Vicentino's 31+5 tuning (was Re: Desperately Seeking Just justice justness justitude Justinia)

Graham wrote:

> The first tuning, and the one most associated him, was
> essentially 31 note equal temperament spread over the two
> manuals. The 19 note manual was tuned to a meantone
> chain. (The exact tuning isn't specified, so you could
> call it 1/4 comma meantone or a subset of 31-equal.) 12
> notes of the other manual were tuned to a different
> meantone chain that you can think of as an extension of the
> 31-equal tuning. His notation distinguished between the
> two manuals, but he also used equivalent spellings that
> show he was thinking of a 31 note scale. The tuning of the
> other 5 keys of the second manual is unclear.

From what I've read so far, I've come to the following conclusion:

Similarly to describing the chromatic tetrachord consisting of a minor third and two different sizes of semitones (which one comes first is unclear), Vicentino describes similarly the enharmonic tetrachord as a major third and two dieses of different sizes (which he calls "major" and "minor" diesis), giving the major diesis the same size as the minor semitone. It's never obvious, though, which of the two comes first or if their order is of any particular significance. Anyway, he goes on to call the syntonic comma "insignificant" and effectively tempers it out by not bothering about major or minor whole tones. This makes Vicentino's enharmonic tetrachords depictable as, for example, "A_F_E#_E" (with the minor diesis on top followed by the major one) or "A_F_Fb_E" (with the dieses swapped). Later, he describes what looks like some rough approximations to his tuning and claims that two minor dieses add up to one major diesis. This statement seems to imply 31-tone equal temperament if taken literally but the context in which it appears makes me think that the two "actual" minor dieses don't necessarily have to be of one size -- and there's no great deal of stuff which would clearly support that hypothesis. However, he was, of course, very aware that 31-tet comes pretty close to quarter-comma meantone (this was well documented in the radio show we were discussing some time ago). Anyway, when applying his concept to the whole system, he apparently takes a 31-tone chain of quarter-comma tempered fifths (again, no clue where the chain begins or ends) and adds a quarter-comma higher copy of "G, D, A, E, B", which makes 36 tones in an octave. As far as I could observe, he used his "half-sharps" and "half-flats" mostly in situations where there would be double-flats or double-sharps otherwise. Now let's recall that lots of Renaissance musicians felt very uncomfortable with double accidentals. So he found it much more convenient and more appropriate to write (if I denote half-sharps and half-flats with + and -, respectively) something like C+ rather than Dbb or something like B- rather than A##. What I'm not sure about is how he dealt with triads where some tones would either require sesqui-accidentals or double accidentals -- like "A#_C##_E#" which equates in 31-tet to "Bb-_D-_F-" (maybe these are the situations of 31-tet equivalence spellings Graham is talking about). Nevertheless, the general context makes me think that he regarded 31-tet as the ideal approximation and the half-accidentals didn't actually mean a pitch change of exactly 1/2 of a "chroma" (or augmented prime). Of course, there were other people after Vicentino and some of them really did propose semi-/sesqui-alterations but they did so for representing 7-limit intervals, which makes perfect sense since if quarter-comma meantone is the starting point, then "C_Bb-" is away from 7/4 by about 1/140 of a cent! However, Vicentino never refers to 7-limit intervals. For me personally, this looks a bit puzzling -- neither do I find the evidence clearly in favor of the 31-tet view, nor do I find it clearly in favor of the view that an interval like "C+_C#" would be very close to "C_C+" but not exactly the same (and that the extra symbols were used primarily to avoid double-accidentals). Nevertheless, call me whatever you want but despite all that, I still think there are more "bits and pieces of information" supporting the latter than those supporting the former.

Petr

Petr Parízek <petrparizek2000@...> wrote:

> From what I've read so far, I've come to the following
> conclusion:

It's all in one paragraph, so I'm going to have to break it
up. That's generally bad netiquette in that it leads to
nitpicking.

This is also a complicated issue, which is why I held off
for a while. And I have to go on my memory because I don't
have the book with me.

