Yahoo Tuning Groups Ultimate Backup tuning Marchetto

For the second time I've been 'scooped' on Marchetto!
It's amazing, but it seems that every time I have something
important and new to say about him, someone else steps
in and beats me to it.

Margo Schulter's arguing against my interpretation
of Marchetto's tuning system as pseudo-5-limit couldn't
have come at a more appropriate time. Margo, all the
evidence seems to completely support your statements.

This morning, just a few hours before receiving my Tuning
Digest and reading your posting, I have had a major new
insight into Marchetto's theories, so everything in my
webpage is possibly incorrect!

I will not delete it, however, because there is no way to
be certain, and there is still a possibility that what I
have there is correct. But my new interpretation of
Marchetto's 'fifth-tones' fits all the evidence better,
and strongly supports a 17-tone Pythagorean chromatic scale
(I can almost see you grinning!).

Since my translations of the selections of Euler's Latin
in my recent work on that webpage turned out so well (IMO),
and revealed subtle shades of meaning to me that I did not
get from Professor Bailhache's translations, I decided to
translate Marchetto's Latin for myself too.

I am glad that I decided to do this. I roundly criticize
Jan Herlinger for the interpretation of the 'fifth-tones'
as an equal-tempered division of the 'whole tone' [= (9/8)^(1/5)],
and had a suspicion that the English translation of Marchetto's
_Lucidarium_ by Herlinger, in the critical edition, may not
pick up on some of the finer points of tuning.

I was certainly correct about that.

Marchetto, in Treatise 2 chapter 4, is discussing the way
that the continuum keeps getting smaller as one takes
8/9, and then 8/9 of 8/9, and so on, which was the standard
Pythagorean way of building a scale by monochord division.

So I applied this method to the division of the whole tone,
instead of the 9 equal string-length divisions which I
present on my webpage now, and I discovered something amazing.

With this method of division, Marchetto got a diesis at
the end which was much larger than all the rest (92 cents!
- the others are 22, 37, 30, and 24 cents), and ratios
with some very high prime factors: his 'enharmonic semitone'
is 182/173 [= 2^1 * 7^1 * 13^1 * 173^-1] and his 'chromatic
semitone' is 577/541 [= 541^-1 * 577^1].

However, they describe perfectly the method of Pythagorean
chromaticism in the early 1300s, with only ~2 cents error.

It is important to remember, as I point out on the webpage,
that accidentals at that time period did not mean the same
thing they mean today. They signified not an absolute
pitch value or pitch-class, but rather a variety of different
'mutations' which produced chromatic melodic progressions.

According to my new rational interpretation, the uses of
Marchetto's four different small intervals can be summarized
as follows, using regular Pythagorean terminology for
'natural', 'flat', and 'sharp':

diatonic semitone (~116 cents)
= from flat to natural above, or natural to flat below

enharmonic semitone (~88 cents)
= from natural to flat above, or flat to natural below

chromatic semitone (~112 cents)
= from natural to sharp above, or sharp to natural below

diesis (~92 cents)
= from sharp to natural above, or natural to sharp below

It turns out that the 'chromatic' and 'diatonic' semitones
are very close in size: only ~4 cents difference. But they
were differentiated by Marchetto because his perspective
on accidentals was different from ours: the diatonic could
only appear in conjunction with the enharmonic, and the
chromatic only with the diesis, in specified melodic settings
which Marchetto illustrates in the remainder of _Lucidarium_.

The vertical intervals (harmonies/ dyads) produced thru use
of the chromatic semitone are not the unusual ones expanded
by approximately a 'quarter-tone', but simply the regular
Pythagorean *major* '3rds' and '6ths' minus ~2 cents (in places
where these intervals would normally be of the 'minor' variety).

The 1962 article 'On the development of musical systems',
by Kraehenbuehl and Schmidt, [Journal of Music Theory, vol 6,
p 32-65] describes how Prosdocimus calculated the 17-tone
Pythagorean scale by inserting a sharp *and* flat between
all 7 of the regular Pythagorean diatonic notes.

Prosdocimus wrote his book about 100 years after Marchetto,
and was bitterly attacking Marchetto's theories. But I can
see, using this interpretation of Marchetto's divisions, that
Marchetto was describing a very close approximation to exactly
the same process which Prosdocimus 'corrected' (i.e., made
strictly 3-limit).

I think this gives further validation to this new interpretation.

So, if this interpretation of his divisions is correct,
and I think it is, he was not theorizing about 5-limit (JI)
intervals at all, but merely describing the introduction
of chromaticism into the music of his day within the
Pythagorean system.

Nor did his 'chromatic semitone' indicate some pitch
sharpened by about a 'quarter-tone'.

I personally am a little disappointed about both of those,
but it is still a very clever theory.

And the accepted view on Marchetto's 'fifth-tones',
Herlinger's, is in serious need of correction.

Also, Jay Rahn published his article on Marchetto in
Music Theory Online last December, with exactly the same
measurements I discuss on my webpage now. This view
also needs to be emended.

This new information will be added to the Marchetto
webpage over the next few days, as well as much of the
Latin and my translation.

-Joe

Joseph L. Monzo....................monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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🔗monz@xxxx.xxx

I wrote:

> [I] had a suspicion that the English translation of Marchetto's
> _Lucidarium_ by Herlinger, in the critical edition, may not
> pick up on some of the finer points of tuning.
>
> I was certainly correct about that.

In all fairness to Herlinger, I must say that the translation
of Marchetto is actually quite good and very accurate.

The only real improvement, at least regarding Treatise 2,
is that following Marchetto's wording and punctuation more
closely brings out the manner of his monochord division
more clearly.

This is decisively important, because he does not quantify
any of these measurements with numbers other than the
discussion of the 8:9 proportion and successive division
by 9.

Herlinger clearly failed to understand this method,
calling Marchetto's division of the 5 dieses "numerological
considerations, not quantitative measurements" [footnote g,
p 135].

On the contrary, they are exactly quantified measurements.
They were just described very vaguely in words by Marchetto,
and more vaguely by Herlinger's translation.

Marchetto probably *could* have learned to express the
manner of his division more exactly, but Latin theorists
of this period generally resorted to one-word Latin names
for particular (actually, superparticular < {;^) >)
ratios, and Marchetto was no exception.

And if my new interpretation turns out to be incorrect,
then my old one is certainly right, and Herlinger's is still
wrong. (The tradition of viewing Marchetto's dieses as
equal or quasi-equal goes back several centuries before
that, BTW.)

-monz

visit Marchetto at:
http://www.ixpres.com/interval/monzo/marchet/marchet.htm

🔗Afmmjr@xxx.xxx

🔗Ozan Yarman <ozanyarman@...>

🔗robert thomas martin <robertthomasmartin@...>

🔗monz <joemonz@...>

Hi Oz and Robert,

--- In tuning@yahoogroups.com, "robert thomas martin"
<robertthomasmartin@...> wrote:
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
> >
> > I was just re-reading monz' article on Marchetto on Padua:
> >
> > http://sonic-arts.org/monzo/marchet/marchet.htm

To Oz and everyone else:

_Please_ refer to the newer versions of my webpages
at the Tonalsoft website, and not to the old Sonic Arts
pages!

http://tonalsoft.com/monzo/marchetto/marchetto.aspx

The old Sonic Arts pages are mostly still lurking on
the internet, and eventually i'll make all of them into
redirection pages which point to the Tonalsoft pages.

I also realize that the old Sonic Arts pages have all
of the internal links intact, and most of the links in
the newer Tonalsoft pages are missing. But whenever i
make any updates, i make them only to the Tonalsoft pages.

To Oz:

> > Following the first diagram showing the division of the whole tone
> > into 9 parts, monz states that Marchetto advocated a wide variety
> of
> > small intervals. However, I think the passage:
> >
> > Any fifth part as it is desired, may be called a diesis, whether
> the
> > lowest (smallest?) or the highest (largest?) division, this is the
> > most important division that can be obtained in singing a tone."
> >
> > should be understood to mean the arithmetical division of 9:8 into
> > five parts:
> >
> > 40:41:42:43:44:45
> >
> > 0: 1/1 0.000 unison, perfect prime
> > 1: 45/44 38.906 1/5-tone
> > 2: 45/43 78.706 (enharmonic semitone, whose
> > half is a diesis or made up of 2 dieses)
> > 3: 15/14 119.443 major diatonic semitone (3
> > dieses, larger than 18:17. Plus diesis makes chromatic semitone)
> > 4: 45/41 161.161 (chromatic semitone or
> chroma,
> > completes to whole tone when diesis is added)
> > 5: 9/8 203.910 major whole tone
> >
> > This should clear up much issues! How about the neutral second
> > peculiar to maqams then?
> >
> > Oz.
> >

Yet another interesting possible interpretation of
Marchetto's theory! Will the speculation about this
ever end? ;-)

Thanks to your post piquing my interest in Marchetto
yet again, i've done some more thinking about it.

I'm still very much undecided as to exactly which
mathematics may accurately describe what Marchetto
wrote about in his theory ... and here is yet another
possibility that i've just thought of:

It has always bothered me that Marchetto first divides
the whole-tone into 9 parts, _then_ uses only the
odd-numbered portions of that to achieve his 5-part
division. What makes it really irksome is that he
writes "the first part is the first diesis", and then
describes the other four dieses as "from the first to
the third", "from the third to the fifth", etc.

So in a flash of sudden insight, i realized that perhaps
he was thinking of the integers the old-fashioned way,
that is, _without zero as part of the set_!

If he started by counting the whole monochord string
as the "first part" -- instead of the "zeroeth part"
as we would do it today (and as i did in my webpage
diagrams) -- then he could have a 9-part division
of the whole-tone by labeling the divisions thus:

(use the "Option" and "Use Fixed Width Font" links if
viewing this on the Yahoo website)

72 . 71 . 70 . 69 . 68 . 67 . 66 . 65 . 64 -- arithmetic part
1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 --- Marchetto "part"
1 ....... 2 ....... 3 ....... 4 ....... 5 --- Marchetto diesis
0 ....... 49 ...... 99 ...... 151 ..... 204 - ~cents

If this is the proper interpretation, then all Marchetto
was doing after all, was describing what are essentially
quarter-tones.

No great argument from me yet in support of this
interpretation -- it's just another of the many
possibilities i and others have already thought up.

To Robert:

> From Robert. I'm not sure if it clears up anything
> at all. The harmonic series keeps doubling the denominator
> of the ratios so that 2,4,8,16 etc are significant.
> 144/128=204cents. Why divide by five? Why divide by
> any number? Why not accept the harmonic series as
> it stands? From a scientific point of view these are
> legitimate questions.

You personally may not see any point in this kind of
division, and that's ok. But my page and the work of
my colleagues is intended to study and shed light on
an interesting theory published in 1318. Marchetto's
theory manifestly describes a whole-tone being divided
into 5 parts.

Exactly what he meant, we're not sure ... but he
definitely wrote about a 5-part division which extended
the chromatic possibilities beyond the usual two-sizes-
of-semitone.

