Yahoo Tuning Groups Ultimate Backup tuning circulating meantones

Hello, there, Joe Pehrson, and before getting into 31-tET and
1/4-comma meantone, please let me congratulate you on your Moscow trip
as reported in the new issue of TMA, and thank you for being such a
microtonal ambassador of goodwill to a country with its own
fascinating microtonal and "ultrachromatic" traditions.

----------------------------------------
1. Paul Erlich's point: thinking "5 + 2"
----------------------------------------

Please let me open by saying that Paul Erlich has made a very
important point, one that makes my own explanation maybe a bit
simpler.

In tunings based on 12n-tET, it's natural to take as an automatic
assumption that six whole-tones equal a pure 2:1 octave.

However, for other tuning systems involving the diatonic scale, and
very specifically for the systems of Gothic and Renaissance Europe and
current offshoots, I'd echo Paul's wise advice: "Think 5 plus 2!"

That is, think of an octave as 5 tones and 2 semitones. I tend to
write "5T + 2S"; you can go by "5L + 2s" ("five large, two small"), or
any other notation you like, but the basic rule is a very important
one.

Incidentally, if you find the idea that "six tones equals an octave"
is an attractive one, you're not alone historically either: this
opinion was associated in medieval times with Aristoxenos, whatever
the actual nature of his tuning systems, and some theorists pointed
out how it did not hold under Pythagorean tuning.

Before getting into 31-tET and 31-note meantone with pure major thirds
(1/4-comma), let's quickly consider why 5 + 2 is the way to go.

If six tones are equal to a 2:1 octave -- and here we're talking about
"eventone" tunings where all regular whole-tones have the same size --
then a tone must be equal to precisely 1/6 octave, or 200 cents.

However, it's easy to show that for any Gothic or neo-Gothic eventone,
Pythagorean tuning or a modern temperament with fifths wider than
pure, a tone will be larger than 200 cents -- about 204 cents in
Pythagorean (9:8), 207 cents in 29-tET, 212 cents in 17-tET, or 218
cents in 22-tET, etc. Thus six of these tones will be _larger_ than an
octave.

How about Renaissance meantones, another form of eventone tuning? Here
we find that the tones of these tunings are _smaller_ than 200 cents,
for example around 193 cents for 1/4-comma or 31-tET. Thus six of
these tones will be _smaller_ than an octave.

-------------------------------------------------------------
2. Positive and negative tunings: prelude to the "31" systems
-------------------------------------------------------------

In fact, this basic difference has given rise to one very useful
method for classifying these regular diatonic tunings. If a system has
a whole tone larger than 200 cents, so that six whole-tones exceed an
octave, we call it a _positive_ tuning. If the tones are smaller than
200 cents, so that six fall short of an octave, then it is a
_negative_ system.

From this viewpoint, we could describe 12n-tET, where six tones are
precisely equal to a 2:1 octave or 1200 cents, as the limit of either
a positive or a negative tuning.

With Renaissance meantones, we're talking about negative tunings.
Here there are two more general points to note before we get into the
"31" tunings, 31-tET or 1/4-comma.

Since five tones plus two semitones equal an octave in any eventone
tuning, and six tones are _less_ than an octave in a negative tuning
such as a Renaissance meantone, this tells us something interesting.
In this tunings two semitones -- diatonic semitones, that is
(e.g. E-F, B-C) -- make a _larger_ interval than a tone. In other
words, a diatonic semitone is larger than half of a tone.

In 1/4-comma meantone, for example, a tone is around 193.16 cents --
exactly half of a pure major third at 5:4, or ~386.31 cents. A
diatonic semitone is around 117.11 cents, considerably larger than
half of this tone.

In 31-tET, as Paul has pointed out, we can use simple fractions. A
diatonic semitone is equal to precisely 3/5-tone, or 3/31 octave.

Our second general point is that since a diatonic semitone plus a
chromatic semitone make a tone (e.g. F-F#-G), in negative tunings such
as Renaisance meantones the diatonic semitone is larger than the
chromatic semitone.

In 31-tET, the chromatic semitone F-F# is exactly 2/5-tone, and the
diatonic semitone F#-G 3/5-tone. This makes 5/5-tone, F-G, in all.

