Yahoo Tuning Groups Ultimate Backup tuning Boethius

Dear Monz,

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 13 Kasï¿½m 2005 Pazar 0:44
Subject: [tuning] Re: Boethius

> Hi Ozan,
>
>
> --- In tuning@yahoogroups.com, <oyarman@o...> wrote:
>
> > Thank you for the promise Monz, it will be a relief to
> > witness the clearing up of the confusion on this matter,
> > however late. Seeing as there is no consensus as to what
> > he meant, even by Ecclesiastical theorists after him, I
> > would like to know why you and Harold Powers define modes
> > of Boethius in mutually exclusive ways... and more
> > importantly, howcome Powers says modes of B. are
> > `transposition keys that generated those octave species
> > wthin the characteristic octave` while you attribute
> > additional elements not originally a part of his treatise:
> >
> > http://tonalsoft.com/enc/m/mode.aspx
>
>
> What are those "additional elements"? The only thing i
> attribute to Boethius in that Encyclopedia entry is the
> Greek names of the modes.
>

You also attribute protus, deuterus... and authentus-plagis categories to
Boethius. Does he use these terms or imply that his hypo modes have anything
to do with plagal modes of Ecclesiastical theory centuries after him?
Moreover, you specifiy reciting tones... I'm not sure if Boethius specified
anything but the finalis of each mode... Then again, I may be a total
ignorant in this field and misconstrue Powers' article.

> I should point out that papers i've read discussing
> Boethius's description of modes have interpreted his
> work in differing ways, and i too found it confusing.
>

I find this not only confusing but annoying. I face similar issues when
dealing with Maqam Music treatises. A consensus must be reached on
matters as critical as this after much deliberation, don't you think?

> That's exactly why i said i'd have to invest the time
> in reading Boethius again, to be more sure about it.
>

I'm sure you will do fine once you find the time.

> Have you actually read Boethius's treatise?

Ah, I wish I could. There is no English translation within reach, and
Latin-English online translators are no good for the task at hand.

Here are
> links to three versions of the relevant book 4 (the
> first a manuscript and the other two published versions)
> -- his descriptions of the modal system are in chapters
> 15, 16, and 17:
>
>
> http://www.music.indiana.edu/tml/6th-8th/BOEMUS4C_MCTC944.html
>
> http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_TEXT.html
>
> http://www.music.indiana.edu/tml/6th-8th/BOEDIM4_TEXT.html
>
>
> These are in Latin ...

Yes, I had accessed them. Especially this section in chapter 15 does seem
critical:

(From the octave, therefore consonant of species, comes into existence what
are called modes, which same are named tropes or tones...)

"XV. Ex diapason igitur consonantiae speciebus existunt, qui appellantur
modi, quos eosdem tropos vel tonos nominant. Sunt autem tropi constitutiones
in totis vocum ordinibus vel gravitate vel acumine differentes. Constitutio
vero est plenum veluti modulationis corpus ex consonantiarum coniunctione
consistens quale est vel diapason vel diapason et diatessaron vel bis
diapason. Est enim diapason [-342-] constitutio a proslambanomeno in mesen
ceteris quae sunt mediae vocibus adnumeratis, vel a mese rursus in neten
hyperboleon cum vocibus interiectis, vel ab hypate meson in neten
diezeugmenon cum his, quas extremae voces medias claudunt. Synemmenon vero
constitutio ea est, quae a proslambanomeno in neten synemmenon cum his, quae
mediae interiectae sunt, constat. Bis diapason autem a proslambanomeno in
neten hyperboleon cum his, quae in medio sunt interpositae, consideratur.
Has igitur constitutiones si quis totas faciat acutiores, vel in gravius
totas remittat secundum supradictas diapason consonantiae species, efficiet
modos .VII., quorum nomina sunt haec: hypodorius, hypophrygius, hypolydius,
dorius, phrygius, lydius, mixolydius.

Horum vero sic ordo procedit. Sit in diatonico genere vocum ordo dispositus
a proslambanomeno in neten hyperboleon atque hic sit hypodorius modus. Si
quis igitur proslambanomenon in acumen intendat tono hypatenque hypaton
eodem tono adtenuet ceterasque omnes tono faciat acutiores, acutior totus
ordo proveniet, quam fuit priusquam toni susciperet intentionem. Erit igitur
tota constitutio acutior effecta hypophrygius modus. Quod si in hypophrygio
toni rursus intentionem voces acceperint, hypolydii modulatio nascetur. At
si hypolydium quis semitonio intendat, dorium faciet. Et in aliis quidem
similis est in acumen intentionemque processus, quorum non ut intellegentia
solum ratio conprehendatur, verum oculis quoque forma possit agnosci, ab
antiquis tradita [-343-] musicis descriptio subponenda est. Sed quoniam per
singulos modos a veteribus musicis unaquaeque vox diversis notulis insignita
est, descriptio prius notularum videtur esse ponenda, ut his primum per se
cognitis in modorum descriptione facilis possit esse dispectio. (Vide
descriptionem primam in tabula seorsum addita.)"

the English version is:
>
> Bower, Calvin M (transl., ed., and notes). 1989.
> New Have: Yale U. Press
> ISBN: 0300039433
>
>
> The search i did shows that it is available ...
> unfortunately, it's $249.95.
>
>

Ouch! That is extortion... Although I do not doubt the quality of the
translation, it seems a much better option to learn Latin
enough to master the translations of necessary parts. That is a
cheaper and surer way out for the likes of me at least!

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

Cordially,
Ozan

🔗monz <monz@tonalsoft.com>

Hi Ozan,

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:
>
> Please please please do translate also the portion on
> Chapter XV I mentioned... the one on modes.

OK, here it is ... not a portion, but the whole chapter.

The text is Bower's English translation, but with the
text broken into smaller paragraphs by me, and i've also
added in brackets the Roman-alphabet nomenclature to make
Boethius's explanation clear to the modern reader.

Note that in the previous chapter (14), Boethius explained
the species of consonances: the 4th [for example: A B C D],
5th [A B C D E], and 8ve [A B C D E F G a].

-----------

Boethius, _de institutione musica_, Book 4, chapter 15:

"Concerning the origins of the modes and the disposition
of notational signs for individual modes and pitches"

From the species of the consonance of the diapason arise
what are called "modes". They are also called "tropes"
or "tones".

[Here Bower supplies a footnote on Boethius's terminology:
"mode" is the English rendering of Latin "modus", a literal
translation of the Greek word spelled with the letters
tau-rho-omicron-pi-omicron-sigma, which i can render here
as "tropos"; "trope" is the English version of Latin "tropus",
which is the Latin transliteration of the Greek "tropos",
and "tone" is the English version of Latin "tonus" which
is the Latin transliteration of the Greek word "tonos"
(tau-omicron-nu-omicron-sigma). Greek music-theorists used
both "tonos" and "tropos" to describe "mode".]

Tropes are systems that differ according to highness or
lowness throughout entire sequences of pitches.

[Bower notes that for Boethius, "a mode is a transposition
of a whole system, not merely a segment of a system".
This is an important distinction, and the source of
the confusion i wrote about earlier.]

A system is, as it were, an entire collection of pitches,
brought together within the framework of a consonance such
as the diapason [8ve], diapason-plus-diatessaron [8ve + 4th,
or 11th], or the bis-diapason [double-8ve].

* A system of the diapason consists of the proslambanomenos
to the mese, along with the other pitches counted between
them [A B C D E F G a], or the mese to the nete hyperboleon,
with the intervening pitches [a b c d e f g a'], or the
hypate meson to the nete diezeugmenon, with those pitches
enclosed by these [E F G a b c d e].

* The synemmenon system is that which is found between the
proslambanomenos and the nete synemmenon, along with those
pitches which these surround [A B C D E F G a b c d].

* Finally, the system of the bis-diapason is seen from the
proslambanomenos to the nete hyperboleon, along with the
intervening pitches [A B C D E F G a b c d e f g a'].

If these entire systems were made higher or lower in
accordance with the species of the consonance of the
diapason discussed above, this would bring about seven
modes, which are named Hypodorian, Hypophrygian, Hypolydian,
Dorian, Phrygian, Lydian, and Mixolydian.

The arrangement of the modes proceeds in the following
manner.

Set out the succession of pitches in the diatonic genus
from the proslambanomenos to the nete hyperboleon. Let
this be the Hypodorian mode [A B C D E F G a b c d e f g a'].

If one were to raise the proslambanomenos by one tone, and
further raise the hypate hypaton by the same tone, thereby
making the whole disposition higher by a tone, then the
entire succession would turn out higher than it was before
it was raised by a tone [B C# D E F# G a b c# d e f# g a' b'].
Thus this whole system, having been made higher, forms the
Hypophrygian mode.

Now if the pitches of the Hypophrygian mode were similarly
raised by a tone, the collection of pitches for the Hypolydian
mode would be born [C# D# E F# G# a b c# d# e f# g# a' b' c#'].

If someone raised the Hypolydian by a semitone, he would make
the Dorian [D E F G a b-flat c d e f g a' b-flat' c' d'].

The progression to higher pitch is similar in the other modes.

So that the theory of these modes might be grasped by the eye
as well as the intellect, a diagram, a visual representation
of the modes, handed down from ancient musicians is presented
below. But since each single pitch for the individual modes
is recorded with different written symbols by ancient musicians,
it seems necessary to present a diagram of the written symbols
first; then, with these understood in themselves, consideration
of the diagram of modes should be easy.

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

Below are the links to the Latin texts of 3 versions of
this chapter (one manuscript and two published editions),
with the associated diagram mentioned by Boethius:

manuscript: Cambridge, Trinity College, R.15.22 (944)
http://www.music.indiana.edu/tml/6th-8th/BOEMUS4C_MCTC944.html
http://www.music.indiana.edu/tml/6th-8th/BOEMUS4C_17GF.gif

Godofredus Friedlein (Leipzig: B. G. Teubner, 1867)
http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_TEXT.html
http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_08GF.gif

J. P. Migne, 221 vols. (Paris: Garnier, 1844-1904)
http://www.music.indiana.edu/tml/6th-8th/BOEDIM4_TEXT.html
http://www.music.indiana.edu/tml/6th-8th/BOEDIM4_17GF.gif

Boethius writes a lot more about the modal system in
chapters 16 and 17 of Book 4. I'll post the translations
when i have time. Chapter 14 needs to introduce all this
as well.

Eventually, i'll have all of Boethius's modes set up
as Tonescape tuning files ... in each of the different
genera (diatonic, chromatic, and enharmonic).

Years ago, i also wrote a paper which i've never published
in which i believe i discovered that Boethius's Greek-letter
notation recognized the difference of a syntonic-comma
(~22 cents) but equated pairs of notes which were a schisma
apart (~2 cents), thus in effect presenting the ancient
modal system as a schismic temperament.

I wrote a lot about that paper here at various times,
and if you have the patience to do a search, you'll find
my posts on it. But the typed paper is the one which has
the Greek-letter notation, which is the essential evidence
proving my points. I do plan to someday make it into
a webpage.

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

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

Splendid Monz! My sincerest appreciation for your kind assistance... my
comments are below:

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 16 Kasï¿½m 2005 ï¿½arï¿½amba 8:06
Subject: [tuning] Boethius modal system, chapter 15 of Book 4

SNIP

>
> The text is Bower's English translation, but with the
> text broken into smaller paragraphs by me, and i've also
> added in brackets the Roman-alphabet nomenclature to make
> Boethius's explanation clear to the modern reader.
>
> Note that in the previous chapter (14), Boethius explained
> the species of consonances: the 4th [for example: A B C D],
> 5th [A B C D E], and 8ve [A B C D E F G a].
>
> -----------
>
> Boethius, _de institutione musica_, Book 4, chapter 15:
>
>
> "Concerning the origins of the modes and the disposition
> of notational signs for individual modes and pitches"
>
>
> >From the species of the consonance of the diapason arise
> what are called "modes". They are also called "tropes"
> or "tones".
>
> [Here Bower supplies a footnote on Boethius's terminology:
> "mode" is the English rendering of Latin "modus", a literal
> translation of the Greek word spelled with the letters
> tau-rho-omicron-pi-omicron-sigma, which i can render here
> as "tropos"; "trope" is the English version of Latin "tropus",
> which is the Latin transliteration of the Greek "tropos",
> and "tone" is the English version of Latin "tonus" which
> is the Latin transliteration of the Greek word "tonos"
> (tau-omicron-nu-omicron-sigma). Greek music-theorists used
> both "tonos" and "tropos" to describe "mode".]
>
>
> Tropes are systems that differ according to highness or
> lowness throughout entire sequences of pitches.
>
> [Bower notes that for Boethius, "a mode is a transposition
> of a whole system, not merely a segment of a system".
> This is an important distinction, and the source of
> the confusion i wrote about earlier.]
>
>
>
> A system is, as it were, an entire collection of pitches,
> brought together within the framework of a consonance such
> as the diapason [8ve], diapason-plus-diatessaron [8ve + 4th,
> or 11th], or the bis-diapason [double-8ve].
>
> * A system of the diapason consists of the proslambanomenos
> to the mese, along with the other pitches counted between
> them [A B C D E F G a], or the mese to the nete hyperboleon,
> with the intervening pitches [a b c d e f g a'], or the
> hypate meson to the nete diezeugmenon, with those pitches
> enclosed by these [E F G a b c d e].
>
> * The synemmenon system is that which is found between the
> proslambanomenos and the nete synemmenon, along with those
> pitches which these surround [A B C D E F G a b c d].
>

I do believe it should have been a b-flat in the a b c d tetrachord.

> * Finally, the system of the bis-diapason is seen from the
> proslambanomenos to the nete hyperboleon, along with the
> intervening pitches [A B C D E F G a b c d e f g a'].
>
>

This clarifyies a lot of things.

>
> If these entire systems were made higher or lower in
> accordance with the species of the consonance of the
> diapason discussed above, this would bring about seven
> modes, which are named Hypodorian, Hypophrygian, Hypolydian,
> Dorian, Phrygian, Lydian, and Mixolydian.
>
>
> The arrangement of the modes proceeds in the following
> manner.
>
> Set out the succession of pitches in the diatonic genus
> from the proslambanomenos to the nete hyperboleon. Let
> this be the Hypodorian mode [A B C D E F G a b c d e f g a'].
>
> If one were to raise the proslambanomenos by one tone, and
> further raise the hypate hypaton by the same tone, thereby
> making the whole disposition higher by a tone, then the
> entire succession would turn out higher than it was before
> it was raised by a tone [B C# D E F# G a b c# d e f# g a' b'].
> Thus this whole system, having been made higher, forms the
> Hypophrygian mode.
>

I am suspecting whether he meant B C D E F G A b c d e f g a' b' instead?
Boethius actually mentioned being in accord with the species of the
consonance of the octave, and direct transposition is certainly not
synonymous with his previous statement, modulation is.

> Now if the pitches of the Hypophrygian mode were similarly
> raised by a tone, the collection of pitches for the Hypolydian
> mode would be born [C# D# E F# G# a b c# d# e f# g# a' b' c#'].
>

C D E F G a b c d e f g a' b' c' ?

The implication that both proslambanomenos and hypate hypaton could be
raised by a whole tone leads to the first tetrachord being composed in this
manner: 9/8 + 9/8 + 256/243 which means that the Cs, Fs and Gs are actually
sharp to begin with.

> If someone raised the Hypolydian by a semitone, he would make
> the Dorian [D E F G a b-flat c d e f g a' b-flat' c' d'].
>

D E F G a b c d e f g a' b' c' d' ?

This scheme fits perfectly if the entire system was:

A B C(#) D E F(#) G(#) a b c(#) d e f(#) g(#) a'

> The progression to higher pitch is similar in the other modes.
>

1. Hypodorian mode [A B C# D E F# G# a b c# d e f# g# a'] ?
2. Hypophyrigian mode [B C# D E F# G# a b c# d e f# g# a'] ?
3. Hypolydian mode [C# D E F# G# a b c# d e f# g# a' b' c'#] ?
4. Dorian mode [D E F# G# a b c# d e f# g# a' b' c'# d'] ?
5. Phyrigian mode [E F# G# a b c# d e f# g# a' b' c'# d' e'] ?
6. Lydian mode [F# G# a b c# d e f# g# a' b' c'# d' e' f'#] ?
7. Mixolydian mode [G# a b c# d e f# g# a' b' c'# d' e' f'# g'#] ?

Transposed to natural tones:

C D E F G A B c d e f g a b c' - HD (major)
D E F G A B c d e f g a b c' d' - HP
E F G A B c d e f g a b c' d' e' - HL
F G A B c d e f g a b c' d' e' f' - D
G A B c d e f g a b c' d' e' f' g' -P
A B c d e f g a b c' d' e' f' g' a' -L (minor)
B c d e f g a b c' d' e' f' g' a' b' -M

Does it make sense?

> So that the theory of these modes might be grasped by the eye
> as well as the intellect, a diagram, a visual representation
> of the modes, handed down from ancient musicians is presented
> below. But since each single pitch for the individual modes
> is recorded with different written symbols by ancient musicians,
> it seems necessary to present a diagram of the written symbols
> first; then, with these understood in themselves, consideration
> of the diagram of modes should be easy.
>

http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_08GF.gif

> -------------
>
> Below are the links to the Latin texts of 3 versions of
> this chapter (one manuscript and two published editions),
> with the associated diagram mentioned by Boethius:
>
> manuscript: Cambridge, Trinity College, R.15.22 (944)
> http://www.music.indiana.edu/tml/6th-8th/BOEMUS4C_MCTC944.html
> http://www.music.indiana.edu/tml/6th-8th/BOEMUS4C_17GF.gif
>
> Godofredus Friedlein (Leipzig: B. G. Teubner, 1867)
> http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_TEXT.html
> http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_08GF.gif
>
> J. P. Migne, 221 vols. (Paris: Garnier, 1844-1904)
> http://www.music.indiana.edu/tml/6th-8th/BOEDIM4_TEXT.html
> http://www.music.indiana.edu/tml/6th-8th/BOEDIM4_17GF.gif
>
>
> Boethius writes a lot more about the modal system in
> chapters 16 and 17 of Book 4. I'll post the translations
> when i have time. Chapter 14 needs to introduce all this
> as well.
>
>
> Eventually, i'll have all of Boethius's modes set up
> as Tonescape tuning files ... in each of the different
> genera (diatonic, chromatic, and enharmonic).
>

That would be most necessary. Keep up the good work Monz!

>
> Years ago, i also wrote a paper which i've never published
> in which i believe i discovered that Boethius's Greek-letter
> notation recognized the difference of a syntonic-comma
> (~22 cents) but equated pairs of notes which were a schisma
> apart (~2 cents), thus in effect presenting the ancient
> modal system as a schismic temperament.
>

Fascinating.

> I wrote a lot about that paper here at various times,
> and if you have the patience to do a search, you'll find
> my posts on it. But the typed paper is the one which has
> the Greek-letter notation, which is the essential evidence
> proving my points. I do plan to someday make it into
> a webpage.
>
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>
>

Cordially,
Ozan

🔗monz <monz@tonalsoft.com>

Hi Ozan,

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:

> > [Boethius/Bower/monz]
> > * The synemmenon system is that which is found between the
> > proslambanomenos and the nete synemmenon, along with those
> > pitches which these surround [A B C D E F G a b c d].
>
>
> I do believe it should have been a b-flat in the
> a b c d tetrachord.

Sorry! Yes, you are absolutely correct.
The synemmenon system always has b-flat and never b-natural.

> > The arrangement of the modes proceeds in the following
> > manner.
> >
> > Set out the succession of pitches in the diatonic genus
> > from the proslambanomenos to the nete hyperboleon. Let
> > this be the Hypodorian mode [A B C D E F G a b c d e f g a'].
> >
> > If one were to raise the proslambanomenos by one tone, and
> > further raise the hypate hypaton by the same tone, thereby
> > making the whole disposition higher by a tone, then the
> > entire succession would turn out higher than it was before
> > it was raised by a tone [B C# D E F# G a b c# d e f# g a' b'].
> > Thus this whole system, having been made higher, forms the
> > Hypophrygian mode.
> >
>
> I am suspecting whether he meant
> B C D E F G A b c d e f g a' b' instead?
> Boethius actually mentioned being in accord with the
> species of the consonance of the octave, and direct
> transposition is certainly not synonymous with his
> previous statement, modulation is.

My opinion is that it's very difficult to determine
which of these two interpretations is correct.

When he says "in accord with the species of the consonance
of the octave", it could be interpreted to mean that
all of the relative pitch relationships involving the
tone/semitone steps are to be retained, which gives
B C# D E F# G a b c# d e f# g a' b'. He also says
"the entire succession would turn out higher than it was
before it was raised by a tone", which could also be used
to argue that this interpretation is correct.

On the other hand, since at this time there was no
absolute pitch nomenclature and everything was relative,
it makes more sense that B C D E F G A b c d e f g a' b'
would be correct.

This points out why it's important to reach an
understanding of exactly what the original Latin text
means. It also illustrates why there has been confusion
over this point for centuries, and why i'd have to really
dig back in to the the previous and subsequent chapters
to put this one in context.

> This scheme fits perfectly if the entire system was:
>
> A B C(#) D E F(#) G(#) a b c(#) d e f(#) g(#) a'
>
>
> > The progression to higher pitch is similar in the
> > other modes.
> >
>
>
> 1. Hypodorian mode [A B C# D E F# G# a b c# d e f# g# a'] ?
> 2. Hypophyrigian mode [B C# D E F# G# a b c# d e f# g# a'] ?
> 3. Hypolydian mode [C# D E F# G# a b c# d e f# g# a' b' c'#] ?
> 4. Dorian mode [D E F# G# a b c# d e f# g# a' b' c'# d'] ?
> 5. Phyrigian mode [E F# G# a b c# d e f# g# a' b' c'# d' e'] ?
> 6. Lydian mode [F# G# a b c# d e f# g# a' b' c'# d' e' f'#] ?
> 7. Mixolydian mode [G# a b c# d e f# g# a' b' c'# d' e' f'# g'#] ?
>
>
> Transposed to natural tones:
>
> C D E F G A B c d e f g a b c' - HD (major)
> D E F G A B c d e f g a b c' d' - HP
> E F G A B c d e f g a b c' d' e' - HL
> F G A B c d e f g a b c' d' e' f' - D
> G A B c d e f g a b c' d' e' f' g' -P
> A B c d e f g a b c' d' e' f' g' a' -L (minor)
> B c d e f g a b c' d' e' f' g' a' b' -M
>
>
> Does it make sense?

I would have to find my old paper on this, but IIRC
this last tabulation does agree with what i wrote.
In particular, i think i remember the Lydian being
equivalent to our A-natural-minor scale.

> > Years ago, i also wrote a paper which i've never published
> > in which i believe i discovered that Boethius's Greek-letter
> > notation recognized the difference of a syntonic-comma
> > (~22 cents) but equated pairs of notes which were a schisma
> > apart (~2 cents), thus in effect presenting the ancient
> > modal system as a schismic temperament.
> >
>
>
> Fascinating.

Quite a bit more than "fascinating", if you ask me.

There's apparently a long tradition of Arabic music using
these schismic (schismatic?) equivalences while preserving
commatic (syntonic and/or pythagorean) differences, and
i believe i see it also in classical Indian music-theory.

http://tonalsoft.com/monzo/indian/indian.htm

If my interpretation of Boethius is correct, then this
is probably evidence of cross-cultural influence between
the ancient Greeks, Arabs, and Indians.

Of course, several centuries before Boethius, Alexander
had made much of the Arab and Hindu lands part of his
Greek empire, so it wouldn't surprise me at all to see
this cross-cultural infiltration of ideas and practices.

But AFAIK no-one has pointed out this specific one before,
and even Boethius sabotages our knowledge of it, because
he presents the diatonic system in its ancient pythagorean
tuning, and this schismatic equivalence only becomes apparent
if one tunes the diatonic genus according to the ratios
advocated by Didymus (c.100 AD).

That, in turn, would be evidence that Didymus was providing
an accurate analysis of actual musical practice of his time
-- which is something that i think no music historian ever
thought would be found.

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

🔗oyarman@ozanyarman.com

Dear Monz,

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 16 Kasï¿½m 2005 ï¿½arï¿½amba 16:58
Subject: [tuning] Re: Boethius modal system, chapter 15 of Book 4

SNIP

> > >
> >
> > I am suspecting whether he meant
> > B C D E F G A b c d e f g a' b' instead?
> > Boethius actually mentioned being in accord with the
> > species of the consonance of the octave, and direct
> > transposition is certainly not synonymous with his
> > previous statement, modulation is.
>
>
> My opinion is that it's very difficult to determine
> which of these two interpretations is correct.
>
> When he says "in accord with the species of the consonance
> of the octave", it could be interpreted to mean that
> all of the relative pitch relationships involving the
> tone/semitone steps are to be retained, which gives
> B C# D E F# G a b c# d e f# g a' b'. He also says
> "the entire succession would turn out higher than it was
> before it was raised by a tone", which could also be used
> to argue that this interpretation is correct.
>

Is it not evident that the interpretation which considers modes to be direct
transpositions of the greater perfect system is fundamentally lacking, since
the idea of octave species is ruined and there is nothing at all special
when the scale remains unrotated?

> On the other hand, since at this time there was no
> absolute pitch nomenclature and everything was relative,
> it makes more sense that B C D E F G A b c d e f g a' b'
> would be correct.
>

It is edifying to know that I'm not always rambling.

> This points out why it's important to reach an
> understanding of exactly what the original Latin text
> means. It also illustrates why there has been confusion
> over this point for centuries, and why i'd have to really
> dig back in to the the previous and subsequent chapters
> to put this one in context.
>

A similar consideration must be given for Maqam Music treatises as I keep
saying.

