Re: [tuning] Re: More on 29-tet -- a beautiful tuning

"M. Schulter" wrote:

>
> >From my musical perspective, 29-tet is a subtly accentuated version of
> a standard medieval Pythagorean tuning. The fifths are tempered by
> around 1.49 cents in the wide direction (and fourths in the narrow
> direction), leaving these intervals quite close to pure.
>

I got around this problem of purity of 4ths and 5ths by stacking two symmtrical keyboards on
top of each other. I literally hacked off the bottom of one keyboard so that it could sit at
about a 45 degree angle to a more horizontal keyboard below. I used to have a Proteus that
could go as far as 23 tones per octave and stacking two of these made for a fine hybrid 46
tone. I am still in awe of the beauty of that hybrid scale. Maybe if you were up to it, you
could get your 29 tone and make a stacked hybrid setup that would make for a hybrid 58 tone
scale.

I don't add anything new below. I am not currently reading in depth any technical discussions
because I broke my Proteus and I am waiting for a new computer based sampler before I start
testing the technical discussions of the egroup.

>
> If asked for an overall impression of the tuning, I might say, "rather
> mild, gentle, and charming." As a close neighbor of Pythagorean,
> 29-tet offers some intriguing shades of difference; at the same time,
> it's on the mild side of 17-tet, which is a bit more of a clearly
> _neo_-Gothic tuning.
>
> Major thirds in 29-tet at ~413.38 cents are not too far from
> Pythagorean, and about midway between a Pythagorean 81:64 (~407.82
> cents) or a 19:15 (~409.24 cents) and a 14:11 (~417.51 cents). I like
> them very much in certain voicelike timbres on a Yamaha TX-802
> synthesizer, and also for example in a mixture of two harpischord-like
> timbres.
>
> A slight timbral nuance: while I tend in Pythagorean to play the lower
> voices with the richer harpsichord-like timbre, in 29-tet I tend to
> use this timbre for the upper parts, and a more "harp-like" or
> "lute-like" registration for the lower parts. Possibly this preference
> reflects the somewhat greater acoustical tension involving the fifth
> partial of the richer timbre in 29-tet. The goal in either case is a
> "semi-concordant" major third, at once pleasant and intriguing in
> itself, and moving toward some expected resolution.
>
> Interestingly, the 29-tet minor third at ~289.66 cents is an excellent
> approximation of 13:11 (~289.21 cents), and this interval very nicely
> contracts to a unison -- or expands to a fifth also in 13th-century
> styles.
>
> Major seconds at ~206.90 cents and minor sevenths at ~993.10 cents are
> around 2.99 cents from their pure Pythagorean counterparts at 9:8 and
> 16:9. These intervals and the near-pure fifths and fourths produce
> very nice Gothic or neo-Gothic sonorities such as 4:6:9, 9:12:16,
> 6:8:9, and 8:9:12.
>
> A pleasant feature of this tuning, both in its mathematical simplicity
> and in its aural effect, is the division of the whole-tone into five
> equal parts. The usual diatonic semitone or limma is 2/5-tone or
> ~82.76 cents, a very nice size for usual cadences, and narrower than
> the already compact Pythagorean interval of 256:243 (~90.22 cents) or
> about 4/9-tone. The chromatic semitone or apotome is 3/5-tone or
> ~124.14 cents, an interval lending its flavor to a neat cadential
> variation.
>
> This variation brings us to the topic of alternative thirds (augmented
> seconds and diminished fourths) in 29-tet, another neat nuance of this
> tuning. While Pythagorean versions of these intervals ("schisma
> thirds") very closely approach 5-limit ratios, 29-tet gives them their
> own distinct character. Our alternative major third or diminished
> fourth is ~372.41 cents, and our alternative minor third or augmented
> second is ~331.03 cents.
>
> These intervals differ from 5:4 and 6:5 by amounts roughly comparable
> to regular major and minor thirds in 12-tet -- but in the opposite
> directions, of course, maybe approaching the "zones of influence" of
> intervals such as 21:17 (~365.83 cents) or 17:14 (~336.13 cents). A
> tuning like my "e-based hypermeantone" (with the ratio between the
> whole-tone and diatonic semitone equal to Euler's e, ~2.71828) takes
> us more clearly into such zones of influence, but the intermediate
> realm of 29-tet can have its own special attractions.
>
> Thus in 29-tet both the regular and alternative thirds have an active
> quality, and the latter intervals invite resolutions like the
> following involving the wide chromatic semitone or apotome, with C4 as
> middle C and higher note numbers showing higher octaves. Vertical
> intervals in cents are shown in parentheses, while signed numbers show
> melodic intervals in the ascending (positive) or descending (negative)
> directions:
>
> Bb3 -- ~+124.14 -- C4
> (~331.03) (~703.45)
> G3 -- ~-206.90 -- F3
> (~372.41) (0.00)
> Eb3 -- ~+124.14 -- F3
>
> (m3-1 + M3-5)
>
> The difference between the diatonic and chromatic semitones is