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

🔗robert thomas martin <robertthomasmartin@...>

--- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:
>
> Hi Oz and Robert,
>
>
> --- In tuning@yahoogroups.com, "robert thomas martin"
> <robertthomasmartin@> wrote:
> >
> > --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
> > >
> > > I was just re-reading monz' article on Marchetto on Padua:
> > >
> > > http://sonic-arts.org/monzo/marchet/marchet.htm
>
>
> To Oz and everyone else:
>
> _Please_ refer to the newer versions of my webpages
> at the Tonalsoft website, and not to the old Sonic Arts
> pages!
>
> http://tonalsoft.com/monzo/marchetto/marchetto.aspx
>
> The old Sonic Arts pages are mostly still lurking on
> the internet, and eventually i'll make all of them into
> redirection pages which point to the Tonalsoft pages.
>
> I also realize that the old Sonic Arts pages have all
> of the internal links intact, and most of the links in
> the newer Tonalsoft pages are missing. But whenever i
> make any updates, i make them only to the Tonalsoft pages.
>
>
> To Oz:
>
> > > Following the first diagram showing the division of the whole
tone
> > > into 9 parts, monz states that Marchetto advocated a wide
variety
> > of
> > > small intervals. However, I think the passage:
> > >
> > > Any fifth part as it is desired, may be called a diesis,
whether
> > the
> > > lowest (smallest?) or the highest (largest?) division, this is
the
> > > most important division that can be obtained in singing a tone."
> > >
> > > should be understood to mean the arithmetical division of 9:8
into
> > > five parts:
> > >
> > > 40:41:42:43:44:45
> > >
> > > 0: 1/1 0.000 unison, perfect prime
> > > 1: 45/44 38.906 1/5-tone
> > > 2: 45/43 78.706 (enharmonic semitone,
whose
> > > half is a diesis or made up of 2 dieses)
> > > 3: 15/14 119.443 major diatonic semitone
(3
> > > dieses, larger than 18:17. Plus diesis makes chromatic semitone)
> > > 4: 45/41 161.161 (chromatic semitone or
> > chroma,
> > > completes to whole tone when diesis is added)
> > > 5: 9/8 203.910 major whole tone
> > >
> > > This should clear up much issues! How about the neutral second
> > > peculiar to maqams then?
> > >
> > > Oz.
> > >
>
>
> Yet another interesting possible interpretation of
> Marchetto's theory! Will the speculation about this
> ever end? ;-)
>
> Thanks to your post piquing my interest in Marchetto
> yet again, i've done some more thinking about it.
>
> I'm still very much undecided as to exactly which
> mathematics may accurately describe what Marchetto
> wrote about in his theory ... and here is yet another
> possibility that i've just thought of:
>
> It has always bothered me that Marchetto first divides
> the whole-tone into 9 parts, _then_ uses only the
> odd-numbered portions of that to achieve his 5-part
> division. What makes it really irksome is that he
> writes "the first part is the first diesis", and then
> describes the other four dieses as "from the first to
> the third", "from the third to the fifth", etc.
>
> So in a flash of sudden insight, i realized that perhaps
> he was thinking of the integers the old-fashioned way,
> that is, _without zero as part of the set_!
>
> If he started by counting the whole monochord string
> as the "first part" -- instead of the "zeroeth part"
> as we would do it today (and as i did in my webpage
> diagrams) -- then he could have a 9-part division
> of the whole-tone by labeling the divisions thus:
>
> (use the "Option" and "Use Fixed Width Font" links if
> viewing this on the Yahoo website)
>
> 72 . 71 . 70 . 69 . 68 . 67 . 66 . 65 . 64 -- arithmetic part
> 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 --- Marchetto "part"
> 1 ....... 2 ....... 3 ....... 4 ....... 5 --- Marchetto diesis
> 0 ....... 49 ...... 99 ...... 151 ..... 204 - ~cents
>
>
> If this is the proper interpretation, then all Marchetto
> was doing after all, was describing what are essentially
> quarter-tones.
>
> No great argument from me yet in support of this
> interpretation -- it's just another of the many
> possibilities i and others have already thought up.
>
>
> To Robert:
>
> > From Robert. I'm not sure if it clears up anything
> > at all. The harmonic series keeps doubling the denominator
> > of the ratios so that 2,4,8,16 etc are significant.
> > 144/128=204cents. Why divide by five? Why divide by
> > any number? Why not accept the harmonic series as
> > it stands? From a scientific point of view these are
> > legitimate questions.
>
>
> You personally may not see any point in this kind of
> division, and that's ok. But my page and the work of
> my colleagues is intended to study and shed light on
> an interesting theory published in 1318. Marchetto's
> theory manifestly describes a whole-tone being divided
> into 5 parts.
>
> Exactly what he meant, we're not sure ... but he
> definitely wrote about a 5-part division which extended
> the chromatic possibilities beyond the usual two-sizes-
> of-semitone.
>
>
> -monz
> http://tonalsoft.com/tonescape.aspx
> Tonescape microtonal music software
>

From Robert. I see the point better now that I have familiarised
myself more with the article at Tonalsoft. It seems to me that if
Marchetto was using some form of mean-tone temperament which
roughly corresponded to 31tet then the 5 division of the whole tone
makes perfect sense. The problem appears to involve the initial
condition of defining the value of the whole tone as about 204 or
about 192 cents. (144/128 or 143/128)

🔗robert thomas martin <robertthomasmartin@...>

--- In tuning@yahoogroups.com, "robert thomas martin"
<robertthomasmartin@...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <joemonz@> wrote:
> >
> > Hi Oz and Robert,
> >
> >
> > --- In tuning@yahoogroups.com, "robert thomas martin"
> > <robertthomasmartin@> wrote:
> > >
> > > --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
> > > >
> > > > I was just re-reading monz' article on Marchetto on Padua:
> > > >
> > > > http://sonic-arts.org/monzo/marchet/marchet.htm
> >
> >
> > To Oz and everyone else:
> >
> > _Please_ refer to the newer versions of my webpages
> > at the Tonalsoft website, and not to the old Sonic Arts
> > pages!
> >
> > http://tonalsoft.com/monzo/marchetto/marchetto.aspx
> >
> > The old Sonic Arts pages are mostly still lurking on
> > the internet, and eventually i'll make all of them into
> > redirection pages which point to the Tonalsoft pages.
> >
> > I also realize that the old Sonic Arts pages have all
> > of the internal links intact, and most of the links in
> > the newer Tonalsoft pages are missing. But whenever i
> > make any updates, i make them only to the Tonalsoft pages.
> >
> >
> > To Oz:
> >
> > > > Following the first diagram showing the division of the whole
> tone
> > > > into 9 parts, monz states that Marchetto advocated a wide
> variety
> > > of
> > > > small intervals. However, I think the passage:
> > > >
> > > > Any fifth part as it is desired, may be called a diesis,
> whether
> > > the
> > > > lowest (smallest?) or the highest (largest?) division, this
is
> the
> > > > most important division that can be obtained in singing a
tone."
> > > >
> > > > should be understood to mean the arithmetical division of 9:8
> into
> > > > five parts:
> > > >
> > > > 40:41:42:43:44:45
> > > >
> > > > 0: 1/1 0.000 unison, perfect prime
> > > > 1: 45/44 38.906 1/5-tone
> > > > 2: 45/43 78.706 (enharmonic semitone,
> whose
> > > > half is a diesis or made up of 2 dieses)
> > > > 3: 15/14 119.443 major diatonic
semitone
> (3
> > > > dieses, larger than 18:17. Plus diesis makes chromatic
semitone)
> > > > 4: 45/41 161.161 (chromatic semitone or
> > > chroma,
> > > > completes to whole tone when diesis is added)
> > > > 5: 9/8 203.910 major whole tone
> > > >
> > > > This should clear up much issues! How about the neutral
second
> > > > peculiar to maqams then?
> > > >
> > > > Oz.
> > > >
> >
> >
> > Yet another interesting possible interpretation of
> > Marchetto's theory! Will the speculation about this
> > ever end? ;-)
> >
> > Thanks to your post piquing my interest in Marchetto
> > yet again, i've done some more thinking about it.
> >
> > I'm still very much undecided as to exactly which
> > mathematics may accurately describe what Marchetto
> > wrote about in his theory ... and here is yet another
> > possibility that i've just thought of:
> >
> > It has always bothered me that Marchetto first divides
> > the whole-tone into 9 parts, _then_ uses only the
> > odd-numbered portions of that to achieve his 5-part
> > division. What makes it really irksome is that he
> > writes "the first part is the first diesis", and then
> > describes the other four dieses as "from the first to
> > the third", "from the third to the fifth", etc.
> >
> > So in a flash of sudden insight, i realized that perhaps
> > he was thinking of the integers the old-fashioned way,
> > that is, _without zero as part of the set_!
> >
> > If he started by counting the whole monochord string
> > as the "first part" -- instead of the "zeroeth part"
> > as we would do it today (and as i did in my webpage
> > diagrams) -- then he could have a 9-part division
> > of the whole-tone by labeling the divisions thus:
> >
> > (use the "Option" and "Use Fixed Width Font" links if
> > viewing this on the Yahoo website)
> >
> > 72 . 71 . 70 . 69 . 68 . 67 . 66 . 65 . 64 -- arithmetic part
> > 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 --- Marchetto "part"
> > 1 ....... 2 ....... 3 ....... 4 ....... 5 --- Marchetto diesis
> > 0 ....... 49 ...... 99 ...... 151 ..... 204 - ~cents
> >
> >
> > If this is the proper interpretation, then all Marchetto
> > was doing after all, was describing what are essentially
> > quarter-tones.
> >
> > No great argument from me yet in support of this
> > interpretation -- it's just another of the many
> > possibilities i and others have already thought up.
> >
> >
> > To Robert:
> >
> > > From Robert. I'm not sure if it clears up anything
> > > at all. The harmonic series keeps doubling the denominator
> > > of the ratios so that 2,4,8,16 etc are significant.
> > > 144/128=204cents. Why divide by five? Why divide by
> > > any number? Why not accept the harmonic series as
> > > it stands? From a scientific point of view these are
> > > legitimate questions.
> >
> >
> > You personally may not see any point in this kind of
> > division, and that's ok. But my page and the work of
> > my colleagues is intended to study and shed light on
> > an interesting theory published in 1318. Marchetto's
> > theory manifestly describes a whole-tone being divided
> > into 5 parts.
> >
> > Exactly what he meant, we're not sure ... but he
> > definitely wrote about a 5-part division which extended
> > the chromatic possibilities beyond the usual two-sizes-
> > of-semitone.
> >
> >
> > -monz
> > http://tonalsoft.com/tonescape.aspx
> > Tonescape microtonal music software
> >
>
> From Robert. I see the point better now that I have familiarised
> myself more with the article at Tonalsoft. It seems to me that if
> Marchetto was using some form of mean-tone temperament which
> roughly corresponded to 31tet then the 5 division of the whole
tone
> makes perfect sense. The problem appears to involve the initial
> condition of defining the value of the whole tone as about 204 or
> about 192 cents. (144/128 or 143/128)
>

More from Robert. Here is a possible solution of dividing a mean
whole tone into 5 parts using ratios from the same denominator
family:

128/128 = 0 cents
131/128 = 40
134/128 = 79
137/128 = 118
140/128 = 155
143/128 = 192.

If this is not helpful to your enquiry then please disregard.

🔗monz <joemonz@...>

Hi Robert,

--- In tuning@yahoogroups.com, "robert thomas martin"
<robertthomasmartin@...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <joemonz@> wrote:
> >
> > --- In tuning@yahoogroups.com, "robert thomas martin"
> > <robertthomasmartin@> wrote:
> >
> > > From Robert. I'm not sure if it clears up anything
> > > at all. The harmonic series keeps doubling the denominator
> > > of the ratios so that 2,4,8,16 etc are significant.
> > > 144/128=204cents. Why divide by five? Why divide by
> > > any number? Why not accept the harmonic series as
> > > it stands? From a scientific point of view these are
> > > legitimate questions.
> >
> >
> > You personally may not see any point in this kind of
> > division, and that's ok. But my page and the work of
> > my colleagues is intended to study and shed light on
> > an interesting theory published in 1318. Marchetto's
> > theory manifestly describes a whole-tone being divided
> > into 5 parts.
> >
> > Exactly what he meant, we're not sure ... but he
> > definitely wrote about a 5-part division which extended
> > the chromatic possibilities beyond the usual two-sizes-
> > of-semitone.
> >
>
> From Robert. I see the point better now that I have
> familiarised myself more with the article at Tonalsoft.
> It seems to me that if Marchetto was using some form
> of mean-tone temperament which roughly corresponded
> to 31tet then the 5 division of the whole tone makes
> perfect sense. The problem appears to involve the initial
> condition of defining the value of the whole tone as
> about 204 or about 192 cents. (144/128 or 143/128)

No, Marchetto was not describing any form of meantone.
First of all, the earliest known possible description
of meantone tuning is by Arnold Schlick in 1511, and
it wasn't described accurately until Zarlino in 1558.

It's almost certainly true that meantone would have
been used in practice before it was described in theory,
and possibly long before -- but not 2 centuries before!

Anyway, if you read the Tonalsoft page more carefully,
you'll see that Marchetto explicitly describes the
whole-tone as the 9:8 ratio, several times in fact.
There is a whole chapter of his treatise explaining
that ratio, and that is the ratio which he proceeds to
divide into "5 parts". No meantone there.

However, it _is_ correct to point out that 31-edo and
1/4-comma-meantone both divide the (mean) whole-tone
into 5 parts.

But again, my latest interpretation shows that Marchetto
may have actually been trying to describe a system
which closely resembles quarter-tones, if by "first part"
he really didn't mean a measureable interval at all,
but only the whole string.

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

🔗robert thomas martin <robertthomasmartin@...>

--- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:
>
> Hi Robert,
>
>
> --- In tuning@yahoogroups.com, "robert thomas martin"
> <robertthomasmartin@> wrote:
> >
> > --- In tuning@yahoogroups.com, "monz" <joemonz@> wrote:
> > >
> > > --- In tuning@yahoogroups.com, "robert thomas martin"
> > > <robertthomasmartin@> wrote:
> > >
> > > > From Robert. I'm not sure if it clears up anything
> > > > at all. The harmonic series keeps doubling the denominator
> > > > of the ratios so that 2,4,8,16 etc are significant.
> > > > 144/128=204cents. Why divide by five? Why divide by
> > > > any number? Why not accept the harmonic series as
> > > > it stands? From a scientific point of view these are
> > > > legitimate questions.
> > >
> > >
> > > You personally may not see any point in this kind of
> > > division, and that's ok. But my page and the work of
> > > my colleagues is intended to study and shed light on
> > > an interesting theory published in 1318. Marchetto's
> > > theory manifestly describes a whole-tone being divided
> > > into 5 parts.
> > >
> > > Exactly what he meant, we're not sure ... but he
> > > definitely wrote about a 5-part division which extended
> > > the chromatic possibilities beyond the usual two-sizes-
> > > of-semitone.
> > >
> >
> > From Robert. I see the point better now that I have
> > familiarised myself more with the article at Tonalsoft.
> > It seems to me that if Marchetto was using some form
> > of mean-tone temperament which roughly corresponded
> > to 31tet then the 5 division of the whole tone makes
> > perfect sense. The problem appears to involve the initial
> > condition of defining the value of the whole tone as
> > about 204 or about 192 cents. (144/128 or 143/128)
>
>
> No, Marchetto was not describing any form of meantone.
> First of all, the earliest known possible description
> of meantone tuning is by Arnold Schlick in 1511, and
> it wasn't described accurately until Zarlino in 1558.
>
> It's almost certainly true that meantone would have
> been used in practice before it was described in theory,
> and possibly long before -- but not 2 centuries before!
>
> Anyway, if you read the Tonalsoft page more carefully,
> you'll see that Marchetto explicitly describes the
> whole-tone as the 9:8 ratio, several times in fact.
> There is a whole chapter of his treatise explaining
> that ratio, and that is the ratio which he proceeds to
> divide into "5 parts". No meantone there.
>
> However, it _is_ correct to point out that 31-edo and
> 1/4-comma-meantone both divide the (mean) whole-tone
> into 5 parts.
>
> But again, my latest interpretation shows that Marchetto
> may have actually been trying to describe a system
> which closely resembles quarter-tones, if by "first part"
> he really didn't mean a measureable interval at all,
> but only the whole string.
>
>
> -monz
> http://tonalsoft.com/tonescape.aspx
> Tonescape microtonal music software
>

From Robert. If so, then this suggests some form of 29 notes to the
octave which is what Ozan and Margo suggested. Please forgive the
digression.