A possible distinction between negative or positive tunings and
12n-tET is that in the former tunings, unequal semitones are
"hard-wired" into the basic diatonic structure, so to speak.

------------------------------------------------------
2. The simpler case: 31-tET with one flavor of "quark"
------------------------------------------------------

We can refer to both 31-tET and the almost identical 1/4-comma
meantone as "31-tone" systems, or cycles: 31 notes make a circulating
temperament.

Please let me emphasize that although 1/4-comma meantone is what I use
myself, often I find it very convenient to count in "fifthtones,"
ignoring the small mathematical asymmetries which only seem relevant
musically in certain special contexts, or for people concerned with
very small distinctions of intonation.

Thus 31-tET seems a logical place to begin.

There are various systems for notating the steps of a 31-tone system,
but here for the most part I'll follow the oldest known, Vicentino's
of 1555.

The rules are as follows:

A flat (b) lowers a note by 2/5-tone, the chromatic
semitone of this tuning;

A sharp (#) raises a note by the same 2/5-tone;

A diesis or fifthtone sign (*) raises a note by
1/5-tone.

In 31-tET, the basic "particle" or smallest interval of the system is
the diesis or fifthtone of 1/31-octave or 1/5-tone, about 38.71
cents. The tone consists of 5 such identical "quarks," and the octave
of 31 such quarks.

Here's a 31-tone gamut in 31-tET, moving from C3 to C4 (middle C):

T=5 T=5 S=3 T=5
|---------------|--------------|-------|--------------|
C3 C* C# Db Db* D D* D# Eb Eb* E E* E# F F* F# Gb Gb* G

T=5 T=5 S=3
|---------------|--------------|-------|
(G) G* G# Ab Ab* A A* A# Bb Bb* B B* B# C4

Thanking "5 + 2," we confirm that five tones of 5 "quarks" each, plus
two semitones of 3 quarks each, add up to 31 equal quarks per octave,
as advertised.

Incidentally, while I find Vicentino's diesis or fifthtone sign very
convenient, we can also write this system using only flats and sharps:

T=5 T=5 S=3 T=5
|----------------|---------------|-------|---------------|
C3 Dbb C# Db C## D Ebb D# Eb D## E Fb E# F Gbb F# Gb F## G

T=5 T=5 S=3
|----------------|---------------|-------|
(G) Abb G# Ab G## A Bbb A# Bb A## B Cb B# C4

Note that a double flat lowers a note by 4/5 tone (e.g. Dbb), while a
double sharp raises it by 4/5-tone (e.g. C##).

Personally I find the idea of C*, the note a diesis or 1/5-tone higher
than C, usually more intuitive than Dbb, the note 4/5-tone lower than
D; but each concept can have its uses.

The important point is that C*=Dbb, and so forth, precisely and
symmetrically: our 31 quarks, or fifthtones, are identical in size.

Now we come to 1/4-comma meantone, which as a 31-tone system often
very conveniently fits this model for lots of practical purposes, but
has its asymmetries on a fine scale.

----------------------------------------------------
3. A 1/4-comma meantone cycle: two flavors of quarks
----------------------------------------------------

For maybe 95% of practical musical purposes, we can regard a 31-note
cycle of 1/4-comma meantone as conceptually identical to 31-tET; each
tone is divided into "roughly equal" fifthtones, 31 of which make an
octave.

At a fine scale, however, we find that this model is useful but
approximate, just as the Earth is _almost_ a sphere, but in fact more
like a pear with slight asymmetries. This doesn't prevent anyone from
circumnavigating it -- either the Earth, or a circulating 31-tone
version of 1/4-comma meantone.

For the most part, we can use Vicentino's notation for either system;
and people still debate exactly how his own instruments were tuned.
His own instructions call for a common-practice tuning of the first 12
or 19 notes of his 31-note cycle, with 1/4-comma meantone appearing
(at least to me) a likelier interpretation than a systematic procedure
for 31-tET which might be difficult to tune by ear.

To show the _precise_ intonational structure of 1/4-comma meantone,
however, we may conveniently introduce a new symbol to Vicentino's
very useful notation. Let's first consider the asymmetry involved, and
then the new symbol to show this asymmetry in action.

