Artusi's dilemma: a "just" solution?

In his writings of 1600 and 1603, the theorist Giovanni Maria Artusi
proposed what is literally an impossible task: defining an equal
division of the tone into two parts (and of the octave into 12) using
only "known and specified rational numbers."[1]

Almost 400 years later, I would like to propose a just 5-limit tuning
that seems _virtually_ to solve Artusi's dilemma -- if we are
permitted a very small quantity of imprecision which I propose below
might be named the "Wilsonian comma" in honor of Ervin Wilson -- if
there is not already another ratio bearing this designation, and if
Erv likes the idea. We shall also encounter an even smaller quantity
for which I tentatively propose the name of "Artusi's scintilla," if
it doesn't already have a name familiar to a specialist in
nomenclature such as Manuel Op de Coul.

At the outset, please let me caution that this tuning may be more fun
as a thought experiment than as a scheme to implement on a harpsichord
or the like, due to the very large number of just intervals which
would have to be tuned with requisite precision.

Also, while I'm not sure if the complete 5-limit scale given below has
been published, once one is familiar with the concept of a "schisma
fifth" (described by Owen Jorgensen, for example), the rest of the
scheme is easy to formulate -- although not necessarily to tune on an
actual instrument.


-----------------------------
1. The solution in a nutshell
-----------------------------

To devise a just 5-limit tuning that divides the octave into 12
_virtually_ equal parts, we need only two just intervals as basic
ingredients. The first is the Pythagorean schisma major third
(8192:6561, about 384.36 cents) derived from a chain of eight pure
fourths up or fifths down. This 3-limit interval is smaller than a
pure major third (5:4, about 386.31 cents) by precisely a schisma
(32805:32768, about 1.95 cents).

Our second interval is a 5-limit minor third (6:5, about 315.64
cents).

When combined, these intervals generate a "schisma fifth" of
16384:10935 -- about 700.00128 cents. For the slight discrepancy of
some 0.00128 cents between this fifth and a true 700 cents, I propose
the term "scintilla of Artusi" -- a scintilla, possibly a new category
in tuning theory, being somewhat smaller than a typical comma or
schisma.

Using this schisma fifth as a generating interval, we can achieve an
_almost_ closed chain of 12 such fifths which will produce a
_virtually_ pure octave. The slight variance of this octave from a
true 2:1 -- the proposed "Wilsonian comma" -- is equal to an interval
of

2923003274661805836407369665432566039311865085952:
2922977339492680612451840826835216578535400390625

or approximately 0.015361 cents -- just 12 times the size of Artusi's
scintilla, of course.[2]


-----------------------
2. Intervals and ratios
-----------------------

Here are the intervals for this 5-limit just tuning, listed in the
order they arise from a chain of 12 fifths. Note that in a 12-note as
opposed to open spiral tuning, where octaves are tuned to a pure 2:1,
our 12th fifth would be a Wilsonian comma narrow of 700 cents.


---------------------------------------------------------------------
Note Interval Ratio/
(Cents)
---------------------------------------------------------------------
c Unison 1:1
(0.000000)

g 5 16384:10935
(700.001280)

d M2 134217728:
119574225
(200.002560)

a M6 2199023255552:
1307544150375
(900.003840)

e M3 18014398509481984:
14297995284350625
(400.005120)

b M7 295147905179352825856:
156348578434374084375
(1100.006400)

f# A4/d5 2417851639229258349412352:
1709671705179880612640625
(600.007680)

c# A1/m2 19807040628566084398385987584:
18695260096141994499225234375
(100.008961)

g# A5/m6 324518553658426726783156020576256:
204432669151312709849027937890625
(800.010241)

d# A2/m3 2658455991569831745807614120560689152:
2235471237169604482199120500833984375
(300.011521)

a# A6/m7 43556142965880123323311949751266331066368:
24444877978449625012847382676619619140625
(1000.012801)

e# A3/4 356811923176489970264571492362373784095686656:
267304740694346649515486129568835535302734375
(500.014081)

b# 8 5846006549323611672814739330865132078623730171904:
2922977339492680612451840826835216578535400390625
(1200.015361)
_____________________________________________________________________


Maybe these intervals, like the Pythagorean schisma third of 8192:6561
which plays a vital role in making this tuning possible, suggest a new
trend for just intonation in the 21st century: "Large integer ratios
are beautiful."[3]


----------
Notes
----------

1. See Mark Lindley, _Lutes, viols and temperaments_ (Cambridge:
Cambridge University Press, 1984), pp. 84-92, for a very witty account
of Artusi's intonational views expressed in the course of his
controversy with Claudio Monteverdi, including translations of some
choice passages. Artusi's demand that valid divisions of the tone for
voices be made "with known and specified rational numbers" (_con
certi, & determinati numeri rationali_) is quoted and translated at
pp. 91-92.

2. Artusi's scintilla may be formally defined as the difference
between a schisma (about 1.95372 cents) and 1/12 of a Pythagorean
comma (about 1.95500 cents, the amount by which fifths are narrowed in
12-tet).

3. While Pythagorean schisma thirds occur in Arabic and Persian as
well as medieval European theory and practice, our schisma fifth
scheme would require an amount of tuning calculated to make extended
Pythagorean tunings look simple. To derive _each_ of the 11 schisma
fifths in a 12-note scheme, we would need to tune a series of 8 pure
fifths defining a schisma M3 up or m6 down, plus a pure m3 up or M6
down. These alternatives would permit one to remain within a chosen
bearing octave such as c'-c''.


Most respectfully,

Margo Schulter
mschulter@value.net

------------------------------

End of TUNING Digest 1511
*************************

Thanks to Margo Schulter for the paper in TD 1513! I found it thought
provoking and informative, especially the parts about regarding the consonance
of an interval as a point on a spectrum rather than a binary choice.

That's helped me make more sense of a set of 7-(prime)limit, 12-tone/2:1
tunings, the heptatonic subsets of which I've been exploring for use with what
might be thought of as neo-/Xeno-Medieval/Gothic/Rennaissance compositions
(for lack of more concise term :-). Those tunings are built of five
alternating 21:20s and 15:14s, with two 256:243s sandwiched in here and there
to fill out the octave. Note that 9:8 can be divided into 21:20 and 15:14.

Here's my 2-cents-worth on a few minor points from the paper:

>Thus while both 13th-century and 18th-century Western European music
>are harmonically oriented, they are "harmonic" in very different ways,
>differing in their historical tunings, stable and unstable interval
>categories, and directed cadences. Curiously, certain scales such as
>53-tet can provide an excellent fit with _either_ harmonic system.

That may not be as surprising as it seems. I think that the more finely you
divide the octave, the more likely it is that you will have "an excellent fit"
with a set of relatively smaller (in terms of tones per octave) systems when
one system's prime limit encompasses the other's, as in the case of Western
18th century 5-limit and 13th century Pythagorean.

>most complex prime number

Is it accepted practice to refer to primes as "complex"? Aren't primes, by
definition, the simplest whole numbers? Perhaps "highest prime number" would
be a better way to state it?

>the septimal minor
>seventh (7:4) also becomes literally and figuratively a "prime factor"
>in defining the scale,

Figuratively? How do you mean?

Thanks again, Margo, for an inspiring work!

David J. Finnamore
Just tune it!

------------------------------

End of TUNING Digest 1516
*************************