🔗Ozan Yarman <ozanyarman@...>

monz,

Your new suggestion is ingenious, but Marchetto appears to imply an interval, not degree, from diesis. In your interpretation, we take the whole tone as an interval and dieses as degrees of the whole tone division instead of intervals. In an age of Pythagoreanism, I do not think defending quarter-tones makes much sense.

Let us analyze the quotes I copied from your website:

"Because the aforesaid chromatic, diatonic, and enharmonic [species] cannot be fully treated until the whole tone is examined (since they are semitones), we shall first study the nature of the whole tone and how it is divided by numbers."

The first critical information we glean from this statement is that the chromatic, diatonic and enharmonic are all semitones, meaning half the whole tone.

"We acknowledge therefore that the parts of itself will have to be inequalities, so that 1 is the first part; from 1 to 3, the second; from the 3 to 5, the third; from 5 to 7, the fourth; from 7 to 9, the fifth; and such 5th part is the fifth odd number of the total 9."

As you demonstrated:

1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 --- Marchetto "part"
1 ....... 2 ....... 3 ....... 4 ....... 5 --- Marchetto diesis

However, the word "inequalities" clearly implies that the parts are to be unequal in size. I believe, contrary to my previous assumption, that the 1-3 and 3-5 parts are smaller than 7-9 and 5-7 is even smaller. This means that Marchetto is concentrating the semitones in the center region of half the wholetone and that the 5th diesis is very large.

"Any fifth part as it is desired, may be called a diesis, whether the lowest (smallest?) or the highest (largest?) division, this is the most important division that can be obtained in singing a tone."

Marchetto ascribes great importance to the diesis, which seems to change drastically in size. He says, "maior divisio". It is translated here as the most important division, but I dare say he is quite literal about it. I think he means that the first, second and fifth diese are fairly large, considering that he is talking about singers and given the previous criterion on attaining semitones.

"It is true that from three of these dieses a diatonic semitone is made. This is the larger [semitone] which is called the major apotome. It is the larger part of the tone divided in two."

He clearly divides the whole tone into two (possibly arithmetically) and wants us to recognize the major apotome/diatonic semitone as the larger (but not the largest) semitone part composed of three dieses. He says major apatome, not apatome plainly, which means that we are talking about an interval larger than 2187:2048.

"The diatonic semitone occurs when a permutation is made from round b to square b or vice versa, whether in ascent or descent, as here [musical example]:"

a..bb..b..c
a..G...E..C

In the musical example above, I dare say that the interval between bb and b is none other than 15:14.

"Note that the nature of the diesis is best understood thru comparison with the chromatic semitone; much concerning the diesis will become clear as we demonstrate the nature of the chromatic semitone."

I think he means that the diesis will reveal itself when a chromatic semitone is sung.
"The Chromatic Semitone is that which includes 4 of the 5 dieses of the whole tone, and, as said earlier, it completes a whole tone when a diesis is added to it."

Here comes the confusing part. Remembering that the sizes of dieses are likely unequal, and that the chromatic semitone will demonstrate their nature, I believe that this must be the largest semitone at 27:25, rendering a very large fifth diesis with the ratio 25:24.

"It results when some whole tone is divided in 2 so as to color some dissonance such as a 3rd, a 6th, or a 10th striving toward some consonance."

I understand from this statement that this interval serves as a leading-tone, by yielding 25:24 when a tone is subtracted from it!

"The first part of a tone thus divisible will be larger if the melody ascends, and is called a chroma; the part that remains is a diesis, as here: [musical example]"

c..c#.d | f..f#.g | G..G#.A

F..E..D | F..D..C | C..E..D

This perfectly gives us a chroma of 25:24 between c#-d, f#-g and G#-A!

"from the diatonic and diesis arise the chromatic."

The fourth diesis is only 14 cents large and occuring between 15:14 and 27:25 in support of all the previous speculations.

"2 of these 5 intervals [the dieses] joined together make up the enharmonic semitone, which is the smaller."

The diesis here is 35 cents and is the tritonic diesis at 50:49, which is half the chroma with the ratio 25:24. So the whole tone division of Marchetto becomes:

1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 --- Marchetto "part"
1 ....... 2 ....... 3 ....... 4 ....... 5 --- Marchetto diesis
50:49 ....25:24 ... 15:14 ... 27:25 ... 9:8 -- ratios of pitches
35 ...... 70 ...... 119 ..... 133 ..... 204 -- tones in cents
35 ...... 35 ...... 49 ...... 14 ...... 71 --- consecutive intervals in cents

Cordially,
Oz.

On Aug 15, 2008, at 2:24 AM, monz wrote:

> Hi Oz and Robert,
>
>
> --- In tuning@yahoogroups.com, "robert thomas martin"
> <robertthomasmartin@...> wrote:
>>
>> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
>>>
>>> I was just re-reading monz' article on Marchetto on Padua:
>>>
>>> http://sonic-arts.org/monzo/marchet/marchet.htm
>
>
> To Oz and everyone else:
>
> _Please_ refer to the newer versions of my webpages
> at the Tonalsoft website, and not to the old Sonic Arts
> pages!
>
> http://tonalsoft.com/monzo/marchetto/marchetto.aspx
>
> The old Sonic Arts pages are mostly still lurking on
> the internet, and eventually i'll make all of them into
> redirection pages which point to the Tonalsoft pages.
>
> I also realize that the old Sonic Arts pages have all
> of the internal links intact, and most of the links in
> the newer Tonalsoft pages are missing. But whenever i
> make any updates, i make them only to the Tonalsoft pages.
>
>
> To Oz:
>
>>> Following the first diagram showing the division of the whole tone
>>> into 9 parts, monz states that Marchetto advocated a wide variety
>> of
>>> small intervals. However, I think the passage:
>>>
>>> Any fifth part as it is desired, may be called a diesis, whether
>> the
>>> lowest (smallest?) or the highest (largest?) division, this is the
>>> most important division that can be obtained in singing a tone."
>>>
>>> should be understood to mean the arithmetical division of 9:8 into
>>> five parts:
>>>
>>> 40:41:42:43:44:45
>>>
>>> 0: 1/1 0.000 unison, perfect prime
>>> 1: 45/44 38.906 1/5-tone
>>> 2: 45/43 78.706 (enharmonic semitone, whose
>>> half is a diesis or made up of 2 dieses)
>>> 3: 15/14 119.443 major diatonic semitone (3
>>> dieses, larger than 18:17. Plus diesis makes chromatic semitone)
>>> 4: 45/41 161.161 (chromatic semitone or
>> chroma,
>>> completes to whole tone when diesis is added)
>>> 5: 9/8 203.910 major whole tone
>>>
>>> This should clear up much issues! How about the neutral second
>>> peculiar to maqams then?
>>>
>>> Oz.
>>>
>
>
> Yet another interesting possible interpretation of
> Marchetto's theory! Will the speculation about this
> ever end? ;-)
>
> Thanks to your post piquing my interest in Marchetto
> yet again, i've done some more thinking about it.
>
> I'm still very much undecided as to exactly which
> mathematics may accurately describe what Marchetto
> wrote about in his theory ... and here is yet another
> possibility that i've just thought of:
>
> It has always bothered me that Marchetto first divides
> the whole-tone into 9 parts, _then_ uses only the
> odd-numbered portions of that to achieve his 5-part
> division. What makes it really irksome is that he
> writes "the first part is the first diesis", and then
> describes the other four dieses as "from the first to
> the third", "from the third to the fifth", etc.
>
> So in a flash of sudden insight, i realized that perhaps
> he was thinking of the integers the old-fashioned way,
> that is, _without zero as part of the set_!
>
> If he started by counting the whole monochord string
> as the "first part" -- instead of the "zeroeth part"
> as we would do it today (and as i did in my webpage
> diagrams) -- then he could have a 9-part division
> of the whole-tone by labeling the divisions thus:
>
> (use the "Option" and "Use Fixed Width Font" links if
> viewing this on the Yahoo website)
>
> 72 . 71 . 70 . 69 . 68 . 67 . 66 . 65 . 64 -- arithmetic part
> 1 .. 2 .. 3 .. 4 .. 5 .. 6 .. 7 .. 8 .. 9 --- Marchetto "part"
> 1 ....... 2 ....... 3 ....... 4 ....... 5 --- Marchetto diesis
> 0 ....... 49 ...... 99 ...... 151 ..... 204 - ~cents
>
>
> If this is the proper interpretation, then all Marchetto
> was doing after all, was describing what are essentially
> quarter-tones.
>
> No great argument from me yet in support of this
> interpretation -- it's just another of the many
> possibilities i and others have already thought up.
>
>
> To Robert:
>
>> From Robert. I'm not sure if it clears up anything
>> at all. The harmonic series keeps doubling the denominator
>> of the ratios so that 2,4,8,16 etc are significant.
>> 144/128=204cents. Why divide by five? Why divide by
>> any number? Why not accept the harmonic series as
>> it stands? From a scientific point of view these are
>> legitimate questions.
>
>
> You personally may not see any point in this kind of
> division, and that's ok. But my page and the work of
> my colleagues is intended to study and shed light on
> an interesting theory published in 1318. Marchetto's
> theory manifestly describes a whole-tone being divided
> into 5 parts.
>
> Exactly what he meant, we're not sure ... but he
> definitely wrote about a 5-part division which extended
> the chromatic possibilities beyond the usual two-sizes-
> of-semitone.
>
>
> -monz
> http://tonalsoft.com/tonescape.aspx
> Tonescape microtonal music software
>
>
>
>

🔗Dave Keenan <d.keenan@...>

🔗Ozan Yarman <ozanyarman@...>

Dave, Marchetto seems to take the odd numbers from 1 to 9, where the first part is the first diesis. So, the diagram should be:

| 0-1 | 1-2 | 2-3 | 3-4 | 4-5 | 5-6 | 6-7 | 7-8 | 8-9 | Marchetto ninths of a tone
|--1--| |-----3-----| |-----5-----| Marchetto "parts" (later dieses)
|-----2-----| |-----4-----| " " " "

I interpret this to mean:

Type of interval Size in dieses Ratio Cents
-----------------------------------------------------
whole tone 5 dieses 9:8 204
chromatic semitone 4 dieses 27:25 133
diatonic semitone 3 dieses 15:14 119
enharmonic semitone 2 dieses 25:24 70

x......X...x....X.....x.....X.x..X.......x........X (ninths of a tone)
0......1........2...........3....4................5 (dieses)
0......35......70..........119..133...............204 cents

Oz.

On Aug 16, 2008, at 7:01 AM, Dave Keenan wrote:

> I think that Marchetto was one confused dude, mathematically.
>
> When I read this:
>
> "We acknowledge therefore that the parts of itself will have to be
> inequalities, so that 1 is the first part; from 1 to 3, the second;
> from the 3 to 5, the third; from 5 to 7, the fourth; from 7 to 9, the
> fifth; and such 5th part is the fifth odd number of the total 9."
>
> I get this:
>
> | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Marchetto ninths of a tone
> |-1-| |-----3-----| |-----5-----| Marchetto "parts" (later dieses)
> |-----2-----| |-----4-----| " " " "
>
> When he says the 5th part is the fifth odd number, that implies that
> _all_ the parts were odd numbers. In what sense can a part _be_ an odd
> number. At least in this interpretation we have an obvious answer: by
> it's size in ninth-tones. None of the other interpretations I've seen
> have such an obvious sense in which all the parts _are_ odd numbers.
>
> However I completely fail to match that up with this table:
>
> Type of interval Size in dieses Ratio Cents
> -----------------------------------------------------
> whole tone 5 dieses 9/8 204
> chromatic semitone 4 dieses -- --
> diatonic semitone 3 dieses 17/16 105
> enharmonic semitone 2 dieses 18/17 99
>
> It seems extremely odd that he should have
>
> diatonic_semitone + enharmonic_semitone = whole tone
>
> instead of the usual
>
> chromatic_semitone + diatonic_semitone = whole tone
>
> The fact that he's talking about singers makes me think "5-limit" if
> there was any harmony involved.
>
> -- Dave Keenan

🔗monz <joemonz@...>

Hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> I think that Marchetto was one confused dude, mathematically.

I'm not so sure about that. I think he just described
his mathematical ideas very poorly, and now it's difficult
for us to figure out exactly what he meant.

