Yahoo Tuning Groups Ultimate Backup tuning-math "Vicentino" temperament

I'm proposing "Vicentino" as a name for the 7 and 11 limit
temperaments (with identical TOP generators, so they should get the
same name) one obtains from the 9/31 generator. The obvious objection
to this is that 7-limit, and certainly 11-limit, is not what he had in
mind. Paul will probably have a strong opinion one way or another; if
that opinion is "no" I hope he offers an alternative.

Vicentino 7-limit
Wedgie: <2 8 -11 8 -23 -48|
TM commas: {81/80, 6144/6125}
Mapping: [<1 1 9 6|, <0 2 8 -11|]

Vicentino 11-limit
Wedgie: <2 8 -11 5 8 -23 1 -48 -16 52|
TM commas: {81/80, 121/120, 176/175}
Mapping: [<1 1 0 6 2|, <0 2 8 -11 5|]

TOP generators: [1201.698520 348.7821945]
TOP tuning: [1201.698521 1899.262909 2790.257556 3373.586984 4147.308013]

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:

> > Actually, I'm not so sure this is true--didn't someone say Vicentino
> > considered 11/9 to be a consonance?
>
> No -- at least not if you mean he made any reference to any ratios of
> 11. He did find neutral thirds somewhat more consonant than most of
> the other novel intervals of his 31-tone system, though.

Half of a 1/4-comma meantone fifth is 5^(1/8), which is less than a
cent away from 11/9, so this seems like a pretty good reason to give
the 11-limit temperament with this generator and 81/80 as a comma his
name. If we halve a 1/3-comma meantone fifth instead we get a truly
tiny difference--0.0148 cents--related to the comma (11/9)^6/(10/3) =
1171561/1171470, but this seems more relevant to microtemperaments
than anything using 81/80. Someone wanting to try out methods for
getting periods and generators might try 342&494, with this comma and
a smaller than ordinary period.

Of course, we could just use the no-sevens temperament from
121/120 and 81/80 I suppose.

🔗Graham Breed <graham@microtonal.co.uk>

Paul Erlich wrote:

> No -- at least not if you mean he made any reference to any ratios of > 11. He did find neutral thirds somewhat more consonant than most of > the other novel intervals of his 31-tone system, though.

There is one reference to a ratio of 11, in Book II, Chapter 4 (p.124) where he gives the major tenth as 11:8, that is an octave above 6:5. Possibly he's adding 6:5 to 5:3.

Otherwise, along with the usual extended 5-limit ratios, he gives the following for intervals in his enharmonic scale:

14:13 for a diatonic semitone (Book V, Chapter 60, p.433)

21:20 for a chromatic semitone (Book V, Chapter 60, p.433)

13:12 for a neutral tone (Book V, Chapter 61, p.434 and again on p.435)

8:7 for a supermajor second (Book V, Chapter 61, p.436)

5.5:4.5 for a neutral third (Book V, Chapter 62, p.437)

4.5:3.5 for a supermajor third (Book V, Chapter 62, p.439)

I'd have to check through to see if he ever gives 16:15 and 25:24 for the semitones. Certainly in this section of Book V he doesn't.

14:13 and 13:12 probably come from dividing the interval 7:6 arithmetically.

5.5:4.5 is probably the average of 5:4 and 6:5. If he mulitiplied through by 2, he'd get 11:9. But he doesn't, and says it's irrational.

4.4:3.5 is probably the average of 5:4 and 4:3, and is the same as 9:7.

Neutral thirds aren't used as vertical intervals in any of his example compositions. However, because he uses implicit accidentals (and there are misprints) it'd be difficult to know if he did intend one. Still, the implication of such a method is that 5-limit harmony is assumed.

The examples in Book III, Chapter 50 (p.208) are probably intended to show neutral thirds (accidentals are implied). In each case, a step of a chromatic semitone is broken into two dieses, so the progression is minor -> neutral -> major.

There are two passages concerning neutral thirds in music. Book I, Chapter 28 compares the neutral third to the major third, but only in melody. Book V, Chapter 8 (pp.336-7) has the famous paragraph>

"If a player fails to pay attention to the proximate and most proximate consonances, he will be deceived by them, for they are so proximate to imperfect consonances that they seem identical to them. Thus, when playing the archicembalo, you may use the third larger than the minor third, that is, the proximate third that is one minor diesis larger than the minor third. This step resembles the major third without being a major third, and the minor third without being a minor third. The minor third we use below the low A la mi re [1A] is the second G sol re ut [2F#]. Its proximate is on the third F fa ut on the fourth rank [4F^], and it seems better than the minor third because it is not as weak as the minor third in comparison to the major third. Still, the proximate is somewhat weaker than the major third because it is smaller by one enharmonic diesis. Thus, the proximate or most proximate to the minor third sounds acceptable and can be played. I believe that some people sing proximate and most proximate thirds as they sharpen these minor and major consonances when performing compositions, and they do not create discords despite the fact that the former are not the same size as the latter."

There's a translator or editor's note to the effect that he might be talking about the second tuning of the archicembalo here. So the (most) proximate minor third will be an exact 6:5 (is that right?) rather than a neutral third. That would make him a tad more obtuse than usual, but I wouldn't rule it out.

Graham