> > This scheme fits perfectly if the entire system was:
> >
> > A B C(#) D E F(#) G(#) a b c(#) d e f(#) g(#) a'
> >
> >
> > > The progression to higher pitch is similar in the
> > > other modes.
> > >
> >
> >
> > 1. Hypodorian mode [A B C# D E F# G# a b c# d e f# g# a'] ?
> > 2. Hypophyrigian mode [B C# D E F# G# a b c# d e f# g# a'] ?
> > 3. Hypolydian mode [C# D E F# G# a b c# d e f# g# a' b' c'#] ?
> > 4. Dorian mode [D E F# G# a b c# d e f# g# a' b' c'# d'] ?
> > 5. Phyrigian mode [E F# G# a b c# d e f# g# a' b' c'# d' e'] ?
> > 6. Lydian mode [F# G# a b c# d e f# g# a' b' c'# d' e' f'#] ?
> > 7. Mixolydian mode [G# a b c# d e f# g# a' b' c'# d' e' f'# g'#] ?
> >
> >
> > Transposed to natural tones:
> >
> > C D E F G A B c d e f g a b c' - HD (major)
> > D E F G A B c d e f g a b c' d' - HP
> > E F G A B c d e f g a b c' d' e' - HL
> > F G A B c d e f g a b c' d' e' f' - D
> > G A B c d e f g a b c' d' e' f' g' -P
> > A B c d e f g a b c' d' e' f' g' a' -L (minor)
> > B c d e f g a b c' d' e' f' g' a' b' -M
> >
> >
> > Does it make sense?
>
>
> I would have to find my old paper on this, but IIRC
> this last tabulation does agree with what i wrote.
> In particular, i think i remember the Lydian being
> equivalent to our A-natural-minor scale.
>
>

Strike!

>
> > > Years ago, i also wrote a paper which i've never published
> > > in which i believe i discovered that Boethius's Greek-letter
> > > notation recognized the difference of a syntonic-comma
> > > (~22 cents) but equated pairs of notes which were a schisma
> > > apart (~2 cents), thus in effect presenting the ancient
> > > modal system as a schismic temperament.
> > >
> >
> >
> > Fascinating.
>
>
>
> Quite a bit more than "fascinating", if you ask me.
>
>
> There's apparently a long tradition of Arabic music using
> these schismic (schismatic?) equivalences while preserving
> commatic (syntonic and/or pythagorean) differences, and
> i believe i see it also in classical Indian music-theory.
>
> http://tonalsoft.com/monzo/indian/indian.htm
>
>

Please... this music does not belong solely to Arabs, but to all nations
within the cultural perimeters of the Middle East, which is predominantly
Islamic still. The music you mention is more deserving of the cross-cultural
term `Maqam Music`. Also, the greatest contributors to this genre were
muslim theorists, some of whom were not at all Arabs in origin. Furthermore,
there were Byzantine and Armenian scholars of high repute as well. The
Islamic Civilization made no discriminations regarding ethnicity and
benefitted equally from the collective wisdom of all.

The 79 MOS 159-tET proposal of mine attempts to unite all Maqam Music
varieties throughout this geography in just the way you said: Using schismic
equivalances and preserving commatic differences.

> If my interpretation of Boethius is correct, then this
> is probably evidence of cross-cultural influence between
> the ancient Greeks, Arabs, and Indians.
>

And Turks, Armenians, Persians, Azeris, Afghanis...

> Of course, several centuries before Boethius, Alexander
> had made much of the Arab and Hindu lands part of his
> Greek empire, so it wouldn't surprise me at all to see
> this cross-cultural infiltration of ideas and practices.
>

Dhulkarneyn is esteemed highly in our Holy Scripture and is oft associated
with Alexander the Great. There is a strong bearing in what you say.

> But AFAIK no-one has pointed out this specific one before,
> and even Boethius sabotages our knowledge of it, because
> he presents the diatonic system in its ancient pythagorean
> tuning, and this schismatic equivalence only becomes apparent
> if one tunes the diatonic genus according to the ratios
> advocated by Didymus (c.100 AD).
>

Also the ratios propounded by Al-Farabi much after him. Yet the general
tendency in theory was, how shall I put it, Boethian(?) in this geography...

> That, in turn, would be evidence that Didymus was providing
> an accurate analysis of actual musical practice of his time
> -- which is something that i think no music historian ever
> thought would be found.
>
>
>

Yekta says that the Greater Perfect System of Pythagoras was performed in
this way:

A B C# D E F# G a b c# d e f# g a

by Eastern musicians. But he neglects the fact that the second tetrachord
requires G to be sharp for symmetry to be preserved and thus cannot
understand why Guido d'Arezzo required a gamma for his major scale. He also
mentions a pitch named by Meibomius as hypoproslambanomenos. In a schismic
temperament where the above mentioned major gamut is the harmonic major
(Rast) scale extended to two octaves, I would feel at a liberty to equate
Greek relative-pitches with Maqam Music main relative-pitches thusly:

Hypoproslambanomenos _ Nerm Chargah
Proslambanomenos _ Yegah
Hypate hypaton _ Ashiran
Parhypate hypaton _ Araq
Lichanos hypaton _ Rast
Hypate meson _ Dugah
Parhypate meson _ Segah
Lichanos meson _ Chargah
Mese _ Neva
Paramese _ Huseyni
Trite diezeugmenon _ Evdj
Paranete diezeugmenon _ Gerdaniye
Nete diezeugmenon _ Muhayyer
Trite hyperbolaion _ Tiz Segah
Paranete hyperbolaion _ Tiz Chargah
Nete hyperbolaion _ Tiz Neva

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

Cordially,
Ozan

🔗monz <monz@tonalsoft.com>

Hi Ozan,

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:

> Is it not evident that the interpretation which considers
> modes to be direct transpositions of the greater perfect
> system is fundamentally lacking, since the idea of octave
> species is ruined and there is nothing at all special
> when the scale remains unrotated?

Yes, it's evident ... i just can't put my mind back into
this place right now, to really look at all this.

> > There's apparently a long tradition of Arabic music using
> > these schismic (schismatic?) equivalences while preserving
> > commatic (syntonic and/or pythagorean) differences, and
> > i believe i see it also in classical Indian music-theory.
> >
> > http://tonalsoft.com/monzo/indian/indian.htm
>
>
> Please... this music does not belong solely to Arabs, but
> to all nations within the cultural perimeters of the
> Middle East, which is predominantly Islamic still. The
> music you mention is more deserving of the cross-cultural
> term `Maqam Music`. Also, the greatest contributors to
> this genre were muslim theorists, some of whom were not
> at all Arabs in origin. Furthermore, there were Byzantine
> and Armenian scholars of high repute as well. The Islamic
> Civilization made no discriminations regarding ethnicity
> and benefitted equally from the collective wisdom of all.

This just points out my ignorance of the whole subject
of Maqam. I've dabbled in studying some Arab and Indian
music-theory, and that's why i always use those cultural
terms -- i don't want to include Turkish, Persian, and
others without knowing whether or not i'm correct, and
then find out that i did statement something wrong.

I mostly skip past comments here in the Maqam threads.
Much as i'd like to learn about it and watch the theory
evolve, my head is just too full of other things right now.

However, please note that my studies of music in the
geographical Middle-East go much farther back in history
than the advent of Islam ... all the way to the Sumerians,
which is as far as one can go.

And what i see is that many ideas which i believe were
part of Sumerian music and music-theory 5000 years ago
have survived, and i believe i can trace their transmission
thus (the time-scale on the left is very rough):

years
BC
3000 ..............._______ Sumerian _____
.................. / ......... | .........\
2000 ........ Babylonian .. Egyptian ... Indian
............... / ...... \ ....____|_______/
1000 ........ Greek .... Persian . |
............. / ... \______ | ____/
0 AD .. Byzantine ....... Arab

Apart from the obvious direct transmissions, i don't
know how clearly i've illustrated the more complicated
ones. Babylonian and Indian both lead to Persian.
Greek, Persian, and Egyptian all lead to Arab.
I'm sure that Persian is also supposed to connect
to Byzantine.

I don't know where Turkish and Armenian fit in -- feel
free to add them to the diagram ... and also to correct
anything i've done which you know to be wrong.

> The 79 MOS 159-tET proposal of mine attempts to unite
> all Maqam Music varieties throughout this geography in
> just the way you said: Using schismic equivalances and
> preserving commatic differences.

I look forward to learning more about it as soon as i
have time!

Also, it would be great to get you as a Tonescape tester,
and for you to create Tonescape tuning files of all this
stuff ... and also Tonescape pieces illustrating the
tunings, which we can put up on our website.

> > If my interpretation of Boethius is correct, then this
> > is probably evidence of cross-cultural influence between
> > the ancient Greeks, Arabs, and Indians.
> >
>
> And Turks, Armenians, Persians, Azeris, Afghanis...

Right ... my deliberately leaving people out is because
of my ignorance of the subject.

> > But AFAIK no-one has pointed out this specific one before,
> > and even Boethius sabotages our knowledge of it, because
> > he presents the diatonic system in its ancient pythagorean
> > tuning, and this schismatic equivalence only becomes apparent
> > if one tunes the diatonic genus according to the ratios
> > advocated by Didymus (c.100 AD).
>
>
> Also the ratios propounded by Al-Farabi much after him.
> Yet the general tendency in theory was, how shall I put it,
> Boethian(?) in this geography...

Absolutely. As the last great Roman scholar (and perhaps
the greatest), Boethius's work commanded a tremendous prestige
all during the medieval period, not only in Europe, but in
all lands that had been part of the Roman Empire, which
extended all the way to Mesopotamia and Northern Africa.

> Yekta says that the Greater Perfect System of Pythagoras
> was performed in this way:
>
> A B C# D E F# G a b c# d e f# g a
>
> by Eastern musicians. <snip> In a schismic temperament
> where the above mentioned major gamut is the harmonic major
> (Rast) scale extended to two octaves, I would feel at a
> liberty to equate Greek relative-pitches with Maqam Music
> main relative-pitches thusly:
>
> Hypoproslambanomenos _ Nerm Chargah
> Proslambanomenos _ Yegah
> Hypate hypaton _ Ashiran
> Parhypate hypaton _ Araq
> Lichanos hypaton _ Rast
> Hypate meson _ Dugah
> Parhypate meson _ Segah
> Lichanos meson _ Chargah
> Mese _ Neva
> Paramese _ Huseyni
> Trite diezeugmenon _ Evdj
> Paranete diezeugmenon _ Gerdaniye
> Nete diezeugmenon _ Muhayyer
> Trite hyperbolaion _ Tiz Segah
> Paranete hyperbolaion _ Tiz Chargah
> Nete hyperbolaion _ Tiz Neva

Wow, you certainly are fearless ... my quoting of a
Latin text got me labelled as a "supercillious [sic] jerk",
and now you're bold enough to post a Greek <-> Turkish
glossary. (Good show!)

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

🔗monz <monz@tonalsoft.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@tonalsoft.com>

Hi Ozan and Hans,

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:

[Ozan]
> I am suspecting whether he [Boethius] meant
> B C D E F G A b c d e f g a' b' instead?
> Boethius actually mentioned being in accord with the
> species of the consonance of the octave, and direct
> transposition is certainly not synonymous with his
> previous statement, modulation is.

Yes, upon further reflection, i think you're right.

In order to preserve the intervals of the octave-species,
and rotate the modes thru all forms without using any
flats or sharps in the nomenclature, the reference Hypodorian
mode must use either G or C as the reference pitch.
The example Ozan gaving using "natural pitches" was
based on C as the reference, but i think G is more
likely, as will become clear later.

This is interesting, because i've read some other analyses
of the modal system where the reference pitch was F.
In particular, i think i recall that Kathleen Schlesinger
based the whole modal system on Proslambanomenos as a low F.
But using G as the reference brings Boethius's system
exactly into agreement with the standard early medieval
gamut beginning on a low-G notated as the letter gamma.

This is fascinating, because while Boethius did use
Roman letters in his descriptions, they did not have
the same pitch-meaning as we use today, but rather were
simply labels for points along his arithmetic measurements.
The descriptions of the ancient Greeks were later labeled
with Roman letters using A as the name for Mese, and it
was Hucbald who shifted the whole system down a tone to G,
which brings his system in line with Boethius. Hmm...

Since Boethius only gives a detailed explanation of
the first four modes in this chapter, and just dismisses
the rest of his explanation with "similarly for the rest",
it's impossible to know whether the reference pitch is
G or C, because both labels work for the first four modes.
I'd have to study the other chapters to see if this can
be resolved.

In fact this is a very intriguing topic, because the
_music enchiriadis_ and _scholia enchiriadis_ treatises,
which i think date from the early 800's, use the unique
daseain notation to specify where the semitone should
be placed, and it too leaves unresolved the question
of whether we would label the reference pitch as G or C.
Similiarly, the next really influential theorist after
Boethius was Guido d'Arezzo, and he too used a similar
procedure.

The later "fasola" system also simplifies modal rotation
to using a reference tetrachord containing two tones and
one semitone, and this became the basis of American
"shape-note" notation which results in just-intonation
singing.

So anyway, here are some relevant sections from Boethius
again, with the Latin and English together, and my revised
illustrative scales using both C and G as reference pitches.
I append numbers to the letters here, to bring the
illustrations into conformity with the standard MIDI
pitch nomenclature where C4 = middle-C and each
new octave numbering begins on C ascending.

>> [Latin]
>> EX diapason igitur speciebus consonantiae existunt. qui
>> appellantur modi. quos eosdem tropos uel tonos nominant.
>>
>> [English - Bower]
>> From the species of the consonance of the diapason
>> arise what are called "modes". They are also called
>> "tropes" or "tones".

>> Has igitur constitutiones [[constitutiones]] si quis
>> totas faciat acutiores. uel in graues totas remittat:
>> secundum supradictas diapason consonantiae species
>> efficiet modos septem. quorum nomina sunt haec.
>> Hypodorius. hypofrigius. hypolidius. dorius. phygius
>> [phrygius corr. supra lin]. lidius. mixolidius.
>>
>> If these entire systems were made higher or lower in
>> accordance with the species of the consonance of the
>> diapason discussed above, this would bring about seven
>> modes, which are named Hypodorian, Hypophrygian,
>> Hypolydian, Dorian, Phrygian, Lydian, and Mixolydian.

>> Horum uero sic ordo procedit.
>>
>> The arrangement of the modes proceeds in the following
>> manner.

>> Sit in diatonico genere uocum ordo dispositus a
>> proslambanomenon in netenhyperboleon. atque hic
>> sit hypodorius modus.
>>
>> Set out the succession of pitches in the diatonic genus
>> from the proslambanomenos to the nete hyperboleon. Let
>> this be the Hypodorian mode

[G2 A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4] or
[C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5]

>> Si quis igitur proslambanomenon in acumen intendit
>> [intendat corr. supra lin.] tono hypatenque hypaton
>> eodem tono attenuet. caeterasque omnes tono faciat
>> acutiores: acutior totus ordo proueniat quam fuit
>> prius quam toni susciperet intensionem. Erit igitur
>> tota constitutio acutior effecta hypofrigius modus.
>>
>> If one were to raise the proslambanomenos by one tone,
>> and further raise the hypate hypaton by the same tone,
>> thereby making the whole disposition higher by a tone,
>> then the entire succession would turn out higher than it
>> was before it was raised by a tone. Thus this whole system,
>> having been made higher, forms the Hypophrygian mode.

[A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4] or
[D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5]

The first version of this would be the "normal" system of
labeling, with "A" for proslambanomenos and mese.

>> [-f.87r-] Quod si in hypophrigio toni rursus intensionem
>> uoces acceperint. hypolidii modulatio nascitur.
>>
>> Now if the pitches of the Hypophrygian mode were similarly
>> raised by a tone, the collection of pitches for the
>> Hypolydian mode would be born.

[B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4] or
[E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5]

>> At si hypolidium quis semitonio intendat: dorium faciet.
>>
>> If someone raised the Hypolydian by a semitone,
>> he would make the Dorian

[C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5] or
[F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 F5]

>> Et in aliis quidem similis est in acumen intensionemque
>> processus;
>>
>> The progression to higher pitch is similar in the
>> other modes.

So the whole system looks like this:

Hypodorian
[G2 A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4] or
[C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5]

Hypophrygian
[A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4] or
[D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5]

Hypolydian
[B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4] or
[E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5]

Dorian
[C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5] or
[F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 F5]

Phrygian
[D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5] or
[G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 F5 G5]

Lydian
[E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5] or
[A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 F5 G5 A5]

Mixolydian
[F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 F5] or
[B3 C4 D4 E4 F4 G4 A4 B4 C4 D4 E4 F4 G4 A4 B4]

I have serious doubts if the Mixolydian mode was
really meant to illustrate the octave-species of
natural notes based on B, considering that the
tonic 5th is diminished instead of perfect, and
also considering the duplicity of B's which arose
later when Roman letters were applied to the system.

Thus, my current interpretation of Boethius's modal
system equates it to modern pitch notation as follows:

G2 A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 hypodorian
A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 hypophrygian
B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 hypolydian
C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 dorian
D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 phrygian
E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 lydian
F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 F5 mixolydian

>> quorum non ut intelligentia solum modo ratio
>> comprehendatur. uerum oculis quoque forma possit
>> agnosci. ab antiquis tradita musicis descriptio
>> supponenda est.
>>
>> So that the theory of these modes might be grasped
>> by the eye as well as the intellect, a diagram,
>> a visual representation of the modes, handed down
>> from ancient musicians is presented below.

>> Sed quoniam per singulos modos a ueteribus musicis
>> unaquaeque uox diuersis notulis insignita est: descriptio
>> prius notularum uidetur esse ponenda. ut his primum per
>> se cognitis in modorum. descriptione facilis possit esse
>> inspectio.
>>
>> But since each single pitch for the individual modes
>> is recorded with different written symbols by ancient
>> musicians, it seems necessary to present a diagram of
>> the written symbols first; then, with these understood
>> in themselves, consideration of the diagram of modes
>> should be easy.

Hans, can you verify that Bower's translation of the
first phrase here is accurate?

Is Boethius really saying "each single *pitch* for the
individual modes is recorded with different written symbols"?

Or is he saying "each individual mode"?, which is what
it looks like to me.

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

🔗monz <monz@tonalsoft.com>

🔗klaus schmirler <KSchmir@online.de>

🔗monz <monz@tonalsoft.com>

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

> The diagrams in the numerous manuscripts of Boethius's
> treatise on music are notorious for their confusion and
> inaccuracies. I've found something very weird here:
>
> http://www.music.indiana.edu/tml/6th-8th/BOEMUS4C_16GF.gif

In fact, a reading of the text shows that Boethius used
letters from the Greek alphabet in the Greek order.
The Cambridge manuscript uses the normal Roman ordering
of the first 14 letters, which is a later manuscript
tradition. I have no access to any manuscripts which
use the older Greek ordering, but it is apparent from
the text.

----------

>> liber IV, caput XIV
>> book 4, chapter 14

>> De consonantiarum speciebus.
>> Concerning the species of consonances

>> NVNC de speciebus primarum consonantiarum tractandum est.
>>
>> The species of the primary consonances must now be discussed.

>> Primae autem consonantiae sunt diapason. diapente.
>> diatessaron.
>>
>> The primary consonances are the diapason [8ve],
>> the diapente [5th], and the diatessaron [4th].

>> Species autem [est add. supra lin.] quaedam positio
>> propriam habens formam secundum unumquodque genus.
>>
>> A species is a particular segment of notes in one
>> of the genera with a unique pattern of intervals;

>> in uniuscuiusque terminis proportionis consonantiam
>> facientis terminis constituta.
>>
>> the segment is arranged within the terms of some ratio
>> yielding a consonance.

>> ut in diatonico [-f.85r-] genere.
>>
>> Take the diatonic genus, for example.

>> Nam si diezeugmenon tetracordum inter hyperboleon
>> tetracordum mesonque ponamus. [subtracto scilicet
>> sinemmenon tetracordo: add. supra in marg.] erunt
>> .XV. nerui.
>>
>> If we should place the diezeugmenon tetrachord between
>> the hyperboleon tetrachord and the mese, with the
>> synemmenon tetrachord taken away, there would be 15 notes.

My illustration on the PIS webpage makes this clear:
http://tonalsoft.com/enc/p/pis.aspx

>> At si ab his prolambanomenos detrahatur: erunt .XIIII.
>>
>> And if the proslambanomenos is removed, there would be 14.

>> Hi ergo disponantur hoc modo.
>>
>> These may be set out in this manner.

>> Sit .A. hypatehypaton. .B. parhypatehypaton
>> .C. hypatelicanos [hipaton add. in marg.] .D. hypatemeson
>> .E. parhypatemeson .F. lichanosmeson .G. mese .H. paramese
>> .I. tritediezeugmenon .K. paranetediezeugmenon
>> .L. netediezeugmenon M. tritehyperboleon
>> .N. paranetehyperboleon .O. netehyperboleon.

Here i think it's best to simply translate the Latin text
into a diagram, with the highest pitch at the top:

modern
note .. Boethius description

a' .. O. nete hyperboleon
g ... N. paranete hyperboleon
f ... M. trite hyperboleon
e ... L. nete diezeugmenon
d ... K. paranete diezeugmenon
c ... I. trite diezeugmenon
b ... H. paramese
a ... G. mese
G ... F. lichanos meson
F ... E. parhypate meson
E ... D. hypate meson
D ... C. hypate licanos [hipaton] ("lichanos hypaton" is correct)
C ... B. parhypate hypaton
B ... A. hypate hypaton.

This diagram presents the correct tonal relationships
as described in Boethius's text, and supercedes the
strange diagram i submitted in my previous post.

>> Ab hypate igitur ad mesen [paramesen corr. supra lin.]
>> diapason consonantia est.
>>
>> There is a consonance of a diapason [8ve] between
>> the hypate [hypaton] and the paramese [B-b];

>> A[[b eadem]] uero mese ad lichanonhypaton. diapente.
>>
>> between the same paramese and the hypate meson,
>> a diapente [5th, b-E],

>> A [Ab eadem corr. supra lin.] mese uero ad hypatemeson
>> diatessaron.
>>
>> and between the mese and the hypate meson, a
>> diatessaron [4th, a-E].

>> Erit igitur diapason quidem octo cordarum.
>> Diatessaron uero quattuor. Diapente uero. quinque.
>>
>> Thus the diapason [8ve] consists of 8 strings,
>> the diatessaron [4th] of 4, and the diapente [5th] of 5.

>> Ac per hoc habebit diatessaron quidem species tres.
>> Diapente uero species quattuor. Diapason uero species .VII.
>>
>> Because of this, the diatessaron [4th] has 3 species,
>> the diapente [5th] has 4 species, and the diapason [8ve]
>> has 7 species.

>> semperque una minus species erit quam fuerint uoces.
>>
>> There is always one fewer species than there are pitches.

>> Vt enim a mese caeteras ordiamur: diatessaron consonantia
>> [consonantiae corr. supra lin.] species sunt tres hoc modo.
>>
>> Beginning the enumeration of species from the mese, there
>> are 3 species of the consonance of the diatessaron [4th],
>> according to this pattern.

>> Vna quidem species erit. ab .G. ad .D.
>> One species will be from G to D,

pitches a to E, descending interval pattern t-t-s.

>> Secunda ab .F. ad .C.
>> a second [species] from F to C,

pitches G to D, descending interval pattern t-s-t.

>> Tertia ab .E. ad .B.
>> a third [species] from E to B.

pitches F to C, descending interval pattern s-t-t.

>> Et huc usque diatessaron species progrediuntur.
>> iccirco quia huc usque species binos continent
>> neruos eiusdem diatessaron.
>>
>> The species of the diatessaron [4th] proceed up to
>> this point, because up to this point the species contain
>> two notes of the same diatessaron [4th]:

>> ut .G.D. quidem eos qui sunt .E.F.
>>
>> GD contains EF [_sic_: ED is correct -- in pitches,
>> A-E contains F-E]

>> et .F.C. eos qui sunt .E.D
>>
>> FC contains ED [in pitches, G-D contains F-E]

>> et EB. eos qui sunt [ED. uel add. supra lin.].C.D.
>>
>> EB contains CD [_sic_; "ED" written above is correct -
>> in pitches, F-C contains F-E]

Bower's footnote explains that medieval scribes had great
difficulty with this passage, and the manuscripts all
show the same confusion that is present here in the
Cambridge manuscript.

"ED" in Boethius's letters represents the semitone
between the pitches F and E, and his point is to demonstrate
that all three species-of-4th contain the semitone, and
that its position rotates according to the bounding
pitches of the 4th.

>> Si uero his adiecero diatessaron .D.A: diuersa erit ab
>> ea quae est .G.D. Vnum enim solum .G.D. consonantiae
>> neruum continebit. id est .D. solum.
>>
>> But if I add the diatessaron [4th] DA [pitches E-B] to
>> these, it will have little in common with GD [pitches a-E],
>> for it will contain only one string of the consonance GD
>> -- that is, only D [pitch E] alone. [i.e., it does not
>> contain the F-E semitone]

>> Excessit igitur .G.D. consonantiam. atque ideo
>> diatessaron tres species habere perhibetur.
>>
>> Thus, DA [pitches E-B] has gone beyond the consonance
>> GD [pitches a-E], and for that reason the diatessaron [4th]
>> is said to have 3 species.

>> Et in caeteris quidem consonantiis idem est.
>>
>> The same occurs in the other consonances.

I'll complete this when i have time.
That's enough to get started.

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

🔗oyarman@ozanyarman.com

Hey there again!

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 16 Kasï¿½m 2005 ï¿½arï¿½amba 23:23
Subject: [tuning] Re: Boethius modal system, chapter 15 of Book 4

> Hi Ozan and Hans,
>
>
> --- In tuning@yahoogroups.com, <oyarman@o...> wrote:
>
>
> [Ozan]
> > I am suspecting whether he [Boethius] meant
> > B C D E F G A b c d e f g a' b' instead?
> > Boethius actually mentioned being in accord with the
> > species of the consonance of the octave, and direct
> > transposition is certainly not synonymous with his
> > previous statement, modulation is.
>
>
> Yes, upon further reflection, i think you're right.
>

Can you see my grin? :)))

> In order to preserve the intervals of the octave-species,
> and rotate the modes thru all forms without using any
> flats or sharps in the nomenclature, the reference Hypodorian
> mode must use either G or C as the reference pitch.
> The example Ozan gaving using "natural pitches" was
> based on C as the reference, but i think G is more
> likely, as will become clear later.
>

Let's beg to differ on this, shall we?

