source file: m1511.txt Date: Fri, 21 Aug 1998 18:59:26 -0700 (PDT) Subject: Re: Artusi's dilemma: a "just" solution? From: "M. Schulter" 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 *************************