Re: some 2/7 comma experiments on harpsichord
> Spurred by Margo's recent postings, I've tried some experimentationDear Brad,
> with 2/7 syntonic comma on harpsichord. Here are some preliminary
> comments from that. These are all from working it by ear at the
> instrument, not calculating anything.
Thank you again for devoting this kind of creative energy to a tuning
system in a very hands-on way, exploring its nuances and indeed
customizing it to your own very experienced tastes.
In my response, I'll try to keep to this spirit by avoiding too much
mathematical complexity and focusing on the sound of the intervals and
their intended musical effect. Your impressions are fascinating to
read, and come from someone who can illuminate all kinds of practical
angles that my experience simply doesn't cover.
While it appears that our recipes for tuning the _extraordinaire_ may
have varied a bit, as explained below, most of your comments are
equally applicable to my recipe. They neatly highlight some aspects of
the tuning, and questions about its purpose, which I'll try to address
here, and ask for your feedback on how well I'm doing.
I've decided to rewrite my original long response, because I realize
that the biggest issue needing clarification here may be the _purpose_
of the tuning. As will appear below, it is a hybrid circulating system
meant mainly for _new_ music often alternating between or juxtaposing
Renaissance/Manneristic meantone styles in the nearer portion of the
cycle with medieval or especially neomedieval styles using transposed
modes focusing on the remote portion of the circle.
Here's a piece I did in a mostly 16th-century style emphasizing the
nearer portion of the circle, with a few septimal touches:
Here are some neomedieval pieces and improvisations focusing on the
remote portion of the circle:
Following your admirable lead of giving first place to actual sound
and musical experience, I'm offering this sample of how the
_temperament extraordinaire_ sounds with music designed for or adapted
to it. In some of what follows, I'll refer back to these pieces to
illustrate some of your very well-taken points and explain why I find
certain compromises acceptable or even attractive for my purposes --
but not necessarily for others!
> I set up regular 2/7 all the way around from Eb up to G#. The majorBrad, you've hit on one of the attractions of 2/7-comma: if the idea
> 3rds are a little bit narrow, the minor 3rds (6:5) scarcely wide.
> The F# to Bb, B to Eb, C# to F, and G# to C work out to hit almost
> on a 9:7, and the F to G#, Bb to C#, Eb to F# are almost on 7:6.
is to optimize both 9:7 and 7:6, this is a "sweet spot." A small
clarification: in 2/7-comma, the minor thirds are also "a little bit
narrow" of 6:5, as you have noted the major ones are of 5:4.
> With this installed I played through a bunch of random pages ofFor Renaissance music, I would indeed say that a regular 2/7-comma (or
> medieval and Renaissance music, mostly vocal, from the _Historical
> Anthology of Music_ (HAM). Margo's point, as I recall, is to set up
> something suitable for accompanying this type of repertoire.
1/4-comma) is hard to outdo, in comparison to _any_ 12-note
circulating scheme! For medieval music, I'd likewise recommend the
usual period tuning of a regular Pythagorean -- or possibly, for a
discreetly "modern" variation, something like Peppermint where all
regular fifths are a tad more than 2 cents (66 TU) wide of 3:2.
My circle, in comparison, can hardly be ideal if we are aiming
consistently for either type of period sound. As compared with a
regular meantone, some very important pure or near-pure thirds are
compromised; as compared with Pythagorean or even a modern variant
such as the gentle Peppermint, the fifths are impure by 6-7 cents!
This will come out more in your remarks below and my responses.
The purpose the _temperament extraordinaire_, rather, is primarily to
accommodate new music which is free to transpose any mode to any step,
or cadence on any of the 12 fifths, suiting the style to a given
region of the circle -- now Renaissance meantone, now neomedieval,
sometimes with striking transitional colors or "fringe effects."
