Sent: 29 Mart 2007 Per�embe 0:28
Subject: [tuning] Re: more on the "Arabian comma"

> --- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains of fifths, being
> 16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
> alternative would be to use it as a generator; you are then in the 11-
> limit version of semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to object.
>
>

Where is this temperament?

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@sbcglobal.net>
To: <tuning@yahoogroups.com>
Sent: 29 Mart 2007 Per�embe 0:28
Subject: [tuning] Re: more on the "Arabian comma"

> --- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains of fifths, being
> 16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
> alternative would be to use it as a generator; you are then in the 11-
> limit version of semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to object.
>
>

Where is this temperament?

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

> If Murat *does* mean 55-edo, then he seems to be using
> it not only as a meantone representation of 5-limit JI,
> but rather to represent 11-limit JI, which seems to be
> the interpretation you yourself understood him to mean.
> The 55-edo "16.comma" (~349.09 cents) does give a very
> good representation not only of 27:22 as you note, but
> also of the 11:9 "neutral third".

The 16deg55 neutral third is complex if you use chains of fifths, being
16/32 = 1/2 = -27 fifths, as complex as is possible. A logical
alternative would be to use it as a generator; you are then in the 11-
limit version of semififths temperament, the 11-limit 31&55 temperament.
I wonder if this is a good enough excuse to call this
temperament "murat"? I like it, and Paul isn't here to object.

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > If Murat *does* mean 55-edo, then he seems to be using
> > it not only as a meantone representation of 5-limit JI,
> > but rather to represent 11-limit JI, which seems to be
> > the interpretation you yourself understood him to mean.
> > The 55-edo "16.comma" (~349.09 cents) does give a very
> > good representation not only of 27:22 as you note, but
> > also of the 11:9 "neutral third".
>
> The 16deg55 neutral third is complex if you use chains
> of fifths, being 16/32 = 1/2 = -27 fifths, as complex as
> is possible. A logical alternative would be to use it as
> a generator; you are then in the 11-limit version of
> semififths temperament, the 11-limit 31&55 temperament.
> I wonder if this is a good enough excuse to call this
> temperament "murat"? I like it, and Paul isn't here to
> object.

Exactly. I created a Tonescape 3-5-11 JI Tonespace with
a 55-tone periodicity-block using TM-basis, to get the
ratios i used in my response to Ozan. From there, it's
a simple matter to temper out the commas and get 55-edo.

... and i have no objection to "murat".

-monz
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:

> Where is this temperament?

I'm not sure what you mean by "where", but this is it:

semififths/murat

<<2 8 -11 5 8 -23 1 -48 -16 52||

[<1, 1, 0, 6, 2|, <0, 2, 8, -11, 5|]

generators: 9/31, 16/32 neutral third from flat fifth, ~11/9

commas: 81/80, 121/120, 176/175, 243/242

Where is the list of cents?

----- Original Message -----
From: "Gene Ward Smith" <genewardsmith@sbcglobal.net>
To: <tuning@yahoogroups.com>
Sent: 30 Mart 2007 Cuma 0:15
Subject: [tuning] Re: more on the "Arabian comma"

> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> > Where is this temperament?
>
> I'm not sure what you mean by "where", but this is it:
>
> semififths/murat
>
> <<2 8 -11 5 8 -23 1 -48 -16 52||
>
> [<1, 1, 0, 6, 2|, <0, 2, 8, -11, 5|]
>
> generators: 9/31, 16/32 neutral third from flat fifth, ~11/9
>
> commas: 81/80, 121/120, 176/175, 243/242
>
>

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@...> wrote:
>
> Where is the list of cents?

That's a scale, not a temperament.

Here are some murat MOS in 55. My guess is that while 31
might be better for some purposes, 55 would be better
for maqm nusic--if it can be played in these scales.

! murat17.scl
Murat[17] with 16/55 generator
17
!
43.636364
152.727273
196.363636
305.454545
349.090909
392.727273
501.818182
545.454545
654.545455
698.181818
807.272727
850.909091
894.545455
1003.636364
1047.272727
1156.363636
1200.000000

! murat24.scl
Murat[24] with 16/55 generator
24
!
43.636364
109.090909
152.727273
196.363636
240.000000
305.454545
349.090909
392.727273
458.181818
501.818182
545.454545
589.090909
654.545455
698.181818
741.818182
807.272727
850.909091
894.545455
960.000000
1003.636364
1047.272727
1090.909091
1156.363636
1200.000000