> When I read this:
>
> "We acknowledge therefore that the parts of itself will have
> to be inequalities, so that 1 is the first part; from 1 to 3,
> the second; from the 3 to 5, the third; from 5 to 7, the
> fourth; from 7 to 9, the fifth; and such 5th part is the
> fifth odd number of the total 9."
>
> I get this:
>
> | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | Marchetto ninths of a tone
> |-1-| |-----3-----| |-----5-----| Marchetto "parts" (later dieses)
> |-----2-----| |-----4-----| " " " "
>
> When he says the 5th part is the fifth odd number, that
> implies that _all_ the parts were odd numbers. In what
> sense can a part _be_ an odd number. At least in this
> interpretation we have an obvious answer: by it's size
> in ninth-tones. None of the other interpretations I've seen
> have such an obvious sense in which all the parts _are_
> odd numbers.

But as i wrote on my webpage, in 1318 i'm absolutely
certain that Marchetto was thinking in terms of arithmetic
divisions of a monochord string, and not in terms of
a logarithmic division of pitch-space.

> However I completely fail to match that up with this table:
>
> Type of interval Size in dieses Ratio Cents
> -----------------------------------------------------
> whole tone 5 dieses 9/8 204
> chromatic semitone 4 dieses -- --
> diatonic semitone 3 dieses 17/16 105
> enharmonic semitone 2 dieses 18/17 99

That's what i find most intriguing about my latest
interpretation of his theory, as meaning to represent
~quarter-tones. The arithmetic division of the
9:8 whole-tone into 72:71:70:69:68:67:66:65:64 _does_
in fact perfectly produce the 18:17:16 division.

And it also makes a little sense of his saying that
the "first part is the first diesis", if in fact
you take his use of "diesis" to mean an actual pitch
instead of an interval.

> It seems extremely odd that he should have
>
> diatonic_semitone + enharmonic_semitone = whole tone
>
> instead of the usual
>
> chromatic_semitone + diatonic_semitone = whole tone

Yes, i've always remarked that that is unusual.
Marchetto's "chromatic semitone" is the larger one:

chromatic_semitone + diesis = whole-tone

> The fact that he's talking about singers makes me think
> "5-limit" if there was any harmony involved.

Yes, i think you're on the right track there. I mention
that on my webpage too. Marchetto's _Lucidarium_ appeared
right around the same time as Walter Odington's treatise,
which, as Partch noted, is the earliest mention in post-Greek
theory of the two different types of major-3rds, along with
the exact ratios for them: 81:64 and 80:64 (= 5:4).
The timing is right.

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

🔗Dave Keenan <d.keenan@...>

--- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:
> But as i wrote on my webpage, in 1318 i'm absolutely
> certain that Marchetto was thinking in terms of arithmetic
> divisions of a monochord string, and not in terms of
> a logarithmic division of pitch-space.

Oh sure. I don't have a problem with that. I didn't intend to imply
that the "ninth tones" were necessarily logarithmically equal. But
when you divide a tone into nine parts, be it arithmetically,
geometrically or harmonically, they will be _nearly_ equal
logarithmically in any case.

> > However I completely fail to match that up with this table:
> >
> > Type of interval Size in dieses Ratio Cents
> > -----------------------------------------------------
> > whole tone 5 dieses 9/8 204
> > chromatic semitone 4 dieses -- --
> > diatonic semitone 3 dieses 17/16 105
> > enharmonic semitone 2 dieses 18/17 99
>
>
> That's what i find most intriguing about my latest
> interpretation of his theory, as meaning to represent
> ~quarter-tones. The arithmetic division of the
> 9:8 whole-tone into 72:71:70:69:68:67:66:65:64 _does_
> in fact perfectly produce the 18:17:16 division.

That certainly is attractive. But how do you then have
3 dieses in the "diatonic" semitone 68:67:66:65:64, and
2 dieses in the "enharmonic" semitone 72:71:70:69:68
while corresponding to that business of 1, 1-3, 3-5, 5-7, 7-9.

And how do you make sense of the 5th part [i.e. 7-9] being the fifth
odd number. As Oz noted, an obvious meaning of "the fifth odd number"
is the number 9. But in what sense could we say "the 5th part is 9"
since we've just been told that the fifth part is 7 to 9?

> And it also makes a little sense of his saying that
> the "first part is the first diesis", if in fact
> you take his use of "diesis" to mean an actual pitch
> instead of an interval.

I find that "if" to be far-fetched. And if a diesis is a pitch here,
then why not elsewhere? e.g. In what sense could a diatonic semitone
be 3 _pitches_?

Regarding the semitone names: I guess if you're going to have 3 kinds
of semitone and name them after the chromatic, diatonic and enharmonic
tetrachord genera in which they occur, then the smallest has to be
named "enharmonic" and the largest "chromatic". But in the
tetrachords, these would normally be something like 67, 90 and 114
cents respectively, whereas Marchetto's would seem to be 99, 105 and
something in the range 125 to 165 cents.

-- Dave Keenan

🔗Ozan Yarman <ozanyarman@...>

Dave and monz, I do not think the interpretation you are talking about (en.99, dia.105, chr.125-165 cents) is correct. What do you make of the diatonic semitone being called the major apotome and that being the larger part of the whole tone divided in two then? I don't think Marchetto was a non-conformist at all, and the correct range for enharmonic is somewhere about 60-90 cents, for diatonic, 110-120 cents, and for enharmonic, 130-145 cents. For the sake of keeping the intervals between them homogeneous, how about the suggestion below?

Diesis 37 cents 47:46
Enharmonic semitone 74 cents 24:23
Diatonic semitone 109 cents 49:46
Chromatic semitone 144 cents 25:23
Whole tone 204 cents 9:8

........S........E........D........C..............W

| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|---1---| |---3----| |------5-------|
|---2----| |---4----|

In which case, Marchetto seems to have taken the number 184 as the nut and 207 as the whole tone.

Oz.

On Aug 16, 2008, at 1:50 PM, Dave Keenan wrote:

> --- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:
>> But as i wrote on my webpage, in 1318 i'm absolutely
>> certain that Marchetto was thinking in terms of arithmetic
>> divisions of a monochord string, and not in terms of
>> a logarithmic division of pitch-space.
>
> Oh sure. I don't have a problem with that. I didn't intend to imply
> that the "ninth tones" were necessarily logarithmically equal. But
> when you divide a tone into nine parts, be it arithmetically,
> geometrically or harmonically, they will be _nearly_ equal
> logarithmically in any case.
>
>>> However I completely fail to match that up with this table:
>>>
>>> Type of interval Size in dieses Ratio Cents
>>> -----------------------------------------------------
>>> whole tone 5 dieses 9/8 204
>>> chromatic semitone 4 dieses -- --
>>> diatonic semitone 3 dieses 17/16 105
>>> enharmonic semitone 2 dieses 18/17 99
>>
>>
>> That's what i find most intriguing about my latest
>> interpretation of his theory, as meaning to represent
>> ~quarter-tones. The arithmetic division of the
>> 9:8 whole-tone into 72:71:70:69:68:67:66:65:64 _does_
>> in fact perfectly produce the 18:17:16 division.
>
> That certainly is attractive. But how do you then have
> 3 dieses in the "diatonic" semitone 68:67:66:65:64, and
> 2 dieses in the "enharmonic" semitone 72:71:70:69:68
> while corresponding to that business of 1, 1-3, 3-5, 5-7, 7-9.
>
> And how do you make sense of the 5th part [i.e. 7-9] being the fifth
> odd number. As Oz noted, an obvious meaning of "the fifth odd number"
> is the number 9. But in what sense could we say "the 5th part is 9"
> since we've just been told that the fifth part is 7 to 9?
>
>> And it also makes a little sense of his saying that
>> the "first part is the first diesis", if in fact
>> you take his use of "diesis" to mean an actual pitch
>> instead of an interval.
>
> I find that "if" to be far-fetched. And if a diesis is a pitch here,
> then why not elsewhere? e.g. In what sense could a diatonic semitone
> be 3 _pitches_?
>
> Regarding the semitone names: I guess if you're going to have 3 kinds
> of semitone and name them after the chromatic, diatonic and enharmonic
> tetrachord genera in which they occur, then the smallest has to be
> named "enharmonic" and the largest "chromatic". But in the
> tetrachords, these would normally be something like 67, 90 and 114
> cents respectively, whereas Marchetto's would seem to be 99, 105 and
> something in the range 125 to 165 cents.
>
> -- Dave Keenan

🔗Dave Keenan <d.keenan@...>

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>
> Dave and monz, I do not think the interpretation you are talking about
> (en.99, dia.105, chr.125-165 cents) is correct. What do you make of
> the diatonic semitone being called the major apotome and that being
> the larger part of the whole tone divided in two then? I don't think
> Marchetto was a non-conformist at all, and the correct range for
> enharmonic is somewhere about 60-90 cents, for diatonic, 110-120
> cents, and for enharmonic, 130-145 cents. For the sake of keeping the
> intervals between them homogeneous, how about the suggestion below?
>
> Diesis 37 cents 47:46
> Enharmonic semitone 74 cents 24:23
> Diatonic semitone 109 cents 49:46
> Chromatic semitone 144 cents 25:23
> Whole tone 204 cents 9:8
>
> ........S........E........D........C..............W
>
> | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
> |---1---| |---3----| |------5-------|
> |---2----| |---4----|
>
> In which case, Marchetto seems to have taken the number 184 as the nut
> and 207 as the whole tone.
>
> Oz.

Hi Oz,

That's a good point about Marchetto calling the diatonic semitone the
"major apotome". The etymology of "apotome" is "offcut", i.e. a
remainder. In the musical case it is usually the remainder when a
limma is cut from a whole tone, which leaves what we now call a
_chromatic_ semitone, not a diatonic one as Marchetto says. So I'm
willing to take apotome in its more general sense here and take it
that he is just telling us that his diatonic semitone is larger than
half a tone. Halving a tone in string-length terms means dividing the
tone 18:17:16. So I take it that he's telling us his diatonic semitone
is larger than 17:18 (99 cents), or maybe larger than 16:17 (105 cents).

Like almost every other commentator, I can only think he made a
mistake (or bad approximation) in describing his enharmonic and
diatonic semitones as 17:18 and 16:17 (and therefore as adding to make
a whole tone). 16:17 and 17:18 semitones are so close together in size
(6 cents difference) that there would be little point in giving them
different names, and one could hardly expect singers to hear or sing
them differently. And as I mentioned, they correspond to an _equal_
2-fold division of the whole tone in string-length terms, and he
apparently spent a lot of words telling is why he didn't want to
divide the tone into 2 or 4 or 8 etc equal parts.

Apart from the 16:17:18 thing, does Marchetto clearly say that
diatonic + enharmonic = whole tone? Or are we just inferring that from
his saying that diatonic = 3 dieses, enharmonic = 2 dieses and tone =
5 dieses, while forgetting that one of those dieses is much smaller
than the others (really a comma).

To be heard and sung distinctly, and to be worthy of the names
enharmonic, diatonic and chromatic, I feel they would need to differ
by _at_least_ a syntonic comma (at least 20 cents). But I find it far
more likely that they all differ by one of the larger of Marchetto's
dieses.

So I really like two aspect of your interpretation:
1. The fact that all three semitones can be demonstrated on the
monochord with the same starting pitch, that of the open string.
2. The fact that enharmonic, diatonic and chromatic semitones are in
approximately their normal ranges in relation to the tetrachordal
genera of the same name.

However it makes no sense to me that Marchetto would first derive an
_unequal_ 9-fold division of the tone in order to derive an unequal
5-fold division. He would just have gone directly to the unequal
5-fold division. The only point in doing a 9-fold division first, is
that the 9-fold division is _equal_ (in terms of string lengths).

I like Monz's original interpretation (shared by others): having
monochord movable-bridge positions (more like fret positions in modern
terms) at
72 thru 80 1/81-ths of the open string length.

In modern terms, Marchetto actually divides the tone into one comma
and four dieses. If we put the comma nearest the nut then with your
arrangment we get:

Enharmonic semitone 65 cents 78:81 26:27
Diatonic semitone 110 cents 76:81
Chromatic semitone 156 cents 74:81
Whole tone 204 cents 72:81 9:8

............E.......D.......C.......W
81 80 79 78 77 76 75 74 73 72
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
|-1-| |---3---| |---5---|
|---2---| |---4---|

This preserves E D and C having sizes close to their traditional
(ancient greek) sizes and makes them sufficiently different (45 cents
different) to be worth distinguishing for singers. And preserves them
being 2, 3 and 4 of Marchetto's "dieses". But no two of these
semitones add to give a whole tone.

Here are two online sources of Marchetto translations:
http://tonalsoft.com/monzo/marchetto/marchetto.aspx
http://mto.societymusictheory.org/issues/mto.98.4.6/mto.98.4.6.rahn.html

-- Dave Keenan

🔗monz <joemonz@...>

Hi Dave,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <joemonz@> wrote:
> >
> > That's what i find most intriguing about my latest
> > interpretation of his theory, as meaning to represent
> > ~quarter-tones. The arithmetic division of the
> > 9:8 whole-tone into 72:71:70:69:68:67:66:65:64 _does_
> > in fact perfectly produce the 18:17:16 division.
>
> That certainly is attractive. But how do you then have
> 3 dieses in the "diatonic" semitone 68:67:66:65:64, and
> 2 dieses in the "enharmonic" semitone 72:71:70:69:68
> while corresponding to that business of 1, 1-3, 3-5, 5-7, 7-9.
>
> And how do you make sense of the 5th part [i.e. 7-9] being
> the fifth odd number. As Oz noted, an obvious meaning of
> "the fifth odd number" is the number 9. But in what sense
> could we say "the 5th part is 9" since we've just been
> told that the fifth part is 7 to 9?