Actually I _do_ use this tuning for playing historical Renaissance
pieces in the nearer portion of the circle (Bb-G#, with Eb calling for
discretion as discussed below), and like the result -- but a regular
meantone would, as you note, be more idiomatic and consistent both
harmonically and melodically with a period ethos.
> Then I tried her recommendation of converting this to _temperamentPlease let me join you in noting the difference between the French
> extraordinaire_ (her nomenclature, not mine; I'd already co-opted
> the phrase "extraordinary temperament" to mean something different,
> a couple of years ago...).
term _temperament extraordinaire_, which I've been using since 2003
describe a modified 1/4-comma or 2/7-comma meantone with eight narrow
and four equally wide fifths; and "Bach's Extraordinary Temperament,"
the title of your articles of 2005.
I've never thought of translating my French term literally into
English, and your personal trademark now clearly stamped on the phrase
"extraordinary temperament" is another reason not to do so.
> That is: stretch F-Bb-Eb wide each, and F#-C#-G# wide each, so itPlease let me clarify that my recipe for the _extraordinaire_ is a bit
> eradicates the wolf G#-Eb and reduces it to four other lesser
> canines, all the same as one another. In practice at harpsichord
> this is simply a process of cranking the Bb and Eb downward in turn,
> and the C# and G# upward in turn, until we have similar 5ths
different in that only three notes are adjusted: Bb, Eb, and G#. The
following diagram using our beloved TU may make this clearer:
F C G D A E B F# C# G# D#/Eb A# F
-189 -189 -189 -189 -189 -189 -189 -189 +198 +198 +198 +198
One aural recipe for use at the harpsichord might be to tune the eight
fifths F-C# in a regular 2/7-comma, thus setting nine of the 12 notes.
Then, _without changing any of these notes_, temper G#, D#/Eb, and Bb
so that the near-9:7 major third or diminished fourth C#/Db-F already
set is derived from a chain of four equally wide fifths.
Specifically, in my recipe, C# is not adjusted, so that A-C# stays a
regular meantone third and C#/Db-F stays at its near-9:7 size. This
changes the details of the compromises involved, but not their basic
nature, which you well describe.
You make a very important point about the wide fifths:
> They're all a little bit rougher than the 2/7 comma 5ths, and ofThank you for this invaluable aural feedback, which for me is really a
> course they're rough in the opposite direction, but given that all
> these 5ths/4th are growly anyway they don't sound too bad in
> context...playing them as open 5ths/4ths, without any 3rds (whether
> major or minor) in them.
critical point: in a medieval or neomedieval styles with their remote
modal transpositions, those wide fifths are indeed going continually
to be used as stable consonances without any thirds, in this context
definitely unstable intervals.
> But, does it work? I played some more from HAM, some of the sameHere I might be curious as to whether these were mainly medieval or
> compositions and some others randomly. To the extent that the notes
> Bb, Eb, G#, and C# came up, which was rarely, they always stood out
> to me as not fitting their context when played in major or minor
> 3rds. Neither fish nor fowl. Especially in the case of Bb, the
> most frequent of these four notes, it seemed to me there's too much
> difference in character between the Bb major triad and the F major
> triad, and between the A-Bb semitone vs the E-F semitone.
Renaissance pieces from HAM, or some mixture. I'll focus here on a
Renaissance style, since typically with medieval pieces I would make
remote transpositions (e.g. D Dorian to Eb or Bb) that would bring
other intervals and issues into play, some addressed below.
So that people can hear some of the things you discuss in relation
both to altered third sizes and melodic semitones in a Renaissance
meantone kind of style, here's a link repeated for convenience to my
_Intrada in F Lydian_, which prominently uses lots of the accidentals
and intervals you mention. Apart from the matter of C#, which as I
noted is not modified in my recipe, all of your comments apply:
Thus while usual meantone thirds are at 383 cents, a tad narrow of 5:4
as you have said, Bb-D is around 396 cents, and Bb-D-F often occurs
close to a near-pure sonority such as F-A-C or D-F#-A. While I might
call this "a stimulating touch of modal color," it's definitely a
compromise for a period sound where the uniformly pure or near-pure
thirds of regular meantone are the norm.
