zarlino, ptolemy, tense

In antique and medieval discussions, the species of 4th and octave were
numbered from A, and species of 5th from D, i.e. A-B-C-D was a 1st-species
4th; D-E-F-G-A-B-C-D was a 4th-species octave; F-G-A-B-C was a 3rd-species
5th. One thing Zarlino did, after the original 1558 edition of _Le
istitutioni harmoniche_, was to propose a new ordering, starting
consistently from C. He also proposed new names for the modes in the
_Dimostrationi_, 1571. He dropped the new names, but kept the revised
numbering, starting on C, in the 1573 edition of _istitutione_.

Why the focus on C? Because it allowed an optimum harmonic division of the
octave, with harmonic divisions of the endpoints with the arithmetic and
harmonic means of the whole octave. There's a diagram, in book 1 I think,
that shows all of this:

octave divided harmonically
______________________________________
| | |

5th divided harmonically
______________________
| | |
C E F G A C
|____________|________|
5th divided harmonically

|________________|_____________________|
octave divided arithmetically

You can't get this starting on any other note. The relation to Ptolemy's
writings can be found in the middle of this scale - we get

E F G A
16:15 9:8 10:9

which is one of several tetrachordal divisions Ptolemy gives, including
five for the diatonic genus. This one, compared to most of the other
diatonic divisions, has a higher F and G, so the strings for those notes
had to be tenser. Thus the name syntonic, and Zarlino, having just read
a newly available Latin translation of Ptolemy, gave the name syntonic
diatonic to his scale of a whole octave that incorporated Ptolemy's
tetrachord.

As for Zarlino talking about major and minor, as far as I remember the
only thing along these lines is that he says you could classify the octave
modes according to whether they had a major or minor third above the
final. He doesn't seem to place as much importance on it as people
sometimes say he did. (By the way, the modes on C and A had been
described earlier, in Glarean's _Dodekahedron_ of 1547).

There was quite a debate between Zarlino and Galilei (Galileo's father)
about whether singers and instrumentalists actually use the syntonic
diatonic or not - Galilei points out some of the harsh consonances that
would arise. Zarlino ends up fudging it a bit, and says that singers
really do use the syntonic diatonic, but because of the flexibility of the
voice they can get around the harsh intervals (which amounts to admitting
they don't use it at all).

Someone also mentioned Didymus - his diatonic tetrachord was divided this
way:

E F G A
16:15 10:9 9:8

and so could be called the first true 5-limit division - he wrote maybe a
century earlier than Ptolemy. (Earlier still you get the strange 7-limit
suggestions by Archytas, like 28:27 - 8:7 - 9:8)

Bye -Jon

--- In tuning@egroups.com, jon wild <wild@f...> wrote:

http://www.egroups.com/message/tuning/15319

The relation to Ptolemy's
> writings can be found in the middle of this scale - we get
>
> E F G A
> 16:15 9:8 10:9
>
> which is one of several tetrachordal divisions Ptolemy gives,
including five for the diatonic genus. This one, compared to most of
the other diatonic divisions, has a higher F and G, so the strings
for
those notes had to be tenser. Thus the name syntonic, and Zarlino,
having just read a newly available Latin translation of Ptolemy, gave
the name syntonic diatonic to his scale of a whole octave that
incorporated Ptolemy's tetrachord.
>

Thanks, Jon! That pretty much answers my question!

____________ ____ __ _
Joseph Pehrson

--- In tuning@egroups.com, jon wild <wild@f...> wrote:
>

>
> octave divided harmonically
> ______________________________________
> | | |
>
> 5th divided harmonically
> ______________________
> | | |
> C E F G A C
> |____________|________|
> 5th divided harmonically
>
> |________________|_____________________|
> octave divided arithmetically
>
>
> You can't get this starting on any other note.

I beg to differ: you could do this starting on G (the notes would be G B C D E G).

>
> As for Zarlino talking about major and minor, as far as I remember the
> only thing along these lines is that he says you could classify the octave
> modes according to whether they had a major or minor third above the
> final. He doesn't seem to place as much importance on it as people
> sometimes say he did. (By the way, the modes on C and A had been
> described earlier, in Glarean's _Dodekahedron_ of 1547).

_Dodekachordon_ -- the modes on A were #9 and #10, and the modes on C were #11 and
#12.