> This is interesting, because i've read some other analyses
> of the modal system where the reference pitch was F.
> In particular, i think i recall that Kathleen Schlesinger
> based the whole modal system on Proslambanomenos as a low F.
> But using G as the reference brings Boethius's system
> exactly into agreement with the standard early medieval
> gamut beginning on a low-G notated as the letter gamma.
>

If that be the case, the discovery belongs to Rauf Yekta. Please find his
entry in French in the 1922 edition of the Lavignac Encyclopedia. There,
under the article `Turquie`, not too far down, you will see that he
explains:

..................................Execution by
Westerners..................Execution by Easterners

Proslambanomenos..............la............................................
.............la....................
Hypate
hypaton......................si.............................................
............si....................
Parhypate hypaton................do
(natural).........................................do#................
Lichanos
hypaton...................re................................................
........re...................
Hypate
meson........................mi.............................................
..........mi...................
Parhypate meson...................fa
(natural).........................................fa#..................
Lichanos
meson....................sol................................................
.......sol (natural)....
Mese........................................la..............................
...........................la....................
Paramese................................si..................................
......................si.....................
Trite diezeugmenon................do
(natural)........................................do#.................
Paranete
diezeugmenon........re......................................................
..re....................
Nete
diezeugmenon................mi..............................................
.........mi....................
Trite hyperbolaion....................fa
(natural).........................................fa#...................
Paranete
hyperbolaion............sol.................................................
......sol (natural)....
Nete
hyperbolaion....................la..........................................
...............la.....................

He then goes on to say that transposing the diatonic scale within the octave
over to Sol (G), we will find the Arezzian scale:

G4 A B C D E F G5

Whereby Yekta explains: rumor had it that Gui himself did not like the minor
scale La Si Do Re Mi Fa Sol La and added a Sol below and called it the
hypoproslambanomenos which he notated with the letter Gamma. He says that
this is the origin of the term "gamut" and also the origin of the faulty
execution of Westerners. He comments that Alphonse Heegmann in his "Examen
de la theorie musicale des Grecs" became almost aware of the situation where
he explicated that Guido made the A-proslambanomenos LA which he notated
between the first and second lines of an F-cleffed staff. Equating
proslambanomenos with LA, according to Yekta, is the single greatest mistake
made on this subject, because it was only necessary to equate
proslambanomenos with SOL (G) for everything to fit together.

But he then proceeds to ask himself: `why would Guido feel the need to add a
Gamma below if the existing Greek system was already major?`

Perhaps, the answer lies with the faulty Hucbald interpretation of Boethius.

He also makes fun of the numerous mistakes that lead to this division
between the East and the West, criticizes severely those accountable for
this state of affairs and comments that even 5 volumes of the Lavignac
encyclopedia would not suffice to review and analyze the entirety of the
grave errors committed by those responsible. ROFL

> This is fascinating, because while Boethius did use
> Roman letters in his descriptions, they did not have
> the same pitch-meaning as we use today, but rather were
> simply labels for points along his arithmetic measurements.
> The descriptions of the ancient Greeks were later labeled
> with Roman letters using A as the name for Mese, and it
> was Hucbald who shifted the whole system down a tone to G,
> which brings his system in line with Boethius. Hmm...
>

Are you sure it was Hucbald? Yekta says it was done in Guido's time.

> Since Boethius only gives a detailed explanation of
> the first four modes in this chapter, and just dismisses
> the rest of his explanation with "similarly for the rest",
> it's impossible to know whether the reference pitch is
> G or C, because both labels work for the first four modes.
> I'd have to study the other chapters to see if this can
> be resolved.
>

Please confer to my other reply for a detailed interpretation by myself.

> In fact this is a very intriguing topic, because the
> _music enchiriadis_ and _scholia enchiriadis_ treatises,
> which i think date from the early 800's, use the unique
> daseain notation to specify where the semitone should
> be placed, and it too leaves unresolved the question
> of whether we would label the reference pitch as G or C.
> Similiarly, the next really influential theorist after
> Boethius was Guido d'Arezzo, and he too used a similar
> procedure.
>

He notated HYPOproslambanomenos with GAMMA according to Yekta. I know not if
this is true. What sayest thou?

> The later "fasola" system also simplifies modal rotation
> to using a reference tetrachord containing two tones and
> one semitone, and this became the basis of American
> "shape-note" notation which results in just-intonation
> singing.
>
>

Shape note?

> So anyway, here are some relevant sections from Boethius
> again, with the Latin and English together, and my revised
> illustrative scales using both C and G as reference pitches.
> I append numbers to the letters here, to bring the
> illustrations into conformity with the standard MIDI
> pitch nomenclature where C4 = middle-C and each
> new octave numbering begins on C ascending.
>
>

Ach! You should have used octave designator numbers just once per octave!
Look at that clutter...

> >> [Latin]
> >> EX diapason igitur speciebus consonantiae existunt. qui
> >> appellantur modi. quos eosdem tropos uel tonos nominant.
> >>
> >> [English - Bower]
> >> From the species of the consonance of the diapason
> >> arise what are called "modes". They are also called
> >> "tropes" or "tones".
>
>
> >> Has igitur constitutiones [[constitutiones]] si quis
> >> totas faciat acutiores. uel in graues totas remittat:
> >> secundum supradictas diapason consonantiae species
> >> efficiet modos septem. quorum nomina sunt haec.
> >> Hypodorius. hypofrigius. hypolidius. dorius. phygius
> >> [phrygius corr. supra lin]. lidius. mixolidius.
> >>
> >> If these entire systems were made higher or lower in
> >> accordance with the species of the consonance of the
> >> diapason discussed above, this would bring about seven
> >> modes, which are named Hypodorian, Hypophrygian,
> >> Hypolydian, Dorian, Phrygian, Lydian, and Mixolydian.
>
>
>
> >> Horum uero sic ordo procedit.
> >>
> >> The arrangement of the modes proceeds in the following
> >> manner.
>
>
> >> Sit in diatonico genere uocum ordo dispositus a
> >> proslambanomenon in netenhyperboleon. atque hic
> >> sit hypodorius modus.
> >>
> >> Set out the succession of pitches in the diatonic genus
> >> from the proslambanomenos to the nete hyperboleon. Let
> >> this be the Hypodorian mode
>
> [G2 A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4] or
> [C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5]
>

G according to Yekta
C according to Oz.

>
> >> Si quis igitur proslambanomenon in acumen intendit
> >> [intendat corr. supra lin.] tono hypatenque hypaton
> >> eodem tono attenuet. caeterasque omnes tono faciat
> >> acutiores: acutior totus ordo proueniat quam fuit
> >> prius quam toni susciperet intensionem. Erit igitur
> >> tota constitutio acutior effecta hypofrigius modus.
> >>
> >> If one were to raise the proslambanomenos by one tone,
> >> and further raise the hypate hypaton by the same tone,
> >> thereby making the whole disposition higher by a tone,
> >> then the entire succession would turn out higher than it
> >> was before it was raised by a tone. Thus this whole system,
> >> having been made higher, forms the Hypophrygian mode.
>
> [A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4] or
> [D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5]
>

Ditto.

> The first version of this would be the "normal" system of
> labeling, with "A" for proslambanomenos and mese.
>
>
> >> [-f.87r-] Quod si in hypophrigio toni rursus intensionem
> >> uoces acceperint. hypolidii modulatio nascitur.
> >>
> >> Now if the pitches of the Hypophrygian mode were similarly
> >> raised by a tone, the collection of pitches for the
> >> Hypolydian mode would be born.
>
> [B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4] or
> [E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5]
>
>

Ditto.

> >> At si hypolidium quis semitonio intendat: dorium faciet.
> >>
> >> If someone raised the Hypolydian by a semitone,
> >> he would make the Dorian
>
> [C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5] or
> [F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 F5]
>
>

Ditto.

> >> Et in aliis quidem similis est in acumen intensionemque
> >> processus;
> >>
> >> The progression to higher pitch is similar in the
> >> other modes.
>
> So the whole system looks like this:
>

Hypodorian

[C3 D E F G A B c4 d e f g a b c'5]

[G3 A B C D E F g4 a b c d e f g'5]

Hypophrygian

[D3 E F G A B C d4 e f g a b c' d'5]

[A3 B C D E F g a4 b c d e f g' a'5]

Hypolydian

[E3 F G A B C d e4 f g a b c' d' e'5]

[B3 C D E F g a b4 c d e f g' a' b'5]

Dorian

[F3 G A B C d e f4 g a b c' d' e' f'5]

[C4 D E F g a b c5 d e f g' a' b' c'6]

Phrygian

[G3 A B C d e f g4 a b c' d' e' f' g'5]

[D4 E F g a b c d5 e f g' a' b' c' d'6]

Lydian

[A3 B C d e f g a4 b c' d' e' f' g' a'5]

[E4 F g a b c d e5 f g' a' b' c' d' e'6]

Mixolydian

[B3 C d e f g a b4 c' d' e' f' g' a' b'5]

[F4 g a b c d e f5 g' a' b' c' d' e' f'6]

> I have serious doubts if the Mixolydian mode was
> really meant to illustrate the octave-species of
> natural notes based on B, considering that the
> tonic 5th is diminished instead of perfect, and
> also considering the duplicity of B's which arose
> later when Roman letters were applied to the system.
>

Why? Mixo means part Greek, part barbaric. Literally Mixolydian would mean
`Barbaric Lydian`. What better mode to ascribe the tonic diminished fifth?
Besides, your dissonant fifth lies with Hypolydian. Why should that be more
preferable? And look... here are your pitches:

>
> Thus, my current interpretation of Boethius's modal
> system equates it to modern pitch notation as follows:
>
> G2 A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 hypodorian
> A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 hypophrygian
> B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 hypolydian
> C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 dorian
> D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 phrygian
> E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 lydian
> F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 F5 mixolydian
>
>

G2 is just too low a pitch for Proslambanomenos, impossible for even tenors
to sing, while my C3 is just right.

SNIPPED the rest.

Cordially,
Ozan

🔗oyarman@ozanyarman.com

Monz, you should be commended for your excellent efforts! My comments are
below:

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 17 Kasï¿½m 2005 Perï¿½embe 2:33
Subject: [tuning] Re: Boethius modal system, chapter 14 of Book 4

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > The diagrams in the numerous manuscripts of Boethius's
> > treatise on music are notorious for their confusion and
> > inaccuracies. I've found something very weird here:
> >
> > http://www.music.indiana.edu/tml/6th-8th/BOEMUS4C_16GF.gif
>
>
> In fact, a reading of the text shows that Boethius used
> letters from the Greek alphabet in the Greek order.
> The Cambridge manuscript uses the normal Roman ordering
> of the first 14 letters, which is a later manuscript
> tradition. I have no access to any manuscripts which
> use the older Greek ordering, but it is apparent from
> the text.
>
> ----------
>
> >> liber IV, caput XIV
> >> book 4, chapter 14
>
> >> De consonantiarum speciebus.
> >> Concerning the species of consonances
>
>
> >> NVNC de speciebus primarum consonantiarum tractandum est.
> >>
> >> The species of the primary consonances must now be discussed.
>
>
> >> Primae autem consonantiae sunt diapason. diapente.
> >> diatessaron.
> >>
> >> The primary consonances are the diapason [8ve],
> >> the diapente [5th], and the diatessaron [4th].
>
>
> >> Species autem [est add. supra lin.] quaedam positio
> >> propriam habens formam secundum unumquodque genus.
> >>
> >> A species is a particular segment of notes in one
> >> of the genera with a unique pattern of intervals;
>
>

My interpretation based on this and subsequent explanations is that, while
the assigned frequencies of the fixed tuning of `stabilis` pitches of the
greater perfect system do not change, `mobilis` pitches are altered in order
to yield possible genera of a consonant interval. For example:

A B C D is one species,
A Bb C D is another species,
A B C# D is yet another species,

Species would then be a term confined to permutations of pitches within a
consonant interval (especially the octave) with a fixed tonic, whereby
certain notes are switched up or down, or in other words, altered. Such a
function would then be synonymous with mode of a consonant interval
(rotation of the scale) transposed back to a point of reference that would
match the alterations made.

For the sake of clarity, I shall consign the term `trope` to this manner of
transposed modes of the consonance of the octave, hence `octave species`,
and reserve the term `genera` for the species of the fourth and the fifth,
since according to Boethius: "Tropes are systems that differ according to
highness or lowness throughout entire sequences of pitches."

Mode would then be the specie carried over to a pitch whereby the default
disposition of the notes remain unperturbed (i. e. zero alteration). For
example:

If C D E F G A B C is the default disposition

unperturbed: A B C D E F G A, this is a mode.
perturbed: C D Eb F G Ab Bb C, this is a trope (octave specie).

Therefore, `species` are not equivalent to `modes`, but to `transposed
modes`.

I do believe this interpretation of Boethius is much more valid compared to
the alternatives, and sheds light on the confusing stemming therefrom. Judge
for yourself:

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

`From the species of the consonance of the diapason arise what are called
"modes". They are also called "tropes" or "tones". Tropes are systems that
differ according to highness or lowness throughout entire sequences of
pitches. If these entire systems were made higher or lower in accordance
with the species of the consonance of the diapason discussed above, this
would bring about seven
modes, which are named Hypodorian, Hypophrygian, Hypolydian, Dorian,
Phrygian, Lydian, and Mixolydian.`

> >> in uniuscuiusque terminis proportionis consonantiam
> >> facientis terminis constituta.
> >>
> >> the segment is arranged within the terms of some ratio
> >> yielding a consonance.
>
>
> >> ut in diatonico [-f.85r-] genere.
> >>
> >> Take the diatonic genus, for example.
>
>
> >> Nam si diezeugmenon tetracordum inter hyperboleon
> >> tetracordum mesonque ponamus. [subtracto scilicet
> >> sinemmenon tetracordo: add. supra in marg.] erunt
> >> .XV. nerui.
> >>
> >> If we should place the diezeugmenon tetrachord between
> >> the hyperboleon tetrachord and the mese, with the
> >> synemmenon tetrachord taken away, there would be 15 notes.
>
>
> My illustration on the PIS webpage makes this clear:
> http://tonalsoft.com/enc/p/pis.aspx
>
>
> >> At si ab his prolambanomenos detrahatur: erunt .XIIII.
> >>
> >> And if the proslambanomenos is removed, there would be 14.
>
>
> >> Hi ergo disponantur hoc modo.
> >>
> >> These may be set out in this manner.
>
>
> >> Sit .A. hypatehypaton. .B. parhypatehypaton
> >> .C. hypatelicanos [hipaton add. in marg.] .D. hypatemeson
> >> .E. parhypatemeson .F. lichanosmeson .G. mese .H. paramese
> >> .I. tritediezeugmenon .K. paranetediezeugmenon
> >> .L. netediezeugmenon M. tritehyperboleon
> >> .N. paranetehyperboleon .O. netehyperboleon.
>
>
> Here i think it's best to simply translate the Latin text
> into a diagram, with the highest pitch at the top:
>
> modern
> note .. Boethius description
>
> a' .. O. nete hyperboleon
> g ... N. paranete hyperboleon
> f ... M. trite hyperboleon
> e ... L. nete diezeugmenon
> d ... K. paranete diezeugmenon
> c ... I. trite diezeugmenon
> b ... H. paramese
> a ... G. mese
> G ... F. lichanos meson
> F ... E. parhypate meson
> E ... D. hypate meson
> D ... C. hypate licanos [hipaton] ("lichanos hypaton" is correct)
> C ... B. parhypate hypaton
> B ... A. hypate hypaton.
>
>
> This diagram presents the correct tonal relationships
> as described in Boethius's text, and supercedes the
> strange diagram i submitted in my previous post.
>
>

It looks very clear to me. A new Grove entry is in order perhaps?

> >> Ab hypate igitur ad mesen [paramesen corr. supra lin.]
> >> diapason consonantia est.
> >>
> >> There is a consonance of a diapason [8ve] between
> >> the hypate [hypaton] and the paramese [B-b];
>
>
> >> A[[b eadem]] uero mese ad lichanonhypaton. diapente.
> >>
> >> between the same paramese and the hypate meson,
> >> a diapente [5th, b-E],
>
>
> >> A [Ab eadem corr. supra lin.] mese uero ad hypatemeson
> >> diatessaron.
> >>
> >> and between the mese and the hypate meson, a
> >> diatessaron [4th, a-E].
>
>
> >> Erit igitur diapason quidem octo cordarum.
> >> Diatessaron uero quattuor. Diapente uero. quinque.
> >>
> >> Thus the diapason [8ve] consists of 8 strings,
> >> the diatessaron [4th] of 4, and the diapente [5th] of 5.
>
>
> >> Ac per hoc habebit diatessaron quidem species tres.
> >> Diapente uero species quattuor. Diapason uero species .VII.
> >>
> >> Because of this, the diatessaron [4th] has 3 species,
> >> the diapente [5th] has 4 species, and the diapason [8ve]
> >> has 7 species.
>
>

So, there are 3 tetrachordal genera, 4 pentachordal genera and 7 tropes of
the octave from which arise 7 modes when the tonic is transposed over so
that all pitches become unaltered.

> >> semperque una minus species erit quam fuerint uoces.
> >>
> >> There is always one fewer species than there are pitches.
>
>

Most interesting, but that seems more a convention than a rule of thumb.

> >> Vt enim a mese caeteras ordiamur: diatessaron consonantia
> >> [consonantiae corr. supra lin.] species sunt tres hoc modo.
> >>
> >> Beginning the enumeration of species from the mese, there
> >> are 3 species of the consonance of the diatessaron [4th],
> >> according to this pattern.
>
>
> >> Vna quidem species erit. ab .G. ad .D.
> >> One species will be from G to D,
>
> pitches a to E, descending interval pattern t-t-s.
>
>
> >> Secunda ab .F. ad .C.
> >> a second [species] from F to C,
>
> pitches G to D, descending interval pattern t-s-t.
>
>
> >> Tertia ab .E. ad .B.
> >> a third [species] from E to B.
>
> pitches F to C, descending interval pattern s-t-t.
>
>
> >> Et huc usque diatessaron species progrediuntur.
> >> iccirco quia huc usque species binos continent
> >> neruos eiusdem diatessaron.
> >>
> >> The species of the diatessaron [4th] proceed up to
> >> this point, because up to this point the species contain
> >> two notes of the same diatessaron [4th]:
>
> >> ut .G.D. quidem eos qui sunt .E.F.
> >>
> >> GD contains EF [_sic_: ED is correct -- in pitches,
> >> A-E contains F-E]
>
> >> et .F.C. eos qui sunt .E.D
> >>
> >> FC contains ED [in pitches, G-D contains F-E]
>
> >> et EB. eos qui sunt [ED. uel add. supra lin.].C.D.
> >>
> >> EB contains CD [_sic_; "ED" written above is correct -
> >> in pitches, F-C contains F-E]
>
>
> Bower's footnote explains that medieval scribes had great
> difficulty with this passage, and the manuscripts all
> show the same confusion that is present here in the
> Cambridge manuscript.
>
> "ED" in Boethius's letters represents the semitone
> between the pitches F and E, and his point is to demonstrate
> that all three species-of-4th contain the semitone, and
> that its position rotates according to the bounding
> pitches of the 4th.
>

But dear Monz, have we not agreed that for his explanations to make better
sense, C, F and G should be sharps, or that the proslambanomenos start with
C according to our modern conception?

>
> >> Si uero his adiecero diatessaron .D.A: diuersa erit ab
> >> ea quae est .G.D. Vnum enim solum .G.D. consonantiae
> >> neruum continebit. id est .D. solum.
> >>
> >> But if I add the diatessaron [4th] DA [pitches E-B] to
> >> these, it will have little in common with GD [pitches a-E],
> >> for it will contain only one string of the consonance GD
> >> -- that is, only D [pitch E] alone. [i.e., it does not
> >> contain the F-E semitone]
>
>
> >> Excessit igitur .G.D. consonantiam. atque ideo
> >> diatessaron tres species habere perhibetur.
> >>
> >> Thus, DA [pitches E-B] has gone beyond the consonance
> >> GD [pitches a-E], and for that reason the diatessaron [4th]
> >> is said to have 3 species.
>
>
> >> Et in caeteris quidem consonantiis idem est.
> >>
> >> The same occurs in the other consonances.
>
>
>
> I'll complete this when i have time.
> That's enough to get started.
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
>
>
>
>

Still struggling to understand with all due excitement.

Cordially,
Ozan

🔗oyarman@ozanyarman.com

Hey there again!

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 16 Kasım 2005 Çarşamba 23:23
Subject: [tuning] Re: Boethius modal system, chapter 15 of Book 4

Can you see my grin? :)))

Let's beg to differ on this, shall we?

..............Execution by Westerners..........Execution by Easterners

Proslambanomenos..............la..........................la
Hypate hypaton......................si..........................si
Parhypate hypaton................do (natural)..........do#
Lichanos hypaton...................re.........................re
Hypate meson........................mi........................mi
Parhypate meson...................fa (natural)..........fa#
Lichanos meson....................sol........................sol (natural)
Mese........................................la.........................la
Paramese...............................si..........................si
Trite diezeugmenon...............do (natural).........do#
Paranete diezeugmenon.......re.........................re
Nete diezeugmenon..............mi.........................mi
Trite hyperbolaion..................fa (natural)...........fa#
Paranete hyperbolaion..........sol........................sol (natural)
Nete hyperbolaion..................la..........................la

He then goes on to say that transposing the diatonic scale within the octave
over to Sol (G), we will find the Arezzian scale:

G4 A B C D E F G5

But he then proceeds to ask himself: `why would Guido feel the need to add a
Gamma below if the existing Greek system was already major?`

Perhaps, the answer lies with the faulty Hucbald interpretation of Boethius.

Are you sure it was Hucbald? Yekta says it was done in Guido's time.

Please confer to my other reply for a detailed interpretation by myself.

He notated HYPOproslambanomenos with GAMMA according to Yekta. I know not if
this is true. What sayest thou?

Shape note?

Ach! You should have used octave designator numbers just once per octave!
Look at that clutter...

G according to Yekta
C according to Oz.

Ditto.

> >> Et in aliis quidem similis est in acumen intensionemque
> >> processus;
> >>
> >> The progression to higher pitch is similar in the
> >> other modes.
>
> So the whole system looks like this:
>

Hypodorian

[C3 D E F G A B c4 d e f g a b c'5]

[G3 A B C D E F g4 a b c d e f g'5]

Hypophrygian

[D3 E F G A B C d4 e f g a b c' d'5]

[A3 B C D E F g a4 b c d e f g' a'5]

Hypolydian

[E3 F G A B C d e4 f g a b c' d' e'5]

[B3 C D E F g a b4 c d e f g' a' b'5]

Dorian

[F3 G A B C d e f4 g a b c' d' e' f'5]

[C4 D E F g a b c5 d e f g' a' b' c'6]

Phrygian

[G3 A B C d e f g4 a b c' d' e' f' g'5]

[D4 E F g a b c d5 e f g' a' b' c' d'6]

Lydian

[A3 B C d e f g a4 b c' d' e' f' g' a'5]

[E4 F g a b c d e5 f g' a' b' c' d' e'6]

Mixolydian

[B3 C d e f g a b4 c' d' e' f' g' a' b'5]

[F4 g a b c d e f5 g' a' b' c' d' e' f'6]

G2 is just too low a pitch for Proslambanomenos, impossible for even tenors
to sing, while my C3 is just right.

SNIPPED the rest.

Cordially,
Ozan

🔗oyarman@ozanyarman.com

It's me again.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 16 Kasï¿½m 2005 ï¿½arï¿½amba 19:23
Subject: [tuning] Re: Boethius modal system, chapter 15 of Book 4

SNIP

> >
> > Please... this music does not belong solely to Arabs, but
> > to all nations within the cultural perimeters of the
> > Middle East, which is predominantly Islamic still. The
> > music you mention is more deserving of the cross-cultural
> > term `Maqam Music`. Also, the greatest contributors to
> > this genre were muslim theorists, some of whom were not
> > at all Arabs in origin. Furthermore, there were Byzantine
> > and Armenian scholars of high repute as well. The Islamic
> > Civilization made no discriminations regarding ethnicity
> > and benefitted equally from the collective wisdom of all.
>
>
> This just points out my ignorance of the whole subject
> of Maqam. I've dabbled in studying some Arab and Indian
> music-theory, and that's why i always use those cultural
> terms -- i don't want to include Turkish, Persian, and
> others without knowing whether or not i'm correct, and
> then find out that i did statement something wrong.
>

Oh, alright then. Still, we have common roots with the others in this
geography, it's only natural that we share our wisdow.

> I mostly skip past comments here in the Maqam threads.
> Much as i'd like to learn about it and watch the theory
> evolve, my head is just too full of other things right now.
>

Ach. Back to the drawing board then.

> However, please note that my studies of music in the
> geographical Middle-East go much farther back in history
> than the advent of Islam ... all the way to the Sumerians,
> which is as far as one can go.
>

Right.

