L. Perretti's translation of Zarlino -- enharmonic tones?

Hi there,

I've found this webpage: http://tonalsoft.com/monzo/zarlino/1558/zarlino1558-2.aspx
There Zarlino describes 2/7-comma meantone and how you can arrive at particular tones of the temperament. One of the sections says "About thickening the shown diatonic monochord, with the strings of the chromatic genus. Cap. 46". I suspect this has something to do with filling the 7-tone diatonic scale with 5 other tones to get the 12-tone chromatic scale, although I'm not sure.
Another section says "How we can thicken the mentioned monochord with the enharmonic strings. Chap. 47". The description is full of strange letters and symbols which I don't understand and I have no idea what he could have meant by "enharmonic strings" here -- perhaps his idea was to fill the 12-tone chromatic scale with 7 other tones leading to a 19-tone scale? I'm pretty unsure here but this is what I would do, at least.
Anyway, if you have any suggestions, these are welcome.

Petr

2010/4/11 Petr Pařízek <p.parizek@...>:
> Hi there,
>
> I've found this webpage:
> http://tonalsoft.com/monzo/zarlino/1558/zarlino1558-2.aspx
> There Zarlino describes 2/7-comma meantone and how you can arrive at
> particular tones of the temperament. One of the sections says "About
> thickening the shown diatonic monochord, with the strings of the chromatic
> genus. Cap. 46". I suspect this has something to do with filling the 7-tone
> diatonic scale with 5 other tones to get the 12-tone chromatic scale,
> although I'm not sure.

While we're waiting for an expert to show up, let's assume that's what it means.

> Another section says "How we can thicken the mentioned monochord with the
> enharmonic strings. Chap. 47". The description is full of strange letters
> and symbols which I don't understand and I have no idea what he could have
> meant by "enharmonic strings" here -- perhaps his idea was to fill the
> 12-tone chromatic scale with 7 other tones leading to a 19-tone scale? I'm
> pretty unsure here but this is what I would do, at least.
> Anyway, if you have any suggestions, these are welcome.

It sounds like a 19 note scale. He talks about "major semitone
divided in two Diesis" [sic] and there are 7 of the larger semitones
in the meantone chromatic.

His usage of "enharmonic" is surely related to Vicentino's. They were
both students of Willaert and that BBC program said the Zarlino was
writing in opposition to Vicentino's treatise of a few years earlier.
(Or something like that. You can listen back if you want.)

So for the word "diesis" we can turn to Vicentino, who used it about
various intervals, including the steps of the 24 note enharmonic
scale. These would be 1 and 2 steps of 31-equal, and correspond to a
division of the diatonic semitione. Search for other occurrences of
"enharmonic" and you'll find Zarlino talking about splitting both
semitones. This would give Vicentino's enharmonic scale.

He also talks about tetrachords of the ancients. The Greek enharmonic
had a semitone divided into two roughly equal parts. The basic tuning
would have been Pythagorean, and Vicentino was playing fast and loose
with history by adapting this to a meantone tuning. But, still,
Zarlino seems to be doing the same kind of thing.

The term "enharmonic" doesn't, at least, mean the same as the modern
"enharmonic equivalence". Enharmonic equivalents would normally
differ by an enharmonic diesis. Zarlino is talking about adding these
intervals in although some today assume "enharmonics" are tempered
out.

Some of the other terminology is in John Chalmers' "Divisions of the
Tetrachord" which is now available as a free download from somewhere.
That may well give you a better understanding of the Greek theories
than Zarlino had.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> Some of the other terminology is in John Chalmers' "Divisions of the
> Tetrachord" which is now available as a free download from somewhere.
> That may well give you a better understanding of the Greek theories
> than Zarlino had.
>
>
> Graham
>

Here

http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf

with OCR so you can search it.

-Carl

thank you for this Carl.

Chris

On Sun, Apr 11, 2010 at 3:49 PM, Carl Lumma <carl@...> wrote:

>
>
>
>
> Here
>
> http://lumma.org/tuning/chalmers/DivisionsOfTheTetrachord.pdf
>
> with OCR so you can search it.
>
> -Carl
>
>
>

Hi, Petr and Graham,

Petr wrote:
> I've found this webpage:
> http://tonalsoft.com/monzo/zarlino/1558/zarlino1558-2.aspx
> There Zarlino describes 2/7-comma meantone and how you can arrive at
> particular tones of the temperament. One of the sections says "About
> thickening the shown diatonic monochord, with the strings of the chromatic
> genus. Cap. 46". I suspect this has something to do with filling the 7-tone
> diatonic scale with 5 other tones to get the 12-tone chromatic scale,
> although I'm not sure.
> Another section says "How we can thicken the mentioned monochord with the
> enharmonic strings. Chap. 47". The description is full of strange letters
> and symbols which I don't understand and I have no idea what he could have
> meant by "enharmonic strings" here -- perhaps his idea was to fill the
> 12-tone chromatic scale with 7 other tones leading to a 19-tone scale? I'm
> pretty unsure here but this is what I would do, at least.
> Anyway, if you have any suggestions, these are welcome.

You are right; in chap. 46, Zarlino adds the chromatic notes to the diatonic scale he had described in a previous chapter. In actual fact, the Bb had been included with the diatonic scale, so in this chapter Zarlino adds the 4 remaining notes.
In chapter 47, he adds 7 more notes, which he calls "enharmonic strings", since he derives them someway from the ancient Enharmonic Genus, although he drifts away substantially from the antique construction.
The final result is a scale of 19 notes, as you and Graham state correctly. In chap. 47, Zarlino includes an image (which has not been included in Encyclopedia's page) representing an harpsichord with a span of 2 octaves, demonstrating his entire system, with additional keys between B-C and E-F, and with the black keys split in the direction of their length. He also states that the chromatic split keys are "colored" (painted black), while the enharmonic ones are painted red. You can find it here:
http://euromusicology.cs.uu.nl:6334/dynaweb/tmiweb/z/zarih58/@ebt-link;cs=default;ts=default;pt=919?book=zarih58;collection=z;target=IDMATCH(PART2)

Graham wrote:
>He also talks about tetrachords of the ancients. The Greek enharmonic
>had a semitone divided into two roughly equal parts. The basic tuning
>would have been Pythagorean, and Vicentino was playing fast and loose
>with history by adapting this to a meantone tuning. But, still,
>Zarlino seems to be doing the same kind of thing.

Exact. Zarkino built his system following the same procedures of the antique theorists, where the tetrachords are the "building blocks". He specially relies on Ptolemy, although he shows a deep knowledge of most of the ancient texts, which he appears to be able to read in the original Greek and Latin languages.
Anyway, Zarlino's basic tuning was not Pythagorean, rather he used two different quantities for the tones of the diatonic tetrachord (9/8 and 10/9) from which the "Coma" arose, that is fundamental for his development of the meantone temperament. He calls this tetrachord "Diatonico Sintono".

About Graham's comments on Vicentino, Zarlino does not mention him at all, at least in this part of the book (as far as I can see). The term "Diesis" is the word used by the ancients to designate the two quarter-tone intervals obtained from the division of the lowest semitone in the Enharmonic tetrachord, as Graham correctly states. Here again, Zarlino drifts away from the antique theory, and makes them differently sized, so as to obtain the desired pure intervals with the other notes.

If you think I could be of any help, feel free to ask.

Finally, many thanks to Graham and Carl Lumma for having addressed and provided John Chalmers' book. I have been searching for it unsuccessfully for a long time before.

Regards
Leonardo