> Similarly to describing the chromatic tetrachord
> consisting of a minor third and two different sizes of
> semitones (which one comes first is unclear), Vicentino
> describes similarly the enharmonic tetrachord as a major
> third and two dieses of different sizes (which he calls
> "major" and "minor" diesis), giving the major diesis the
> same size as the minor semitone. It's never obvious,
> though, which of the two comes first or if their order is
> of any particular significance.

He doesn't give an enharmonic octave. We can, however,
infer one from the enharmonic fourths and fifths. It would
have 24 notes. 12 of those are taken from the upper
manual. He only described the tuning of 12 notes in the
upper manual so he had no choice over these. The other 12
come from the lower manual. Each set of 12 notes is a
single chain of fifths. It isn't possible for those two
chains to join to give a single chain of fifths. The
result wouldn't be an MOS in 31-equal -- it couldn't be
described with two step sizes.

The pattern of major and minor dieses is therefore
determined by the choice of interlocking chromatics.
(Correct any of the above that's wrong.)

Note that in the ancient music Vicentino was extrapolating
from, the tuning would have been Pythagorean, and so the
two semitones were much closer in size. Either semitone
could then be roughly equally divided into dieses with no
counterparts on the chain of fifths.

> Anyway, he goes on to
> call the syntonic comma "insignificant" and effectively
> tempers it out by not bothering about major or minor
> whole tones. This makes Vicentino's enharmonic
> tetrachords depictable as, for example, "A_F_E#_E" (with
> the minor diesis on top followed by the major one) or
> "A_F_Fb_E" (with the dieses swapped). Later, he describes
> what looks like some rough approximations to his tuning
> and claims that two minor dieses add up to one major
> diesis. This statement seems to imply 31-tone equal
> temperament if taken literally but the context in which
> it appears makes me think that the two "actual" minor
> dieses don't necessarily have to be of one size -- and
> there's no great deal of stuff which would clearly
> support that hypothesis. However, he was, of course, very
> aware that 31-tet comes pretty close to quarter-comma
> meantone (this was well documented in the radio show we
> were discussing some time ago).

He doesn't say much at all about the tuning. I don't think
he was a tuning mathematician but he knew how to tune
meantone, and he based the archicembalo on that.

> Anyway, when applying his
> concept to the whole system, he apparently takes a
> 31-tone chain of quarter-comma tempered fifths (again, no
> clue where the chain begins or ends) and adds a
> quarter-comma higher copy of "G, D, A, E, B", which makes
> 36 tones in an octave.

If you don't know where the chain begins or ends, that
suggests it's being treated as an equal temperament. Does
he ever talk about a single chain of fifths? Does he say
anything about the difference in size of the minor dieses?

> As far as I could observe, he used
> his "half-sharps" and "half-flats" mostly in situations
> where there would be double-flats or double-sharps
> otherwise. Now let's recall that lots of Renaissance
> musicians felt very uncomfortable with double
> accidentals. So he found it much more convenient and more
> appropriate to write (if I denote half-sharps and
> half-flats with + and -, respectively) something like C+
> rather than Dbb or something like B- rather than A##.

Of course Renaissance musicians were uncomfortable with
double accidentals -- they couldn't be played on standard
instruments of the day! Vicentino's music published
outside the treatise uses neither the enharmonic
notation nor double accidentals.

He used his minor-diesis dots to refer to the upper
manual. He didn't say so but Margo Schulter noticed it.
The bottom manual has a standard 19 note gamut and as such
requires no double sharps or flats. The notes he used from
the upper manual constitute a 12 note gamut. Because he
wasn't using the other 5 keys on the upper manual, the
spelling of each note is uniquely determined.

Maybe those 5 notes could be tuned to meantone/31-equal
duplicates and he simplified the treatise by leaving that
detail out. When he talks about the different intervals
playable on the archicembalo, these keys seem to be tuned
to give pure fifths, like in the second tuning.

> What I'm not sure about is how he dealt with triads where
> some tones would either require sesqui-accidentals or
> double accidentals -- like "A#_C##_E#" which equates in
> 31-tet to "Bb-_D-_F-" (maybe these are the situations of
> 31-tet equivalence spellings Graham is talking about).