> And what i see is that many ideas which i believe were
> part of Sumerian music and music-theory 5000 years ago
> have survived, and i believe i can trace their transmission
> thus (the time-scale on the left is very rough):
>
> years
> BC
> 3000 ..............._______ Sumerian _____
> .................. / ......... | .........\
> 2000 ........ Babylonian .. Egyptian ... Indian
> ............... / ...... \ ....____|_______/
> 1000 ........ Greek .... Persian . |
> ............. / ... \______ | ____/
> 0 AD .. Byzantine ....... Arab
>
>
> Apart from the obvious direct transmissions, i don't
> know how clearly i've illustrated the more complicated
> ones. Babylonian and Indian both lead to Persian.
> Greek, Persian, and Egyptian all lead to Arab.
> I'm sure that Persian is also supposed to connect
> to Byzantine.
>
> I don't know where Turkish and Armenian fit in -- feel
> free to add them to the diagram ... and also to correct
> anything i've done which you know to be wrong.
>
>

Akkadians, Assyrians, Hittites, Urartus and Scythians are missing from this
list, as well as Parthians and Turks. Assyria would be somewhere between
Sumer and Babylon. Hittites and Akkadians would be trade partners up north,
with the Urartus, Scythians and Parthians coming into the picture later on.

>
> > The 79 MOS 159-tET proposal of mine attempts to unite
> > all Maqam Music varieties throughout this geography in
> > just the way you said: Using schismic equivalances and
> > preserving commatic differences.
>
>
> I look forward to learning more about it as soon as i
> have time!
>

Glad to hear it!

> Also, it would be great to get you as a Tonescape tester,
> and for you to create Tonescape tuning files of all this
> stuff ... and also Tonescape pieces illustrating the
> tunings, which we can put up on our website.
>

That is a most interesting suggestion. I would like to give it a try, but
cannot promise anything in advance. Also, would you consider nominating the
notayaz group including Dr. Can Akkoc and others as beta-testers?

>
> > > If my interpretation of Boethius is correct, then this
> > > is probably evidence of cross-cultural influence between
> > > the ancient Greeks, Arabs, and Indians.
> > >
> >
> > And Turks, Armenians, Persians, Azeris, Afghanis...
>
>
> Right ... my deliberately leaving people out is because
> of my ignorance of the subject.
>
>

Agreed.

>
> > > But AFAIK no-one has pointed out this specific one before,
> > > and even Boethius sabotages our knowledge of it, because
> > > he presents the diatonic system in its ancient pythagorean
> > > tuning, and this schismatic equivalence only becomes apparent
> > > if one tunes the diatonic genus according to the ratios
> > > advocated by Didymus (c.100 AD).
> >
> >
> > Also the ratios propounded by Al-Farabi much after him.
> > Yet the general tendency in theory was, how shall I put it,
> > Boethian(?) in this geography...
>
>
> Absolutely. As the last great Roman scholar (and perhaps
> the greatest), Boethius's work commanded a tremendous prestige
> all during the medieval period, not only in Europe, but in
> all lands that had been part of the Roman Empire, which
> extended all the way to Mesopotamia and Northern Africa.
>
>
>

We must keep up this intellectual dialogue, we are really getting somewhere.

>
>
> > Yekta says that the Greater Perfect System of Pythagoras
> > was performed in this way:
> >
> > A B C# D E F# G a b c# d e f# g a
> >
> > by Eastern musicians. <snip> In a schismic temperament
> > where the above mentioned major gamut is the harmonic major
> > (Rast) scale extended to two octaves, I would feel at a
> > liberty to equate Greek relative-pitches with Maqam Music
> > main relative-pitches thusly:
> >
> > Hypoproslambanomenos _ Nerm Chargah
> > Proslambanomenos _ Yegah
> > Hypate hypaton _ Ashiran
> > Parhypate hypaton _ Araq
> > Lichanos hypaton _ Rast
> > Hypate meson _ Dugah
> > Parhypate meson _ Segah
> > Lichanos meson _ Chargah
> > Mese _ Neva
> > Paramese _ Huseyni
> > Trite diezeugmenon _ Evdj
> > Paranete diezeugmenon _ Gerdaniye
> > Nete diezeugmenon _ Muhayyer
> > Trite hyperbolaion _ Tiz Segah
> > Paranete hyperbolaion _ Tiz Chargah
> > Nete hyperbolaion _ Tiz Neva
>
>
> Wow, you certainly are fearless ... my quoting of a
> Latin text got me labelled as a "supercillious [sic] jerk",
> and now you're bold enough to post a Greek <-> Turkish
> glossary. (Good show!)
>
>

Once you turn into a 250 pound feline with razor sharp claws and iron
mandibles, you do become fearless! Or was that the other way around? LOL

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

Cordially,
Ozan

🔗monz <monz@tonalsoft.com>

Hi Oz,

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:

> > --- In tuning@yahoogroups.com, <oyarman@o...> wrote:

> > In order to preserve the intervals of the octave-species,
> > and rotate the modes thru all forms without using any
> > flats or sharps in the nomenclature, the reference Hypodorian
> > mode must use either G or C as the reference pitch.
> > The example Ozan gaving using "natural pitches" was
> > based on C as the reference, but i think G is more
> > likely, as will become clear later.
> >
>
>
> Let's beg to differ on this, shall we?

I ultimately came to the same conclusion you did:
that the lowest Boethian mode (hypodorian) would be
represented in modern nomenclature as beginning on "C".

I wanted to explain why, but my work on his chapter 14
was such a flurry of activity, and then i had to go out
and teach piano lessons ... and now i don't remember
why i came to that conclusion. :(

Hopefully it will come back to me, because i want to
explain my reasoning.

> > This is interesting, because i've read some other analyses
> > of the modal system where the reference pitch was F.
> > In particular, i think i recall that Kathleen Schlesinger
> > based the whole modal system on Proslambanomenos as a low F.
> > But using G as the reference brings Boethius's system
> > exactly into agreement with the standard early medieval
> > gamut beginning on a low-G notated as the letter gamma.
>
>
> If that be the case, the discovery belongs to Rauf Yekta.
> Please find his entry in French in the 1922 edition of
> the Lavignac Encyclopedia. There, under the article
> `Turquie`, not too far down, you will see that he
> explains:

Thanks very much for that, Oz, but ... while i see that
you commendably used strings of periods to circumvent
the stupid Yahoo space-deleting "feature" as i always do,
your spacing still didn't come out correctly when viewing
in a non-proportional font. Can you please redo the diagram?

> He then goes on to say that transposing the diatonic
> scale within the octave over to Sol (G), we will find
> the Arezzian scale:
>
> G4 A B C D E F G5

Just a semantic note: Western music-theorists always
use the adjective "Guidonian" when referring to the
work of Guido d'Arezzo. Do Turks normally use "Arezzian"?

> Whereby Yekta explains: rumor had it that Gui himself
> did not like the minor scale La Si Do Re Mi Fa Sol La and
> added a Sol below and called it the hypoproslambanomenos
> which he notated with the letter Gamma. He says that
> this is the origin of the term "gamut" and also the
> origin of the faulty execution of Westerners.

Yekta is wrong about that one. It was Hucbald (early 900's)
who transposed the Greek system down one whole-tone so
that the tetrachordal structure of tones and semitones
was, in ascending order, t-s-t, instead of the ancient
Greek order s-t-t.

> He comments that Alphonse Heegmann in his "Examen
> de la theorie musicale des Grecs" became almost aware
> of the situation where he explicated that Guido made the
> A-proslambanomenos LA which he notated between the first
> and second lines of an F-cleffed staff. Equating
> proslambanomenos with LA, according to Yekta, is the
> single greatest mistake made on this subject, because
> it was only necessary to equate proslambanomenos with
> SOL (G) for everything to fit together.

No, Guido was not the first theorist who called
proslambanomenos "A". That ocurred just a few decades
(or perhaps a century) earlier, in the _dialogus_ by
an anonymous author now usually called "pseudo-Odo",
because for a long time it was attributed to Odo of
Arezzo, who was often confused with Odo of Cluny.

http://www.music.indiana.edu/tml/9th-11th/ODODIA_TEXT.html

I just posted something about this recently ... it's
the first instance i'm aware of which used the Roman
alphabet as nomenclature for musical pitches.

I beg to differ with the esteemed Prof. Yekta that
this was a mistake. The use of "A" for proslambanomenos,
which is exactly one octave below mese, stems from
the fact that mese was considered by all ancient and
classical theorists to be the reference pitch, and "A"
was applied to mese, and thus in recognizing octave
equivalence, also became the name for proslambanomenos.

It totally makes sense to call the reference "A" if
you're going to employ the built-in ordering of the
alphabet to represent pitches.

The real deal is that ancient Greek theorists recognized
that some melodies required a whole-tone above mese, and
some required a semitone: thus, the Greater and Lesser
Perfect Systems, respectively.

When Western theorists tried to apply the Roman alphabet
as pitch nomenclature to the ancient Greek system, it
was only logical that they would call mese "A".

The fact that the step above mese could be either a
whole-tone or semitone was complication, which they
got around by using two differently-shaped "B's", which
eventually evolved into our sharp, natural, and flat
accidental symbols.

Anyway ... you can see that Boethius was already using
the Roman alphabet to designate pitches on his geometrical
diagrams. It was Hucbald who redesigned the Greek system
into one which fit into accordance with the medieval Frankish
modal system, and then pseduo-Odo who finally used the
Roman alphabet to designate pitches which recognized
octave-equivalence, which is still the system in use today.

> But he then proceeds to ask himself: `why would Guido
> feel the need to add a Gamma below if the existing Greek
> system was already major?`
>
> Perhaps, the answer lies with the faulty Hucbald
> interpretation of Boethius.

On the contrary, Hucbald's interpretation of Boethius
is not at all faulty -- it's actually brilliant.

In order to properly understand the evolution of
Western music-theory and notation, one must read the
source texts in proper chronological order. It's
very easy to look at these ancient texts with
modern eyes and see "mistakes".

One of the biggest stumbling-blocks is that it's
very difficult for us moderns, with our ingrained
sense of absolute pitch representation, to become
comfortable with the ancients's relative representation.

During the earliest part of the Frankish music-theory
tradition (c.800-1000), one can see how theorists
grappled with the difficulty of discussing musical
pitches without having any form of absolute representation.
See, for example, the treatise of Aurelian of Reome:

http://www.music.indiana.edu/tml/9th-11th/AURMD_TEXT.html

in which Aurelian had to use the beginnings of familiar
chant melodies to pin down the intervals he discussed.

This is the equivalent of saying something like: "OK,
think of the first 3 notes of 'Mary had a little lamb'.
Those first two intervals are whole-tones. That's what
we're talking about here." That's how complicated it was.

Hucbald's great achievement was to incorporate the
familiarity with the existing chant repertoire into
a synthesis with the ancient Greek *and* contemporary
Byzantine theoretical systems.

Hucbald reorganized the whole pitch structure so that
the finals of the 4 main modes agreed with the most
often used modes in the chant repertoire, and it was
easy for him to do that by simply moving the _hestotes_
(fixed bounding notes of each tetrachord) down by a
whole-tone.

(I remember posting something about this just within
the last couple of weeks, but Yahoo's search facility
is so shitty that i can't find it now. Good luck.)

> He also makes fun of the numerous mistakes that lead
> to this division between the East and the West,
> criticizes severely those accountable for this state
> of affairs and comments that even 5 volumes of the
> Lavignac encyclopedia would not suffice to review and
> analyze the entirety of the grave errors committed by
> those responsible. ROFL

Yes, well, he might be correct about that amount of
paper that could be wasted on writing about this stuff
... but as i've tried to point out, Hucbald's theory
was actually a brilliant way of bringing the illiterate
Frankish practice in line with established ancient Greek
theory.

> > This is fascinating, because while Boethius did use
> > Roman letters in his descriptions, they did not have
> > the same pitch-meaning as we use today, but rather were
> > simply labels for points along his arithmetic measurements.
> > The descriptions of the ancient Greeks were later labeled
> > with Roman letters using A as the name for Mese, and it
> > was Hucbald who shifted the whole system down a tone to G,
> > which brings his system in line with Boethius. Hmm...
>
>
> Are you sure it was Hucbald? Yekta says it was done in Guido's time.

Yekta is wrong. It was Hucbald who shifted the whole
ancient Greek system down by a whole-tone.

http://www.music.indiana.edu/tml/9th-11th/HUCHAR_TEXT.html

> > Since Boethius only gives a detailed explanation of
> > the first four modes in this chapter, and just dismisses
> > the rest of his explanation with "similarly for the rest",
> > it's impossible to know whether the reference pitch is
> > G or C, because both labels work for the first four modes.
> > I'd have to study the other chapters to see if this can
> > be resolved.
> >
>
>
> Please confer to my other reply for a detailed interpretation
> by myself.

I too have come to the conclusion that the lowest mode,
hypodorian, used C as a reference.

> > In fact this is a very intriguing topic, because the
> > _music enchiriadis_ and _scholia enchiriadis_ treatises,
> > which i think date from the early 800's, use the unique
> > daseain notation to specify where the semitone should
> > be placed, and it too leaves unresolved the question
> > of whether we would label the reference pitch as G or C.
> > Similiarly, the next really influential theorist after
> > Boethius was Guido d'Arezzo, and he too used a similar
> > procedure.
> >
>
>
> He notated HYPOproslambanomenos with GAMMA according to
> Yekta. I know not if this is true. What sayest thou?

http://tonalsoft.com/enc/g/gamma.aspx

(definition #3)

> > The later "fasola" system also simplifies modal rotation
> > to using a reference tetrachord containing two tones and
> > one semitone, and this became the basis of American
> > "shape-note" notation which results in just-intonation
> > singing.
>
>
> Shape note?

http://fasola.org/introduction/introduction.html

> Why? Mixo means part Greek, part barbaric. Literally
> Mixolydian would mean `Barbaric Lydian`. What better
> mode to ascribe the tonic diminished fifth? Besides,
> your dissonant fifth lies with Hypolydian. Why should
> that be more preferable? And look... here are your pitches:
>
>
> > Thus, my current interpretation of Boethius's modal
> > system equates it to modern pitch notation as follows:
> >
> > G2 A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 hypodorian
> > A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 hypophrygian
> > B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 hypolydian
> > C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 dorian
> > D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 phrygian
> > E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 lydian
> > F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 F5 mixolydian
>
>
> G2 is just too low a pitch for Proslambanomenos, impossible
> for even tenors to sing, while my C3 is just right.

As i just posted in another message, i've agreed that the
Lydian mode was based on A and not on E.

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

🔗monz <monz@tonalsoft.com>

Hi Oz,

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:

> > And what i see is that many ideas which i believe were
> > part of Sumerian music and music-theory 5000 years ago
> > have survived, and i believe i can trace their transmission
> > thus (the time-scale on the left is very rough):
> >
> > years
> > BC
> > 3000 ..............._______ Sumerian _____
> > .................. / ......... | .........\
> > 2000 ........ Babylonian .. Egyptian ... Indian
> > ............... / ...... \ ....____|_______/
> > 1000 ........ Greek .... Persian . |
> > ............. / ... \______ | ____/
> > 0 AD .. Byzantine ....... Arab
> >
> >
> > Apart from the obvious direct transmissions, i don't
> > know how clearly i've illustrated the more complicated
> > ones. Babylonian and Indian both lead to Persian.
> > Greek, Persian, and Egyptian all lead to Arab.
> > I'm sure that Persian is also supposed to connect
> > to Byzantine.
> >
> > I don't know where Turkish and Armenian fit in -- feel
> > free to add them to the diagram ... and also to correct
> > anything i've done which you know to be wrong.
> >
> >
>
>
> Akkadians, Assyrians, Hittites, Urartus and Scythians
> are missing from this list, as well as Parthians and Turks.
> Assyria would be somewhere between Sumer and Babylon.
> Hittites and Akkadians would be trade partners up north,
> with the Urartus, Scythians and Parthians coming into the
> picture later on.

"Akkadians" essentially means "Babylonians", at least
in the earlier periods. Beyond that ...

I'm one of those people who grasps things much better
when they're presented in visual format (hence, my
penchant for Lattice Diagrams). So please, by all means,
add the Assyrians, Hittites, Urartus, Scythians, Parthians,
and Turks to my diagram!

> > > The 79 MOS 159-tET proposal of mine attempts to unite
> > > all Maqam Music varieties throughout this geography in
> > > just the way you said: Using schismic equivalances and
> > > preserving commatic differences.
> >
> >
> > I look forward to learning more about it as soon as i
> > have time!
> >
>
>
> Glad to hear it!

The idea is that you become a Tonescape tester and
create files for all these tunings!

>
> > Also, it would be great to get you as a Tonescape tester,
> > and for you to create Tonescape tuning files of all this
> > stuff ... and also Tonescape pieces illustrating the
> > tunings, which we can put up on our website.
> >
>
>
> That is a most interesting suggestion. I would like to
> give it a try, but cannot promise anything in advance.
> Also, would you consider nominating the notayaz group
> including Dr. Can Akkoc and others as beta-testers?

I consider Dr. Akkoc to be a very good friend, and would
welcome his participation in my project!

> > > Also the ratios propounded by Al-Farabi much after him.
> > > Yet the general tendency in theory was, how shall I put it,
> > > Boethian(?) in this geography...
> >
> >
> > Absolutely. As the last great Roman scholar (and perhaps
> > the greatest), Boethius's work commanded a tremendous prestige
> > all during the medieval period, not only in Europe, but in
> > all lands that had been part of the Roman Empire, which
> > extended all the way to Mesopotamia and Northern Africa.
>
>
> We must keep up this intellectual dialogue, we are really
> getting somewhere.

Agreed! (most emphatically!)

I'm a bit surprised that Kriag Grady hasn't had anything
to say so far ... this stuff seems right up his alley.

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

🔗oyarman@ozanyarman.com

I really like the Oz way better when pursuing intellectual endeavours...

If you are truly looking for a cause,
As to why there is confusion and chaos
On all the tropes, tones and modes,
By which are written doleful odes,
Sang in lament by woeful hordes,
In miserably detuned hexachords,
And elegies long forgotten since
The times of Machiavelli's Prince,
Open up a treatise composed in latin,
Bound with rutilant gilded satin,
The text of which is hopelessly jumbled,
The theorist rather obviously fumbled!
Have no fear, though, help is near,
Just for a moment lend me your ear:
Sing a pitch and name it LA,
See how it resonates without a flaw?
Sound all such notes up to eight,
Reach now heaven's golden gate,
Each such segment makes a trope,
Evoking hatred, sorrow or hope,
Haven't you got it still you dope?
Acknowledged these even the pope!
Now move this over to natural tones,
And listen to how it rattles your bones!
Every such thing does make a mode,
Each now in its own abode.
These are what we `species` call,
Whose fixed pitches rise and fall,
Does it anymore a problem pose?
Resolved this matter, indeed, Oz!

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 17 Kasï¿½m 2005 Perï¿½embe 8:02
Subject: [tuning] Re: Boethius modal system, chapter 15 of Book 4

> Hi Oz,
>
>
> --- In tuning@yahoogroups.com, <oyarman@o...> wrote:
>
> > > --- In tuning@yahoogroups.com, <oyarman@o...> wrote:
>
> > > In order to preserve the intervals of the octave-species,
> > > and rotate the modes thru all forms without using any
> > > flats or sharps in the nomenclature, the reference Hypodorian
> > > mode must use either G or C as the reference pitch.
> > > The example Ozan gaving using "natural pitches" was
> > > based on C as the reference, but i think G is more
> > > likely, as will become clear later.
> > >
> >
> >
> > Let's beg to differ on this, shall we?
>
>
> I ultimately came to the same conclusion you did:
> that the lowest Boethian mode (hypodorian) would be
> represented in modern nomenclature as beginning on "C".
>
> I wanted to explain why, but my work on his chapter 14
> was such a flurry of activity, and then i had to go out
> and teach piano lessons ... and now i don't remember
> why i came to that conclusion. :(
>
> Hopefully it will come back to me, because i want to
> explain my reasoning.
>
>

Ah! My grin is even wider now. :))))) Is it early to hope that you will at
last give my poor self a reference in your tonal encyclopedia now that the
issue of Boethian modes are resolved to our satisfaction?

> >
> > If that be the case, the discovery belongs to Rauf Yekta.
> > Please find his entry in French in the 1922 edition of
> > the Lavignac Encyclopedia. There, under the article
> > `Turquie`, not too far down, you will see that he
> > explains:
>
>
> Thanks very much for that, Oz, but ... while i see that
> you commendably used strings of periods to circumvent
> the stupid Yahoo space-deleting "feature" as i always do,
> your spacing still didn't come out correctly when viewing
> in a non-proportional font. Can you please redo the diagram?
>

Harumph... ok let's try this one more time:

..............Execution by Westerners..........Execution by Easterners

Proslambanomenos............la..........................la
Hypate hypaton....................si..........................si
Parhypate hypaton..............do (natural)..........do#
Lichanos hypaton.................re.........................re
Hypate meson......................mi........................mi
Parhypate meson.................fa (natural)..........fa#
Lichanos meson..................sol........................sol (natural)
Mese......................................la.........................la
Paramese.............................si..........................si
Trite diezeugmenon.............do (natural).........do#
Paranete diezeugmenon.....re.........................re
Nete diezeugmenon............mi.........................mi
Trite hyperbolaion................fa (natural)...........fa#
Paranete hyperbolaion........sol........................sol (natural)
Nete hyperbolaion................la..........................la

>
> > He then goes on to say that transposing the diatonic
> > scale within the octave over to Sol (G), we will find
> > the Arezzian scale:
> >
> > G4 A B C D E F G5
>
>
> Just a semantic note: Western music-theorists always
> use the adjective "Guidonian" when referring to the
> work of Guido d'Arezzo. Do Turks normally use "Arezzian"?
>
>

The only Turk I know of who did that is myself. In truth, it should have
been more properly spelled `d'Arezzian`.

>
>
> > Whereby Yekta explains: rumor had it that Gui himself
> > did not like the minor scale La Si Do Re Mi Fa Sol La and
> > added a Sol below and called it the hypoproslambanomenos
> > which he notated with the letter Gamma. He says that
> > this is the origin of the term "gamut" and also the
> > origin of the faulty execution of Westerners.
>
>
> Yekta is wrong about that one. It was Hucbald (early 900's)
> who transposed the Greek system down one whole-tone so
> that the tetrachordal structure of tones and semitones
> was, in ascending order, t-s-t, instead of the ancient
> Greek order s-t-t.
>
>

Ah, now I see that he quoted J. J. Rousseau in his `Dictionnaire de
Musique`: Hypoproslambanomenos. Not a very convincing reference if I do say
so myself, but that at least clears his name.

>
> > He comments that Alphonse Heegmann in his "Examen
> > de la theorie musicale des Grecs" became almost aware
> > of the situation where he explicated that Guido made the
> > A-proslambanomenos LA which he notated between the first
> > and second lines of an F-cleffed staff. Equating
> > proslambanomenos with LA, according to Yekta, is the
> > single greatest mistake made on this subject, because
> > it was only necessary to equate proslambanomenos with
> > SOL (G) for everything to fit together.
>
>
> No, Guido was not the first theorist who called
> proslambanomenos "A". That ocurred just a few decades
> (or perhaps a century) earlier, in the _dialogus_ by
> an anonymous author now usually called "pseudo-Odo",
> because for a long time it was attributed to Odo of
> Arezzo, who was often confused with Odo of Cluny.
>
> http://www.music.indiana.edu/tml/9th-11th/ODODIA_TEXT.html
>
>
> I just posted something about this recently ... it's
> the first instance i'm aware of which used the Roman
> alphabet as nomenclature for musical pitches.
>
>
> I beg to differ with the esteemed Prof. Yekta that
> this was a mistake. The use of "A" for proslambanomenos,
> which is exactly one octave below mese, stems from
> the fact that mese was considered by all ancient and
> classical theorists to be the reference pitch, and "A"
> was applied to mese, and thus in recognizing octave
> equivalence, also became the name for proslambanomenos.
>

He did not have an academical title in all his life, so that would make him
Rauf Yekta plainly. But he is not a man to be trifled with, even in his
grave. You've been warned. ;)

As for your arguments, I concur wholeheartedly. But know that Maqam Music
notation today is based on the interpretations of the Greek system by Yekta.
Prove its falsity, and its foundations will crumble to dust.

> It totally makes sense to call the reference "A" if
> you're going to employ the built-in ordering of the
> alphabet to represent pitches.
>
>

I concur.

> The real deal is that ancient Greek theorists recognized
> that some melodies required a whole-tone above mese, and
> some required a semitone: thus, the Greater and Lesser
> Perfect Systems, respectively.
>

Agreed.

> When Western theorists tried to apply the Roman alphabet
> as pitch nomenclature to the ancient Greek system, it
> was only logical that they would call mese "A".
>

Agreed.

> The fact that the step above mese could be either a
> whole-tone or semitone was complication, which they
> got around by using two differently-shaped "B's", which
> eventually evolved into our sharp, natural, and flat
> accidental symbols.
>

Affirmative.

> Anyway ... you can see that Boethius was already using
> the Roman alphabet to designate pitches on his geometrical
> diagrams. It was Hucbald who redesigned the Greek system
> into one which fit into accordance with the medieval Frankish
> modal system, and then pseduo-Odo who finally used the
> Roman alphabet to designate pitches which recognized
> octave-equivalence, which is still the system in use today.
>

So Boethius used letters A thru O?

>
> > But he then proceeds to ask himself: `why would Guido
> > feel the need to add a Gamma below if the existing Greek
> > system was already major?`
> >
> > Perhaps, the answer lies with the faulty Hucbald
> > interpretation of Boethius.
>
>
> On the contrary, Hucbald's interpretation of Boethius
> is not at all faulty -- it's actually brilliant.
>

Is there not rumors that the Carolingian clergy mistranslated his latin?

> In order to properly understand the evolution of
> Western music-theory and notation, one must read the
> source texts in proper chronological order. It's
> very easy to look at these ancient texts with
> modern eyes and see "mistakes".
>

I see `mistakes` committed in the name of modernity. I hate to be seen in
alliance with them, especially after elucidating the matter of Boethian
modes based on the translations and comments you were kind enough to
provide.

