Many thanks to Ava Creative, who raised a question about mathematical measures for just tunings or also what I took to be near-just temperaments, especially those which have three or more generators.
Here I'll suggest a few objective and quantitative measures, generally available in Manuel Op de Coul's Scala, as well as some important qualitative measures which often involve asking what styles or genres of music one may be pursuing, and knowing current or historical musical traditions where relevant. Please note that this does mean a need to know "everything" about a tuning system or the music one will play on it before designing that system: serendipity is a very pleasant kind of artistic "known unknown."
But having certain resources in a tuning system can lend a bit of help to serendipity in exploring it, a consideration which also plays a role in what follows.
1. Economy, accuracy, and efficiency
Suppose a goal of a tuning is to get some septimal intervals, meaning ratios of primes 2-3-7 (e.g. 7/6, 9/7, 7/4). Tuning a temperament with fifths somewhere around the region of 709 cents offers economy, that is, lots of these intervals with very compact chains of fifths (-2 fifths for 7/4, -3 fifths for 7/6, +4 fifths for 9/7). However, while 9/7 will be just or near-just in this region, neither the fifths (about 7 cents wide) nor 7/6 and especially 7/4 will be comparably accurate. As the _Hitchhiker's Guide to the Galaxy_ might put it: "The Archytan or septimal comma at 64/63 (27.264 cents) is big, really big." Trying to disperse it in only two, three, or four fifths is very economic, but not, overall, especially accurate.
Other solutions, the kind especially of interest here, are more accurate but less economic. A JI solution would be Pythagorean tuning, where 7/6 (-15 fifths), 9/7 (+16 fifths), and 7/4 (-14 fifths) are all impure by 3.804 cents. That means a 24-note tuning would have 10 locations for 7/4, 9 for 7/6, and 8 for 9/7. A 3-D tempered solution is the parapyth family, for example Simplemint, with fifths at an even 704 cents and two 12-note chains of these fifths spaced at an even 58 cents (this makes the math easier). We get 7/6 at 266 cents from a regular 208-cent tone plus spacing; 7/4 at 970 cents from a regular 912-cent major sixth plus spacing; and 9/7 at 438 cents from a regular 496-cent fourth less spacing. They're all within 3 cents of just. However, taking a spacing generator as equivalent to a chain of 12 fifths, we get mappings of -13 fifths for 9/7, +14 fifths for 7/6, and +15 fifths or 7/4. That means that in a 24-note system, we get 11 locations for 9/7, 10 for 7/6, and 9 for 7/4.
We might think of "efficiency" as a combination of economy and accuracy: the ideal would be getting lots of highly accurate intervals using relatively small tuning sets. Sometimes there are shortcuts, or more efficient solutions.
For example, suppose we want virtually just fifths and ratios of primes 2-3-7-11-13. By tuning fifths minutely wide, we can get all of these primes near-just: one solution I like is 702.139 cents. Here the main complication is economy (or lack thereof): it takes +20 fifths for a near-just 13/8, and +23 for a near-just 11/8.
However, a neat solution is Erv Wilson's Rast-Bayyati or Jacques Dudon's Mohajira: we can use as our generators a larger middle third at a just 59/48 (357.217 cents), and a smaller middle third at 344.923 cents, a virtually just 72/59. A 17-note set gives us 8 locations for 13/8 and 6 for 11/8, as well as 2 for 7/4, 3 for 7/6, and 4 for 9/7.
Scala commands such as SHOW INTERVALS, SHOW LOCATIONS, and SHOW TEMPERINGS can help in measuring economy and accuracy, and getting a sense of how "efficient" a given tuning system is for this or that category of desired intervals.
2. Fifths and diversity
In systems where fifths and fourths at or close to 3:2 and 4:3 are important (e.g. for European and Near Eastern traditions), there's a balance between seeking these fifths and fourths at as many locations as possible, and seeking a nice diversity of interval sizes within a reasonably compact system. There's also a balance between desiring a specific size of interval such as a middle sixth at a just or near-just 13/8, and seeking a variety of sizes for middle sixths and other middle intervals. Near-just 3-D systems offer greater variety, at some cost in the number of locations for fifths and other desired intervals.