In 1/4-comma meantone, as mentioned above, a whole-tone is equal to
precisely half of a 5:4 major third, about 193.16 cents; and a
diatonic semitone to about 117.11 cents. The difference between these
two intervals is the chromatic semitone of around 76.05 cents.

One way of defining a fifthtone in this system is through the diesis,
the difference in size between these two semitones. As it happens,
this type of fifthtone has a ratio of precisely 128:125, or ~41.06
cents.

Let's, for our present purposes, define our diesis sign (*) as raising
a note specifically by this diesis, a rounded 41 cents.

As it happens, this diesis is also the amount by which 12 fifths fall
short of 7 pure octaves -- or six tones fall short of one such octave
(this is a negative tuning).

This means that the diesis is equal to the interval we get by moving
12 fifths _down_, or in the flat direction, on the tuning circle.

To build the whole 31-note cycle, we need 30 such fifths in all. One
approach is to have 15 fifths down from C, and another 15 fifths up
from C. Let's arbitrarily build our gamut in that way.

Here we'll find an extra symbol helpful, one actually familiar from
some modern notation systems: the double sharp abbreviated as "x". For
our purposes, this symbol raises a note by _two chromatic semitones_,
or about 152.1 cents.

Using Vicentino's signs, plus this "x," here's the part of our 31-note
meantone gamut between C and D, with C* identical to Dbb (12 fifths
down from C):

(Dbb)
C C* C# Db Cx D
0 41 76 117 152 193
LF SF LF SF LF
41 35 41 35 41

The important point here is that in our 1/4-comma meantone cycle, we
have _two_ distinct flavors of fifthtone "quarks," rather than one as
in 31-tET. Instead of every quark at 1/31 octave, or ~38.71 cents, we
have two distinct sizes around 41 cents and 35 cents.

We can prove this by noting that C-C#, a chromatic semitone, is a
rounded 76 cents; C-C*, or C-Dbb on the circle of fifths, is a 41-cent
diesis. This means that the remaning interval of C*-C# must be equal
to the difference between chromatic semitone and diesis, 35 cents or
so.

More precisely, this "small fifthtone" is equal to about 34.99 cents.
People who like superparticular approximations may note that this is
very close to 50:49.

Anyway, the basic pattern to note is that a chromatic semitone
consists of a large fifthtone (LF) plus a small fifthtone (SF), or as we
might abbreviate, LF + SF.

Since a diatonic semitone equals a chromatic semitone plus a diesis
(e.g. C-C#-Db), the latter identical to our LF, that means that a
diatonic semitone is equal to two large fifthtones plus one small
fifthtone (2 LF + SF).

Now let's look at our complete 31-note gamut:

T=~193.16 T=~193.16 S=~117.11 T=~193.16
|-------------------|--------------------|----------|------------------|
C3 C* C# Db Cx D D* D# Eb Eb* E E* E# F F* F# Gb Fx G
LF SF LF SF LF LF SF LF LF SF LF SF LF LF SF LF SF LF

T=~193.16 T=~193.16 S=117.11
|--------------------|--------------------|----------|
(G) G* G# Ab Gx A A* A# Bb Bb* B B* B# C4
LF SF LF SF LF LF SF LF LF SF LF SF LF

If we break our octave down into the smallest intervals or particles
of the tuning, or large and small "quarks" or fifthtones, then we have
19 large fifthtones and 12 small fifthtones (19 LF + 12 SF). This
makes 31 in all, showing the kinship to 31-tET with its single uniform
size of quark or fifthtone.

-------------------------------------------
4. Regular and "odd" intervals -- in quarks
-------------------------------------------

Here are some intervals expressed in terms of their constituent
"quarks," with T for whole-tone, S for diatonic semitone, and C for
chromatic semitone:

8 = 19 LF + 12 SF
5 = 11 LF + 7 SF
4 = 8 LF + 5 SF
T = 3 LF + 2 SF
S = 2 LF + 1 SF
C = 1 LF + 1 SF
M3 = 6 LF + 4 SF
m3 = 5 LF + 3 SF

You may recall that in some earlier messages, we discussed the slight
asymmetries which occur in certain concords in this 31-note cycle, and
our analysis of flavors of "quarks" can show where these "odd"
intervals occur -- quite musically acceptable, but what we might call
variations on the regular sizes of fifths and thirds, etc.

