Re: circulating meantones -- that ~6.07 cents

Hello, there, Joe Pehrson, and thank you for a question about
circulating meantones which not only gives me the opportunity to
clarify a point made in my "New Music in 62 Tones," but also has
provided me with an occasion to look at the problem in a way that I'll
discuss in another article.

First, you're quite correct that _a few_ major and minor thirds in a
circulating 31-note version of 1/4-comma meantone differ in size by
about 6 cents from the others -- more specifically four major thirds
and three minor thirds.

Paul, you on the other hand are, of course, correct that the _regular_
major thirds of 1/4-comma meantone (27 out of 31) at a pure 5:4
(~386.31 cents) differ by less than one cent from those of the
perfectly symmetrical 31-tET (all 31 at ~387.10 cents).

Actually, this discussion might also relate to the thread about
"microtemperament," although a 31-note version of 1/4-comma meantone
with pure 5:4 major thirds, or a 19-note version of 1/3-comma meantone
with pure 6:5 minor thirds, is already a temperament.

In these circulating temperaments, we get a few "odd" intervals
because 31 fifths at 1/4-comma narrow do not quite equal 18 pure
octaves, falling short by around 6.07 cents; and 19 fifths at
1/3-comma narrow exceed 11 pure octaves by around 0.94 cents.

If we "microtemper" the first system to 31-tET, or the second to
19-tET, then these slight asymmetries are spread uniformly. As the
microtemperament thread suggests, this can be a matter of taste.

If we choose to keep the pure major thirds of 1/4-comma meantone, or
the pure minor thirds of 1/3-comma meantone, then what kinds of
asymmetries result?

Here's a table to summarize the discussion which follows:

-------------------------------------------------------------------
31 notes: 1/4-comma meantone or 31-tET
-------------------------------------------------------------------
1/4-comma 31-tET
...................................................................
31-comma ~-6.07 cents 0 cents
(31 5ths - 18 8ves)
...................................................................
Fifths 30 at ~696.57 cents ~696.77 cents
(~5.38 cents narrow) (~5.18 cents narrow)
....................
1 at ~702.65 cents
(~0.69 cents wide)
...................................................................
Major 3rds 27 at ~386.31 cents ~387.10 cents
(pure 5:4) (~0.78 cents wide)
,,,,,,,,,,,,,,,,,,,
4 at ~392.38 cents
(~6.07 cents wide)
...................................................................
minor 3rds 28 at ~310.26 cents ~309.68 cents
(~5.38 cents narrow) (~5.96 cents narrow)
....................
3 at ~304.20 cents
(~11.45 cents narrow)
-------------------------------------------------------------------
19 notes: 1/3-comma meantone or 19-tET
-------------------------------------------------------------------
1/3-comma 19-tET
...................................................................
19-comma ~+0.94 cents 0 cents
(19 5ths - 11 8ves)
...................................................................
Fifths 18 at ~694.79 cents ~694.74 cents
(~7.17 cents narrow) (~7.22 cents narrow)
....................
1 at ~693.85 cents
(~8.11 cents wide)
...................................................................
Major 3rds 15 at ~379.14 cents ~387.10 cents
(~7.17 cents narrow) (~0.78 cents wide)
,,,,,,,,,,,,,,,,,,,
4 at ~378.21 cents
(~8.11 cents narrow)
...................................................................
minor 3rds 16 at ~315.64 cents ~315.79 cents
(pure 6:5) (~0.15 cents wide)
....................
3 at ~316.58 cents
(~0.94 cents wide)
-------------------------------------------------------------------

For 1/4-comma meantone in 31 notes, a quick answer focuses on the
amount of about 6.07 cents by which 31 fifths would fall short of 18
pure octaves.

In practice, what we do is tune a chain of 30 fifths (or 31 notes) in
a given octave, and then tune an octave of the starting note, in
effect letting the last "odd" fifth absorb the difference: this makes
it about 6.07 cents larger than the others.

Since a regular fifth in this tuning is 1/4-comma narrow, about 5.38
cents, this means that our "odd" fifth is actually very slightly
_wider_ than pure by about (6.07 - 5.38) cents, or ~0.69 cents, at
around 702.65 cents. As it turns out, our odd fifth is almost just,
much closer to pure than the regular ones.