Thus for 13/8, we might tune a single 24-note chain of fifths at around 704.7-705.0 cents. If we tune fifths at an even 705 cents, then +8 fifths will be 840 cents, and well get 16 locations for these virtually just 13/8 middle sixths. We'll also get 15 larger middle sixths at 855 cents from -9 fifths, close to 18/11 or 64/39. So we get these two sizes of middle sixths, with 23 fifths at 3.045 cents wide.
Suppose we tune a 3-D system, Simplemint, with generators (2/1, 704.0, 58.0). Now a 13/8 is less economic, involving a regular 784-cent minor sixth (-4 fifths) plus spacing, at 842 cents, giving us 8 locations out of 24 (effectively a chain length equivalent to 16 fifths). We also get 9 locations for a larger middle sixth equal to a regular major sixth (+3 fifths) at 912 cents less spacing, or 854 cents, close to 18/11.
Thus for 13/8 and 18/11 or 64/39 alone, the 2-D solution is more efficient (much more economic, and at least comparably accurate for these ratios). However, we are tempering fifths in the 3-D solution by 2.045 rather than 3.045 cents, a full cent less.
What the 3-D solution also offers is diversity: more sizes of middle sixths. In addition to a near-just 13/8 and 18/11, we get 4 locations on each chain of fifths, or 8 in all, for a small middle sixth at 832 cents (+8 fifths), a near-just 21/13 also close to Phi (833.090 cents). In each chain we also get 3 large middle sixths at 864 cents, a virtually just 28/17, from +9 fifths, or 6 locations in all.
While the 2-D system offers perfect fifths at 23 of 24 locations, with the 3-D system we get 22 out of 24, 11 fifths for each chain -- but a "semiwolf" fifth at 714 cents connecting these two chains (e.g. G#-Eb^). This odd fifth is the price of diversity, but can be very useful in near-just gamelan tunings, for example, or the Iranian "Old Bayat-e Tork," where a fourth around 484 cents (here at 486 cents) is idiomatic.
While 3-D regular near-just systems are one approach in seeking diversity with lots of locations for fifths, another is irregular temperaments like George Secor's 17-tone well-temperament (17-WT). This provides lots of diversity, and perfect fifths at all 17 locations, with some compromise in accuracy (since the fifths must be impure by an average of 3.927 cents to make the system circulate in 17 notes).
Again, Scala SHOW INTERVALS can give a useful overview, and also SHOW /LINE INTERVALS, which gives a table for all the intervals available from each location in the tuning.
3. Commas and comma pairs.
A major motivation for tuning 3-D near-just systems, not to speak of JI systems, is more or less accurately to observe commas that define both melodic nuances and differences in vertical or harmonic color. It's nice to be able to play melodic divisions of either 12:13:14 (with the larger 13:12 step first) or 14:13:12 (with the smaller 14:13 step first), maybe at different locations, or even at the same location in one JI system I've designed. It's also nice to have a choice from the same location of either 9:12:16 or 4:6:7, which means having steps available at both 16/9 and 7/4.
In Near Eastern music, and the related Byzantine tradition, it's also nice to be able to vary the inflection of a step by a comma or so, for example moving a 9/8 step up to 8/7 when cadencing up to a step around 16/13 (thus reducing the leading tone step from around 12:11 to 14:13); or, in an Ottoman Rast, to lower the usual third at around 21/16 to an 11/9 when descending toward a final cadence. Byzantine musicians call this the "law of attraction," known in Turkish as _cazibe_ (with the modern Turkish "c" equivalent to English "j").
In just or near-just systems, there are two kinds of useful and important commas. The first are "direct commas" or "step commas," which appear as actual melodic steps in the system, e.g. in Simplemint, the 22-cent comma step between Eb in B-Eb in an Ottoman Rast at 368 cents (e.g. 26/21, 369.7 cents) and D^ in B-D^ at 346 cents (e.g. 11/9, 347.4 cents). While in certain xenharmonic contexts we might actually want to play this comma as a melodic step, the point here is that we can play either a 26/21 or an 11/9 above the same note B.