Again, i think that if one assumes that Marchetto was
beginning his counting with 1 rather than with 0, it
starts to make more sense. I think perhaps he was counting
the monochord divisions (i.e., his "dieses" and "parts")
according to the bridges which pinpoint pitches, rather
than according to the actual intervals between the pitches
as we would do today.

The Hindu-Arabic numeral system was really only first
written about in Europe by Fibonacci in 1202, and its
wide acceptance in Europe was mostly due to the spread
of the printing press and printed books in the 1400s. See:

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

So at the time Marchetto wrote _Lucidarium_ it is quite
likely that he would still have been starting his counting
with 1 instead of 0. That's the only way i can make sense
of his statement that "the first part is the first diesis".

But yes, there certainly are some complications in the
measurement of the various semitones according to numbers
of dieses.

> Regarding the semitone names: I guess if you're going to
> have 3 kinds of semitone and name them after the chromatic,
> diatonic and enharmonic tetrachord genera in which they
> occur, then the smallest has to be named "enharmonic"
> and the largest "chromatic". But in the tetrachords,
> these would normally be something like 67, 90 and 114
> cents respectively, whereas Marchetto's would seem to be
> 99, 105 and something in the range 125 to 165 cents.

Now _that_ is a useful observation! I'll keep that in mind
the next time i ponder Marchetto's semitones, and see if
i can find some way to fit his descriptions to those numbers.

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

🔗monz <joemonz@...>

Hi Dave and Oz,

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> That's a good point about Marchetto calling the diatonic
> semitone the "major apotome". The etymology of "apotome"
> is "offcut", i.e. a remainder. In the musical case it is
> usually the remainder when a limma is cut from a whole tone,
> which leaves what we now call a _chromatic_ semitone,
> not a diatonic one as Marchetto says. So I'm willing to
> take apotome in its more general sense here and take it
> that he is just telling us that his diatonic semitone is
> larger than half a tone. Halving a tone in string-length
> terms means dividing the tone 18:17:16. So I take it that
> he's telling us his diatonic semitone is larger than 17:18
> (99 cents), or maybe larger than 16:17 (105 cents).
>
> <snip>
>
> Apart from the 16:17:18 thing, does Marchetto clearly say
> that diatonic + enharmonic = whole tone? Or are we just
> inferring that from his saying that diatonic = 3 dieses,
> enharmonic = 2 dieses and tone = 5 dieses, while forgetting
> that one of those dieses is much smaller than the others
> (really a comma).

Marchetto does specify that enharmonic + diatonic = whole-tone,
represented as 2 + 3 = 5 dieses. And he also says that
chromatic + diesis = whole-tone, represented as 4 + 1 = 5 dieses.

Marchetto's musical illustrations show his enharmonic semitone
ascending as the one between A -> Bb (the pythagorean "limma"),
and his diatonic semitone ascending as Bb -> B (the pythagorean
"apotome").

He gives three illustrations of his chromatic semitone
ascending as C -> C#, F -> F#, and G -> G#. According to
traditional pythagorean theory, these are also all instances
of the "apotome", but Marchetto invokes his new larger
"chromatic semitone" as a special tuning used in cadences,
as has been most admirably explained by Margo.

You can see the musical illustrations on my webpage:
http://tonalsoft.com/monzo/marchetto/marchetto.aspx

(Note that the illustration of the diatonic semitone has
an incorrect clef on the lower staff -- i'm not sure if
i copied this error from Herlinger's translation of
_Lucidarium_, or if he has it correct and i introduced
it myself ... i don't have access to my copy right now.)

What i find most interesting is that this seems to preserve
a "hard-coded" link to ancient Greek theory. The ancient
Greek theory posited a gamut of pitches thus:
A B C D E F G A Bb B C D E F G A

The reason why it included a Bb is that the Lesser Perfect
System incorporated the _tetrachord synemmenon_ (conjunct
tetrachord) above the central note _mese_ ("A"), and
this required a note a semitone above _mese_. The Greater
Perfect System used the _tetrachord diezeugmenon_
(disjunct tetrachord) and thus the two tetrachords above
_mese_ exactly replicated the two below it, an octave
higher. See my webpages:

http://tonalsoft.com/enc/l/lps.aspx
http://tonalsoft.com/enc/g/gps.aspx
http://tonalsoft.com/enc/p/pis.aspx

This system was taken over by the medieval European theorists,
and the Greek name _systema teleion ametabolon_ ("Perfect
Immutable System" was apparently accepted entirely at
face-value by the medieval theorists -- that is, from the
earliest resurfacing of ancient Greek theory (in the
_enchiriadis_ treatises, c. 900) all the way up to c.1300,
theorists presented this set of pitches as _the_ set of
musical pitch resources, without any question and without
any options to include or consider any other pitches.

As noted by Margo, in Europe, the early 1300s was a time
of great experimentation in many aspects of music. It's
typical for theorists of today to say that the addition
of the other notes with flats, and also those with sharps,
was an extension of the pythagorean system.

I myself have explained the history this way. I.e.,
since Bb->B represents 3^7 (in terms of the pythagorean
lattice), and gives the primary example of _mutation_,
and a mutation of F->F# also represents 3^7, a mutation
of any nominal (A B C D E F G) to one with either a
sharp or flat would always represent 3^7. See:

http://tonalsoft.com/enc/p/pythagorean.aspx
http://tonalsoft.com/enc/m/mutation.aspx

http://tonalsoft.com/monzo/article/prime-factor-notation.aspx
(the "Extended 3-Limit Systems" section)

That last link shows a 17-tone pythagorean system posited
by Prosdocimus, who was a vehement critic of Marchetto.

But according to Marchetto, the C->C#, F->F#, G->G# intervals,
at least when used in a cadence, are all considered to be
the same interval, but he also considers them all to be
considerably wider than 3^7. Also note that Marchetto
invented a new symbol, which eventually became rendered
as our "#" (sharp) to represent these "chromatic semitones".

So while later theorists consider Bb and C#, F#, G# to all be
instances of the same type pitch inflection, apparently, to
Marchetto, C#, F#, G# were quite different from the "normal"
Bb.

For those who want to go straight to the source, the Latin
text of Treatise 2 is here:

http://www.chmtl.indiana.edu/tml/14th/MARLUC2_TEXT.html

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

🔗Ozan Yarman <ozanyarman@...>

O Dave,

The size of the chromatic semitone (4 dieses) suggests a super-Pythagorean augmented prime (apotome), while the size of the diatonic semitone (3 dieses) suggests a 5-limit JI or meantone minor second. Confusingly enough, Marchetto calles the latter major apotome. If his natural diatonic scale is Pythagorean, as it seems to be, then the actual diatonic semitone is the limma. But Marchetto calls the "larger remainder" the diatonic semitone, which would produce the dileimma (pun intended) that we have 90 cents as well as 114 cents as the diatonic semitone. To solve the issue, we must conclude that the diatonic natural scale is not a Pythagorean major, but JI or close to JI, perhaps the Rast scale suggested by Safi al-din Urmavi who lived a century prior to Marchetto:

G 1
A 9/8
B 8192/6561
C 4/3
D 3/2
E 27/16
F 16/9
G 2

In which case, the diatonic semitone becomes 2187:2048 and is the major remainder of the whole tone and the larger part of the tone divided into two (16:17:18 as you demonstrated). For this to be composed of three "dieses" that are not equalities, we need two quarter-tones and a comma, which arises if we divide the whole tone into 9 arithmetical parts (just as we divided it into two):

72/72 1/1 0 cents 0
73/72 24 cents 1 1st part/diesis
74/72 37/36 47 cents 2
75/72 25/24 71 cents 3 2nd part/diesis
76/72 19/18 94 cents 4
77/72 116 cents 5 3rd part/diesis
78/72 13/12 139 cents 6
79/72 161 cents 7 4th part/diesis
80/72 10/9 182 cents 8
81/72 9/8 204 cents 9 5th part/diesis

Notice, that we have a comma of 24 cents as the first diesis, a quarter-tone of 47 cents as the second diesis and 46 cents as the third diesis. This solves the matter of the diatonic semitone.

What about the chromatic semitone then? Marchetto says that it is the chroma that completes to a whole tone with the addition of the diesis, is a diatonic semitone plus a diesis, and is composed of 4 of the 5 dieses mentioned. This could mean nothing other then 161 cents given above. Though I would have much preferred 13:12 as the chroma, but Marchetto seems to insist on taking only the primes within the number 9.

This leaves us with 25/24 as the enharmonic semitone, if we take a comma plus a quarter-tone, or 19/18 as the enharmonic semitone if we take two quarter-tones. This interpretation favours 53-LEDO as a basis for "Marchettan harmony"

Oz.

On Aug 17, 2008, at 5:58 AM, Dave Keenan wrote:

> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>
>> Dave and monz, I do not think the interpretation you are talking >> about
>> (en.99, dia.105, chr.125-165 cents) is correct. What do you make of
>> the diatonic semitone being called the major apotome and that being
>> the larger part of the whole tone divided in two then? I don't think
>> Marchetto was a non-conformist at all, and the correct range for
>> enharmonic is somewhere about 60-90 cents, for diatonic, 110-120
>> cents, and for enharmonic, 130-145 cents. For the sake of keeping the
>> intervals between them homogeneous, how about the suggestion below?
>>
>> Diesis 37 cents 47:46
>> Enharmonic semitone 74 cents 24:23
>> Diatonic semitone 109 cents 49:46
>> Chromatic semitone 144 cents 25:23
>> Whole tone 204 cents 9:8
>>
>> ........S........E........D........C..............W
>>
>> | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
>> |---1---| |---3----| |------5-------|
>> |---2----| |---4----|
>>
>> In which case, Marchetto seems to have taken the number 184 as the >> nut
>> and 207 as the whole tone.
>>
>> Oz.
>
> Hi Oz,
>
> That's a good point about Marchetto calling the diatonic semitone the
> "major apotome". The etymology of "apotome" is "offcut", i.e. a
> remainder. In the musical case it is usually the remainder when a
> limma is cut from a whole tone, which leaves what we now call a
> _chromatic_ semitone, not a diatonic one as Marchetto says. So I'm
> willing to take apotome in its more general sense here and take it
> that he is just telling us that his diatonic semitone is larger than
> half a tone. Halving a tone in string-length terms means dividing the
> tone 18:17:16. So I take it that he's telling us his diatonic semitone
> is larger than 17:18 (99 cents), or maybe larger than 16:17 (105 > cents).
>
> Like almost every other commentator, I can only think he made a
> mistake (or bad approximation) in describing his enharmonic and
> diatonic semitones as 17:18 and 16:17 (and therefore as adding to make
> a whole tone). 16:17 and 17:18 semitones are so close together in size
> (6 cents difference) that there would be little point in giving them
> different names, and one could hardly expect singers to hear or sing
> them differently. And as I mentioned, they correspond to an _equal_
> 2-fold division of the whole tone in string-length terms, and he
> apparently spent a lot of words telling is why he didn't want to
> divide the tone into 2 or 4 or 8 etc equal parts.
>
> Apart from the 16:17:18 thing, does Marchetto clearly say that
> diatonic + enharmonic = whole tone? Or are we just inferring that from
> his saying that diatonic = 3 dieses, enharmonic = 2 dieses and tone =
> 5 dieses, while forgetting that one of those dieses is much smaller
> than the others (really a comma).
>
> To be heard and sung distinctly, and to be worthy of the names
> enharmonic, diatonic and chromatic, I feel they would need to differ
> by _at_least_ a syntonic comma (at least 20 cents). But I find it far
> more likely that they all differ by one of the larger of Marchetto's
> dieses.
>
> So I really like two aspect of your interpretation:
> 1. The fact that all three semitones can be demonstrated on the
> monochord with the same starting pitch, that of the open string.
> 2. The fact that enharmonic, diatonic and chromatic semitones are in
> approximately their normal ranges in relation to the tetrachordal
> genera of the same name.
>
> However it makes no sense to me that Marchetto would first derive an
> _unequal_ 9-fold division of the tone in order to derive an unequal
> 5-fold division. He would just have gone directly to the unequal
> 5-fold division. The only point in doing a 9-fold division first, is
> that the 9-fold division is _equal_ (in terms of string lengths).
>
> I like Monz's original interpretation (shared by others): having
> monochord movable-bridge positions (more like fret positions in modern
> terms) at
> 72 thru 80 1/81-ths of the open string length.
>
> In modern terms, Marchetto actually divides the tone into one comma
> and four dieses. If we put the comma nearest the nut then with your
> arrangment we get:
>
> Enharmonic semitone 65 cents 78:81 26:27
> Diatonic semitone 110 cents 76:81
> Chromatic semitone 156 cents 74:81
> Whole tone 204 cents 72:81 9:8
>
> ............E.......D.......C.......W
> 81 80 79 78 77 76 75 74 73 72
> | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
> |-1-| |---3---| |---5---|
> |---2---| |---4---|
>
> This preserves E D and C having sizes close to their traditional
> (ancient greek) sizes and makes them sufficiently different (45 cents
> different) to be worth distinguishing for singers. And preserves them
> being 2, 3 and 4 of Marchetto's "dieses". But no two of these
> semitones add to give a whole tone.
>
> Here are two online sources of Marchetto translations:
> http://tonalsoft.com/monzo/marchetto/marchetto.aspx
> http://mto.societymusictheory.org/issues/mto.98.4.6/mto.98.4.6.rahn.html
>
> -- Dave Keenan
>
>
> ------------------------------------
>
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>

🔗Ozan Yarman <ozanyarman@...>

monz, I finally came back to adopting your original interpretation of Marchetto, with the minor difference that I think the natural diatonic scale he adopted was Urmavi's Rast. Also, if what I glean from your site is correct, the 17-tone scale you attribute to Beldemandis Prosdocimus is not originally his, but Abbasid music theorist Safi al-din Urmavi's (1216-1294). Please do not disregard the Islamic connection in Medieval European music theory. The fact that Prosdocimus replicated Urmavi's 17-tone Pythagorean scale in the 15th century lends credibility to my assumption that Marchetto might have known of Urmavi and made use of Rast as the natural diatonical scale, from which he found the diatonic semitone as the larger part of the whole tone divided in two.