I don't think he ever used them. His chords were taken
from either the upper or lower chromatic. Because they
weren't a single chain of fifths, they couldn't be mixed
without producing dissonances. Occasionally (maybe only
once) he ran out of notes on the upper manual and so had to
extend its fifths onto the lower manual.

> Nevertheless, the general context makes me think that he
> regarded 31-tet as the ideal approximation and the
> half-accidentals didn't actually mean a pitch change of
> exactly 1/2 of a "chroma" (or augmented prime). Of
> course, there were other people after Vicentino and some
> of them really did propose semi-/sesqui-alterations but
> they did so for representing 7-limit intervals, which
> makes perfect sense since if quarter-comma meantone is
> the starting point, then "C_Bb-" is away from 7/4 by
> about 1/140 of a cent! However, Vicentino never refers to
> 7-limit intervals. For me personally, this looks a bit
> puzzling -- neither do I find the evidence clearly in
> favor of the 31-tet view, nor do I find it clearly in
> favor of the view that an interval like "C+_C#" would be
> very close to "C_C+" but not exactly the same (and that
> the extra symbols were used primarily to avoid
> double-accidentals). Nevertheless, call me whatever you
> want but despite all that, I still think there are more
> "bits and pieces of information" supporting the latter
> than those supporting the former.

His harmony is entirely 5-limit. He only used the
enharmonic for melodic purposes. In that he was guided by
the ancient literature.

Graham

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

> His harmony is entirely 5-limit. He only used the
> enharmonic for melodic purposes. In that he was guided by
> the ancient literature.

Which ancient literature? It was always my impression that
Vicentino thought he was reviving ancient ideas, or at least
used it as a sales pitch, but really, you know, wasn't.

-Carl

"Carl Lumma" <carl@...> wrote:
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...>
> wrote:
>
> > His harmony is entirely 5-limit. He only used the
> > enharmonic for melodic purposes. In that he was guided
> > by the ancient literature.
>
> Which ancient literature? It was always my impression
> that Vicentino thought he was reviving ancient ideas, or
> at least used it as a sales pitch, but really, you know,
> wasn't.

You'll have to read the preface. I haven't suddenly
mutated into an expert on the sources for Renaissance
music theory.

Graham

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

> His harmony is entirely 5-limit. He only used the
> enharmonic for melodic purposes. In that he was guided by
> the ancient literature.

Costeley was quite different on that score. Wikipedia sez:

A peculiarity of Costeley's style – and his notation – is that he specified the accidentals he wanted applied to his music with great care and precision, something which was unusual prior to the middle of the 16th century, but which began to occur thereafter. He was fond of unusual melodic intervals, such as the diminished third, and probably wanted to make sure they were performed correctly. Some of his chansons, for example the earthy Grosse garce noire et tendre, use this interval prominently: in this work he uses it in an imitative passage. In other pieces he uses augmented intervals, including seconds, fourths, fifths, and sixths.

Graham wrote:

> > > His harmony is entirely 5-limit. He only used the
> > > enharmonic for melodic purposes. In that he was guided
> > > by the ancient literature.
> >
> > Which ancient literature? It was always my impression
> > that Vicentino thought he was reviving ancient ideas, or
> > at least used it as a sales pitch, but really, you know,
> > wasn't.
>
> You'll have to read the preface. I haven't suddenly
> mutated into an expert on the sources for Renaissance
> music theory.

In the preface he only mentions Boethius. The notion that
the revival of ancient practice was a deceit I believe I
took from the BBC documentary. I note this problem also
arose in modern times in the 'American JI' school, where I
think it more a mistake or perhaps self-deceit. Perhaps
this was the case for Vicentino too.

Another remarkable portent of our modern movement is his
mention of the terminology problem, and his insistence on
microtonal theory being a bona fide science, yet one likely
to be reproached by the mainstream.

-Carl

This may be of good interest to read:
/tuning/topicId_26884.html#26884

Petr