A "difference comma" is also important, but may be invisible in this sense: we don't have the comma as a direct melodic step, but have intervals at different locations that differ by this comma. For example, Simplemint has middle thirds at 346 cents (11/9) and 358 cents (16/13, 359.5 cents), although not from the same location. Rather as in an unequal well-temperament, we can choose different locations or transpositions to get either size, although not both from any one location.
Comma pairs can make a tuning system look less tidy by certain standard measures, especially if used mechanically, but are a valuable resource. As long as the chain of fifths isn't seriously disrupted, it's often true that the more difference and step commas, the merrier.
4. Constant Structures
A Constant Structure is a tuning set or system where any instance of a given interval size subtends the same number of scale steps. Note that "a given interval size" means precisely that: having different step sizes for fifths at 3/2 and 176/117 (352:351 or 4.925 cents larger than 3/2) is consistent with a Constant Structure. As Kraig Grady has explained, the Constant Structure (CS) concept is especially concerned with melody, and various compromises may arise when dealing with both melody and vertical sonorities, as in counterpoint or harmony.
Having the same mapping for each interval size in an overall system can be convenient, and I've found that it makes it easy to navigate subsets which may not themselves be CS sets. A generalized keyboard is ideal, although I've found this true with two standard 12-note keyboards for a 24-note tuning also.
5. Ajnasality or genera-lity
The Arabic _ajnas_ is the plural of _jins_, a loanword from the Greek _genus_, so that _ajnas_ could be translated in Greek or English as _genera_: and thus ajnasality or genera(-)lity, the number and variety of genera, trichords, tetrachords, and pentachords that a tuning system offers, especially those of interest for a given style or set of styles.
The use of the plural, _ajnas_ or genera, is intentional: we are focusing on having a variety of useful tetrachords and other genera, and having them in places where they can combine in desired modes, inflections, and mutations or modulations.
While some measures of "tetrachordality" or "omnitetrachordality" focus, for example, on 7-note sets with as many modes with perfectly symmetrical conjunct or disjunct tetrachords as possible, in Near Eastern music we also want pleasingly asymmetric modes, and the ability to alter one note in a given tetrachord or pentachord in order to modulate from one maqam or pattern in the modal network to another.
It's a question of having the right notes and steps at the right sizes in the right places. This is where Scala's SHOW /LINE INTERVALS can be immensely helpful.
For example, we might want a choice above the 1/1 of Maqam Rast of a usual Rast tetrachord around 1/1-9/8-16/13-4/3, or a Hijaz tetrachord with the same Rast third at around 1/1-22/21-16/13-4/3 for Maqam Hijazkar, or an alternative Hijazkar with a tetrachord around Qutb al-Din al-Shirazi's 1/1-12/11-14/11-4/3. In Simplemint, it happens that with D on the upper chain of fifths we score a trifecta: Rast at 0-208-358-496 cents; our first Hijazkar at 0-80-358-496 cents (changing the 9/8 step to a regular diatonic semitone at a near-just 22/21, with the other notes the same as Rast); and a Qutb al-Din variety of Hijaz tetrachord for Hijazkar at 0-150-416-498 cents.
While 24 notes may be a large tuning size for many people, we can also design just or near-just sets with 12 notes with 9 or t0 locations having perfect fifths and lots of useful ajnas or genera of various kinds -- and at least a few opportunities for creative inflections and modulations, e.g. from Rast to Hijazkar, or from one permutation of the Archytas Diatonic to another.
There's also the consideration that sometimes a functional and pleasing "tetrachord," at least for purely melodic purposes, may have its boundaries vary a bit from 4:3. Thus 9:8-28:27-9:8 or the like, with a fourth at a narrow 21:16 (470.8 cents) instead of 4:3 (498.0 cents), can be very handy in certain contexts.
But whether we're looking for conventional tetrachords and pentachords, or something a bit offbeat, having lots of variety in interval sizes can help.
6. Incongruity: a real-world issue (at times)
Theorists of equal temperaments speak of "inconsistency" when it isn't possible to use the best approximations for each interval in a given sonority; in 3-D tempered systems and JI systems, we may speak of "incongruity" when the same kind of situation arises.