Oz.

On Aug 18, 2008, at 1:03 AM, monz wrote:

> Hi Dave and Oz,
>
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>>
>> That's a good point about Marchetto calling the diatonic
>> semitone the "major apotome". The etymology of "apotome"
>> is "offcut", i.e. a remainder. In the musical case it is
>> usually the remainder when a limma is cut from a whole tone,
>> which leaves what we now call a _chromatic_ semitone,
>> not a diatonic one as Marchetto says. So I'm willing to
>> take apotome in its more general sense here and take it
>> that he is just telling us that his diatonic semitone is
>> larger than half a tone. Halving a tone in string-length
>> terms means dividing the tone 18:17:16. So I take it that
>> he's telling us his diatonic semitone is larger than 17:18
>> (99 cents), or maybe larger than 16:17 (105 cents).
>>
>> <snip>
>>
>> Apart from the 16:17:18 thing, does Marchetto clearly say
>> that diatonic + enharmonic = whole tone? Or are we just
>> inferring that from his saying that diatonic = 3 dieses,
>> enharmonic = 2 dieses and tone = 5 dieses, while forgetting
>> that one of those dieses is much smaller than the others
>> (really a comma).
>
>
> Marchetto does specify that enharmonic + diatonic = whole-tone,
> represented as 2 + 3 = 5 dieses. And he also says that
> chromatic + diesis = whole-tone, represented as 4 + 1 = 5 dieses.
>
> Marchetto's musical illustrations show his enharmonic semitone
> ascending as the one between A -> Bb (the pythagorean "limma"),
> and his diatonic semitone ascending as Bb -> B (the pythagorean
> "apotome").
>
> He gives three illustrations of his chromatic semitone
> ascending as C -> C#, F -> F#, and G -> G#. According to
> traditional pythagorean theory, these are also all instances
> of the "apotome", but Marchetto invokes his new larger
> "chromatic semitone" as a special tuning used in cadences,
> as has been most admirably explained by Margo.
>
> You can see the musical illustrations on my webpage:
> http://tonalsoft.com/monzo/marchetto/marchetto.aspx
>
> (Note that the illustration of the diatonic semitone has
> an incorrect clef on the lower staff -- i'm not sure if
> i copied this error from Herlinger's translation of
> _Lucidarium_, or if he has it correct and i introduced
> it myself ... i don't have access to my copy right now.)
>
>
> What i find most interesting is that this seems to preserve
> a "hard-coded" link to ancient Greek theory. The ancient
> Greek theory posited a gamut of pitches thus:
> A B C D E F G A Bb B C D E F G A
>
> The reason why it included a Bb is that the Lesser Perfect
> System incorporated the _tetrachord synemmenon_ (conjunct
> tetrachord) above the central note _mese_ ("A"), and
> this required a note a semitone above _mese_. The Greater
> Perfect System used the _tetrachord diezeugmenon_
> (disjunct tetrachord) and thus the two tetrachords above
> _mese_ exactly replicated the two below it, an octave
> higher. See my webpages:
>
> http://tonalsoft.com/enc/l/lps.aspx
> http://tonalsoft.com/enc/g/gps.aspx
> http://tonalsoft.com/enc/p/pis.aspx
>
> This system was taken over by the medieval European theorists,
> and the Greek name _systema teleion ametabolon_ ("Perfect
> Immutable System" was apparently accepted entirely at
> face-value by the medieval theorists -- that is, from the
> earliest resurfacing of ancient Greek theory (in the
> _enchiriadis_ treatises, c. 900) all the way up to c.1300,
> theorists presented this set of pitches as _the_ set of
> musical pitch resources, without any question and without
> any options to include or consider any other pitches.
>
> As noted by Margo, in Europe, the early 1300s was a time
> of great experimentation in many aspects of music. It's
> typical for theorists of today to say that the addition
> of the other notes with flats, and also those with sharps,
> was an extension of the pythagorean system.
>
> I myself have explained the history this way. I.e.,
> since Bb->B represents 3^7 (in terms of the pythagorean
> lattice), and gives the primary example of _mutation_,
> and a mutation of F->F# also represents 3^7, a mutation
> of any nominal (A B C D E F G) to one with either a
> sharp or flat would always represent 3^7. See:
>
> http://tonalsoft.com/enc/p/pythagorean.aspx
> http://tonalsoft.com/enc/m/mutation.aspx
>
> http://tonalsoft.com/monzo/article/prime-factor-notation.aspx
> (the "Extended 3-Limit Systems" section)
>
> That last link shows a 17-tone pythagorean system posited
> by Prosdocimus, who was a vehement critic of Marchetto.
>
> But according to Marchetto, the C->C#, F->F#, G->G# intervals,
> at least when used in a cadence, are all considered to be
> the same interval, but he also considers them all to be
> considerably wider than 3^7. Also note that Marchetto
> invented a new symbol, which eventually became rendered
> as our "#" (sharp) to represent these "chromatic semitones".
>
> So while later theorists consider Bb and C#, F#, G# to all be
> instances of the same type pitch inflection, apparently, to
> Marchetto, C#, F#, G# were quite different from the "normal"
> Bb.
>
> For those who want to go straight to the source, the Latin
> text of Treatise 2 is here:
>
> http://www.chmtl.indiana.edu/tml/14th/MARLUC2_TEXT.html
>
>
>
> -monz
> http://tonalsoft.com/tonescape.aspx
> Tonescape microtonal music software
>

🔗Aaron Krister Johnson <aaron@...>

🔗Ozan Yarman <ozanyarman@...>

Aaron, apparently you are misinformed and misguided about the so-called destruction of Alexandria by the Muslims during the reign of Caliph Omar. All sorts of filth and lies have been disseminated via internet for the past decades to taint Islam. What a disturbing deja vu this is. Please read these articles:

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

http://www.crystalinks.com/libraryofalexandria.html

The intellectual part of the Muslim World has done everything to preserve and further knowledge, including even those by heathen Greeks and pagan Hindus. If anything, Islam has been the instrument of preserving the wisdom of Antiquity. When I read about books on Western philosophy, there is a disturbing two thousand year gap from Zeno to Descartes, as if Islamic philosophers, including members of Mutazilah, had nothing to say on metaphysics and morality. The contributions to the theory of music by scholars such as Kindi, Farabi, Ibn Sina, Urmevi, Shirazi, Meragi, Jami are immense.

Oz.

On Aug 18, 2008, at 5:12 AM, Aaron Krister Johnson wrote:

> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>> Please do not disregard the Islamic
>> connection in Medieval European music theory.
>
> It makes sense to me that Islamic scholars really fed this.
>
> This would be consistent with so many other areas of intellectual
> pursuit at the time--the Islamic connection in Medieval Europe
> intellectual life in general was one of straight lineage to ancient
> Greece through Muslims preserving Greek manuscripts...(outside of
> incidents like the destruction of Alexandria by Muslims)
>
> From what I understand, Al-Farabi was the earliest source of Greek
> music theory, and obviously made his on contributions firmly rooted in
> the Greek tradition.
>
> In any event, I would not be at all surprised to find that the most
> authentic way, in a majority of cases, to do much medieval music would
> be to have a 'near eastern sound'...my initial experiments in retuning
> medieval music this way seem very convincing to me, at least.
>
> -AKJ.

🔗Carl Lumma <carl@...>

🔗monz <joemonz@...>

🔗Daniel Wolf <djwolf@...>

The contributions by Muslim and Islamicate scholars to both the transmission of older theory and theoretical innovations are immense but the idea that there was Greek Music theory, which got lost, and was transmitted to Europe only via Islam is misleading, not least because it reduces Islamic theory to a way station on Europe's march towards polyphony, rather than identifying an important conjunction among traditions with very different trajectories. (Another, profound, connection is in the continuum of Ud and Lute, fretted and unfretted, monodic and polyphonic – and even a very important impetus to the development of keyboard music in the West —, with a geographical stretch so wide that drawing borders among them is an essentially meaningless exercise.)

In classical times, there were thriving cultivated musical traditions throughout the mediterranean and adjoining cultures, and the traces we have vividly suggest very deep connections and exchange as well as major differences among these cultures. The largest corpus of theory which survives classical times (at least in a known and accessible form -- the extent of music theory in the fertile crescent region being still a large unknown) is Greek and Hellenistic, and that scholarly corpus was maintained to a large extant in the huge area Byzantine music theory, before, during, and after the advent of Islam, while in the much more limited Roman theoretical tradition, represented by Boethius, was in fact continuous in the west (so continuous, in fact, that Boethius's errors, in classical terms, are often "correct", or at least more appropriate, in medieval European terms.) It may be argued that both of these theoretical traditions were seriously circumscribed versions of Classical and Hellenistic theory, but I would argue instead that they simply became closely adapted to contemporary practice in the respective traditions. In any case, one must be cautious not to ignore these practical and theoretical traditions in preference for a neat line of transfer of cultural goods and practices.

These theoretical traditions contain, moreover, compelling evidence about other practices in the larger cultural region which did not either maintain or cultivate a theoretical practice of their own, for example in the missing chapter of Boethius, concerning the division of the fourth into two parts, one immediately thinks of the trichordal modes that remain the basis of classical music -- practiced by both Christians and Muslims -- in Ethiopia. And when evidence is lacking, our curiosity is just as great: We know far too little about early musical practice in oriental Christendom, for example from the thriving musical culture of the Copts, whose musical tradition connects to both Ancient Egypt, the Hellenized world, the Upper Nile region and Aksum, and presumably to the Near East, with its own Jewish, Jewish-Christian, and early Christian diversity (and possibly others — Mandean, Samaritan, who knows?).

Likewise, the assimilation of classical and Hellenistic theory to practices throughout the Islamicate world has to be taken in stride, differentiating the science (or more precisely liberal art)of harmonics, with its culturally neutral techniques for measurement and comparison of intervals, and their composition into tones, n-chords, systems, and modes from the description of the specific materials of real musical practice. That practice, in the Islamicate world, includes — often assimilated, often isolated — continuity with its own huge range of additional cultures, radiating from Arabia to India, Persia, Central Asia, the Black Sea and the Caucasus, the Balkans, North Africa, and, yes, Byzantium.

Daniel Wolf

🔗Ozan Yarman <ozanyarman@...>

🔗Carl Lumma <carl@...>

🔗Aaron Krister Johnson <aaron@...>

Hi ozan,

This is very interesting. I had known that the Muslims were an enlightened lot and showed much moral integrity by allowing diversity of thought and belief among those they conquored. Unfortunately, the Christians didn't allow the same freedoms and also tended to mercilessly rape and slaughter even women and childen.

I had not heard the alternate interpretation of what happened to the great library at Alexandria, but it has the ring of truth that the story regarding the Muslims burning all non Koran texts could be mere propaganda, especially considering the broad prevalence of Muslim scholarship at the time.

Alas, would it that such a golden age of Islamic quest for knowledge existed today in the fundamentalist circles -one always hears of these severe madrassas where only the Koran is taught.

But we are straying from tuning, so let's get back to that, eh?

-AKJ

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>
> Aaron, apparently you are misinformed and misguided about the so-
> called destruction of Alexandria by the Muslims during the reign of
> Caliph Omar. All sorts of filth and lies have been disseminated via
> internet for the past decades to taint Islam. What a disturbing deja
> vu this is. Please read these articles:
>
> http://en.wikipedia.org/wiki/Muslim_conquest_of_Alexandria
>
> http://www.crystalinks.com/libraryofalexandria.html
>
> The intellectual part of the Muslim World has done everything to
> preserve and further knowledge, including even those by heathen Greeks
> and pagan Hindus. If anything, Islam has been the instrument of
> preserving the wisdom of Antiquity. When I read about books on Western
> philosophy, there is a disturbing two thousand year gap from Zeno to
> Descartes, as if Islamic philosophers, including members of Mutazilah,
> had nothing to say on metaphysics and morality. The contributions to
> the theory of music by scholars such as Kindi, Farabi, Ibn Sina,
> Urmevi, Shirazi, Meragi, Jami are immense.
>
> Oz.
>
> On Aug 18, 2008, at 5:12 AM, Aaron Krister Johnson wrote:
>
> > --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
> >> Please do not disregard the Islamic
> >> connection in Medieval European music theory.
> >
> > It makes sense to me that Islamic scholars really fed this.
> >
> > This would be consistent with so many other areas of intellectual
> > pursuit at the time--the Islamic connection in Medieval Europe
> > intellectual life in general was one of straight lineage to ancient
> > Greece through Muslims preserving Greek manuscripts...(outside of
> > incidents like the destruction of Alexandria by Muslims)
> >
> > From what I understand, Al-Farabi was the earliest source of Greek
> > music theory, and obviously made his on contributions firmly rooted in
> > the Greek tradition.
> >
> > In any event, I would not be at all surprised to find that the most
> > authentic way, in a majority of cases, to do much medieval music would
> > be to have a 'near eastern sound'...my initial experiments in retuning
> > medieval music this way seem very convincing to me, at least.
> >
> > -AKJ.
>

🔗Ozan Yarman <ozanyarman@...>

Carl, you are telling me to keep faith-based remarks to a minimum, while you yourself question the tolerance of my religion/culture and my convictions with snide and belittling remarks based on heresay and bigotry. I find this attitude hypocritical. Apparently the confusion arises from the fact that Islam is both a system of belief and an amalgamate civilization based, for the most part, on religious codes. In this list, I try to bring forward the importance of the latter concerning music theory. It goes without saying that I am disturbed by accounts that are highlighted to cast aspersions on the humane contributions of Islamic civilization and religion to the world. Whether or not Islam was a power-hungry economic empire upon which tolerance was a requisite just like the Greek, Roman or Persian templates and whether if tolerance was only shown toward Jews and Christians is a discussion that exceeds the scope of this list and the one-paragraph exposition you adulate. Inappropriate? By all means. If you wish to pursue a discourse on the fine points of Islamic history and credo with me, why not communicate with me directly?