Also, the major third Eb-G, used here in the sixth sonority Eb-G-C, is
actually at 408 cents a tad larger than Pythagorean; while I much like
the color of this sonority and also C-Eb-G in a Renaissance as well as
neomedieval context, it is in the former setting a dramatic departure
from the smooth 5:4 sound again offered by a regular meantone.
Your point about the different melodic sizes of A-Bb and E-F is also
right on. While E-F is a regular 2/7-comma semitone at 121 cents, the
altered A-Bb is about 108 cents (comparable to 1/5-1/6 comma), while
Eb-D, featured in some cadences, is 96 cents. One can relish this
variegation, or have more mixed feelings, but it is again a departure
from a regular meantone where all these steps would be 121 cents.
> The Bb-D is scarcely wide as a major 3rd, and pleasant enough, butCertainly the different is there -- 384 vs. 396 cents. From a
> it doesn't have that tight center of gravity like the major 3rds on
> the naturals.
21st-century perspective, I might argue that in a Renaissance kind of
style with restful thirds, Bb is a less likely modal final than the
naturals (other than B), and so, in a circulating system, can better
afford some compromise. However, that argument is double-edged, as
will appear shortly! In any event, for a period sound, regular
meantone would, of course, leave Bb-D near-pure like the naturals.
> Similarly, A-C# is only a little bit wide but it's lostWhile A-C# is in my scheme unaltered, what you say about it definitely
> its gravity. And Eb-G and E-G# are watered down even more, each.
> They're still smaller than Pythagorean ditones and they're pleasant
> enough, in isolation, but they don't seem to fit their context.
could apply to E-G# at 396 cents, the "double edge" of my possible
argument regarding Bb-D, since E _is_ a natural, and the untransposed
final for one of my favorite modes, E Phrygian, which comes up
continually. While some other things are going in the piece which
follows, I'd like people to hear the compromise of closing in Phrygian
on that 396-cent third, which I accept, but wouldn't happen in a
> Worse yet: we haven't really gained any usable B-D#, F#-A#, Ab-C, orCertainly I'd agree, as a septimal fan myself, that having only one
> Db-F. They were kind of OK and attractive in their own way (while
> exotic) when they were hitting nearly 9:7, tuned regularly. But
> now, having these modified 5ths, these misspelled major 3rds are in
> a no-man's-land. They just sound like "wrong notes" giving nothing
> for my ear to grab onto, or to take me past the wincing.
near-9:7 major third at Db-F, rather than four, is a compromise --
although one not so unpleasant for me, since in a neomedieval setting
I also relish thirds around 408 or 421 cents, or anywhere along the
spectrum from Pythagorean to septimal. This can be very much a matter
of taste, with one person's "pleasantly variegated neomedieval thirds"
being another person's "wrong notes."
This more remote terrain in the circle is rather like a distinctively
modern circle of another kind: a 17-tone well-temperament such as
George Secor's of 1978, with major thirds at 418-429 cents. While
Marchettus of Padua (1318) and the modern scholar and performer of
medieval music Christopher Page provide a basis for concluding that
thirds of around this size may have occurred in 14th-century
performances of vocal music, this is above all a 20th-21st century
keyboard concept rather than an historical one!
Why don't I repeat for convenience the links to neomedieval pieces so
that people can hear these thirds at 408 and 421 as well as 434 cents.
It might not hurt for me to emphasize that these thirds are generally
intended for use in a neomedieval context where they contrast with and
often expand to stable fifths; or in a Renaissance context as meantone
diminished fourths (B-Eb, C#-F, F#-Bb, G#-C). Apart from Eb-G at about
408 cents or Pythagorean, which I might use in a meantone setting in
Eb-G-C or C-Eb-G, they are _not_ meant for use in a Renaissance style
as usual thirds.