First, let's address the argument that since an open unequal temperament (other than a closed system such as a well-temperament) or a JI system can always have more notes added, the concept of inconsistency or my "incongruity" really doesn't apply: resources are in principle infinite.
While that's a great mathematical argument, the reality is that tuning systems, especially those for hands-on fixed-pitched instruments, have a finite and often rather modest number of notes, such as 12, 17, 19, 24, or 31, etc. So how well intervals fit together for a given purpose within that size, related to the concept of "economy" (see above), is relevant -- at least at times.
Incongruity can sometimes involve small deviations from JI in a generally "near-just" system that make a difference, at least if and when the difference counts. For example, in JI, a regular Pythagorean minor third at 32/27 plus a septimal semitone or thirdtone step at 28/27 (294.1 + 63.0 cents) yields a largish middle third at 896/729 or 357.1 cents, great for a moderate Arab Rast. Likewise, a Pythagorean diatonic semitone or limma at 256/243 (90.2 cents) plus a pure 7:6 minor third (266.9 cents) yields the same Rast third at 896/729 (a reasonable facsimile for either 27/22 or 16/13, and a virtually just 59/48).
This JI equation means that we can get a fine Rast third for many purposes from a regular semitone plus a pure 7:6 minor sixth.
Now let's look at Simplemint, where we find indeed that 358 cents is a fine approximation of 896/729 at 357.1 cents or 59/48 at 357.2 cents, as well as the more familiar 16/13 at 359.5 cents. We're not too far either from al-Farabi's 27/22 at 354.5 cents. So 0-208-358-496 cents is a good Rast, as expected.
The fun comes when we try to put our JI equation into practice. We take a regular diatonic semitone or limma at 80 cents, and follow it with a near-just 7:6 at 266 cents -- and get a smallish middle third at 346 cents, a near 11/9 (347.4 cents) rather than 27/22, 16/13, etc. Surprise: incongruity has appeared on the "near-just" scene!
Make no mistake: 0-80-346-496 cents, a near-just 22:21-7:6-12:11, is a very beautiful tetrachord, a permutation of either Ptolemy's Intense Chromatic or Qutb al-Din's Hijaz (in fact a "mirror reversal" of the latter). However, the smaller middle third fits better with Ibn Sina's Mustaqim mode (1/1-9/8-39/32-4/3, with the third at 342.5 cents) than with a typical Arab Rast.
To get a Hijazkar compatible with that Arab Rast, we need to use a large middle step in the tetrachord somewhat wider than a just 7:6 -- 0-80-358-498 cents, tuning that large step at 278 cents (e.g. 168/143, 278.9 cents). This hardly a musical calamity, but an example of what incongruity means.
Let's consider another side of this incongruity: the reality that while Simplemint is "near-just" in lots of things, it doesn't have any really accurate 28/27 at 63.0 cents. Our choices are 58 cents (e.g. 91/88 or 121/117) or 70 cents (e.g. 126/121 or 176/169).
A melodic consequence of this is that if we want a really accurate version of the Archytas Chromatic, at 28:27-243:224-32:27 (63.0-140.9-294.1 cents), we might choose a different tuning system.
Our choices are to use a virtually just 13:12 (138 cents, just value 138.6 cents) for the upper 243:224 step dividing the lower 9:8 tone in this tetrachord, thus 70-138-288 cents, which gives a division of this tone not too different in its proportions from the just 63.0-140.9 cents; or 58-150 cents, which gives the best tempered versions of septimal intervals such as 7/6, 9/7, and 7/4, but has 28:27 at almost 5 cents narrow, and 243:224 at a full 9 cents wide. What we have is more like 121:117-12:11-13:11 (117:121:132:156), or 58.2-150.6-289.2 cents -- maybe an interesting footnote for John Chalmers, _Divisions of the Tetrachord_ (it's not listed in Chapter 9 in the edition that John has most generously made available on the web for downloading), but hardly an accurate Archytas Chromatic!