Cordially,
Oz.

On Aug 18, 2008, at 6:55 PM, Carl Lumma wrote:

> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>
>> It does not surprise me in the least that people with little
>> faith in the Maker are just as easily influenced by fables and
>> concocted stories as are members of religions for whose
>> strenghtening of convictions the said myths are fabricated in
>> the first place:
>
> Hi Ozan,
>
> I'm not sure whether your comment was directed at the authors
> of the Wikipedia page (how do you know their faiths?) or at
> myself, but either way it is inappropriate. Does the claimed
> tolerance of Islam limit itself to the 'god of Abraham'
> religions?
>
> In fact, Islam was an empire like all others. Its desire was
> for power, and to achieve that desire best, tolerance was
> prescribed. That empires have generally gotten more tolerant
> with the march of history (Rome was also a huge breakthrough
> in this respect) is due to humanity's ever increasing dependence
> on economic markets (which require the creative participation
> of those involved) for survival.
>
> For the record, I made no claim about the library; I merely
> linked to Wikipedia, as you had also done. Please keep the
> faith-based remarks to a minimum on this list.
>
> -Carl
>
>

🔗monz <joemonz@...>

🔗Ozan Yarman <ozanyarman@...>

Hello Aaron,

Thank you for clarifying your position. It is indeed a tragedy that the Asharite-Hanbalite-Salafi views have spread unhindered throughout the Muslim world following the downfall of the Mutazili school of thought which was a prime agent for learning and scientific progress. Alas, you are right in stating that Madrassas have been reduced to reciting and memorizing the Quran and Hadith.

To tie all this with tuning: In my thesis, I call the four hundred year period from Meragi-Jami (15th century) to Mushaqah (19th century) the "dark ages of maqam theory". Is it a coincidence that music theorists of the Turco-Mongol empires abandoned the pursuit to pinpoint tones of maqamat in mathematical terms? The treatises of the 15th, 17th and 18th centuries are riddled with astrological parapharnalia, mysticism, and confounding descriptions of maqams.

Cordially,
Oz.

On Aug 18, 2008, at 7:36 PM, Aaron Krister Johnson wrote:

> Hi ozan,
>
> This is very interesting. I had known that the Muslims were an > enlightened lot and showed much moral integrity by allowing > diversity of thought and belief among those they conquored. > Unfortunately, the Christians didn't allow the same freedoms and > also tended to mercilessly rape and slaughter even women and childen.
>
> I had not heard the alternate interpretation of what happened to the > great library at Alexandria, but it has the ring of truth that the > story regarding the Muslims burning all non Koran texts could be > mere propaganda, especially considering the broad prevalence of > Muslim scholarship at the time.
>
> Alas, would it that such a golden age of Islamic quest for knowledge > existed today in the fundamentalist circles -one always hears of > these severe madrassas where only the Koran is taught.
>
> But we are straying from tuning, so let's get back to that, eh?
>
> -AKJ
>
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>
>> Aaron, apparently you are misinformed and misguided about the so-
>> called destruction of Alexandria by the Muslims during the reign of
>> Caliph Omar. All sorts of filth and lies have been disseminated via
>> internet for the past decades to taint Islam. What a disturbing deja
>> vu this is. Please read these articles:
>>
>> http://en.wikipedia.org/wiki/Muslim_conquest_of_Alexandria
>>
>> http://www.crystalinks.com/libraryofalexandria.html
>>
>> The intellectual part of the Muslim World has done everything to
>> preserve and further knowledge, including even those by heathen >> Greeks
>> and pagan Hindus. If anything, Islam has been the instrument of
>> preserving the wisdom of Antiquity. When I read about books on >> Western
>> philosophy, there is a disturbing two thousand year gap from Zeno to
>> Descartes, as if Islamic philosophers, including members of >> Mutazilah,
>> had nothing to say on metaphysics and morality. The contributions to
>> the theory of music by scholars such as Kindi, Farabi, Ibn Sina,
>> Urmevi, Shirazi, Meragi, Jami are immense.
>>
>> Oz.
>>
>> On Aug 18, 2008, at 5:12 AM, Aaron Krister Johnson wrote:
>>
>>> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@> wrote:
>>>> Please do not disregard the Islamic
>>>> connection in Medieval European music theory.
>>>
>>> It makes sense to me that Islamic scholars really fed this.
>>>
>>> This would be consistent with so many other areas of intellectual
>>> pursuit at the time--the Islamic connection in Medieval Europe
>>> intellectual life in general was one of straight lineage to ancient
>>> Greece through Muslims preserving Greek manuscripts...(outside of
>>> incidents like the destruction of Alexandria by Muslims)
>>>
>>> From what I understand, Al-Farabi was the earliest source of Greek
>>> music theory, and obviously made his on contributions firmly >>> rooted in
>>> the Greek tradition.
>>>
>>> In any event, I would not be at all surprised to find that the most
>>> authentic way, in a majority of cases, to do much medieval music >>> would
>>> be to have a 'near eastern sound'...my initial experiments in >>> retuning
>>> medieval music this way seem very convincing to me, at least.
>>>
>>> -AKJ.
>>
>
>

🔗Carl Lumma <carl@...>

🔗Charles Lucy <lucy@...>

I am not surprised that Carl is getting flak.
He invites it.
His latest game seems to be to remove any reference that he can find to scalecoding, e.g. Wikipedia, because it doesn't fit his "expert" (narrow?) view of tuning and music theory.

http://www.lucytune.com/new_to_lt/pitch_05.html

Is he suffering another late middle-age crisis, or is this an example of NIH (not invented here);-)

Has anyone else had trouble with his "eccentricities", or are "just" Oz and I getting special treatment?

BTW Carl, where is my $50 Paypal winnings?

On 18 Aug 2008, at 23:40, Carl Lumma wrote:

> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
> >
> > Carl, you are telling me to keep faith-based remarks to a minimum,
> > while you yourself question the tolerance of my religion/culture
> > and my convictions with snide and belittling remarks based on
> > heresay and bigotry.
>
> Ozan, what happenned? - we used to have such a friendly and
> productive relationship. Suddenly, through no part of my own
> that I know of, you started to take issue with things I've said.
> I'm thinking for example of your remarks about my (in reality
> nonexistent) support for GW Bush's policies. And then a
> lone wikipedia link.
>
> > Whether or not Islam was a power-hungry economic empire upon which
> > tolerance was a requisite just like the Greek, Roman or Persian
> > templates and whether if tolerance was only shown toward Jews and
> > Christians is a discussion that exceeds the scope of this list and
> > the one-paragraph exposition you adulate. Inappropriate? By all
> > means. If you wish to pursue a discourse on the fine points of
> > Islamic history and credo with me, why not communicate with me
> > directly?
>
> I was responding to a public comment you made about (seemingly)
> my faith. Stand up and take responsibility for the grandiose,
> religious, and personal comments you continually make here, and
> stop trying to accuse me of "bigotry", which is nowhere to be
> found in my post.
>
> -Carl
>
>
>
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

🔗Carl Lumma <carl@...>

🔗Dave Keenan <d.keenan@...>

--- In tuning@yahoogroups.com, "monz" <joemonz@...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
> >
> > --- In tuning@yahoogroups.com, "monz" <joemonz@> wrote:
> > >
> > > That's what i find most intriguing about my latest
> > > interpretation of his theory, as meaning to represent
> > > ~quarter-tones. The arithmetic division of the
> > > 9:8 whole-tone into 72:71:70:69:68:67:66:65:64 _does_
> > > in fact perfectly produce the 18:17:16 division.

I'm more inclined to believe, as suggested by Margo, that 18:17:16 was
given more as an example of what _not_ to do. That certainly agrees
with Marchetto's preamble on the correctness of cutting things into 3
parts rather than 2.

By the way, I can't help thinking that, when he speaks of "the
continuum" he means "the continuous octave", i.e. the musical (not
mathematical) continuum with octave-equivalence assumed. So that when
he says that dividing the continuum in two doesn't produce anything
new, but dividing it in three does, it makes perfect sense. But then
he uses a little sleight-of-mathematics to convince us that this
should apply also to divisions of the tone.

> Again, i think that if one assumes that Marchetto was
> beginning his counting with 1 rather than with 0, it
> starts to make more sense.

I don't know anyone who starts _counting_ with zero (except perhaps
some computer programmers, but they're weird).

> I think perhaps he was counting
> the monochord divisions (i.e., his "dieses" and "parts")
> according to the bridges which pinpoint pitches, rather
> than according to the actual intervals between the pitches
> as we would do today.

I agree. But I don't believe he would count the nut.

We measure from zero, yes. But most of us still don't _count_ from
zero. But one need not name the origin with a number for it to be an
effective zero. We still speak of "the nut" rather than "the zeroth fret".

> That's the only way i can make sense
> of his statement that "the first part is the first diesis".

But he does not say that. His "parts" in that passage are _the_same_
as what he later calls dieses, i.e. the 5 unequal "parts", not the 9
equally-spaced movable-bridge positions.

Allow me to refer to these equally-spaced movable-bridge positions as
"frets".

So I suggest that in effect he says: "Fret 1 is the first diesis, fret
1 to fret 3 is the second diesis, fret 3 to fret 5 is the third
diesis, fret 5 to fret 7 is the fifth diesis, fret 7 to fret 9 is the
5th diesis."

And I take that last part as just saying, "notice that we use only
(and all of) the odd numbered frets of the 9 frets".

He could have said "The nut to fret 1 is the first diesis", but that
seems unnecessary even to our ears today. That is the totally obvious
interpretation of "Fret 1 is the first diesis".

I think your diagram showing successive divisions of the string into 3
equal parts and then those parts into 3 equal parts etc until we get 9
equal parts within the 8:9 tone, is what Marchetto intended.

I think 16:17:18 is either a mistake or an example of what _not_ to do.

> > Regarding the semitone names: I guess if you're going to
> > have 3 kinds of semitone and name them after the chromatic,
> > diatonic and enharmonic tetrachord genera in which they
> > occur, then the smallest has to be named "enharmonic"
> > and the largest "chromatic". But in the tetrachords,
> > these would normally be something like 67, 90 and 114
> > cents respectively, whereas Marchetto's would seem to be
> > 99, 105 and something in the range 125 to 165 cents.
>
> Now _that_ is a useful observation! I'll keep that in mind
> the next time i ponder Marchetto's semitones, and see if
> i can find some way to fit his descriptions to those numbers.

Given that one "diesis" is around 23 cents (+-1.5c) and the other four
are around 45 cents (+-3c), and that enharmonic + diatonic =
whole_tone, we have either:

[Use fixed width font]

enharmonic semitone ~= 23 + 45 = 68 c (+-3c)
diatonic semitone ~= 45 + 45 + 45 = 136 c (+-3c)

enharmonic semitone ~= 45 + 45 = 91 c (+-3c)
diatonic semitone ~= 45 + 45 + 23 = 113 c (+-4c)

Similarly for the chromatic semitone we have either:

chromatic semitone ~= 23 + 45 + 45 + 45 = 159 c (+-3c)

chromatic semitone ~= 45 + 45 + 45 + 45 = 181 c (+-2c)

I can't conceive of anyone calling something so close to a tone, any
kind of semitone, so I figure the chromatic has to be around the 159
cents.

I don't see any obvious way to decide betwen the two options for
enharmonic and diatonic. 68c + 136c versus 91c + 113c. He calls the
diatonic an "apotome" (offcut), but if he means _the_ (pythagorean)
apotome (114c) then why didn't he call the enharmonic semitone the limma?

A 68 cent is more like what one would find in a tetrachord of the
enharmonic genus. 90 cents is too large for that. But both 113 and 136
cents may be found in diatonic genera. Consider Ptolemy's equable
diatonic as one extreme.

-- Dave Keenan

🔗Ozan Yarman <ozanyarman@...>

🔗Carl Lumma <carl@...>

🔗Jon Szanto <jszanto@...>

🔗Dave Keenan <d.keenan@...>

Thanks Ozan,

I disagree on only two things. I think the non-pythagorean basis
unlikely (and I wouldn't call your scale "JI" but pythagorean apart
from a schismic tempered third) and I find it more likely that the
single-comma "diesis" was at the 81:80 end rather than the 73:72 end.
This is purely because I assume he would begin counting from the nut,
rather than starting partway down the string and counting back towards
the nut. Of course this only changes the results by a few cents,
irrelevant to singers.