Also, your remarks point to a basic consequence of circulation. The
extreme values for major thirds, 383 and 434 cents, or very close to
5:4 and 9:7, are identical to those in a regular 2/7-comma. However,
averaging out that Wolf fifth into four wide fifths results also in
the intermediate values of 396, 408, and 421 cents. While I like these
intermediate shades, they inevitably result in fewer major thirds at
either extreme, at a near-just 5:4 or 9:7.
[On your own preferred modification of 2/7-comma]
> C#: back to its regular 2/7 comma place from F#, or tune it as aThis is very creative: to take my septimal theme, but realize it in a
> pure 5:4 from A-C#, or tune it as pure 7:6 with Bb, or tune it as
> pure 9:7 from F. Or, simply as a pure 3:2 from F#. All five of
> these spots are arguably good for C#. Following Margo's lead I
> favored the two septimal spots. Why hit only near a 7:6 or a 9:7
> when you can hit directly on one of them? It makes the minor
> triads, diminished triads, and tritones lock in that way, too.
different way that leaves the tuning quite close to its regular
historical form! I wonder if your having altered C# in your version of
the _extraordinaire_ recipe for circulation might have facilitated
this happy result.
> All of this writes off the G#-Eb wolf as a loss...but it's not goingHere the different, of course, is that for my remote neomedieval
> to come up in this medieval and Renaissance music anyway.
transpositions Ab-Eb is going to come up all the time in Ab Mixolydian
or Eb Dorian or Gb Lydian, etc. Where this kind of circulation isn't
relevant, however, your solution very subtly optimizes the septimal
intervals while leaving the smooth Renaissance meantone color
> And by going back to regular 2/7, or nearly so (choosing septimalIt's wonderful that your exploration led you to this happy conclusion
> placements on C#, G#, and Eb), we've got a fantastic set of eleven
> minor triads, eight tightly attractive major triads, three septimal
> major triads, and all the other related emoluments. The 5ths
> F#-C#-G# and Eb-Bb work out decently (for context) no matter what we
> do, given that the 2/7 comma 5ths on the naturals are already rough.
bearing your own distinctive stamp!
> If the point is to accompany medieval or Renaissance [vocal] music,Another way of phrasing this question: "What price circulation?" If I
> regular 2/7 works better (at least for me) than the 5th-stretching
> _extraordinaire_ step does. If the point was to get those septimal
> resonances going, why move off them by widening the 5ths?
wanted to play in a Renaissance style around the circle, it wouldn't
make sense. If something like Ab-C-Eb did occur, say in Schlick or an
extended meantone piece assuming more than 12 notes per octave, having
Ab-C at 421 cents would be just as much a barrier as a Wolf at G#-Eb.
The answer is neomedieval styles in remote transpositions -- at the
cost of fewer septimal resonances: Db-F at a near-9:7 as in the
regular meantone, plus F-Ab and Bb-Db at around 275 cents, or not too
far from 7:6. Now we have a 12-note circle where Bb-Db-F, for example,
can cadence beautifully to Ab-Eb in Ab Dorian or Mixolydian, say. I'm
quite happy with Eb-Gb and C-Eb at 287 cents also, another favorite
size, so it's an amenable compromise to me -- but still a compromise!
In short, from the viewpoint of septimal optimizations, I'd consider
this one of the best 12-note circulating systems -- but not so good as
a regular meantone!
> I could change my mind tomorrow, but that's how I hear it today.Again, thank you for your honesty and directness, as well as your
> And all of this might work better (or worse?) on organ as opposed to
> harpsichord. On harpsichord, at least, the septimal intervals are
> clear and pungent.
experience and musical sensitivity. You also bring up an interesting
question regarding timbres: do I tend to select timbres on
synthesizer, for example (which I often consider a kind of organ)
where major thirds at 421 cents, say, might have a different effect
than on harpsichord? Anyway, I'm delighted that your septimal
variation fits your instrument well.