To find where these intervals occur, we might consider the chain of
fifths in this 31-note cycle. Here I'll give both the 31-note notation
above, and conventional spellings for the notes on the flat side of Gb:

Eb* Bb* F* C* G* D* A* E* B* Gb Db Ab Eb Bb F C
Fbb Cbb Gbb Dbb Abb Ebb Bbb Fb Cb

Here the extreme notes of the chain are Eb* or Dbb, and Gx.

Note that the interval Gx-Eb*, or Gx-Fbb, is here something other than
a regular fifth, for example Gx-Dx (the next sharpward fifth in the
chain). Let's look again at our 31-note gamut showing the "quarks."
Since the range of the diagram is C-C, we might find it more
convenient to focus on the fourth Eb*-Gx, or Fbb-Gx.

T=~193.16 T=~193.16 S=~117.11 T=~193.16
|-------------------|-------------------|----------|------------------|
C3 C* C# Db Cx D D* D# Eb Eb* E E* E# F F* F# Gb Fx G
LF SF LF SF LF LF SF LF LF SF LF SF LF LF SF LF SF LF

T=~193.16 T=~193.16 S=117.11
|--------------------|--------------------|----------|
(G) G* G# Ab Gx A A* A# Bb Bb* B B* B# C4
LF SF LF SF LF LF SF LF LF SF LF SF LF

A regular fourth should have 8 LF + 5 SF; taking these quarks as a
rounded 41 cents and 35 cents, this gives us (41*8 + 35*5) or 503
cents -- more precisely, about 503.42 cents.

However, counting the quarks from Eb* to Gx, we actually get an
interval of 7 LF + 6 SF, or around (41*7 + 35*6) or 497 cents -- more
precisely, about 497.35 cents.

This near-pure fourth is smaller than a usual tempered one by the
difference between the two sizes of quarks -- about 6.07 cents.

How about an "odd" major third, a bit different in size than the
regular 5:4 which defines this temperament? Let's try such a third
starting from Gx.

To get a regular major third, we might look for the spelling Gx-Bx.
In 31-tET, we could define the "x" sign or double sharp as raising a
note by precisely 4/5-tone, so that Gx would be identical to Ab*, and
Bx would be four "quarks" or scale steps above B.

Since B-C is a diatonic semitone -- precisely 3/5-tone, or three equal
quarks, in 31-tET -- Bx as a 31-tET step would be one quark higher,
identical to C* or Dbb.

Let's see what happens when we try the major third Gx-C* in our
1/4-comma meantone cycle.

In this meantone, as in any eventone tuning (e.g. Pythagorean), a
regular major third is equal to two tones. Since our tone is here
equal to 3 LF + 2 SF, such a major third will have 6 LF + 4 SF, or
a rounded (6*41 + 4*35) or 386 cents -- our expected pure 5:4.

However, from Gx-C* we actually find 7 LF + 3 SF, making the interval
about 6.07 cents larger than pure -- about 392 cents, or more
precisely around 392.38 cents.

How about an odd minor third? Here we might look for such an interval
at F*-Gx, where a regular spelling would be Ex-Gx.

In 31-tET, where Ex is defined as the step four equal quarks above E,
we have E-E*-E#-F-F*; adding one more quark, F*-F#, would give us five
in all, the regular tone E-F#.

In our 1/4-comma temperament, as in any meantone (or more generally
any eventone), a regular minor third consists of a tone plus a
diatonic semitone, or T + S. This gives us an expected 5 LF + 3 SF, or
about (5*41 + 3*35) or 310 cents, more precisely around 310.26 cents.

Moving from F* to Gx, however, we find 4 LF + 4 SF, an interval of
around (41*4 + 35*4) or 304 cents -- more precisely, about 304.20
cents.

While counting quarks along our gamut is one way to keep track of the
sizes of these intervals, we could also calculate these sizes from the
accidentals, either by dividing the interval into any convenient set
of steps, or by comparing the pitch heights of the notes by reference
to any convenient point on the gamut.

A few definitions of notational signs in terms of cents may be
helpful:

a flat (b) lowers a note by a chromatic semitone
of ~76.05 cents;

a sharp (#) raises it by the same ~76.05 cents;

a diesis (*) raises it by a large fifthtone (LF)
of ~41.06 cents;

a double sharp (x) raises it by two sharps or
chromatic semitones, ~152.10 cents.