Regards,
-- Dave Keenan

--- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>
> O Dave,
>
> The size of the chromatic semitone (4 dieses) suggests a super-
> Pythagorean augmented prime (apotome), while the size of the diatonic
> semitone (3 dieses) suggests a 5-limit JI or meantone minor second.
> Confusingly enough, Marchetto calles the latter major apotome. If his
> natural diatonic scale is Pythagorean, as it seems to be, then the
> actual diatonic semitone is the limma. But Marchetto calls the "larger
> remainder" the diatonic semitone, which would produce the dileimma
> (pun intended) that we have 90 cents as well as 114 cents as the
> diatonic semitone. To solve the issue, we must conclude that the
> diatonic natural scale is not a Pythagorean major, but JI or close to
> JI, perhaps the Rast scale suggested by Safi al-din Urmavi who lived a
> century prior to Marchetto:
>
> G 1
> A 9/8
> B 8192/6561
> C 4/3
> D 3/2
> E 27/16
> F 16/9
> G 2
>
> In which case, the diatonic semitone becomes 2187:2048 and is the
> major remainder of the whole tone and the larger part of the tone
> divided into two (16:17:18 as you demonstrated). For this to be
> composed of three "dieses" that are not equalities, we need two
> quarter-tones and a comma, which arises if we divide the whole tone
> into 9 arithmetical parts (just as we divided it into two):
>
> 72/72 1/1 0 cents 0
> 73/72 24 cents 1 1st part/diesis
> 74/72 37/36 47 cents 2
> 75/72 25/24 71 cents 3 2nd part/diesis
> 76/72 19/18 94 cents 4
> 77/72 116 cents 5 3rd part/diesis
> 78/72 13/12 139 cents 6
> 79/72 161 cents 7 4th part/diesis
> 80/72 10/9 182 cents 8
> 81/72 9/8 204 cents 9 5th part/diesis
>
> Notice, that we have a comma of 24 cents as the first diesis, a
> quarter-tone of 47 cents as the second diesis and 46 cents as the
> third diesis. This solves the matter of the diatonic semitone.
>
> What about the chromatic semitone then? Marchetto says that it is the
> chroma that completes to a whole tone with the addition of the diesis,
> is a diatonic semitone plus a diesis, and is composed of 4 of the 5
> dieses mentioned. This could mean nothing other then 161 cents given
> above. Though I would have much preferred 13:12 as the chroma, but
> Marchetto seems to insist on taking only the primes within the number 9.
>
> This leaves us with 25/24 as the enharmonic semitone, if we take a
> comma plus a quarter-tone, or 19/18 as the enharmonic semitone if we
> take two quarter-tones. This interpretation favours 53-LEDO as a basis
> for "Marchettan harmony"
>
> Oz.

🔗Kraig Grady <kraiggrady@...>

🔗Charles Lucy <lucy@...>

🔗Ozan Yarman <ozanyarman@...>

Very good Daniel, I agree with all your statements.

Oz.

On Aug 18, 2008, at 12:35 PM, Daniel Wolf wrote:

> The contributions by Muslim and Islamicate scholars to both the
> transmission of older theory and theoretical innovations are immense
> but
> the idea that there was Greek Music theory, which got lost, and was
> transmitted to Europe only via Islam is misleading, not least
> because it
> reduces Islamic theory to a way station on Europe's march towards
> polyphony, rather than identifying an important conjunction among
> traditions with very different trajectories. (Another, profound,
> connection is in the continuum of Ud and Lute, fretted and unfretted,
> monodic and polyphonic – and even a very important impetus to the
> development of keyboard music in the West —, with a geographical
> stretch
> so wide that drawing borders among them is an essentially meaningless
> exercise.)
>
> In classical times, there were thriving cultivated musical traditions
> throughout the mediterranean and adjoining cultures, and the traces we
> have vividly suggest very deep connections and exchange as well as
> major
> differences among these cultures. The largest corpus of theory which
> survives classical times (at least in a known and accessible form --
> the
> extent of music theory in the fertile crescent region being still a
> large
> unknown) is Greek and Hellenistic, and that scholarly corpus was
> maintained to a large extant in the huge area Byzantine music theory,
> before, during, and after the advent of Islam, while in the much more
> limited Roman theoretical tradition, represented by Boethius, was in
> fact
> continuous in the west (so continuous, in fact, that Boethius's
> errors, in
> classical terms, are often "correct", or at least more appropriate, in
> medieval European terms.) It may be argued that both of these
> theoretical
> traditions were seriously circumscribed versions of Classical and
> Hellenistic theory, but I would argue instead that they simply became
> closely adapted to contemporary practice in the respective
> traditions. In
> any case, one must be cautious not to ignore these practical and
> theoretical traditions in preference for a neat line of transfer of
> cultural goods and practices.
>
> These theoretical traditions contain, moreover, compelling evidence
> about
> other practices in the larger cultural region which did not either
> maintain or cultivate a theoretical practice of their own, for
> example in
> the missing chapter of Boethius, concerning the division of the fourth
> into two parts, one immediately thinks of the trichordal modes that
> remain
> the basis of classical music -- practiced by both Christians and
> Muslims
> -- in Ethiopia. And when evidence is lacking, our curiosity is just
> as
> great: We know far too little about early musical practice in oriental
> Christendom, for example from the thriving musical culture of the
> Copts,
> whose musical tradition connects to both Ancient Egypt, the Hellenized
> world, the Upper Nile region and Aksum, and presumably to the Near
> East,
> with its own Jewish, Jewish-Christian, and early Christian diversity
> (and
> possibly others — Mandean, Samaritan, who knows?).
>
> Likewise, the assimilation of classical and Hellenistic theory to
> practices throughout the Islamicate world has to be taken in stride,
> differentiating the science (or more precisely liberal art)of
> harmonics,
> with its culturally neutral techniques for measurement and
> comparison of
> intervals, and their composition into tones, n-chords, systems, and
> modes
> from the description of the specific materials of real musical
> practice.
> That practice, in the Islamicate world, includes — often
> assimilated,
> often isolated — continuity with its own huge range of additional
> cultures, radiating from Arabia to India, Persia, Central Asia, the> Black
> Sea and the Caucasus, the Balkans, North Africa, and, yes, Byzantium.
>
>
> Daniel Wolf
>
>

🔗Kraig Grady <kraiggrady@...>

🔗Ozan Yarman <ozanyarman@...>

Kraig, for the sake of ending this argument, let me state my position clearly:

1. There was no Islamic culture/civilization the way we understand today prior to the 7th Century, although the religion's foundations precede Prophet Muhammed sallallahu aleyh's appearance by thousands of years.

2. Muslims (aside from a minority of fanatics found in every culture and period) have always preserved and furthered knowledge up until the rise of Asharite kelaam (Al-Ghazali was both an Ashari and a Sufi, and Nizam al-mulk, vezier to the Seljuki sultans and Sufi-lover, advocated the Nizamiyyah Madrassas based on Shafii jurisprudence that was packaged with Asharite views of the world). This meant that the medieval Islamic quest for science and knowledge came to a gradual halt by the 11th century onward, leading to the collapse of the three major Islamic empires by the 19th-20th Centuries.

3. Islamic or Islamicate corresponds to Christendom in the context of music theory, not Christianity. Surely, Boethius, Odo de Cluny, Guido de Arezzo, Marchetto da Padova and Philippe de Vitry were all Christendom theorists. Calling them and the Hellenes European does great injustice to the historical-cultural context. Who claims that Pythagoras, Archytas and Aristoxenus were European music theorists? Such an outlook is extremely deceptive and detrimental to objective neutrality.

Oz.

On Aug 20, 2008, at 10:47 AM, Kraig Grady wrote:

> My point, Ozan, is that if the library was burned between the third
> century and sixth. it could not have been done by Islamics.
> --> I am not sure there is anything that can be labeled Islamic Theory any
> more than Christian Theory. The inclusion of Greece into European > theory
> is to cover up what the rest of the continent was involved in.
>
>
> The contributions by Muslim and Islamicate scholars to both the
>> transmission of older theory and theoretical innovations are immense
>> but
>> the idea that there was Greek Music theory, which got lost, and was
>> transmitted to Europe only via Islam is misleading, not least
>> because it
>> reduces Islamic theory to a way station on Europe's march towards
>> polyphony,
> -->
> /^_,',',',_ //^ /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/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>

🔗Mike Battaglia <battaglia01@...>

FWIW, I always thought scalecoding was a good idea. The concept of
taking a scale and finding the pattern of meantone fifths that
generates it is something that I had independently discovered myself a
while ago, although Charles has developed the idea more than I had.
The concept is useful in playing jazz especially, because it serves as
a useful tool to find quick chord substitutions from any chord in a
scale.

Another concept that I always thought was cool was to extend the
scalecoding concept to 5-limit JI, and describe common scales as
extending a certain distance into the 3-axis and a certain distance
into the 5-axis. The major scale goes out into the 5-axis one step,
the melodic minor scale goes out one step and then one in the opposite
direction, etc.

But I say keep it on Wikipedia. Certainly it's more useful than the
Lydian Chromatic Concept, which I don't like very much at all.

On Mon, Aug 18, 2008 at 8:57 PM, Charles Lucy <lucy@...> wrote:
> I am not surprised that Carl is getting flak.
>
> He invites it.
> His latest game seems to be to remove any reference that he can find to
> scalecoding, e.g. Wikipedia, because it doesn't fit his "expert" (narrow?)
> view of tuning and music theory.
> http://www.lucytune.com/new_to_lt/pitch_05.html
> Is he suffering another late middle-age crisis, or is this an example of NIH
> (not invented here);-)
> Has anyone else had trouble with his "eccentricities", or are "just" Oz and
> I getting special treatment?
> BTW Carl, where is my $50 Paypal winnings?
>
>
>
>
>
>
> On 18 Aug 2008, at 23:40, Carl Lumma wrote:
>
> --- In tuning@yahoogroups.com, Ozan Yarman <ozanyarman@...> wrote:
>>
>> Carl, you are telling me to keep faith-based remarks to a minimum,
>> while you yourself question the tolerance of my religion/culture
>> and my convictions with snide and belittling remarks based on
>> heresay and bigotry.
>
> Ozan, what happenned? - we used to have such a friendly and
> productive relationship. Suddenly, through no part of my own
> that I know of, you started to take issue with things I've said.
> I'm thinking for example of your remarks about my (in reality
> nonexistent) support for GW Bush's policies. And then a
> lone wikipedia link.
>
>> Whether or not Islam was a power-hungry economic empire upon which
>> tolerance was a requisite just like the Greek, Roman or Persian
>> templates and whether if tolerance was only shown toward Jews and
>> Christians is a discussion that exceeds the scope of this list and
>> the one-paragraph exposition you adulate. Inappropriate? By all
>> means. If you wish to pursue a discourse on the fine points of
>> Islamic history and credo with me, why not communicate with me
>> directly?
>
> I was responding to a public comment you made about (seemingly)
> my faith. Stand up and take responsibility for the grandiose,
> religious, and personal comments you continually make here, and
> stop trying to accuse me of "bigotry", which is nowhere to be
> found in my post.
>
> -Carl
>
>
> Charles Lucy
> lucy@...
> - Promoting global harmony through LucyTuning -
> for information on LucyTuning go to:
> http://www.lucytune.com
> For LucyTuned Lullabies go to:
> http://www.lullabies.co.uk
>
>
>

Marchetto

🔗monz@juno.com

🔗monz@xxxx.xxx

🔗monz@xxxx.xxx

🔗Afmmjr@xxx.xxx

🔗Ozan Yarman <ozanyarman@...>

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🔗monz <joemonz@...>

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🔗robert thomas martin <robertthomasmartin@...>

🔗monz <joemonz@...>

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🔗Ozan Yarman <ozanyarman@...>

🔗monz <joemonz@...>

🔗Dave Keenan <d.keenan@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Dave Keenan <d.keenan@...>

🔗monz <joemonz@...>

🔗monz <joemonz@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Aaron Krister Johnson <aaron@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Carl Lumma <carl@...>

🔗monz <joemonz@...>

🔗monz <joemonz@...>

🔗Daniel Wolf <djwolf@...>

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🔗Carl Lumma <carl@...>

🔗Carl Lumma <carl@...>

🔗Aaron Krister Johnson <aaron@...>

🔗Ozan Yarman <ozanyarman@...>

🔗monz <joemonz@...>

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🔗Carl Lumma <carl@...>

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🔗Carl Lumma <carl@...>

🔗Dave Keenan <d.keenan@...>

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🔗Carl Lumma <carl@...>

🔗Jon Szanto <jszanto@...>

🔗Dave Keenan <d.keenan@...>

🔗Kraig Grady <kraiggrady@...>

🔗Charles Lucy <lucy@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Kraig Grady <kraiggrady@...>

🔗Kraig Grady <kraiggrady@...>

🔗Kraig Grady <kraiggrady@...>

🔗Ozan Yarman <ozanyarman@...>

🔗Mike Battaglia <battaglia01@...>

🔗Carl Lumma <carl@...>

🔗Charles Lucy <lucy@...>

🔗Jon Szanto <jszanto@...>

🔗Carl Lumma <carl@...>

🔗Charles Lucy <lucy@...>

🔗Brian Redfern <brianwredfern@...>

🔗robert thomas martin <robertthomasmartin@...>

🔗Kraig Grady <kraiggrady@...>

🔗monz <joemonz@...>