TU reasoning, and Werck III and Kirnberger III examples

> > I don't see any cosmic significance attached to tuning
> > the Pythagorean comma pure. TOP tuning would make the
> > octave 0.00023 cents flat and the Pythagorean comma
> > 0.00040 cents sharp, which gives you an idea of the
> > small size of the ajustments involved. To get the
> > Pythagorean comma pure you need to tune flatly far out
> > of the optimal range for 5-limit consonances, which seems
> > fairly pointless. I would suggest simply using the integer
> > values and ignoring the tuning.
>
>Gene, (presuming to speaking for Brad) from what i know
>of his work, which is only what i've seen here and on his
>website, i think Brad is really only interested in knowing
>deviations from JI in terms of fractions of the pythagorean
>and syntonic commas, and not specifically in "tuning the
>Pythagorean comma pure".

Right. The Pythagorean comma is never tuned DIRECTLY pure, and it's folly to think that it even could be. Why would anybody ever want to do so? Why would anybody try to tune either comma directly, when either one can be derived accurately enough in less than a minute from a series of pure intervals?

The PC results from cramming 11 pure fifths into an octave (which, axiomatically, is ALWAYS pure), then doing a 12th pure fifth that ends up disagreeing with the unison by a PC. Similarly, one never tunes the syntonic comma directly, but it results from tuning four pure fifths in succession and removing a pure major third. The numbers of this stuff matter only on paper (and in computers and/or slide rules and/or abacuses).

With the measurement system scaled such that 1 PC = 720 and 1 SC = 660 and 1 lesser diesis = 1260, all the issues become very easy to visualize, with only the most basic Diophantine arithmetic and group theory.

- There are -720 little pieces of Schmutz that must be distributed somehow among 12 fifths, in total.

- There are 660 little pieces of Schmutz (but note, in the opposite direction!) that must be distributed somehow among *each* set of four consecutive fifths; there are twelve different such sets of four notes to keep track of, i.e. generating twelve major thirds.

- There are 1260 little pieces of Schmutz that must be distributed somehow within *each* of the four sets of three stacked major thirds; average 420 to each third.

Example, the overly familiar "Werckmeister III" (whose sound I dislike, but it's an easy example).

- The fifths have -180 each in C-G-D-A and B-F#, and 0 in the other fifths. Total -720.
Ab 0 Eb 0 Bb 0 F 0 C -180 G -180 D -180 A 0 E 0 B -180 F# 0 C# 0 G#

- Here are the portions sharp for each of the major thirds; average 420 (the average is always 420 as this is a closed system). They are obtained on paper by adding the tempering amounts of the four intervening fifths, and adding 660, the syntonic comma.
C-E 120
G-B 300
D-F# 300
A-C# 480
E-G# 480
B-D# 480
F#-A# 660
Db-F 660
Ab-C 660
Eb-G 480
Bb-D 300
F-A 120

- Stack those major thirds up and it is seen that each set of three adds to 1260, the lesser diesis.

C-E, E-G#, Ab-C is 120 + 480 + 660 (and it is immediately obvious that Ab-C is more out of tune than E-G# is).

G-B, B-D#, Eb-G is 300 + 480 + 480 (and it is immediately obvious that "D#/Eb" is mean between B and G).

D-F#, F#-A#, Bb-D is 300 + 660 + 300 (and D is mean between Bb and F#).

A-C#, Db-F, F-A is 480 + 660 + 120.

All the deviations from either comma are very easy to compute, for any of these thirds or fifths, by dividing into the TU of the appropriate comma. The tempered fifths are obviously 1/4 PC narrow. Some of the resulting major thirds are 1 SC wide (i.e. they're 1 PC wide, 720 units, less the schisma of 60). The major thirds C-E and F-A are *coincidentally* 1/6 PC wide; it comes from being 1/4 PC wide (180) less the schisma (60); but more accurately they're 2/11 SC wide. That's a clear example where the two commas both contribute to the placement (and to the effect) of E vis-a-vis C. Also, if we look at it from the perspective of the major thirds, the tempered fifths are 3/11 SC narrow and we shouldn't get our minds too firmly stuck on visualizing them as 1/4 PC narrow.

But, try to tell anyone that Werckmeister III is really a 3/11 syntonic comma temperament cross-dressing, and it's "Whadju talkin' about, boy?!"

The observation above about the diesis stacks can also be reduced as follows, to represent Werckmeister III as four little ordered sets of three integers, focusing on the proportional interrelationships of the major thirds, as to their errors from pure 5/4. C is {2,8,11}, G is {5,8,8}, D is {5,5,11}, and F is {2,8,11}. That describes Werckmeister III as uniquely as a line of fifths does. Equal temperament is all 7s. The "Vallotti" on electronic tuners is C {3,9,9}, G {3,11,7}, D {5,11,5}, F {3,7,11}. In some other temperaments we don't necessarily hit integers everywhere, but this exercise makes it immediately obvious how each set of three major thirds stacks up, in relative quality of intonation. In "Vallotti", the major thirds on Eb and A are identical to those of equal temperament, as can be read straight off: they're 7s within a total of 21, and they're 1/3 of a Pythagorean comma too sharp (if the PC meant anything related to thirds, which is only coincidence).