How will this affect the sizes of major and minor thirds? It doesn't
have any effect on the 27 regular major thirds or 28 regular minor
thirds built from four regular fifths up or three regular fifths down
respectively. These intervals remain at their usual sizes: the pure
major thirds at 5:4, of course; and the usual minor thirds at
1/4-comma narrow of a pure 6:5 (~315.64 cents), or ~310.26 cents.

However, four "odd" major thirds will have that near-pure fifth as one
of the four fifths in their tuning chains, and these will be ~6.07
cents _larger_ than a pure 5:4, or ~392.38 cents.

Also, three "odd" minor thirds will have this fifth in their chains,
and these will be ~6.07 cents _narrower_ than their usual tempered
size, for a total temperament of around (6.07 + 5.38) cents or ~11.45
cents, producing a size of around 304.20 cents.

These minor asymmetries are actually much smaller than those which
occur in circulating well-temperaments of the late 17th-19th
centuries, and their main effect is simply to add a bit of
intonational variety. We get one near-pure fifth, four odd major
thirds wide by a bit less than in 1/6-comma meantone, and three odd
minor thirds somewhat more heavily tempered, but still less so than in
12-tET.

In 31-tET, a kind of "microtempered" version of this tuning, all 31
fifths are ~5.18 cents narrow, or ~696.77 cents; all 31 major thirds
are slightly wider than pure, about 0.78 cents, at ~387.10 cents; and
all 31 minor thirds are about 5.96 cents narrow, at ~309.68 cents.

How about a 19-note system based on 1/3-comma meantone? Here 19 fifths
tempered by 1/3-comma would come to about 0.94 cents _more_ than 11
pure octaves.

In practice, this means that we tune a chain of 18 fifths (19 notes)
per octave, and then an octave of the starting note. This time, in
effect, the last "odd" fifth is made _narrower_ than the others, which
are 1/3-comma or ~7.17 cents smaller than pure.

This means that our odd fifth will be _more_ impure than the others,
tempered by ~8.11 cents in the narrow direction, at around 693.85
cents. This might be comparable to the 1/3 _Pythagorean_ comma, or
~7.82 cents, by which a few fifths are tempered in some
well-temperaments. We really might prefer that the adjustment were in
the other direction, but the odd fifth remains "playable" by most
standards.

While this "odd" fifth won't affect the sizes of the 15 major thirds
and 16 minor thirds built from chains of regular fifths, it will
affect the other four major thirds and three minor thirds including it
in their chains.

In 1/3-comma meantone, the usual major thirds are tempered 1/3-comma
narrow of pure, or ~7.17 cents, at around 379.14 cents; the four odd
ones with the extra-narrow fifth in their chains will be ~0.94 cents
narrower, or ~8.11 cents, at ~378.21 cents.

The usual minor thirds in this tuning are a pure 6:5, and the three
odd ones will be _larger_ than pure by ~0.94 cents, or ~316.58 cents.

How does this compare with 19-tET, where all fifths are about 7.22
cents narrow, or about 0.05 cents more than in 1/3-comma meantone --
thus distributing that difference of ~0.94 cents among all the fifths?

In 19-tET, all minor thirds are at around 315.79 cents, or ~0.15 cents
wider than pure, much less than the 0.5 cents which Dave Keenan has
suggested as one possible standard for musical "justness."

Major thirds are all at around 378.95 cents, or ~7.37 cents narrower
than just.

Anyway, I hope that this rather long post may make it clearer what
that asymmetry of ~6.07 cents in a 31-note cycle of 1/4-comma meantone
actually refers to, and also provide a bit more detail on the
distinction between 19-tET and 1/3-comma meantone as circulating
systems also.

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@y..., mschulter <MSCHULTER@V...> wrote:

/tuning/topicId_26932.html#26932

> Hello, there, Joe Pehrson, and thank you for a question about
> circulating meantones which not only gives me the opportunity to
> clarify a point made in my "New Music in 62 Tones," but also has
> provided me with an occasion to look at the problem in a way that
I'll discuss in another article.
>

Thank you very much, Margo, for this careful post showing the
differences between 31 notes of 1/4 comma meantone and 31-tET and
between 19 notes of 1/3 comma meantone and 19-tET.

I wasn't, obviously, clear about that distinction when I first read
your article. Thank you!

_________ _______ ________
Joseph Pehrson