> One of the biggest stumbling-blocks is that it's
> very difficult for us moderns, with our ingrained
> sense of absolute pitch representation, to become
> comfortable with the ancients's relative representation.
>

It is not if you are accustomed to Ney Ahenks, or Key Transposition.

> During the earliest part of the Frankish music-theory
> tradition (c.800-1000), one can see how theorists
> grappled with the difficulty of discussing musical
> pitches without having any form of absolute representation.
> See, for example, the treatise of Aurelian of Reome:
>
> http://www.music.indiana.edu/tml/9th-11th/AURMD_TEXT.html
>

I have no means of understanding latin unfortunately.

> in which Aurelian had to use the beginnings of familiar
> chant melodies to pin down the intervals he discussed.
>
> This is the equivalent of saying something like: "OK,
> think of the first 3 notes of 'Mary had a little lamb'.
> Those first two intervals are whole-tones. That's what
> we're talking about here." That's how complicated it was.
>

Ooo the misery!

> Hucbald's great achievement was to incorporate the
> familiarity with the existing chant repertoire into
> a synthesis with the ancient Greek *and* contemporary
> Byzantine theoretical systems.
>

Namely, the octoechos, which eventually had common links to Boethian modes,
no?

> Hucbald reorganized the whole pitch structure so that
> the finals of the 4 main modes agreed with the most
> often used modes in the chant repertoire, and it was
> easy for him to do that by simply moving the _hestotes_
> (fixed bounding notes of each tetrachord) down by a
> whole-tone.
>

Can you provide a comparison chart here showing the shift?

> (I remember posting something about this just within
> the last couple of weeks, but Yahoo's search facility
> is so shitty that i can't find it now. Good luck.)
>
>

Nooooo, it's cruel of you to leave me out in the dark now. I trust you will
be most efficient if I kindly ask?

> > He also makes fun of the numerous mistakes that lead
> > to this division between the East and the West,
> > criticizes severely those accountable for this state
> > of affairs and comments that even 5 volumes of the
> > Lavignac encyclopedia would not suffice to review and
> > analyze the entirety of the grave errors committed by
> > those responsible. ROFL
>
>
> Yes, well, he might be correct about that amount of
> paper that could be wasted on writing about this stuff
> ... but as i've tried to point out, Hucbald's theory
> was actually a brilliant way of bringing the illiterate
> Frankish practice in line with established ancient Greek
> theory.
>
>

We had a similar pitch transformation during the reign of the Ottoman Sultan
Murad II spearheaded by Khizir bin Abdullah c. 1400. Much confusion has
resulted and still persists due to the change.

>
> > > This is fascinating, because while Boethius did use
> > > Roman letters in his descriptions, they did not have
> > > the same pitch-meaning as we use today, but rather were
> > > simply labels for points along his arithmetic measurements.
> > > The descriptions of the ancient Greeks were later labeled
> > > with Roman letters using A as the name for Mese, and it
> > > was Hucbald who shifted the whole system down a tone to G,
> > > which brings his system in line with Boethius. Hmm...
> >
> >
> > Are you sure it was Hucbald? Yekta says it was done in Guido's time.
>
>
> Yekta is wrong. It was Hucbald who shifted the whole
> ancient Greek system down by a whole-tone.
>
> http://www.music.indiana.edu/tml/9th-11th/HUCHAR_TEXT.html
>
>

More translation is required for me to be able to follow... may I humbly
request a little more of your valuable time on this matter? It is truly
enchanting...

> > > Since Boethius only gives a detailed explanation of
> > > the first four modes in this chapter, and just dismisses
> > > the rest of his explanation with "similarly for the rest",
> > > it's impossible to know whether the reference pitch is
> > > G or C, because both labels work for the first four modes.
> > > I'd have to study the other chapters to see if this can
> > > be resolved.
> > >
> >
> >
> > Please confer to my other reply for a detailed interpretation
> > by myself.
>
>
>
> I too have come to the conclusion that the lowest mode,
> hypodorian, used C as a reference.
>
>

Glad to hear it. It is gratifying to know I'm of some use around here.

>
> > > In fact this is a very intriguing topic, because the
> > > _music enchiriadis_ and _scholia enchiriadis_ treatises,
> > > which i think date from the early 800's, use the unique
> > > daseain notation to specify where the semitone should
> > > be placed, and it too leaves unresolved the question
> > > of whether we would label the reference pitch as G or C.
> > > Similiarly, the next really influential theorist after
> > > Boethius was Guido d'Arezzo, and he too used a similar
> > > procedure.
> > >
> >
> >
> > He notated HYPOproslambanomenos with GAMMA according to
> > Yekta. I know not if this is true. What sayest thou?
>
>
> http://tonalsoft.com/enc/g/gamma.aspx
>
> (definition #3)
>

So he is right after all.

>
> > > The later "fasola" system also simplifies modal rotation
> > > to using a reference tetrachord containing two tones and
> > > one semitone, and this became the basis of American
> > > "shape-note" notation which results in just-intonation
> > > singing.
> >
> >
> > Shape note?
>
>
> http://fasola.org/introduction/introduction.html
>
>

Interesting!

>
> > Why? Mixo means part Greek, part barbaric. Literally
> > Mixolydian would mean `Barbaric Lydian`. What better
> > mode to ascribe the tonic diminished fifth? Besides,
> > your dissonant fifth lies with Hypolydian. Why should
> > that be more preferable? And look... here are your pitches:
> >
> >
> > > Thus, my current interpretation of Boethius's modal
> > > system equates it to modern pitch notation as follows:
> > >
> > > G2 A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 hypodorian
> > > A2 B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 hypophrygian
> > > B2 C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 hypolydian
> > > C3 D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 dorian
> > > D3 E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 phrygian
> > > E3 F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 lydian
> > > F3 G3 A3 B3 C4 D4 E4 F4 G4 A4 B4 C5 D5 E5 F5 mixolydian
> >
> >
> > G2 is just too low a pitch for Proslambanomenos, impossible
> > for even tenors to sing, while my C3 is just right.
>
>
> As i just posted in another message, i've agreed that the
> Lydian mode was based on A and not on E.
>
>
>

So, what about the correction on the encyclopedia entry?

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

Cordially,
Ozan

🔗monz <monz@tonalsoft.com>

Hi Ozan,

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:

> > [monz]
> > I ultimately came to the same conclusion you did:
> > that the lowest Boethian mode (hypodorian) would be
> > represented in modern nomenclature as beginning on "C".
>
> Ah! My grin is even wider now. :))))) Is it early to
> hope that you will at last give my poor self a reference
> in your tonal encyclopedia now that the issue of
> Boethian modes are resolved to our satisfaction?

Not so fast!

After examining chapters 16 and 17 of Boethius's Book 4,
it turns out that my interpretation of his modal system,
as i had first presented it when we started this thread,
is correct after all.

That is, the modes are simply transpositions and not
species rotations. This is in direct contradiction to
what Boethius writes in his text, but his explanation
of the diagram leaves no doubt that i'm correct.

I'm going to respond to your comments in this post, and
then write another separate one showing the entire modal
system.

> > Yekta is wrong about that one. It was Hucbald (early 900's)
> > who transposed the Greek system down one whole-tone so
> > that the tetrachordal structure of tones and semitones
> > was, in ascending order, t-s-t, instead of the ancient
> > Greek order s-t-t.
>
>
> Ah, now I see that he quoted J. J. Rousseau in his
> `Dictionnaire de Musique`: Hypoproslambanomenos. Not
> a very convincing reference if I do say so myself,
> but that at least clears his name.

So then it was Rousseau who mistakenly thought that
Guido shifted the tonal system down by a tone? Or did
he say nothing about the shift, and just claimed that
Guido added the low-G "gamma"?

Anyway, the gamma came into the picture with the
pseudo-Odo _dialogus_, which i mentioned here:

> > No, Guido was not the first theorist who called
> > proslambanomenos "A". That ocurred just a few decades
> > (or perhaps a century) earlier, in the _dialogus_ by
> > an anonymous author now usually called "pseudo-Odo",
> > because for a long time it was attributed to Odo of
> > Arezzo, who was often confused with Odo of Cluny.
> >
> > http://www.music.indiana.edu/tml/9th-11th/ODODIA_TEXT.html
> >
> >
> > I just posted something about this recently ... it's
> > the first instance i'm aware of which used the Roman
> > alphabet as nomenclature for musical pitches.

The whole system here was based on G as the reference
pitch ... and as i said, shifting the entire system
down so that the intervallic structure fit the natural
notes with G as a reference, was the work of Hucbald,
in the early 900's -- but of course Hucbald was not
yet using the Roman letters.

The pseudo-Odo _dialogus_ was written later in the
same century, or perhaps in the early 1000's. Guido's
treatise came along at around 1050.

> As for your arguments, I concur wholeheartedly.
> But know that Maqam Music notation today is based
> on the interpretations of the Greek system by Yekta.
> Prove its falsity, and its foundations will crumble
> to dust.

But aren't you planning to crumble the current Maqam
notation to dust yourself anyway, with your new notation?

> > Anyway ... you can see that Boethius was already using
> > the Roman alphabet to designate pitches on his geometrical
> > diagrams. It was Hucbald who redesigned the Greek system
> > into one which fit into accordance with the medieval Frankish
> > modal system, and then pseduo-Odo who finally used the
> > Roman alphabet to designate pitches which recognized
> > octave-equivalence, which is still the system in use today.
> >
>
>
> So Boethius used letters A thru O?

Boethius used lots of different lettering schemes in
many different places in his treatise on music. None
of them have anything to do with our modern system
which recognizes octave-equivalence. They're all just
ways of labeling geometrical points along a string-length.

Only in certain instances is there consistency of
using the same letter labels from one diagram or
description to the next. Most of the time they have
nothing to do with each other and refer only to one
particular diagram or description.

> > On the contrary, Hucbald's interpretation of Boethius
> > is not at all faulty -- it's actually brilliant.
> >
>
>
> Is there not rumors that the Carolingian clergy
> mistranslated his latin?

Not sure what you mean by that. Hucbald was a monk
during the early 900's, so he *was* part of the
Carolingian clergy! Boethius, Hucbald, and the
rest of the Carolingian clergy all wrote in basically
the same post-Classical Latin, so there's no translation
necessary between any of them.

What *is* true is that Boethius had vast and extensive
knowledge of ancient Greek learning (musical and otherwise),
but the medieval scribes who copied his treatise on music
into the 127 existing manuscripts did *not*. So they
made many, many mistakes. And once an error found its
way into one manuscript, any of three things could
happen to it upon subsequent manuscript copying:

1) an erudite monk who realized it was an error would
correct it ... this is the rarest case;

2) the error would not be recognized and would be copied
into subsequent manuscripts and propagated further as
more copies were made from *those* copies ... this is
the most common case;

3) a copyist would be so confused upon recognizing the
error that he would try to correct it -- with another,
sometimes worse, error! ... this happened more often
than you might expect.

The medieval scribes were dealing with descriptions of
a musical practice that had vanished centuries before,
and of course they didn't know how to read Greek!
("it's all Greek to me")

Even after decades of work culminating in the publication
of his translation in 1989, and despite the incredible
amount of research that he has done on Boethius's treatise,
even Bower missed a few of these "manuscript tradition
errors", or fixed them incorrectly in a few cases. He is
not to be faulted for that -- by focusing so intensively
on only Boethius's treatise, he sometimes doesn't see
the context into which it fits. (But i will still
reiterate the Bower did a phenomenal job!)

The only reason i've noticed these few mistakes which
slipped by Bower, is because i've read so much of both
the ancient Greek theory upon which Boethius's work
is based, and also so many of the Frankish treatises
which in turn were based on Boethius.

> I see `mistakes` committed in the name of modernity.
> I hate to be seen in alliance with them, especially after
> elucidating the matter of Boethian modes based on the
> translations and comments you were kind enough to
> provide.

And as i said at the beginning of this post, more elucidation
is on the way! Your work is not finished yet. ;-)

> Namely, the octoechos, which eventually had common
> links to Boethian modes, no?

Yes. It was Hucbald who created the link between the
Byzantine octoechos and the Boethian modes. BTW, Boethius
based his description of the modal system on Ptolemy.

> > Hucbald reorganized the whole pitch structure so that
> > the finals of the 4 main modes agreed with the most
> > often used modes in the chant repertoire, and it was
> > easy for him to do that by simply moving the _hestotes_
> > (fixed bounding notes of each tetrachord) down by a
> > whole-tone.
>
>
> Can you provide a comparison chart here showing the shift?

The essential feature of Hucbald's change is that
the ascending intervallic structure of each tetrachord
became t-s-t as in the pitches D-E-F-G, which were
the finals of the four main Frankish modes.

.............................................. Greek .. Hucbald

/ nete hyperbolaion .............................. a' .. g
|
| paranete hyperbolaion .......................... g ... f
|
| trite hyperbolaion ............................. f ... e
+ nete diezeugmenon .............................. e ... d
|
| paranete diezeugmenon . / nete synemmenon ...... d ... c
| ....................... |
| trite diezeugmenon .... | ...................... c ... b
| ....................... | paranete synemmenon .. c ... bb
\ paramese .............. | ...................... b ... a
......................... | trite synemmenon ..... bb .. a
/ mese ...................\ mese ................. a ... G
|
| lichanos meson ................................. G ... F
|
| parhypate meson ................................ F ... E
+ hypate meson ................................... E ... D
|
| lichanos hypaton ............................... D ... C
|
| parhypate hypaton .............................. C ... B
\ hypate hypaton ................................. B ... A

.. proslambanomenos ............................... A ... G'

> > (I remember posting something about this just within
> > the last couple of weeks, but Yahoo's search facility
> > is so shitty that i can't find it now. Good luck.)
>
>
> Nooooo, it's cruel of you to leave me out in the dark
> now. I trust you will be most efficient if I kindly ask?

I want to write my next post showing the correct diagram
of Boethius's modal system. After that, i'll try to find
my recent posts on Hucbald.

> > Yekta is wrong. It was Hucbald who shifted the whole
> > ancient Greek system down by a whole-tone.
> >
> > http://www.music.indiana.edu/tml/9th-11th/HUCHAR_TEXT.html
> >
> >
>
>
> More translation is required for me to be able to follow...
> may I humbly request a little more of your valuable time
> on this matter? It is truly enchanting...

I won't be able to spend any time translating Hucbald's
actual treatise ... my if i can find my recent posts on
the subject, they'll explain a lot.

> > I too have come to the conclusion that the lowest mode,
> > hypodorian, used C as a reference.
>
>
> Glad to hear it. It is gratifying to know I'm of some
> use around here.

But that's not the end of the story!
My next post will explain ...

> > > > <snip> ... the next really influential theorist after
> > > > Boethius was Guido d'Arezzo, and he too used a similar
> > > > procedure.
> > >
> > >
> > > He notated HYPOproslambanomenos with GAMMA according to
> > > Yekta. I know not if this is true. What sayest thou?
> >
> >
> > http://tonalsoft.com/enc/g/gamma.aspx
> >
> > (definition #3)
>
>
> So he is right after all.

Yekta is not correct that Guido introduced the gamma.
As far as i can tell, the pseduo-Odo _dialogus_ was the
first treatise to use both the Roman letters in their
modern octave-equivalent form, and the gamma for the
lowest G.

> > > > The later "fasola" system also simplifies modal rotation
> > > > to using a reference tetrachord containing two tones and
> > > > one semitone, and this became the basis of American
> > > > "shape-note" notation which results in just-intonation
> > > > singing.
> > >
> > >
> > > Shape note?
> >
> >
> > http://fasola.org/introduction/introduction.html
> >
> >
>
>
> Interesting!

As noted on the homepage of http://fasola.org,
the recent movie "Cold Mountain" featured shape-note
(also called "sacred harp") singing. I recommend that
you check it out.

http://www.sfgate.com/cgi-bin/article.cgi?file=/chronicle/archive/2004/07/02/PNGOP7D6DA1.DTL

http://www.cnn.com/2004/SHOWBIZ/Music/02/14/sacred.harp.revival.ap/

> > As i just posted in another message, i've agreed that the
> > Lydian mode was based on A and not on E.
>
>
> So, what about the correction on the encyclopedia entry?

Which entry is that?

Anyway, let's have more discussion after my next post
(showing the correct diagram for the modal system), before
we agree on changes to the Encyclopedia.

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

🔗oyarman@ozanyarman.com

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 17 Kasï¿½m 2005 Perï¿½embe 17:20
Subject: [tuning] Re: Boethius modal system, chapter 15 of Book 4

> Hi Ozan,
>
>
> --- In tuning@yahoogroups.com, <oyarman@o...> wrote:
>
> > > [monz]
> > > I ultimately came to the same conclusion you did:
> > > that the lowest Boethian mode (hypodorian) would be
> > > represented in modern nomenclature as beginning on "C".
> >
> > Ah! My grin is even wider now. :))))) Is it early to
> > hope that you will at last give my poor self a reference
> > in your tonal encyclopedia now that the issue of
> > Boethian modes are resolved to our satisfaction?
>
>
> Not so fast!
>
> After examining chapters 16 and 17 of Boethius's Book 4,
> it turns out that my interpretation of his modal system,
> as i had first presented it when we started this thread,
> is correct after all.
>
> That is, the modes are simply transpositions and not
> species rotations. This is in direct contradiction to
> what Boethius writes in his text, but his explanation
> of the diagram leaves no doubt that i'm correct.
>

Oh no you don't... The scheme I delineated was in perfect alignment with his
descriptions up to this point and rightfully prepares the basis of Medieval
Church modes the way we understand them. Does it make sense that Boethius
would permit such a blatant contradiction? You yourself say that the scribes
and copyists have transmitted fatal mistakes. If the signal-to-noise ratio
of historical transmission is so high, how can you rely on one portion of
the book and dismiss the other portion?

> I'm going to respond to your comments in this post, and
> then write another separate one showing the entire modal
> system.
>
>
>

Still waiting, still eager...

>
> > > Yekta is wrong about that one. It was Hucbald (early 900's)
> > > who transposed the Greek system down one whole-tone so
> > > that the tetrachordal structure of tones and semitones
> > > was, in ascending order, t-s-t, instead of the ancient
> > > Greek order s-t-t.
> >
> >
> > Ah, now I see that he quoted J. J. Rousseau in his
> > `Dictionnaire de Musique`: Hypoproslambanomenos. Not
> > a very convincing reference if I do say so myself,
> > but that at least clears his name.
>
>
> So then it was Rousseau who mistakenly thought that
> Guido shifted the tonal system down by a tone? Or did
> he say nothing about the shift, and just claimed that
> Guido added the low-G "gamma"?
>

He claims that Guido did both.

> Anyway, the gamma came into the picture with the
> pseudo-Odo _dialogus_, which i mentioned here:
>
>
> > > No, Guido was not the first theorist who called
> > > proslambanomenos "A". That ocurred just a few decades
> > > (or perhaps a century) earlier, in the _dialogus_ by
> > > an anonymous author now usually called "pseudo-Odo",
> > > because for a long time it was attributed to Odo of
> > > Arezzo, who was often confused with Odo of Cluny.
> > >
> > > http://www.music.indiana.edu/tml/9th-11th/ODODIA_TEXT.html
> > >
> > >
> > > I just posted something about this recently ... it's
> > > the first instance i'm aware of which used the Roman
> > > alphabet as nomenclature for musical pitches.
>
>
> The whole system here was based on G as the reference
> pitch ... and as i said, shifting the entire system
> down so that the intervallic structure fit the natural
> notes with G as a reference, was the work of Hucbald,
> in the early 900's -- but of course Hucbald was not
> yet using the Roman letters.
>

I see.

> The pseudo-Odo _dialogus_ was written later in the
> same century, or perhaps in the early 1000's. Guido's
> treatise came along at around 1050.
>
>

So, Hucbald is responsible for the gestalt shift.

>
> > As for your arguments, I concur wholeheartedly.
> > But know that Maqam Music notation today is based
> > on the interpretations of the Greek system by Yekta.
> > Prove its falsity, and its foundations will crumble
> > to dust.
>
>
> But aren't you planning to crumble the current Maqam
> notation to dust yourself anyway, with your new notation?
>

While preserving the notational signs, yes...

>
>
> > > Anyway ... you can see that Boethius was already using
> > > the Roman alphabet to designate pitches on his geometrical
> > > diagrams. It was Hucbald who redesigned the Greek system
> > > into one which fit into accordance with the medieval Frankish
> > > modal system, and then pseduo-Odo who finally used the
> > > Roman alphabet to designate pitches which recognized
> > > octave-equivalence, which is still the system in use today.
> > >
> >
> >
> > So Boethius used letters A thru O?
>
>
>
> Boethius used lots of different lettering schemes in
> many different places in his treatise on music. None
> of them have anything to do with our modern system
> which recognizes octave-equivalence. They're all just
> ways of labeling geometrical points along a string-length.
>

Reminds me of the Abjed system of Kindi and Safi Al-din much after him.

> Only in certain instances is there consistency of
> using the same letter labels from one diagram or
> description to the next. Most of the time they have
> nothing to do with each other and refer only to one
> particular diagram or description.
>
>

Which is reason enough not to trust our conclusions just yet.

>
> > > On the contrary, Hucbald's interpretation of Boethius
> > > is not at all faulty -- it's actually brilliant.
> > >
> >
> >
> > Is there not rumors that the Carolingian clergy
> > mistranslated his latin?
>
>
> Not sure what you mean by that. Hucbald was a monk
> during the early 900's, so he *was* part of the
> Carolingian clergy! Boethius, Hucbald, and the
> rest of the Carolingian clergy all wrote in basically
> the same post-Classical Latin, so there's no translation
> necessary between any of them.
>

Don't shoot the messenger! So, contrary to what others today claim, there is
no misinterpretation of the text of Boethius's treatise?

>
> What *is* true is that Boethius had vast and extensive
> knowledge of ancient Greek learning (musical and otherwise),
> but the medieval scribes who copied his treatise on music
> into the 127 existing manuscripts did *not*. So they
> made many, many mistakes. And once an error found its
> way into one manuscript, any of three things could
> happen to it upon subsequent manuscript copying:
>
> 1) an erudite monk who realized it was an error would
> correct it ... this is the rarest case;
>
> 2) the error would not be recognized and would be copied
> into subsequent manuscripts and propagated further as
> more copies were made from *those* copies ... this is
> the most common case;
>
> 3) a copyist would be so confused upon recognizing the
> error that he would try to correct it -- with another,
> sometimes worse, error! ... this happened more often
> than you might expect.
>
> The medieval scribes were dealing with descriptions of
> a musical practice that had vanished centuries before,
> and of course they didn't know how to read Greek!
> ("it's all Greek to me")
>

These are alarming news, and requires that we exercize further caution
before we reach spontaneous conclusions.

>
> Even after decades of work culminating in the publication
> of his translation in 1989, and despite the incredible
> amount of research that he has done on Boethius's treatise,
> even Bower missed a few of these "manuscript tradition
> errors", or fixed them incorrectly in a few cases. He is
> not to be faulted for that -- by focusing so intensively
> on only Boethius's treatise, he sometimes doesn't see
> the context into which it fits. (But i will still
> reiterate the Bower did a phenomenal job!)
>

We appreciate it both.

> The only reason i've noticed these few mistakes which
> slipped by Bower, is because i've read so much of both
> the ancient Greek theory upon which Boethius's work
> is based, and also so many of the Frankish treatises
> which in turn were based on Boethius.
>
>

It's fortunate that I am priviledged to communicate with such an erudite
researcher as yourself.

> > I see `mistakes` committed in the name of modernity.
> > I hate to be seen in alliance with them, especially after
> > elucidating the matter of Boethian modes based on the
> > translations and comments you were kind enough to
> > provide.
>
>
> And as i said at the beginning of this post, more elucidation
> is on the way! Your work is not finished yet. ;-)
>

You mean, OUR work, don't you? ;) I suggest we collaborate on an article
entitled: `Modes of Boethius and Carolingians`. That would be an interesting
topic, don't you think?

>
> > Namely, the octoechos, which eventually had common
> > links to Boethian modes, no?
>
>
> Yes. It was Hucbald who created the link between the
> Byzantine octoechos and the Boethian modes. BTW, Boethius
> based his description of the modal system on Ptolemy.
>
>

Then the article in question should start with Ptolemy, move on to Didymus
and then Boethius and Carolingians.

>
> > > Hucbald reorganized the whole pitch structure so that
> > > the finals of the 4 main modes agreed with the most
> > > often used modes in the chant repertoire, and it was
> > > easy for him to do that by simply moving the _hestotes_
> > > (fixed bounding notes of each tetrachord) down by a
> > > whole-tone.
> >
> >
> > Can you provide a comparison chart here showing the shift?
>
>
> The essential feature of Hucbald's change is that
> the ascending intervallic structure of each tetrachord
> became t-s-t as in the pitches D-E-F-G, which were
> the finals of the four main Frankish modes.
>
>
> .............................................. Greek .. Hucbald
>
> / nete hyperbolaion .............................. a' .. g
> |
> | paranete hyperbolaion .......................... g ... f
> |
> | trite hyperbolaion ............................. f ... e
> + nete diezeugmenon .............................. e ... d
> |
> | paranete diezeugmenon . / nete synemmenon ...... d ... c
> | ....................... |
> | trite diezeugmenon .... | ...................... c ... b
> | ....................... | paranete synemmenon .. c ... bb
> \ paramese .............. | ...................... b ... a
> ......................... | trite synemmenon ..... bb .. a
> / mese ...................\ mese ................. a ... G
> |
> | lichanos meson ................................. G ... F
> |
> | parhypate meson ................................ F ... E
> + hypate meson ................................... E ... D
> |
> | lichanos hypaton ............................... D ... C
> |
> | parhypate hypaton .............................. C ... B
> \ hypate hypaton ................................. B ... A
>
> .. proslambanomenos ............................... A ... G'
>
>

I'm not sure if the table came out correctly. But is not G supposed to start
on HYPOproslambanomenos?