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Setting up Werckmeister III directly on a harpsichord, one doesn't need any of the numbers. Start from C and tune C-F-Bb-Eb-Ab-Db-Gb all pure. This establishes all the other ersatz sharps in their final positions. Put D between the existing Bb and F# so the quality (not the beat rate!) of these two major thirds is the same. Temper G from both C (as a fifth) and D (as a fourth) so the quality is the same, i.e. the fourth beats 3/2 as fast as the fifth. So far we've taken care of half of the PC, putting it into C-G-D. The other two quarters of it go into D-A and B-F#. Set B from the F# with the same quality as C-G next to it, i.e. a very slightly slower beat rate (since the frequencies are lower). Pure fifths down B-E-A. The resulting D-A should have the same quality as the other tempered fifths. Another checkpoint is that B-D# and Eb-G should sound equally rough. And we're done, having set all twelve of the notes by ear.

It's more important to know these shapes and relationships on the keyboard (i.e. the fragments that each of the fifths and thirds are out of tune, relative to one another), than to crunch any numbers in particular. The major thirds on F and C are the best. On G, D, and Bb are the next best. On A, E, B, and Eb they're very bright-sounding. On the remaining three they're fuerchtbar (being a full syntonic comma out of tune) but at least they're better than their counterparts in any meantone layout...and that's a reason for the existence of this temperament at all, to improve meantone layouts without really changing the relationships of which keys are worst. All the keys become roughly usable, and the ones with fewest sharps or flats have the best-sounding major and minor triads.

The knowledge of the placement of the pure or tempered fifths/fourths makes it easy to set and test all the octaves out to both ends of the keyboard. If a note has drifted out from weather or from being tapped, it becomes very easy to recognize quickly which one it is, in which direction of error, and fix it.

"Kirnberger III" is even easier. C-F-Bb-Eb-Ab-Db all pure. C-E pure major third, and E-B-F# pure. F#-Db has the schisma narrow. All that remains is to apportion C-G-D-A-E equally within the existing C-E, and that's simply 1/4 syntonic comma meantone. One can either dink around with G, D, A until all four of those fifths have the same quality, by trial and error; or do it exactly. From middle C tune the G fourth temporarily pure, and from E tune the A fifth temporarily pure. From these two tenor-range notes one can get a precisely mean D above middle C: setting it so the G-D fifth beats exactly 2/3 as fast as the A-D fourth. Then, the mean G can be dropped in similarly as a fifth and fourth above middle C and D; and the mean A as a fifth and fourth above D and E. And we're done. (Don't forget to copy those back down to the tenor octave, getting rid of the temporary G and A there!)

"This is not rocket science, or even bottle-rocket science." :) This is novice to intermediate tuning-by-ear stuff. The way to learn it is to buy a C tuning fork, leave all the electronic devices on the shelf, and work at a real harpsichord to learn how to hear these relationships among the notes. The two main commas really need to be heard first-hand, as derivations from simple series of pure intervals.

Didn't anybody wonder how I set 2/7 syntonic comma meantone, as reported recently? C-E-G# pure (E and G# being temporary), G#-C# pure. Then average out all the fifths C-G-D-A-E-B-F#-C#, all being narrow. That's easy to do by experience, knowing that all the resulting major thirds are a smidge narrow, as checkpoints. Then finish off C-F-Bb-Eb and C#-G# with the same narrowness of fifths as all these others. All the correctly-spelled major thirds have the same quality of slightly narrow. The four diminished fourths are somewhat close to 9/7 but they sound more like really awful 4/3s, with beat-rate buzzes from hell. Anyway, the derivation of the 2/7 SC should be plain to see. The use of two pure major thirds ascending makes a drop of one SC each, and then the pure fifth G#-C# backs us down to 7 steps from C instead of 8, by fifths. Ergo, 2/7 syntonic comma getting distributed between C and C#, by fifths.

1/6 syntonic comma meantone is even easier. C-E-B-F# pure. Our drop of one SC is sitting right there from C to F#; average out all six fifths C-G-D-A-E-B-F#.

1/6 Pythagorean comma meantone is also easy. C-F-Bb-Eb-Ab-Db-Gb pure (all temporary) to build the F# which is still one PC below C, since we didn't take out anything yet. Again, average out C-G-D-A-E-B-F#. If lazy, just leave all those temporary flats sitting there and call the result "Young's #2". Or, redo C-F to become one of the tempered fifths and then redo F-Bb-Eb-Ab-Db-Gb-Cb all pure, and call the result "Vallotti". (Sloppy "Vallotti", because Vallotti himself and the 17th century Venetians used SC instead of PC for this layout, but hardly anybody knows that, these days. Absorb or hide the schisma imperceptibly into the ersatz pure fifths.)

1/5 SC and 1/5 PC meantones are also trivially easy, after one has learned the above strategies. I leave that as an exercise for anyone who has actually read the preceding half dozen paragraphs!

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p.s. The Bull "Ut re mi fa sol la" goes from Cb to A#. Bach's inventions and sinfonias use 24 notes: from Bbb to Cx. The F minor sinfonia by itself uses 15 of those.

Brad Lehman

--- In tuning@yahoogroups.com, Brad Lehman <bpl@u...> wrote:

> p.s. The Bull "Ut re mi fa sol la" goes from Cb to A#.

hmm ... that's an 18-note chain of 5ths.

-monz