Another kind of incongruity can emerge when exploring some of Erv Wilson's Marwa permutations and Purvi modulations, for example. when an interval comes up like a 99/56 minor seventh at 986.4 cents (e.g. 33/28 x 3/2, 284.4 + 702.0 cents). In Simplemint, this gets mapped to 992 cents (288 + 704 cents), a reasonable 16/9 or 39/22 (at 996.1 or 991.2 cents), but without the nuance of a real 99/56, which approaches the outskirts of a septimal or 7/4 zone. Thus George Secor's 17-WT represents 7/4 as 985.6 cents. At two locations in Simplemint, as it happens, we have a better match for 99/56 (in terms of septimal shading as well as mathematical accuracy) at 982 cents -- but not in the same structural context where 99/56 arises in Wilson's explorations.
Yet another form of incongruity is this: "Oh, I have a wonderful Hijazkar on this step, but not a Rast to go with it at the same location." This is especially likely to happen in smaller tuning sets or subsets. The moral might be that ideal ajnasality or genera-lity happens -- sometimes, not always, even in one's favorite systems.
This discussion of incongruity may seem a bit detailed, but sometimes the "fine print" on advertised "near-just" systems, including mine, can stand a bit of magnification. That's especially true when lots of the relevant ratios are available in just or near-just forms, but just not in the right locations or relationships for a given sonority or tetrachord or the like.
7. User navigability or comprehensibility
While I was tempted to speak of the "complexity" of a system as it maps to a keyboard or other fixed-pitch instrument, that term is already in use as Graham Breed's complexity, more or less the inverse of economy as used here. The higher the "complexity" of a tuning for a given interval, the longer the chain of generators required to arrive at that interval.
So instead I'll speak of "navigability." As I recall, Adriaan Fokker suggested that 31 notes was a good tradeoff between the resources offered by a tuning system and the intricacy or difficulty of accessing them to a keyboard player. This is the kind of issue we're addressing here, not so much tuning size in itself as size and structure together.
A generalized keyboard is a superb tool for making just or near-just systems, especially larger ones, more navigable.
For my purposes, the regularized 24-note keyboard is another such tool, albeit a cruder one. The idea is to tune each keyboard with the same pattern of steps and intervals, with a spacing generator defining the distance between the keyboards. For a regular temperament like Simplemint (of which my favorite near-just systems are variations), this makes navigability easy, with ergonomics as a practical issue at times (especially moving with one hand from a natural on the lower keyboard to an accidental on the upper keyboard).
Being able to view the two regularized keyboards as a 3-D lattice is something I find very helpful. And I know, for example, that a comma (22 cents in Simplemint) results from a regular diatonic semitone less spacing (80 - 58 = 22 cents), e.g. E^-F or C#^-D. This facilitates choices like playing either A-C#^ for 21/16, or A-D for 4/3; and either A-F#^ for 7/4 or A-G for 16/9 (also 39/22).
Because of the diversity and often rich and intricate ajnasality or genera-lity offered by JI tunings and near-just tunings in three or more dimensions, time and in-depth study is indicated to get the most from these tunings. However, a generalized or at least regularized keyboard can help.
8. Subjectivity and flexibility
While measures like economy and accuracy can be handily assessed using Scala, some of the most important characteristics of a tuning are a matter of artistic experience and judgment, either at the design stage or in playing through and enjoying the possibilities.
For example, I find a rotation of a 7-note tuning based on a tetrachord of al-Farabi (28:24:22:21 or 1/1-7/6-14/11-4/3), another permutation of Ptolemy's Intense Chromatic, see al-farabi_g7.scl in the Scala archive (rotation on step 7), as impressively beautiful. In modern terms, this would be a variation on Athar Kurdi: 1/1-22/21-33/28-11/8-3/2-11/7-11/6-2/1 (0-80.5-284.4-551.3-702.0-782.5-1049.4-1200 cents). In Simplemint, this comes out 0-80-288-554-704-784-1050-1200 cents.
This mode doesn't have a lot of tetrachordality or locations with fifths, but is exquisite as pure melody, at least for me.
A system with lots of variety can offer such beautiful melodic modes, as well as more conventional and familiar structures like the regular diatonic (from a chain of six fifths) or a Rast diatonic (the Arab Fundamental Scale), etc.
Having a good creative mixture can be very pleasing.
Margo Schulter
23 January 2019