>
> > > (I remember posting something about this just within
> > > the last couple of weeks, but Yahoo's search facility
> > > is so shitty that i can't find it now. Good luck.)
> >
> >
> > Nooooo, it's cruel of you to leave me out in the dark
> > now. I trust you will be most efficient if I kindly ask?
>
>
> I want to write my next post showing the correct diagram
> of Boethius's modal system. After that, i'll try to find
> my recent posts on Hucbald.
>
>

Ok.

> > > Yekta is wrong. It was Hucbald who shifted the whole
> > > ancient Greek system down by a whole-tone.
> > >
> > > http://www.music.indiana.edu/tml/9th-11th/HUCHAR_TEXT.html
> > >
> > >
> >
> >
> > More translation is required for me to be able to follow...
> > may I humbly request a little more of your valuable time
> > on this matter? It is truly enchanting...
>
>
> I won't be able to spend any time translating Hucbald's
> actual treatise ... my if i can find my recent posts on
> the subject, they'll explain a lot.
>
>

Ok.

> > > I too have come to the conclusion that the lowest mode,
> > > hypodorian, used C as a reference.
> >
> >
> > Glad to hear it. It is gratifying to know I'm of some
> > use around here.
>
>
> But that's not the end of the story!
> My next post will explain ...
>
>

We'll see...

>
> > > > > <snip> ... the next really influential theorist after
> > > > > Boethius was Guido d'Arezzo, and he too used a similar
> > > > > procedure.
> > > >
> > > >
> > > > He notated HYPOproslambanomenos with GAMMA according to
> > > > Yekta. I know not if this is true. What sayest thou?
> > >
> > >
> > > http://tonalsoft.com/enc/g/gamma.aspx
> > >
> > > (definition #3)
> >
> >
> > So he is right after all.
>
>
> Yekta is not correct that Guido introduced the gamma.
> As far as i can tell, the pseduo-Odo _dialogus_ was the
> first treatise to use both the Roman letters in their
> modern octave-equivalent form, and the gamma for the
> lowest G.
>
>

He is correct that Gamma corresponds to Hypoproslambanomenos, not
proslambanomenos. No?

>
> > > > > The later "fasola" system also simplifies modal rotation
> > > > > to using a reference tetrachord containing two tones and
> > > > > one semitone, and this became the basis of American
> > > > > "shape-note" notation which results in just-intonation
> > > > > singing.
> > > >
> > > >
> > > > Shape note?
> > >
> > >
> > > http://fasola.org/introduction/introduction.html
> > >
> > >
> >
> >
> > Interesting!
>
>
> As noted on the homepage of http://fasola.org,
> the recent movie "Cold Mountain" featured shape-note
> (also called "sacred harp") singing. I recommend that
> you check it out.
>
>
http://www.sfgate.com/cgi-bin/article.cgi?file=/chronicle/archive/2004/07/02
/PNGOP7D6DA1.DTL
>
> http://www.cnn.com/2004/SHOWBIZ/Music/02/14/sacred.harp.revival.ap/
>
>
>

Thank you for the links.

> > > As i just posted in another message, i've agreed that the
> > > Lydian mode was based on A and not on E.
> >
> >
> > So, what about the correction on the encyclopedia entry?
>
>
> Which entry is that?
>

The one one modes of course.

> Anyway, let's have more discussion after my next post
> (showing the correct diagram for the modal system), before
> we agree on changes to the Encyclopedia.
>
>
>

Agreed.

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

Cordially,
Ozan

🔗monz <monz@tonalsoft.com>

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

> > [Ozan]
> > Ah! My grin is even wider now. :))))) Is it early to
> > hope that you will at last give my poor self a reference
> > in your tonal encyclopedia now that the issue of
> > Boethian modes are resolved to our satisfaction?
>
>
> Not so fast!
>
> After examining chapters 16 and 17 of Boethius's Book 4,
> it turns out that my interpretation of his modal system,
> as i had first presented it when we started this thread,
> is correct after all.
>
> That is, the modes are simply transpositions and not
> species rotations. This is in direct contradiction to
> what Boethius writes in his text, but his explanation
> of the diagram leaves no doubt that i'm correct.

OK, here it is, in modern letter notation. I had to
rearrange it to a vertical orientation to fit in all
of the notes.

It's impossible to reproduce Boethius's Greek letters
here, but these modern equivalents exactly match his
detailed description of this diagram, as given in
chapters 16 and 17 of Book 4 (which i will post later
when i have time).

He always uses the Lydian mode as the representative
mode illustration his measurement of ratios, and describes
it as the primary mode. Thus, i have equated the mese of
the Lydian mode to "A". All other letters follow from
that reference.

(It must be viewed in a non-proportional font, such as
"Courier", to make sense.)

LEGEND:

h-m-l .. hypermixolydian
m-l .... mixolydian
lyd .... lydian
phr .... phrygian
dor .... dorian
h-l .... hypolydian
h-p .... hypophrygian
h-d .... hypodorian

h-d .. h-p .. h-l .. dor .. phr ... lyd .. m-l . h-m-l

................................................ |C ... |
................................................ | .... |
......................................... |Bb .. |Bb .. |
.................................. |A ... | .... | .... |
...................................|..... |Ab .. |Ab .. |
........................... |G ... |G ....|..... |G ... |
............................|......|..... |Gb .. | .... |
.................... |F ... |F ... |F ... |F ... |F. F. |
............. |E ....|......|......|E ... | .... | .... |
..............|..... |Eb .. |Eb ...|..... |Eb Eb |Eb Eb |
...... |D ... |D ....|..... |D ... |D. D .|..... |D ... |
.......|......|..... |Db ...|......|..... |Db Db |.. Db |
|C ... |C ... |C ... |C ... |C. C. |C. C. |C ... |C. C. |
|......|..... |B ....|......|..... |B ....|.. Cb | .... |
|Bb .. |Bb ...|..... |Bb Bb |Bb Bb |.. Bb |Bb Bb |Bb .. |
|..... |A ... |A. A .|..... |A ... |A. A. | .... | .... |
|Ab ...|......|..... |Ab Ab |.. Ab |..... |Ab .. |Ab .. |
|G ... |G. G. |G. G. |G ... |G. G. |G ....|..... |G ... |
|......|..... |F# ...|.. Gb |......|..... |Gb .. | .... |
|F. F. |F. F .|.. F. |F. F. |F ... |F ... |F ... |F ... |
|..... |E ... |E. E .|......|..... |E ... | .... | .... |
|Eb Eb |.. Eb |..... |Eb .. |Eb ...|..... |Eb .. |Eb .. |
|D ... |D. D. |D ....|..... |D ... |D ....|..... |D ... |
|.. Db |......|..... |Db ...|......|..... |Db .. | .... |
|C. C. |C ... |C ... |C ... |C ... |C ... |C ... |C ... |
|......|..... |B ....|......|..... |B ... |
|Bb ...|Bb ...|..... |Bb .. |Bb ...|..... |Bb
|..... |A ... |A ....|..... |A ... |A
|Ab ...|......|..... |Ab .. |
|G ... |G ... |G ... |G ... |G
|......|..... |F# .. |
|F ... |F ....|..... |F
|..... |E ... |E
|Eb .. |
|D ... |D
|
|C

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

🔗monz <monz@tonalsoft.com>

Hi Ozan,

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:

> Oh no you don't... The scheme I delineated was in
> perfect alignment with his descriptions up to this point
> and rightfully prepares the basis of Medieval Church
> modes the way we understand them. Does it make sense
> that Boethius would permit such a blatant contradiction?
> You yourself say that the scribes and copyists have
> transmitted fatal mistakes. If the signal-to-noise ratio
> of historical transmission is so high, how can you rely
> on one portion of the book and dismiss the other portion?

Certainly, there is a lot of "noise" in this signal,
as i pointed out -- and it's a big signal: there are
137 manuscripts of Boethius's treatise on music in
existence (i had said 127 before, sorry).

But the diagram i just posted in message 62454 is exactly
the same as the one which appears in Boethius's book 4
chapter 16 (except for the reorientation from horizontal
to vertical, and the replacement of Greek letters with
modern note-names). Here's the version of it from the
Cambridge manuscript:

http://www.music.indiana.edu/tml/6th-8th/BOEMUS4C_18GF.gif

In chapter 17 Boethius gives a *painstakingly detailed*
description of that diagram, indicating precisely how
notes in boxes right next to each other indicates that
there is a semitone between them, but that an empty box
between two notes indicates a whole-tone between them.

This chapter is the climax of the surviving portion
of Boethius's treatise. Everything he has done up till
now -- the explanations of consonance and dissonance,
intervals, genera, and monochord measurement -- leads
to this diagram of the entire modal system.

The small chapter 18 after this, which closes book 4,
is simply an explanation of how to construct a monochord
so that one may actually hear the ratios after reading
4 books of theory.

Only the first 19 chapters of book 5 survive, and
there are titles for 11 more. Boethius's book ends there.

According to Bower, Boethius apparently intended his
treatise to be in 7 books: the first 4 a translation
of a lost major work on music by Nicomachus, and the
last 3 a translation of Ptolemy's _Harmonics_. So
the part representing Nicomachus reaches its conclusion
with the diagram of the modal system and then the
instructions on building a monochord.

Also, i've dug out my paper on "Boethius's Greek-letter
notation" (which BTW was written exactly 9 years ago
today!), and the fresh conclusion i reached in the
diagram i posted in message 62454, after this new round
of speculation involving discussion with you, is
*exactly* the same as the one i presented in my 1996 paper.

And as i wrote before, i discovered that if one uses
Didymus's 5-limit tuning for the diatonic genus, then
the Greek-letter notation used in Boethius's diagram
indicates two important things:

1) notes which our modern notation spells differently
but which are only a schisma (~2 cents) apart in the
Didymus tuning (F#/Gb and B/Cb in my version of Boethius's
diagram), are designated by the *same symbol* in Boethius's
Greek notation;

2) in *every* case, notes which our modern notation spells
the same but which are a syntonic-comma (~22 cents) apart
in the Didymus tuning (Bb, C, Db, Eb, F, G, and Ab in my
version of Boethius's diagram), are designated with
different symbols in Boethius's Greek notation.

So there is no doubt at all in my mind that Boethius's
modal system is one which indicates transposition,
preserving the same intervallic structure from bottom
to top, and not one which indicates rotation thru species.

We'll need further discussion on Boethius's use of
the concept of species, since he blatantly does not
use it here.

> So, Hucbald is responsible for the gestalt shift.

Yes, single-handedly.

> You mean, OUR work, don't you? ;) I suggest we collaborate
> on an article entitled: `Modes of Boethius and Carolingians`.
> That would be an interesting topic, don't you think?

That sounds like a good project, but as i've indicated,
i've already written a fairly comprehensive paper laying
out my interpretation of Boethius's modal system.
I did have some doubts about its validity before, but
after being away from it for many years, and now reaching
exactly the same conclusions anew, i think i'm ready
to publish it.

What *is* an interesting topic for further research is
to trace how the transpositional form diagrammed by
Boethius, was somehow transformed into the rotational
species form which took root during the Frankish era.

> > Yes. It was Hucbald who created the link between the
> > Byzantine octoechos and the Boethian modes. BTW, Boethius
> > based his description of the modal system on Ptolemy.
>
>
> Then the article in question should start with Ptolemy,
> move on to Didymus and then Boethius and Carolingians.

Actually, a history of the modal system should start
with Aristoxenus, as he was the first theorist to write
about it.

Or if you want to go further back:
http://tonalsoft.com/monzo/babylonian/hurrian/monzh6.htm

(the last section on the page), descriptions of the modal
system go all the way back to the Sumerians.

Here's a very good overview of the decipherment of
Babylonian musical texts:
http://www.kingmixers.com/Franklin%20PDF%20files%20copy/Franklin%20Dissertation.%20PDF/6%20Concord%20in%20Babylon.pdf

You'll find Franklin's entire dissertation very interesting:
http://www.kingmixers.com/Terp.html

> I'm not sure if the table came out correctly. But is
> not G supposed to start on HYPOproslambanomenos?

In all my research on this stuff, i've never heard of
a note called "hypoproslambanomenos" until you mentioned it.
Guess i have some reading to do. ;-)

Anyway, no ... remember? -- Hucbald's achievement was
the *gestalt shift*. He explained the old Greek system,
with its ascending s-t-t tetrachordal interval pattern,
then explained how the "modern" modes should be fit into
the system with the t-s-t pattern, *using the same
Greek note names*. So the low G is proslambanomenos.

> > Yekta is not correct that Guido introduced the gamma.
> > As far as i can tell, the pseduo-Odo _dialogus_ was the
> > first treatise to use both the Roman letters in their
> > modern octave-equivalent form, and the gamma for the
> > lowest G.
>
>
> He is correct that Gamma corresponds to Hypoproslambanomenos,
> not proslambanomenos. No?

Again, i know not of this hypoproslambanomenos.

(but i love typing it ... what a great word ...)

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

🔗oyarman@ozanyarman.com

Monz, I have a hunch that Boethius is describing in chapters 16-17 not modes
themselves, but the transpositions of the lydian mode over the first notes
of the other modes.

http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_09GF.gif

In the ladder from Hypodorius to Hypermixolydius above, the first notes of
each mode seem to rise in the following pattern if they are to be considered
natural tones:

Sol___Hypodorius
La____Hypophrygius
Si_____Hypolydius
Do_____Dorius
Re______Phrygius
Mi_______Lydius
Fa________Mixolydius vel Hyperdorius
Sol________Hypermixolydius

My interpretation of modes transposed over to natural tones was:

Hypodorian

[C3 D E F G A B c4 d e f g a b c'5]

Hypophrygian

[D3 E F G A B C d4 e f g a b c' d'5]

Hypolydian

[E3 F G A B C d e4 f g a b c' d' e'5]

Dorian

[F3 G A B C d e f4 g a b c' d' e' f'5]

Phrygian

[G3 A B C d e f g4 a b c' d' e' f' g'5]

Lydian

[A3 B C d e f g a4 b c' d' e' f' g' a'5]

Mixolydian

[B3 C d e f g a b4 c' d' e' f' g' a' b'5]

If we transpose the lot down by a pure fourth so that the first notes of all
modes become the natural tones in the diatonically ascending ladder, then
Lydian alone is a perfect match with the Lydius of Boethius's table:

Hypodorian

[G2 A B C d e f# g3 a b c' d' e' f'# g'4]

Hypophrygian

[A2 B C d e f# g a3 b c' d' e' f'# g' a'4]

Hypolydian

[B2 C d e f# g a b3 c' d' e' f'# g' a' b'4]

Dorian

[C3 D E F# G A B c4 d e f# g a b c'5]

Phrygian

[D3 E F# G A B C d4 e f# g a b c' d'5]

Lydian

[E3 F# G A B C d e4 f(#) g a b c' d' e'5]

Mixolydian

[F#3 G A B C d e f#4 g a b c' d' e' f'#5]

The rest then can be thought of as related tropes of the Lydian mode which
you rightfully consider as a frame of reference, or in other words, the
Lydian mode transposed over to the first notes of other modes:

Hypodorius

[G2 A Bb C d eb f g3 a(b) bb c' d' eb' f' g'4]

Hypophrygius

[A2 B C d e f g a3 b(b) c' d' e' f' g' a'4]

Hypolydius

[B2 C# d e f# g a b3 c(#)' d' e' f'# g' a' b'4]

Dorius

[C3 D Eb F G Ab Bb c4 d(b) eb f g ab bb c'5]

Phrygius

[D3 E F G A Bb C d4 e(b) f g a bb c' d'5]

Lydius

[E3 F# G A B C d e4 f(#) g a b c' d' e'5]

Mixolydius vel Hyperdorius

[F#3 G# A B C# d e f#4 g(#) a b c'# d' e' f'#5]

Hypermixolydius

[G3 A Bb C d eb f g4 a(b) bb c' d' eb' f' g'5]

That's my conjecture so far.

Cordially,
Oz.

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 17 Kasï¿½m 2005 Perï¿½embe 18:44
Subject: [tuning] Boethius modal system, chapters 16-17 of Book 4, correct
diagram

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > > [Ozan]
> > > Ah! My grin is even wider now. :))))) Is it early to
> > > hope that you will at last give my poor self a reference
> > > in your tonal encyclopedia now that the issue of
> > > Boethian modes are resolved to our satisfaction?
> >
> >
> > Not so fast!
> >
> > After examining chapters 16 and 17 of Boethius's Book 4,
> > it turns out that my interpretation of his modal system,
> > as i had first presented it when we started this thread,
> > is correct after all.
> >
> > That is, the modes are simply transpositions and not
> > species rotations. This is in direct contradiction to
> > what Boethius writes in his text, but his explanation
> > of the diagram leaves no doubt that i'm correct.
>
>
>
> OK, here it is, in modern letter notation. I had to
> rearrange it to a vertical orientation to fit in all
> of the notes.
>
> It's impossible to reproduce Boethius's Greek letters
> here, but these modern equivalents exactly match his
> detailed description of this diagram, as given in
> chapters 16 and 17 of Book 4 (which i will post later
> when i have time).
>
> He always uses the Lydian mode as the representative
> mode illustration his measurement of ratios, and describes
> it as the primary mode. Thus, i have equated the mese of
> the Lydian mode to "A". All other letters follow from
> that reference.
>
> (It must be viewed in a non-proportional font, such as
> "Courier", to make sense.)
>
>
> LEGEND:
>
> h-m-l .. hypermixolydian
> m-l .... mixolydian
> lyd .... lydian
> phr .... phrygian
> dor .... dorian
> h-l .... hypolydian
> h-p .... hypophrygian
> h-d .... hypodorian
>
>
>
> h-d .. h-p .. h-l .. dor .. phr ... lyd .. m-l . h-m-l
>
> ................................................ |C ... |
> ................................................ | .... |
> ......................................... |Bb .. |Bb .. |
> .................................. |A ... | .... | .... |
> ...................................|..... |Ab .. |Ab .. |
> ........................... |G ... |G ....|..... |G ... |
> ............................|......|..... |Gb .. | .... |
> .................... |F ... |F ... |F ... |F ... |F. F. |
> ............. |E ....|......|......|E ... | .... | .... |
> ..............|..... |Eb .. |Eb ...|..... |Eb Eb |Eb Eb |
> ...... |D ... |D ....|..... |D ... |D. D .|..... |D ... |
> .......|......|..... |Db ...|......|..... |Db Db |.. Db |
> |C ... |C ... |C ... |C ... |C. C. |C. C. |C ... |C. C. |
> |......|..... |B ....|......|..... |B ....|.. Cb | .... |
> |Bb .. |Bb ...|..... |Bb Bb |Bb Bb |.. Bb |Bb Bb |Bb .. |
> |..... |A ... |A. A .|..... |A ... |A. A. | .... | .... |
> |Ab ...|......|..... |Ab Ab |.. Ab |..... |Ab .. |Ab .. |
> |G ... |G. G. |G. G. |G ... |G. G. |G ....|..... |G ... |
> |......|..... |F# ...|.. Gb |......|..... |Gb .. | .... |
> |F. F. |F. F .|.. F. |F. F. |F ... |F ... |F ... |F ... |
> |..... |E ... |E. E .|......|..... |E ... | .... | .... |
> |Eb Eb |.. Eb |..... |Eb .. |Eb ...|..... |Eb .. |Eb .. |
> |D ... |D. D. |D ....|..... |D ... |D ....|..... |D ... |
> |.. Db |......|..... |Db ...|......|..... |Db .. | .... |
> |C. C. |C ... |C ... |C ... |C ... |C ... |C ... |C ... |
> |......|..... |B ....|......|..... |B ... |
> |Bb ...|Bb ...|..... |Bb .. |Bb ...|..... |Bb
> |..... |A ... |A ....|..... |A ... |A
> |Ab ...|......|..... |Ab .. |
> |G ... |G ... |G ... |G ... |G
> |......|..... |F# .. |
> |F ... |F ....|..... |F
> |..... |E ... |E
> |Eb .. |
> |D ... |D
> |
> |C
>
>
>
>
> -monz
> http://tonalsoft.com
> Tonescape microtonal music software
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🔗klaus schmirler <KSchmir@online.de>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗oyarman@ozanyarman.com

Hey again Monz,

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 17 Kasï¿½m 2005 Perï¿½embe 20:34
Subject: [tuning] Re: Boethius modal system, chapter 15 of Book 4

> Hi Ozan,
>
> --- In tuning@yahoogroups.com, <oyarman@o...> wrote:
>
> > Oh no you don't... The scheme I delineated was in
> > perfect alignment with his descriptions up to this point
> > and rightfully prepares the basis of Medieval Church
> > modes the way we understand them. Does it make sense
> > that Boethius would permit such a blatant contradiction?
> > You yourself say that the scribes and copyists have
> > transmitted fatal mistakes. If the signal-to-noise ratio
> > of historical transmission is so high, how can you rely
> > on one portion of the book and dismiss the other portion?
>
>
> Certainly, there is a lot of "noise" in this signal,
> as i pointed out -- and it's a big signal: there are
> 137 manuscripts of Boethius's treatise on music in
> existence (i had said 127 before, sorry).
>

I understand well.

>
> But the diagram i just posted in message 62454 is exactly
> the same as the one which appears in Boethius's book 4
> chapter 16 (except for the reorientation from horizontal
> to vertical, and the replacement of Greek letters with
> modern note-names). Here's the version of it from the
> Cambridge manuscript:
>
> http://www.music.indiana.edu/tml/6th-8th/BOEMUS4C_18GF.gif
>
>

I have prepared two colored version which I uploaded at
http://www.ozanyarman.com/anonymous/

> In chapter 17 Boethius gives a *painstakingly detailed*
> description of that diagram, indicating precisely how
> notes in boxes right next to each other indicates that
> there is a semitone between them, but that an empty box
> between two notes indicates a whole-tone between them.
>

I would urge you once more to bring forth the translations of the following
chapters.

> This chapter is the climax of the surviving portion
> of Boethius's treatise. Everything he has done up till
> now -- the explanations of consonance and dissonance,
> intervals, genera, and monochord measurement -- leads
> to this diagram of the entire modal system.
>
> The small chapter 18 after this, which closes book 4,
> is simply an explanation of how to construct a monochord
> so that one may actually hear the ratios after reading
> 4 books of theory.
>

That won't concern us at this juncture I believe.

> Only the first 19 chapters of book 5 survive, and
> there are titles for 11 more. Boethius's book ends there.
>
> According to Bower, Boethius apparently intended his
> treatise to be in 7 books: the first 4 a translation
> of a lost major work on music by Nicomachus, and the
> last 3 a translation of Ptolemy's _Harmonics_. So
> the part representing Nicomachus reaches its conclusion
> with the diagram of the modal system and then the
> instructions on building a monochord.
>

Scintillating.

>
> Also, i've dug out my paper on "Boethius's Greek-letter
> notation" (which BTW was written exactly 9 years ago
> today!), and the fresh conclusion i reached in the
> diagram i posted in message 62454, after this new round
> of speculation involving discussion with you, is
> *exactly* the same as the one i presented in my 1996 paper.
>

Do we get to see an online (scanned) version?

> And as i wrote before, i discovered that if one uses
> Didymus's 5-limit tuning for the diatonic genus, then
> the Greek-letter notation used in Boethius's diagram
> indicates two important things:
>
> 1) notes which our modern notation spells differently
> but which are only a schisma (~2 cents) apart in the
> Didymus tuning (F#/Gb and B/Cb in my version of Boethius's
> diagram), are designated by the *same symbol* in Boethius's
> Greek notation;
>
> 2) in *every* case, notes which our modern notation spells
> the same but which are a syntonic-comma (~22 cents) apart
> in the Didymus tuning (Bb, C, Db, Eb, F, G, and Ab in my
> version of Boethius's diagram), are designated with
> different symbols in Boethius's Greek notation.
>

The interesting part then is the relationship of `Maqam Music` to the modal
theory of antiquity.

>
> So there is no doubt at all in my mind that Boethius's
> modal system is one which indicates transposition,
> preserving the same intervallic structure from bottom
> to top, and not one which indicates rotation thru species.
>

Not so fast my colleague... I am still trying to understand why there is so
much contraversy between chapter XV and consequent chapters of the 4th
volume of De Institutione Musica.

>
> > You mean, OUR work, don't you? ;) I suggest we collaborate
> > on an article entitled: `Modes of Boethius and Carolingians`.
> > That would be an interesting topic, don't you think?
>
>
> That sounds like a good project, but as i've indicated,
> i've already written a fairly comprehensive paper laying
> out my interpretation of Boethius's modal system.
> I did have some doubts about its validity before, but
> after being away from it for many years, and now reaching
> exactly the same conclusions anew, i think i'm ready
> to publish it.
>

That is good news. But perhaps you would like to include my interpretation
also?

> What *is* an interesting topic for further research is
> to trace how the transpositional form diagrammed by
> Boethius, was somehow transformed into the rotational
> species form which took root during the Frankish era.
>

That is the crux of the issue! Maybe Boethius used the same terminology to
indicate two very different things: octave species and transpositions of a
single mode? Perhaps he attempted to demonstrate how to change pitch-levels
for singers?

>
> > > Yes. It was Hucbald who created the link between the
> > > Byzantine octoechos and the Boethian modes. BTW, Boethius
> > > based his description of the modal system on Ptolemy.
> >
> >
> > Then the article in question should start with Ptolemy,
> > move on to Didymus and then Boethius and Carolingians.
>
>
> Actually, a history of the modal system should start
> with Aristoxenus, as he was the first theorist to write
> about it.
>

Sounds like a worthwhile endeavour. You are of course the expert on this
issue and all I can do is introduce a new outlook on existing materials.

> Or if you want to go further back:
> http://tonalsoft.com/monzo/babylonian/hurrian/monzh6.htm
>
> (the last section on the page), descriptions of the modal
> system go all the way back to the Sumerians.
>
> Here's a very good overview of the decipherment of
> Babylonian musical texts:
>
http://www.kingmixers.com/Franklin%20PDF%20files%20copy/Franklin%20Dissertat
ion.%20PDF/6%20Concord%20in%20Babylon.pdf
>

Perhaps another article would be a better idea concerning the possible link
between Mesopotamian music cultures and Greek theory.