> And if the notes D#, A#, Ab, and Db aren't going to come up in theYour appreciation of 2/7-comma may be rather like mine: a certain
> music anyway...why not just stick with regular 1/4 instead of 2/7?
> It's sort of a trade-off here between the major 3rds and minor 3rds.
> But if we wanted pure minor 3rds, why not go all the way to 1/3
> comma then? 2/7 goes down between these, giving the strengths of
> both...and because those 3rds aren't pure, they're (arguably) more
> attractive just by virtue of being livelier, not sitting there
> dead-on. Every listener might of course have a different opinion
> about that.
balance for major and minor thirds with both gently beating. Also, I
love those just 25:24 minor semitones (kept at C-C# and F-F# in my
_extraordinaire_). They're beautiful in 16th-century chromatic
progressions, and excellent as regular neomedieval semitones, very
close to George Secor's ideal size of around 67 cents.
Please forgive the length of my response, but take it as a measure of
how thoughtful and significant I find your comments.
- --- In firstname.lastname@example.org, Margo Schulter <mschulter@...> wrote:
>it is also possible to read Werckmeister's #3 pattern
>.... it occurred to me that if the
> "comma" may be either Pythagorean or syntonic, with the schisma
> regarded as not so important, then why not 1/3-syntonic comma
> tempering for the narrow and wide fifths alike?
> ! werckmeisterIV_variant.scl
> Werckmeister IV with 1/3 syntonic comma temperings
> ! WerckmeisterIV_variant_c.scl
> Werckmeister IV variation, 1/3-SC, all intervals in cents
> The 1/3-comma variation seems
> to fit this model -- at least if, like Costeley (1570) and Salinas
> (1577), we are ready to accept fifths tempered by this great a
> quantity, as in a regular 1/3-comma meantone or 19-EDO. Zarlino (1571)
> found 1/3-comma temperament "languid," ....
C~G~D~A E B~F#...C
in 1/3 SC terms:
C 242/243 G 241/242 D 240/241 A E B 32768/32805 F# C# G# D# Bb F C
as refinement of his JI tuning presented in his book:
"Musicae mathematicae hodegus curiosus"
FFM 1687: p.71: a'=400cps
extracted from his "Natï¿½rlich" (natural) scale,
there defined in absolute pitch-frequencies:
c" 480 cps
a" 800 overtaken from Mersenne's reference-tone a'=400Hz
The W3 pattern can be understood as
modification of layout pattern,
in absolute terms,
as cycle of partially tempered 5hts:
Db 1 unison, implicit contained in his absolute "hodegus" tuning
F 81 (>80+2/3 (>80+1/3 (80 40 20 10 5)))
C 243 (>242 (>241 (>240 120 60 30 15)))
G (729 >) 726 (>723 (>720 360 180 90 45))
D 2169 (>2160 1080 540 270 135)
A 405 compare to Chr. Hygens(1629-95) Amsterdam determination:~407 Hz
F# (10935=32805/3 >) 32768/3 ... 1/3
C# 1 returend back unison again
that's relative in chromatically ascending order as Scala-file:
Werckmeister's famous C~G~D-A-E-B~F#...C pattern as 1/3 SC + schisma
!C 242/243 G 241/242 D 240/241 A E B 32768/32805 F# C#=Db Ab Eb Bb F C
256/243 ! Db=C# enharmonics @ absolute Mersenne's 256cps unison
241/216 ! D
32/27 ! Eb
5/4 ! E
4/3 ! F
1024/729 ! F#
121/81 ! G = (11/9)^2 = (3/2)*(243/242)
128/81 ! Ab
5/3 ! A
16/9 ! Bb
15/8 ! B (german H)
That one contains more pure intervals than other interpretations.
if you have some better ratios for W3 -even nearer to JI?-,
please let me know about that.