For example, for F*-Gx we could divide this into steps of F*-F#-G-Gx.
Here F*-F# is a small fifthtone of around 35 cents; F#-G a diatonic
semitone of around 117 cents; and G-Gx a double sharp equal to two
chromatic semitones, or about 152 cents. We get about 304 cents in
all. Incidentally, we describe this slightly smaller minor third as a
"tetrachroma," equal precisely to four chromatic semitones of ~76.05
cents each.

How about our slightly larger major third Gx-C*?

Here we could try Gx-A-C-C*. The step Gx-A is a large fifthtone or
diesis of about 41 cents; A-C, a regular minor third around 310 cents;
and C-C* another large or 41-cent fifthtone, for about 392 cents in
all.

How about using pitch heights, for example from C? For F*-Gx, we find
that F* has a distance from the C below equal to a usual meantone
fourth at a rounded 503 cents plus a 41-cent diesis (C-F-F*), or about
544 cents. We can consider C-Gx as C-G-Gx, a meantone fifth at a
rounded 697 cents plus a 152-cent double sharp, or about 849 cents.
The difference of (849 - 544) gives 305 cents, quite close to ~304.20
cents.

For Gx3-C*4, we again find that C3-G3-Gx3 gives about 849 cents;
C3-C*4 is an octave plus a 41-cent diesis, or around 1241 cents. The
difference of (1241 - 849) gives us an estimate of 392 cents, close to
the ~392.38 cents of this slightly large major third.

----------------------------------
5. In practice -- a bit of balance
----------------------------------

Because the topic of 31-tET and 31-note meantone is often raised, but
not so often presented so as to examine the slightly unequal
fifthtones of the latter tuning, I would like to emphasize a
conclusion stated by Ivor Darreg and others: for most practical
purposes, we can disregard the distinctions and happily perform,
improvise, or compose in either of these 31-tone systems.

Here we might note especially that the asymmetries most notably effect
only a few intervals at the extreme of a 31-note cycle -- one fifth,
four major thirds, and three minor thirds.

As Paul might quip, this doesn't mean that there wouldn't be any
tangible differences: one fifth in the 1/4-comma cycle would be
notably almost pure among the all the others a bit more than 5 cents
narrow.

The "odd" major and minor thirds might also be noticeable -- but the
variations of ~6.07 cents are still quite small by comparison with an
18th-century well-temperament, for example.

If we seek an "operational definition" of the distinction in 1555 or
1618 between tuning "1/4-comma meantone" or "31-tET," it might be
this: were major thirds deliberately made pure, or were they
deliberately and very slightly tempered in the wide direction to get
more even fifthtones?

This may be a moot question, because the instructions of which I know
about from Vicentino and Colonna simply say that the first 12 or 19
notes are tuned as in good common practice. There is no explicit
mention of pure major thirds, or of slightly compromising such thirds
to get a more uniform tuning circle.

Vicentino's fifthtone arithmetic suggests 31-tET -- which is how Lemme
Rossi (1666) interprets the system, giving string lengths accurate to
around a tenth of a cent.

Vicentino's adaptive JI tuning, in contrast, tends to suggest pure
regular major thirds, and somehow such a system seems more congenial
to tuning by ear -- Zarlino comments that 1/4-comma meantone is "not
very difficult to do."

There is a curious passage in the famous treatise of Salinas (1577)
discussing an archicembalo dividing each tone into five parts, and
discussing the sizes of the dieses. Possibly this passage might be
relevant to the kind of "1/4-comma vs. 31-tET" distinction we are
considering.

By 1666, Rossi makes it clear that at least some theorists are making
this distinction; in the Vicentino-Colonna era, the question of such a
conceptual distinction remains an open one.

Here the term "31-note cycle" may communicate a useful ambiguity,
describing a circulating tuning with "virtually equal" fifthtones
while leaving open the issues of precise purity of major thirds, or
symmetry of temperament.

Most appreciatively,

Margo Schulter
mschulter@value.net

circulating meantones

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

🔗mschulter <MSCHULTER@VALUE.NET>

🔗mschulter <MSCHULTER@VALUE.NET>

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com

🔗Paul Erlich <paul@stretch-music.com>

🔗jpehrson@rcn.com