> You'll find Franklin's entire dissertation very interesting:
> http://www.kingmixers.com/Terp.html
>
>

Thank you. I'll see to it in due course.

>
> > I'm not sure if the table came out correctly. But is
> > not G supposed to start on HYPOproslambanomenos?
>
>
> In all my research on this stuff, i've never heard of
> a note called "hypoproslambanomenos" until you mentioned it.
> Guess i have some reading to do. ;-)
>

I quoted Yekta who quotes Heegmann who says that Guido assigned La to
proslambanomenos and added gamma below, and also that Meibomius had declared
the existence of hypoproslambanomenos which was neglected by the Greeks.

> Anyway, no ... remember? -- Hucbald's achievement was
> the *gestalt shift*. He explained the old Greek system,
> with its ascending s-t-t tetrachordal interval pattern,
> then explained how the "modern" modes should be fit into
> the system with the t-s-t pattern, *using the same
> Greek note names*. So the low G is proslambanomenos.
>
>

So you say.

>
> > > Yekta is not correct that Guido introduced the gamma.
> > > As far as i can tell, the pseduo-Odo _dialogus_ was the
> > > first treatise to use both the Roman letters in their
> > > modern octave-equivalent form, and the gamma for the
> > > lowest G.
> >
> >
> > He is correct that Gamma corresponds to Hypoproslambanomenos,
> > not proslambanomenos. No?
>
>
> Again, i know not of this hypoproslambanomenos.
>
> (but i love typing it ... what a great word ...)
>
>

Yea, isn't it though?

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

Cordially,
Ozan

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗oyarman@ozanyarman.com

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗oyarman@ozanyarman.com

🔗monz <monz@tonalsoft.com>

Hi klaus,

--- In tuning@yahoogroups.com, klaus schmirler <KSchmir@o...> wrote:
>
> monz wrote:
>
> > So then it was Rousseau who mistakenly thought that
> > Guido shifted the tonal system down by a tone? Or did
> > he say nothing about the shift, and just claimed that
> > Guido added the low-G "gamma"?
> >
> >
>
> > The whole system here was based on G as the reference
> > pitch ... and as i said, shifting the entire system
> > down so that the intervallic structure fit the natural
> > notes with G as a reference, was the work of Hucbald,
> > in the early 900's -- but of course Hucbald was not
> > yet using the Roman letters.
> >
> >
>
> I don't understand why you (apart from the short mention
> of "adding" above) refer to "shifting the system".
>
> I always saw the Gamma as the one note you sometimes need
> that is lower than the the octoechos provides, similar to
> the ut in the hexachord that is a lower approach to "queant",
> re. In other words, the reference pitch is lowest note in
> the lower tetrachord of the lowest mode (deuterus,
> second). The name as well as the fact that it is the
> only Greek letter imply clearly that it is added, but
> apparently not to Boethius' system.

Hucbald's treatise explicitly sets out the ancient Greek
PIS, with its ascending s-t-t tetrachord intervals, then
explicitly applies the same Greek note-names to the
ascending t-s-t tetrachord structure. Using all "natural"
letter-name notes, this is equivalent to shifting the
reference meson tetrachord from E-F-G-A to D-E-F-G,
which is exactly how Hucbald's system was described only
a few decades after his death, in the pseudo-Odo _dialogus_.

> klaus
>
> who is still convinced that these old tracts needn't
> describe any ancient or contemporary practice -- that
> would mean they were thinking scientifically. Probably
> they did what the writers up to the Renaissance
> expected of them: they were passing on tradition,
> however badly (!value judgement ex poteriori) understood.

Certainly, that's a part of it too. Boethius himself
was clearly transmitting Pythagorean doctrine to post-Roman
Latin readers ... with huge success, i might add -- keep
in mind that his treatise was the foundation of Western
music-theory for *1000 years*!

However, Boethius *does* use the Greek-letter notation
in his diagrams of the modes, and for me this is the
most significant thing in his whole treatise, because
it seems to me that the Greek notation proves that the
tuning of the diatonic genus used Didymus's 5-limit ratios.

Note that Boethius himself gives an elaborate description
of the tuning of the diatonic in its standard 3-limit
pythagorean form. But as i have described, the Greek
notation seems to be obviously a reflection of actual
practice, where a 22-cent difference (subtle, but noticeable)
is notated, but a 2-cent difference (arguably inaudible)
is ignored. This is the kind of practical solution that
a real music notation would evolve.

And of course, if i'm correct, then it proves that
5-limit tuning was used in actual practice by the
Graeco-Romans of c.100-500 AD, which i think is highly
significant.

> Have you come across the modal system of Boethius'
> successor at the Ostrogoth court, Cassiodorus? His
> 15 modes are based on the five semitones within a
> fourth, by transposition apparently
> (http://www.music.indiana.edu/tml/6th-8th/CASINST_TEXT.html,
> 8. paragraph).

Thanks for that ... i'll have a look when i get a chance.
I haven't read anything by Cassiodorus since 1996, and
even then i didn't do a very deep study of his work. My
focus then was on Boethius and the _musica enchiriadis_,
two treatises which have given me plenty to think about.

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

🔗monz <monz@tonalsoft.com>

Hi Ozan,

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:

> > But the diagram i just posted in message 62454 is exactly
> > the same as the one which appears in Boethius's book 4
> > chapter 16 (except for the reorientation from horizontal
> > to vertical, and the replacement of Greek letters with
> > modern note-names). Here's the version of it from the
> > Cambridge manuscript:
> >
> > http://www.music.indiana.edu/tml/6th-8th/BOEMUS4C_18GF.gif
>
>
> I have prepared two colored version which I uploaded at
> http://www.ozanyarman.com/anonymous/

Those are awesome!

But ... what do the different colors represent?

To me, it would make sense to use two different colors:
one to represent the _hestotes_ (fixed notes), and
the other to represent the moveable notes.

Also, i'll post my analysis of the various cents-values
when the Didymus tuning is used ... can you make a
version which uses 19 different colors to code the
different pitch-classes in that tuning?

And in the version which uses modern octave-equivalent
Roman-letter notation, why did you choose to base the
system on "G" rather than on "C"? It's clear that the
Lydian mode was the reference mode, whose mese should
be called "A", which bases the whole system on "C".

> > In chapter 17 Boethius gives a *painstakingly detailed*
> > description of that diagram, indicating precisely how
> > notes in boxes right next to each other indicates that
> > there is a semitone between them, but that an empty box
> > between two notes indicates a whole-tone between them.
> >
>
>
> I would urge you once more to bring forth the translations
> of the following chapters.

Ozan, i've now neglected almost an entire week of important
work that i have to do, because i got caught up in this,
and i can afford to neglect it no longer. Some of this stuff
will just have to wait.

> > So there is no doubt at all in my mind that Boethius's
> > modal system is one which indicates transposition,
> > preserving the same intervallic structure from bottom
> > to top, and not one which indicates rotation thru species.
> >
>
>
> Not so fast my colleague... I am still trying to understand
> why there is so much contraversy between chapter XV and
> consequent chapters of the 4th volume of De Institutione
> Musica.

I believe i've pinpointed it: Boethius gives a detailed
description of species, as his introduction to a discussion
of the modal system ... then promptly ignores it when
he gives the actual description of the modal system!

After explaining clearly and in detail the concept of
modal rotation thru species, he then gives an elaborate
diagram which presents the modal system as transpositional
rather than rotational.

> That is good news. But perhaps you would like to
> include my interpretation also?

Yes, sure ... but as i did more and more research into
the Greek modal system, i stopped working on my paper
on Boethius when i became unsure if my interpretation
was really correct. As i've said, there are many divergent
opinions on the rotation-vs.-transposition issue, among
the various scholars of ancient Greek music-theory.

I finally got so thoroughly confused that i just gave up.
But at least now i've convinced myself that my
transpositional interpretation of that chapter 16
diagram is correct.

In fact, i just came across some other notes, and
there's a sheet of music-paper dated 2004.08.01 on
which i wrote out the entire modal system with the
Greek symbols above the notes, which is again exactly
the same as the interpretation given in my 1996 and
in the diagram i posted yesterday -- i.e., *that* was
the second time i independently arrived at this conclusion,
and yesterday's was the *third* time i did it!

And at the bottom of this sheet of music-paper, i wrote:

>> " -- exactly the same as what i derived for
>> Aristides Q. [Quintilianus] (book 1, ch. 11) in 1998 !!
>> see Barker 1989, p. 420 ff."

So there's a *fourth* interpretation i derived of the
Greek notation of the modal system, this time for
Aristides, which again coincides with this one.

So anyway, i'll have to finish the paper on Boethius
first, before i can include anything by you -- and
that is simply not going to happen until after February.
Too many other committments before then.

> > What *is* an interesting topic for further research is
> > to trace how the transpositional form diagrammed by
> > Boethius, was somehow transformed into the rotational
> > species form which took root during the Frankish era.
> >
>
> That is the crux of the issue! Maybe Boethius used the
> same terminology to indicate two very different things:
> octave species and transpositions of a single mode?
> Perhaps he attempted to demonstrate how to change
> pitch-levels for singers?

Quite possible. I hope i've provided enough material
here to get other folks interested enough to do some
digging of their own.

For me, the fact that Boethius's Greek notation seems
to prove that Didymus's tuning was used in practice,
brings enough satisfaction to me about my own discoveries.
I suppose that's probably another reason why i never
finished writing the paper -- after putting that part
of it together, nothing else seemed as significant.

> > Actually, a history of the modal system should start
> > with Aristoxenus, as he was the first theorist to write
> > about it.
>
>
> Sounds like a worthwhile endeavour. You are of course
> the expert on this issue and all I can do is introduce
> a new outlook on existing materials.

I wish that someone else would be interested enough
to take on the job of editing my Aristoxenus webpage
into something readable.

http://tonalsoft.com/monzo/aristoxenus/318tet.htm

I've only managed to make about the first 1/5 of it,
and the very last section ("Amendment from 2003.07.20"),
look decent.

> > Anyway, no ... remember? -- Hucbald's achievement was
> > the *gestalt shift*. He explained the old Greek system,
> > with its ascending s-t-t tetrachordal interval pattern,
> > then explained how the "modern" modes should be fit into
> > the system with the t-s-t pattern, *using the same
> > Greek note names*. So the low G is proslambanomenos.
>
>
> So you say.

It's not *me* saying it, it's Hucbald!

http://www.music.indiana.edu/tml/9th-11th/HUCHAR_TEXT.html

There is an English translation of Hucbald available,
and i have it. When i get the time, i'll post the
relevant excerpts.

It's out-of-print now, so your best bet is a good library:

http://www.worldcatlibraries.org/wcpa/top3mset/aabe08529f14f206.html

I tried searching Turkey and all of the nearby countries
with no luck ... the closest i've found to you which has
it is here:

Accademia Della Crusca Villa Medical Di Cast, Firenze, Italy

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

🔗monz <monz@tonalsoft.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗oyarman@ozanyarman.com

Dear Monz,

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 18 Kasï¿½m 2005 Cuma 11:33
Subject: [tuning] Re: Boethius modal system, chapter 15 of Book 4

> Hi Ozan,
>
>

What happened to Oz? I liked it better.

> >
> > I have prepared two colored version which I uploaded at
> > http://www.ozanyarman.com/anonymous/
>
>
>
> Those are awesome!
>

Thanx.

> But ... what do the different colors represent?
>

Green=pitches employed just once, regardless of octave equivalance.
Yellow & Red=pitches used more than once.

I used the latter in such a way that adjacent pitches from low to high
seperated by a semitone are colored seperately & consecutively for easier
reading.

Here is another table (tones differing by a comma are marked with "o"):

.............HD....HP....HL....D.......P.....L.....M.....H
G...........x............................................................
A...........x.........x..................................................
Bb.........x.............................................................
B......................x.......x.........................................
C...........x.........x................x.................................
C#............................x..........................................
D...........x.........x.......o.......x.......x........................
Eb.........x...........................x.................................
E......................x.......x................x......x.................
F............x........o................x.......o.............o.........
F#.............................x........................x................
g............x.........x.......o.......x.......x......o......x......x
ab..........x...........................x......................o........
a............x.........x.......x................x.......x..............x
bb..........xo.......x................o......x..............o......x
b.......................x.......x........................x...............
c.............x........ox.....o........x......x......o......x......x
db/c#.........................x........x.....................x.........
d.............x.........x......ox......x.......x......x..............x
eb...........x..........................xo......x............o.......x
e........................x.......x................x......x...............
f..............x.........o................x......ox.....o.....x......x
f#/gb..........................x........................x......x........
g'............x.........x.......o.......x........x.....ox....x......x
a'b.......................................x......................xo....x
a'.......................x.......x.................x......x.............x
b'b.......................................x.......o.............x.....ox
b'................................x.........................x.............
c'..........................................x.......x......o......x.....x
d'b................................................................x.....
d'..................................................x.......x.............x
e'b................................................................x.....x
e............................................................x.............
f'....................................................................x....x
g'.........................................................................x

> Also, i'll post my analysis of the various cents-values
> when the Didymus tuning is used ... can you make a
> version which uses 19 different colors to code the
> different pitch-classes in that tuning?
>
>

Certainly, once I see the tuning you analyze.

> And in the version which uses modern octave-equivalent
> Roman-letter notation, why did you choose to base the
> system on "G" rather than on "C"? It's clear that the
> Lydian mode was the reference mode, whose mese should
> be called "A", which bases the whole system on "C".
>
>

But then, the tonics of transpositions of the modes do not equate to the
natural tones of the diatonic scale we recognize today, which are G A B C D
E F G from Hypodorius to Hypermixolidius in Boethius's diagram. However, it
is possible to transpose all the tones a pure fourth up - to the detriment
of Mixolydius which would then start on Bb - if we were to remain true to
the letters.

>
> > > In chapter 17 Boethius gives a *painstakingly detailed*
> > > description of that diagram, indicating precisely how
> > > notes in boxes right next to each other indicates that
> > > there is a semitone between them, but that an empty box
> > > between two notes indicates a whole-tone between them.
> > >
> >
> >
> > I would urge you once more to bring forth the translations
> > of the following chapters.
>
>
> Ozan, i've now neglected almost an entire week of important
> work that i have to do, because i got caught up in this,
> and i can afford to neglect it no longer. Some of this stuff
> will just have to wait.
>
>

I understand. I am grateful for the time you spared.

> >
> > Not so fast my colleague... I am still trying to understand
> > why there is so much contraversy between chapter XV and
> > consequent chapters of the 4th volume of De Institutione
> > Musica.
>
>
> I believe i've pinpointed it: Boethius gives a detailed
> description of species, as his introduction to a discussion
> of the modal system ... then promptly ignores it when
> he gives the actual description of the modal system!
>

A bit crude.

> After explaining clearly and in detail the concept of
> modal rotation thru species, he then gives an elaborate
> diagram which presents the modal system as transpositional
> rather than rotational.
>
>

So he uses mode very much out of context here to the point of contradicting
his previous - more original and valid? -statements. I wish to know what you
think of my categorization of terms so far:

1. Modes are rotations of a single diatonical scale, so that the tonic is
shifted over to the other available `natural` degrees of that scale .

2. A specie of the octave (hence trope) is a rearrangement of notes in a
fixed octave tuning so that the distribution of small and large intervals
yield different `transposed modes`.

3. Genera are rather reserved for tetrachords, but in principle are no more
different compared to the construction of species.

4. Boethius's diagram represents transpositions of the Lydian mode over the
tonics of the other modes.

5. Tone should be reserved for describing the relationships between modes,
sometimes synonymous with the tonic, a derivative function of KEY.

>
> > That is good news. But perhaps you would like to
> > include my interpretation also?
>
>
> Yes, sure ... but as i did more and more research into
> the Greek modal system, i stopped working on my paper
> on Boethius when i became unsure if my interpretation
> was really correct. As i've said, there are many divergent
> opinions on the rotation-vs.-transposition issue, among
> the various scholars of ancient Greek music-theory.
>
> I finally got so thoroughly confused that i just gave up.
> But at least now i've convinced myself that my
> transpositional interpretation of that chapter 16
> diagram is correct.
>
>
> In fact, i just came across some other notes, and
> there's a sheet of music-paper dated 2004.08.01 on
> which i wrote out the entire modal system with the
> Greek symbols above the notes, which is again exactly
> the same as the interpretation given in my 1996 and
> in the diagram i posted yesterday -- i.e., *that* was
> the second time i independently arrived at this conclusion,
> and yesterday's was the *third* time i did it!
>
> And at the bottom of this sheet of music-paper, i wrote:
>
> >> " -- exactly the same as what i derived for
> >> Aristides Q. [Quintilianus] (book 1, ch. 11) in 1998 !!
> >> see Barker 1989, p. 420 ff."
>
>
> So there's a *fourth* interpretation i derived of the
> Greek notation of the modal system, this time for
> Aristides, which again coincides with this one.
>

Seeing as mode is used both to describe octave species and transpositions,
my estimations would then lead me to suggest that a very complex pattern of
modulation and transposition was already in existence in early Medieval
theory and possibly even before.

>
> So anyway, i'll have to finish the paper on Boethius
> first, before i can include anything by you -- and
> that is simply not going to happen until after February.
> Too many other committments before then.
>
>

Acknowledged.

>
> > > What *is* an interesting topic for further research is
> > > to trace how the transpositional form diagrammed by
> > > Boethius, was somehow transformed into the rotational
> > > species form which took root during the Frankish era.
> > >
> >

Not transformed, perhaps... but simplified to the detriment of available
transpositions?

> > That is the crux of the issue! Maybe Boethius used the
> > same terminology to indicate two very different things:
> > octave species and transpositions of a single mode?
> > Perhaps he attempted to demonstrate how to change
> > pitch-levels for singers?
>
>
> Quite possible. I hope i've provided enough material
> here to get other folks interested enough to do some
> digging of their own.
>

It has been a most productive session for me.

> > > Actually, a history of the modal system should start
> > > with Aristoxenus, as he was the first theorist to write
> > > about it.
> >
> >
> > Sounds like a worthwhile endeavour. You are of course
> > the expert on this issue and all I can do is introduce
> > a new outlook on existing materials.
>
>
> I wish that someone else would be interested enough
> to take on the job of editing my Aristoxenus webpage
> into something readable.
>
> http://tonalsoft.com/monzo/aristoxenus/318tet.htm
>
> I've only managed to make about the first 1/5 of it,
> and the very last section ("Amendment from 2003.07.20"),
> look decent.
>

If only I could, but I have little knowledge on Aristoxenus to aid you in
that direction.

>
>
>
> > > Anyway, no ... remember? -- Hucbald's achievement was
> > > the *gestalt shift*. He explained the old Greek system,
> > > with its ascending s-t-t tetrachordal interval pattern,
> > > then explained how the "modern" modes should be fit into
> > > the system with the t-s-t pattern, *using the same
> > > Greek note names*. So the low G is proslambanomenos.
> >
> >
> > So you say.
>
>
> It's not *me* saying it, it's Hucbald!
>
> http://www.music.indiana.edu/tml/9th-11th/HUCHAR_TEXT.html
>
>
> There is an English translation of Hucbald available,
> and i have it. When i get the time, i'll post the
> relevant excerpts.
>

I'll take your word for it till then.

> It's out-of-print now, so your best bet is a good library:
>
> http://www.worldcatlibraries.org/wcpa/top3mset/aabe08529f14f206.html
>
> I tried searching Turkey and all of the nearby countries
> with no luck ... the closest i've found to you which has
> it is here:
>
> Accademia Della Crusca Villa Medical Di Cast, Firenze, Italy
>
>

Thank you so much!

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

Cordially,
Ozan

🔗oyarman@ozanyarman.com

🔗monz <monz@tonalsoft.com>

Hi Oz,

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:

> > >> [monz, quoting Boethius and Bower's translation]
> > >> Sane si quando dispositionem notarum grecarum litterarum
> > >> nuncupatione descripsero: lector nulla nouitate turbetur.
> > >>
> > >> Surely if i sketch the disposition of notes using the
> > >> names of Greek letters, the reader should not be put off
> > >> by anything unusual,
> > >>
> > >>
> > >> Grecis enim litteris in quamlibet partem nunc imminutis.
> > >> nunc etiam inflexis. tota [uel rata supra lin.] haec
> > >> notarum descriptio constituta est.
> > >>
> > >> for this whole arrangement of notes has been organized
> > >> around Greek letters, which are sometimes altered and
> > >> sometimes even rotated thru various positions
> >
> >
>
>
> Aha. Rotated is the word.
>
>
> > [i.e.: The basic symbols for the moveable notes were for
> > the diatonic genus, and they were rotated in one direction
> > for the chromatic genus, and in the other direction for the
> > enharmonic.]
> >
>
>
> modulations of the diatonic genus of the tetrachord?

DON'T GET CONFUSED ABOUT THIS! (sorry about the screaming)

The rotation of the Greek letters indicates a difference
of *GENUS*, and has nothing to do with our concept of
"modal rotation".

For example:

* a regular kappa "K" represents one of the moveable notes
in its diatonic tuning,

* a kappa lying on its back represents the same note but
in its chromatic tuning,

* a reversed kappa reprsents the same note but in its
enharmonic tuning.

(I'm only quickly checking emails right now and don't have
time to consult Boethius's book or my notes ... so this
is only for the purposes of illustration ... the actual
use of the kappa and its rotations might be different,
but the concept is what i'm explaining.)

> > >> Nos uero cauemus aliquid ab antiquitatis auctoritate
> > >> transuertere.
> > >>
> > >> We, of course, guard against changing anything
> > >> handed down by the authority of antiquity.
>
>
> Meaning: `we are conservative when it comes to the
> lettering of the pitches.`

I believe it means a bit more than just that. Knowing
that in his day the only way for his work to be transmitted
was thru handwritten copies, and also knowing that those
doing the copying probably would have far less knowledge
of Greek than he himself, i think Boethius was sending
out a specific warning to subsequent scribes to be very
careful to copy his diagrams of Greek notation exactly
as he had drawn them.

Unfortunately, as i've already explained, those scribes
paid only intermittent heed to his warning. The diagrams
in Boethius are famous among scholars for their confusion.

> > Thus, if scribes would be especially careful about
> > *anything* in this book, it would be Boethius's handling
> > of the Greek-letter musical notation, despite the fact
> > that they didn't know how to read Greek (which is an
> > argument for the ease with which it could be corrupted).
> >
>
>
> The letters themselves are not as much important as
> their arrangement in the table coupled with his
> previous statements on tropes and modes.

That's basically true, but the letters themselves do
reveal a lot about the ancient scales, mostly because
of the rotations thru the chromatic and enharmonic genera.

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

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

Hi monz,

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 21 Kasï¿½m 2005 Pazartesi 19:11
Subject: [tuning] Re: Boethius modal system, chapter 3 of Book 4

> Hi Oz,
>

SNIP

> >
> > > [i.e.: The basic symbols for the moveable notes were for
> > > the diatonic genus, and they were rotated in one direction
> > > for the chromatic genus, and in the other direction for the
> > > enharmonic.]
> > >
> >
> >
> > modulations of the diatonic genus of the tetrachord?
>
>
> DON'T GET CONFUSED ABOUT THIS! (sorry about the screaming)
>

Hey, no need to shout, I can heaaar you...

> The rotation of the Greek letters indicates a difference
> of *GENUS*, and has nothing to do with our concept of
> "modal rotation".
>
> For example:
>
> * a regular kappa "K" represents one of the moveable notes
> in its diatonic tuning,
>

An example if you please?

> * a kappa lying on its back represents the same note but
> in its chromatic tuning,
>

ditto

> * a reversed kappa reprsents the same note but in its
> enharmonic tuning.
>

ditto

> (I'm only quickly checking emails right now and don't have
> time to consult Boethius's book or my notes ... so this
> is only for the purposes of illustration ... the actual
> use of the kappa and its rotations might be different,
> but the concept is what i'm explaining.)
>
>

An valid example might illustrate your point with utmost clarity.

>
> > > >> Nos uero cauemus aliquid ab antiquitatis auctoritate
> > > >> transuertere.
> > > >>
> > > >> We, of course, guard against changing anything
> > > >> handed down by the authority of antiquity.
> >
> >
> > Meaning: `we are conservative when it comes to the
> > lettering of the pitches.`
>
>
> I believe it means a bit more than just that. Knowing
> that in his day the only way for his work to be transmitted
> was thru handwritten copies, and also knowing that those
> doing the copying probably would have far less knowledge
> of Greek than he himself, i think Boethius was sending
> out a specific warning to subsequent scribes to be very
> careful to copy his diagrams of Greek notation exactly
> as he had drawn them.
>
> Unfortunately, as i've already explained, those scribes
> paid only intermittent heed to his warning. The diagrams
> in Boethius are famous among scholars for their confusion.
>
>

Ach... too bad the scribes were not exactly faithful to the original text.

>
> > > Thus, if scribes would be especially careful about
> > > *anything* in this book, it would be Boethius's handling
> > > of the Greek-letter musical notation, despite the fact
> > > that they didn't know how to read Greek (which is an
> > > argument for the ease with which it could be corrupted).
> > >
> >
> >
> > The letters themselves are not as much important as
> > their arrangement in the table coupled with his
> > previous statements on tropes and modes.
>
>
> That's basically true, but the letters themselves do
> reveal a lot about the ancient scales, mostly because
> of the rotations thru the chromatic and enharmonic genera.
>
>

What are the average intervallic values for these genera of the tetrachord
according to prominent ancient and medival theorists?

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

Cordially,
Ozan

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@tonalsoft.com>

Hi Oz,

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:

> > The rotation of the Greek letters indicates a difference
> > of *GENUS*, and has nothing to do with our concept of
> > "modal rotation".
> >
> > For example:
> >
> > * a regular kappa "K" represents one of the moveable notes
> > in its diatonic tuning,
> >
>
>
> An example if you please?
>
>
> > * a kappa lying on its back represents the same note but
> > in its chromatic tuning,
> >
>
> ditto
>
> > * a reversed kappa reprsents the same note but in its
> > enharmonic tuning.
> >
>
> ditto

Well, unfortunately it's impossible for me to provide
an example in ASCII text, because these are symbols
based on the Greek alphabet and then rotated.

The best i can do right now is:

1) quote Boethius's text, which gives a detailed
description of how the Greek symbols are written
for each genus; i've quoted the entire chapter at
the bottom of this post -- every place where it
says "[signum]", the manuscript has a Greek symbol
which, since it can't be reproduced in ASCII, also
does not appear in the texts of the Thesarus Musicarum
Latinarum, the source where i got the text;

2) direct you to the relevant diagrams in Boethius.

Unfortunately, as i keep emphasizing, these have
been corrupted beyond belief during the centuries of
transmission by scribes illiterate in Greek. Anyway,
here are the ones which are available on the internet:

from the Cambridge manuscript -- barely legible and
thus barely useful:

http://www.music.indiana.edu/tml/6th-8th/BOEMUS4C_06GF.gif

from Friedlin's edition -- much more legible, but again
barely useful because the symbols obviously have been
corrupted and don't show a logical rotational system,
but at least it gives you some idea:

http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_01GF.gif
http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_02GF.gif
http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_02GF.gif

from Migne's edition -- again, not very legible:

http://www.music.indiana.edu/tml/6th-8th/BOEDIM4_06GF.gif

The three main sources of the ancient Greek notation
are Alypius, Aristides Quintilianus, and Boethius.

I highly recommend that you track down modern editions
of all three. Bower has done considerable correction
of the diagrams in his English translation of Boethius.
Andrew Barker and Thomas Mathiesen have both published
modern English editions of Aristides. Alypius is the
best-known source of the Greek notation, and his diagrams
are reproduced in many reference works, such as Groves
and the Oxford History of Music. Try searching the web
-- you'll probably find them.

NOTE: there were *two different* notations in use in
ancient Greece, one called "vocal" and the other called
"instrumental". That's why all of these diagrams
(including the ones you made with the different colors)
each have a *pair* of symbols for each note.

The vocal notation (the top symbol of each pair on
your diagrams) stuck pretty strictly to the Greek alphabet,
and simply used different letters for the different
genera without employing rotation.

The instrumental notation appears to me to be older,
because in addition to Greek letters, some of the
non-Greek symbols resemble letters from the Phoenician
alphabet, which precedes, and in fact was the origin of,
the Greek alphabet. This is the notation whose symbols
were rotated for the different genera.

> > > [Oz]
> > > The letters themselves are not as much important as
> > > their arrangement in the table coupled with his
> > > previous statements on tropes and modes.
> >
> >
> > That's basically true, but the letters themselves do
> > reveal a lot about the ancient scales, mostly because
> > of the rotations thru the chromatic and enharmonic genera.
> >
> >
>
>
> What are the average intervallic values for these genera
> of the tetrachord according to prominent ancient and
> medival theorists?

The most general way of describing it is as follows,
using the notes of the meson tetrachord in descending
order as a reference, as the ancients always did:

legend:

q = quarter-tone
s = semitone
t = tone
m3 = trihemitone ("minor-3rd" in modern terminology)
M3 = ditone ("major-3rd")

........... mese .. lichanos .. parhypate .. hypate
diatonic ........ t ......... t .......... s
chromatic ....... m3 ........ s .......... s
enharmonic ...... M3 ........ q .......... q

If you want to get more specific than that, we'll
have to dig into the ratios specified by numerous
authors, and each one gave different measurements.
Take a look at "diatonic", "chromatic", and "enharmonic"
in the Tonalsoft Encyclopedia.

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

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

chapter 3 of book 4, Boethius _de institutione musica_,
from the Cambridge manuscript:

[-f.70r-]
Musicarum per grecas ac latinas litteras notarum nuncupat.

RESTAT quoniam sumus neruum secundum praedictas consonantias
per regulam diuisuri. quoniamque necessarios sonos tribus
generibus cantilenae exhibebit ista partitio musicas interim
notas apponere. ut cum diuisam lineam isdem notulis
significauerimus: quod unicuique nomen sit facillime possit
agnosci. Veteres enim musici propter compendium scriptionis.
ne integra semper nomina necesse esset apponere: excogitauere
notulas quasdam quibus neruorum uocabula notarentur. easque
per genera tria modosque diuisere. Simul etiam hac breuitate
captantes. ut siquando melos aliquod musicus uoluisset
ascribere. super uersum rithmica metri compositione distentum
has sonorum notulas ascriberet. Ita miro modo reperientes.
ut non tantum carminum uerba quae litteris explicantur:
sed melos quoque ipsum quod iis notulis signaretur. in
memoriam posteritatemque duraret. Sed iis omnibus modis.
unum interim lidium. eiusque notulas per tria genera
disponamus. in reliquis modis idem facere in tempus aliud
differentes. Sane si quando dispositionem notarum grecarum
litterarum nuncupatione descripsero: lector nulla nouitate
turbetur. Grecis enim litteris in quamlibet partem nunc
imminutis. nunc etiam inflexis. tota [uel rata supra lin.]
haec notarum descriptio constituta est. Nos uero cauemus
aliquid ab antiquitatis auctoritate transuertere. Erunt
igitur priores ac superiores notulae dictionis. id est
uerborum. Secundae uero atque inferiores: percussionis.
Proslambamenos [Proslambanomenos corr. supra lin.] qui
adquisitus dici potest. zeta non integrum. et tau iacens
[signum] Hypatehypaton quae est principalis principalium.
gamma conuersum. et gamma rectum [signum] Parhypatehypaton
[-f.70v-]
quae est [principalis add. supra lin.] principalium.
Beta non integrum. et gamma supinum. [signum] Hypaton
enarmonios quae est principalium enarmonios alpha supinum.
et gamma conuersum retro habens uirgulam. [signum] Hypaton
chromaticae quae est principalium [signum] alpha supinum
habens lineam. et gamma conuersum duas habens lineas [signum]
Hypaton diatonos quae est principalium extenta [Phi] grecum
et dygammon. [signum] Hypatemeson quae est principalisis
mediarum. simma et simma [signum] Parhypatemeson quae est
subprincipalis mediarum .RO. et simma supinum [signum]
Meson enarmonios quae est mediarum enarmonios .pi. grecum.
et simma conuersum. [signum] Meson chromaticae quae est
mediarum chromatica .pi. grecum habens uirgulam et simma
conuersum. per medium habens uirgulam. [signum] Meson
diatonos quae est mediarum extensa .mi. grecum. et pi.
grecum deductum. [signum] Mese quae est media. iota et
lapda iacens. [signum] Trite sinemmenon quae est tertia
coniunctarum. theta grecum et lapda supinum. [signum]
Sinemmenon enarmonios quae est coniunctarum enarmonios.
eta grecum. et lapda iacens conuersum per medium habens
uirgulam. [signum] Sinemmenon chromaticae quae est
coniunctarum cromatica [chromatica corr. supra lin.].
eta. grecum habens uirgulam. et lapda conuersum. habens
uirgulam [signum] Sinemmenon diatonos quae est extenta
gamma. et .ni. [signum] Netesinemmenon quae est ultima
coniunctarum. otomega quadratum supinum. et zetam. [signum]
Parameses quae est submedia .zeta. et .pi. grecum iacens.
[signum] Trite diezeugmenon quae est tertia diuisarum
.E. quadratum. et .pi. grecum supinum. [signum] Diezeugmenon
enarmonios. quae est diuisarum [-f.71r-] enarmonios .delta
et .pi. grecum iacens conuersum. [[signum] add. supra lin.].
Diezeugmenon chromatice quae est diuersarum chromatica
.delta. habens uirgulam. et .pi. grecum iacens conuersum.
habens uirgulam angularem. [signum] Diezeugmenon diatonos
quae est diuersarum diatonos. oTomega quadratum supinum.
et zeta. [signum] Netediezeugmenon quae est ultima diuersarum
.pi. iacens. et .ni. inconuersum diductum. [signum]
Tritehyperboleon quae est tertia excellentium .y. deorsum
respiciens dextrum. et semi .a. sinistrum sursum respiciens.
[signum] Hyperboleon enarmonios quae est excellentium
enarmonios .tau. supinum. et semialpha dextrum supinum.
[signum] Hyperboleon chromaticae quae est excellentium
chromatica .tau. supinum habens lineam. et semialpha dextrum
supinum. habens retro lineam. [signum] Hyperboleon diatonos
quae est excellentium extenta .mi. grecum habens acutam.
et pi. deductum habens acutam. [signum] Netehyperboleon.
iota habens acutam. et lapda iacens habens acutam. [signum]

🔗monz <monz@tonalsoft.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> >
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> > <wallyesterpaulrus@y...> wrote:
> > >
> > > --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> > >
> > > > However, Boethius *does* use the Greek-letter notation
> > > > in his diagrams of the modes, and for me this is the
> > > > most significant thing in his whole treatise, because
> > > > it seems to me that the Greek notation proves that the
> > > > tuning of the diatonic genus used Didymus's 5-limit ratios.
> > >
> > > How so? There were many Greek tunings of the diatonic
> > > genus, including of course Pythagorean which dominated
> > > in Europe later (c.800-1450), and myriad others. What
> > > "proves" that Boethius meant Didymus's 5-limit tuning?
> >
> >
> > Paul, you're exasperating me.
>
> The feeling is mutual.

Sorry about that. I was more testy than usual anyway,
because i was already late to go teach a lesson when i
started reading your post, and i should have just waited
until coming home from work to post my response ... i
was in way too much of a hurry.

Anyway, i'm not at all trying to "prove" that *Boethius*
meant Didymus tuning -- on the contrary, Boethius
*specifically* goes thru a laborious procedure of
monochord division to produce the standard pythagorean
tuning of the diatonic genus.

It's the Greek musical notation itself which i think
shows that musicians were using Didymus tuning.

According to Boethius's correlation of the verbose
Greek note-names with the Greek-letter notation, there
are several pairs of notes which are functionally totally
different in the various modes, but have exactly the same
Greek symbols in the notation.

In the Didymus tuning, most of these same-symbol notes
have the same pitch -- but in those which i tabulate below,
there are two triplets of same-symbol notes, two of which
are exactly the same pitch but one of which is separated
from those two by a skhisma:

(I use the ratio 1/1 for the "mese" of the Lydian mode
-- i.e., the central reference pitch of the reference mode.
All other ratios are measured up and down from that.
Modern Roman-letter equivalents are from the chart
which i posted in message 62454.)

/tuning/topicId_62290.html#62454

mode ...... note-name ..... modern.. 2,3,5-monzo ... ratio . cents

hypolydian .paramese .......... F#. [-5 3, 0> ..... 27/32 .. -294
dorian .....trite synemmenon .. Gb. [10 -5, -1> . 1024/1215. -296
mixolydian .parhypate meson ... Gb. [10 -5, -1> . 1024/1215. -296

hypolydian .nete diezeugmenon . B . [-3 2, 0> ...... 9/8 .... 204
lydian .....paramese .......... B . [-3 2, 0> ...... 9/8 .... 204
mixolydian .trite synemmenon .. Cb. [12 -6, -1> . 4096/3645.. 202

> > As i stated, Boethius himself specifies the pythagorean
> > tuning of the diatonic genus. But the Greek notation used
> > in his diagrams shows that -- IF DIDYMUS TUNING IS USED --
> > notes which are only a skhisma apart have the same symbols
> > and notes a comma apart have different symbols. Using our
> > modern notation to give equivalents, exactly the opposite
> > is true.
>
> In my opinion, that's a pretty big "IF". For one thing, the
> Didymus tuning of the diatonic scale contains a perfect
> fifth which is tuned as a "wolf".

I know it's a big "IF", but the notation seems to indicate
that there is no discernible pitch difference between
the notes, and in the Didymus tuning the difference is
indeed so small that for *practical* musical purposes
the same symbol could easily represent both pitches
with no audible harm done.

Refresh my memory: where is the "wolf-5th" in the
Didymus scale?

And anyway, i didn't assume any complete *scale*
using Didymus tuning -- i only used the ratios which
Didymus specified for the reference tetrachord:
descending, 9/8 - 10/9 - 16/15.

> > I'm sure it will finally only become crystal clear
> > when i publish my paper, *with* the Greek symbols.
>
> OK, I look forward to that. You can get a variety of
> different structures by fitting the Didymus scale with
> transposed/rotated versions of itself, depending on just
> how you go about it. The Pythagorean case is far more
> cut-and-dried, since there a tone is a tone is a tone.

Didymus specified this reference tetrachord for the
diatonic genus, descending:

. A ......... G ............ F .......... E (modern equivalents)
mese ..... lichanos .... parhypate .... hypate
...... 9/8 ........ 10/9 ......... 16/15

I didn't transpose or rotate anything -- i simply
used the ratios specified by Didymus exactly as he
stated they were used, and when i measured Boethius's
entire modal system according to that tuning, i found
the skhismic equivalents tabulated above.

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

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 22 Kasï¿½m 2005 Salï¿½ 0:27
Subject: [tuning] Re: Boethius modal system, chapter 3 of Book 4

> Hi Oz,
>
>

That's better.

>
> Well, unfortunately it's impossible for me to provide
> an example in ASCII text, because these are symbols
> based on the Greek alphabet and then rotated.
>

A pity. Nevertheless, I meant examples with latin characters.

> The best i can do right now is:
>
> 1) quote Boethius's text, which gives a detailed
> description of how the Greek symbols are written
> for each genus; i've quoted the entire chapter at
> the bottom of this post -- every place where it
> says "[signum]", the manuscript has a Greek symbol
> which, since it can't be reproduced in ASCII, also
> does not appear in the texts of the Thesarus Musicarum
> Latinarum, the source where i got the text;
>
> 2) direct you to the relevant diagrams in Boethius.
>
> Unfortunately, as i keep emphasizing, these have
> been corrupted beyond belief during the centuries of
> transmission by scribes illiterate in Greek. Anyway,
> here are the ones which are available on the internet:
>
>

I'm counting on your translations and interpretations more than those, next
to my intutions.

> from the Cambridge manuscript -- barely legible and
> thus barely useful:
>
> http://www.music.indiana.edu/tml/6th-8th/BOEMUS4C_06GF.gif
>
>
> from Friedlin's edition -- much more legible, but again
> barely useful because the symbols obviously have been
> corrupted and don't show a logical rotational system,
> but at least it gives you some idea:
>
> http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_01GF.gif
> http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_02GF.gif
> http://www.music.indiana.edu/tml/6th-8th/BOEMUS4_02GF.gif
>
>
> from Migne's edition -- again, not very legible:
>
> http://www.music.indiana.edu/tml/6th-8th/BOEDIM4_06GF.gif
>
>

Urm... These leave something to be desired.

> The three main sources of the ancient Greek notation
> are Alypius, Aristides Quintilianus, and Boethius.
>
> I highly recommend that you track down modern editions
> of all three. Bower has done considerable correction
> of the diagrams in his English translation of Boethius.
> Andrew Barker and Thomas Mathiesen have both published
> modern English editions of Aristides. Alypius is the
> best-known source of the Greek notation, and his diagrams
> are reproduced in many reference works, such as Groves
> and the Oxford History of Music. Try searching the web
> -- you'll probably find them.

Seeing as you are the expert on these sources, would it be too rude to
request that you prepare a handy list with the relevant reference data
concerning those treatises which have been at least translated to English?

>
> NOTE: there were *two different* notations in use in
> ancient Greece, one called "vocal" and the other called
> "instrumental". That's why all of these diagrams
> (including the ones you made with the different colors)
> each have a *pair* of symbols for each note.
>

No wonder people are confused!

> The vocal notation (the top symbol of each pair on
> your diagrams) stuck pretty strictly to the Greek alphabet,
> and simply used different letters for the different
> genera without employing rotation.
>
> The instrumental notation appears to me to be older,
> because in addition to Greek letters, some of the
> non-Greek symbols resemble letters from the Phoenician
> alphabet, which precedes, and in fact was the origin of,
> the Greek alphabet. This is the notation whose symbols
> were rotated for the different genera.
>
>

It would be fantastic if you and I could prepare some tables which
illustrate this point with clarity.

> >
> > What are the average intervallic values for these genera
> > of the tetrachord according to prominent ancient and
> > medival theorists?
>
>
> The most general way of describing it is as follows,
> using the notes of the meson tetrachord in descending
> order as a reference, as the ancients always did:
>
> legend:
>
> q = quarter-tone
> s = semitone
> t = tone
> m3 = trihemitone ("minor-3rd" in modern terminology)
> M3 = ditone ("major-3rd")
>
>
> ........... mese .. lichanos .. parhypate .. hypate
> diatonic ........ t ......... t .......... s
> chromatic ....... m3 ........ s .......... s
> enharmonic ...... M3 ........ q .......... q
>
>

Fabulous, but margins in cents would mean a lot. Perhaps I can assist you in
the creation of some tables with the relevant letter notations?

> If you want to get more specific than that, we'll
> have to dig into the ratios specified by numerous
> authors, and each one gave different measurements.
> Take a look at "diatonic", "chromatic", and "enharmonic"
> in the Tonalsoft Encyclopedia.
>
>

I already have. It's looks nice. Still, it would be a good idea to
incorporate all the genera in one page, including the hyper-enharmonic.

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

Cordially,
Ozan

🔗oyarman@ozanyarman.com

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

--- In tuning@yahoogroups.com, <oyarman@o...> wrote:

> > Well, unfortunately it's impossible for me to provide
> > an example in ASCII text, because these are symbols
> > based on the Greek alphabet and then rotated.
> >
>
>
> A pity. Nevertheless, I meant examples with latin characters.

I can give you HEWM notation if that helps.

> > The best i can do right now is:
> >
> > 1) quote Boethius's text, which gives a detailed
> > description of how the Greek symbols are written
> > for each genus; <snip>
> >
> > 2) direct you to the relevant diagrams in Boethius.
> > <snip>
>
>
> I'm counting on your translations and interpretations
> more than those, next to my intutions.

But as i've continually emphasized, especially to Paul,
the only way you will understand how i discovered the
possibility of schismatic temperament in Didymus tuning,
is if you see how the Greek symbols fit the lattice.

> > The three main sources of the ancient Greek notation
> > are Alypius, Aristides Quintilianus, and Boethius.
> >
> > I highly recommend that you track down modern editions
> > of all three. Bower has done considerable correction
> > of the diagrams in his English translation of Boethius.
> > Andrew Barker and Thomas Mathiesen have both published
> > modern English editions of Aristides. Alypius is the
> > best-known source of the Greek notation, and his diagrams
> > are reproduced in many reference works, such as Groves
> > and the Oxford History of Music. Try searching the web
> > -- you'll probably find them.
>
>
>
> Seeing as you are the expert on these sources, would it
> be too rude to request that you prepare a handy list with
> the relevant reference data concerning those treatises
> which have been at least translated to English?

I don't understand ... i just provided that list,
in the post of mine that you quoted.

OK, here are the complete citations, taken from the
huge Tuning and Temperament Bibliography:

Boethius, Anicius Manlius Torquatus Severinus.
_De institutione musica libri quinque_.
Beginning 6th century.
Godofredus Friedlein (ed.), Leipzig, 1867, Frankfurt a.M., 1966.
Italian translation by G. Marzi, Cremona.
English translation Fundamentals of music by Calvin M. Bower,
.. Yale University Press, New Haven, 1989.

Barker, Andrew (ed.)
_Greek Musical Writings. Vol. 2 Harmonic and Acoustic Theory_.
Cambridge Readings in the Literature of Music,
Cambridge Univ. Press, Cambridge, 1989.
(contains translations of Nicomachus, Aristides Q. and Ptolemy)

Quintilianus, Aristides.
_De musica libri tres_.
2nd cent. AD.
German translation by Arthur Jahn (ed.), Berlin, 1882.
English translation by R.P. Winnington-Ingram (ed.), Leipzig, 1963.
English translation by Thomas J. Mathiesen (ed.),
.. Music Theory Translation Series, New Haven, 1983.
French translation by Marie-Claire Lemoigne-Mussat (ed.),
.. PhD thesis, Paris, 1973, 442 pages.

Nicomachus of Gerasa.
_Manual of Harmonics_.
2nd cent. AD.
English translation and commentary by Flora Rose Levin (ed.),
.. PhD thesis, Columbia University, 1967, 275 pages.
.. Phanes Press, Grand Rapids MI, 1994, 208 pages. Also WWW, 1998.
French translation by Charles Émile Ruelle
.. _Manuel d'harmonie de Nicomaque de Gérase Pithagoricien_,
.. 1880, WWW.

Ptolemaeus, Claudius.
_Klaudiou Ptolemaiou harmoniko vivlia iii_
_Claudii Ptolemaei harmonicorum libri tres_.
John Wallis (ed.), Oxford, e theatro Scheldoniano, 1682.
.. London, 1699.
.. Reprint Ingemar Düring (ed.), Elanders, Göteborg, 1930.
English translation _The Harmonics_ by A. Barker
.. in _Greek Musical Writings vol. 2_,
.. Cambridge University Press, Cambridge, 1989.
Other English translation by Jon Solomon, E.J. Brill, Leiden, 2000.

> It would be fantastic if you and I could prepare some
> tables which illustrate this point with clarity.

The ancient Greek musical notation has now actually
been encoded by the Unicode consortium:

www.unicode.org/charts/PDF/U1D200.pdf

A font for it is available for download here:

http://ist-socrates.berkeley.edu/~pinax/greekkeys/NAUdownload.html

> > legend:
> >
> > q = quarter-tone
> > s = semitone
> > t = tone
> > m3 = trihemitone ("minor-3rd" in modern terminology)
> > M3 = ditone ("major-3rd")
> >
> >
> > ........... mese .. lichanos .. parhypate .. hypate
> > diatonic ........ t ......... t .......... s
> > chromatic ....... m3 ........ s .......... s
> > enharmonic ...... M3 ........ q .......... q
> >
> >
>
>
> Fabulous, but margins in cents would mean a lot.
> Perhaps I can assist you in the creation of some
> tables with the relevant letter notations?

The perfect-4th bounding the reference tetrachord
is generally assumed to be a 4/3 ratio, ~498 cents.
In Aristoxenus's approiximative system it probably
should be ~500 cents.

There's such a range of variability among various
authors, the only thing one can really do is provide
an approximate range of values. Measuring downward,
mese = zero cents and hypate = 500 cents.
The whole table in cents is thus:

........... mese .. lichanos .. parhypate .. hypate
diatonic .... 0 .... -200 ...... -400 ...... -500
chromatic ... 0 .... -300 ...... -400 ...... -500
enharmonic .. 0 .... -400 ...... -450 ...... -500

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

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

Hi monz,

----- Original Message -----
From: "monz" <monz@tonalsoft.com>
To: <tuning@yahoogroups.com>
Sent: 23 Kasï¿½m 2005 ï¿½arï¿½amba 9:39
Subject: [tuning] Re: Boethius modal system, chapter 3 of Book 4

I can give you HEWM notation if that helps.

---------- Possibly, until I am able to scrutinize a complete table of
discernable greek letters as used by Boethius.

-----------I already have a hunch as to what you mean, since I am entangled
with this 79 MOS 159-tET of mine that fits your descriptions. I believe it
represents well enough Ancient Greek theory too besides Maqam Music.

> > The three main sources of the ancient Greek notation
> > are Alypius, Aristides Quintilianus, and Boethius.
> >

-----------Important names...

> > I highly recommend that you track down modern editions
> > of all three. Bower has done considerable correction
> > of the diagrams in his English translation of Boethius.
> > Andrew Barker and Thomas Mathiesen have both published
> > modern English editions of Aristides. Alypius is the
> > best-known source of the Greek notation, and his diagrams
> > are reproduced in many reference works, such as Groves
> > and the Oxford History of Music. Try searching the web
> > -- you'll probably find them.
>
>
>

SNIP!

-------------Thank you for posting the relevant reference data. But where is
the treatise of Alypius?

> It would be fantastic if you and I could prepare some
> tables which illustrate this point with clarity.

The ancient Greek musical notation has now actually
been encoded by the Unicode consortium:
www.unicode.org/charts/PDF/U1D200.pdf

------------Splendid!

A font for it is available for download here:
http://ist-socrates.berkeley.edu/~pinax/greekkeys/NAUdownload.html

-------------Got it!

The perfect-4th bounding the reference tetrachord
is generally assumed to be a 4/3 ratio, ~498 cents.
In Aristoxenus's approiximative system it probably
should be ~500 cents.

----------------Wonderful, and what about the hyperenharmonic genus?

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

Cordially,
Ozan

Boethius

🔗oyarman@ozanyarman.com

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗klaus schmirler <KSchmir@online.de>

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

🔗oyarman@ozanyarman.com

🔗oyarman@ozanyarman.com

🔗oyarman@ozanyarman.com

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

🔗klaus schmirler <KSchmir@online.de>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗oyarman@ozanyarman.com

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗oyarman@ozanyarman.com

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗oyarman@ozanyarman.com

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗oyarman@ozanyarman.com

🔗oyarman@ozanyarman.com

🔗oyarman@ozanyarman.com

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

🔗oyarman@ozanyarman.com

🔗oyarman@ozanyarman.com

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗monz <monz@tonalsoft.com>

🔗monz <monz@tonalsoft.com>

🔗oyarman@ozanyarman.com

🔗oyarman@ozanyarman.com

🔗Dave Seidel <dave@superluminal.com>

🔗oyarman@ozanyarman.com

🔗Jon Szanto <jszanto@cox.net>

🔗oyarman@ozanyarman.com

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Jon Szanto <jszanto@cox.net>

🔗oyarman@ozanyarman.com