> 1/5-tone, or one scale step -- about 41.38 cents. Interestingly this
> is very close to the 128:125 diesis of 1/4-comma meantone (~41.06
> cents), and maybe invites some experiments in neo-Gothic chromaticism
> with melodic figures alternating between these contrasting semitones.
>
> ---------------------------------------------------------------
> 2. Quasi-historical aspects: 29-tet and "ultra-Gothic" cadences
> ---------------------------------------------------------------
>
> As a tuning system which rather closely approximates Pythagorean while
> dividing the whole-tone into five equal parts, 29-tet has another and
> more "radical" side. It permits variants on the usual cadences
> featuring the use of a melodic "diesis" of only 1/5-tone or ~41.38
> cents, and "superwide" major thirds near 13:10 and major sixths near
> 26:15.
>
> Before getting into the possible "quasi-historical" implications of
> these progressions, let's look at a characteristic example, using the
> ASCII asterisk (*) to show a note raised by a diesis or 1/5-tone:
>
> E*4 -- ~+41.38 -- F4
> (~951.72) (1200.00)
> B*3 -- ~+41.38 -- C4
> (~455.17) (~703.45)
> G3 -- ~-206.90 -- F3
>
> (Mx3-5 + Mx6-8)
>
> Here the cadential major third expanding to a fifth is a diesis wider
> than usual, or ~455.17 cents, very close to 13:10 (~454.21 cents); and
> likewise the major sixth (~951.72 cents) expanding to an octave, very
> close to 26:15 (~952.26 cents). To show that these intervals differ
> from usual major thirds and sixths, I call them "maximal" thirds and
> sixths, or Mx3 and Mx6, thus notating this cadence as (Mx3-5 + Mx6-8).
>
> As a feature of neo-medieval style, this kind of "ultra-Gothic"
> cadence can be very striking, and I like the effect especially with
> certain voice-like timbres on synthesizer.
>
> Now for the quasi-historical part, which the Monz (and maybe others)
> may have anticipated. Marchettus of Padua, in his _Lucidarium_ of
> 1318, advocates that singers follow an intonation based on the usual
> 2:1 octaves, 3:2 fifths, 4:3 fourths, and 9:8 whole-tones of
> Pythagorean tuning -- but deriving other intervals, especially certain
> cadential ones, from a division of the whole-tone into "five parts."
>
> Each of these parts is known as a "diesis," and here the
> interpretive debate swiftly gets under way. Are these dieses equal or
> unequal? Is the "fivefold division" of the tone a geometric one
> (calling for the element of a temperament) or possibly an arithmetic
> one (as in a monochord division)? Are the "five parts" of the tone a
> mathematical formulation, or more of a guide for singers -- who are
> not bound to any fixed tuning?
>
> Of special interest is the cadential diesis of Marchettus, which he
> defines as equal to "one of the five parts of a tone." If this is an
> equal or roughly equal division, than a cadence like the above
> "ultra-Gothic" example would result.
>
> While scholars including our own Monz have come up with many ingenious
> interpretations of Marchettus, there is one passage which to me
> suggests that 29-tet may in fact provide one intriguing realization
> of his cadential aesthetics.
>
> In his discussion of the resolution of unstable intervals or
> "dissonances" -- Marchettus refers to thirds and sixths as "tolerable
> dissonances," while 14th-century writers in the French tradition, for
> example, tend to call them "imperfect concords" or the like -- he
> compares two resolutions of the cadential major sixth, one standard
> and the other a "feigned color" (loosely translated, a "deceptive
> cadential inflection"):
>
> D4 C#4 D4 D4 C#4 C4
> D3 E3 D3 D3 E3 F3
>
> 8 M6 8 8 M6 5
>
> In the notation of Marchettus, the C# shows that the note C is raised
> by an extra-wide "chromatic semitone" equal to "four parts of a tone,"
> leaving only "one part of a tone" for the remaining diesis C#-D. The
> problem which Marchettus considers is why the first resolution is
> standard, while the second is less expected and favored.
>
> Having earlier explained that a cadential dissonance should approach
> the consonance which it seeks (e.g. M3-5, M6-8) as closely as possible
> -- thus calling for the cadential diesis of only "one of the five
> parts" of a tone -- he remarks that the major sixth of these two
> progressions is equally far from either the octave or the fifth. Thus
> the question arises: why does the standard resolution to the octave
> seem better to fit the norm of "closest approach"?
>
> One answer, the first offered by Marchettus, is that the octave at 2:1
> is a purer consonance than the fifth at 3:2, and thus a more
> satisfying destination for the major sixth, located as it is "six
> dieses" from either stable goal. However, both the framing of the
> question, and the second answer of Marchettus, are of special
> interest.
>
> In saying that a cadential major sixth is equally "six dieses" from
> either the octave or the fifth, Marchettus may be implying that his
> dieses are at least roughly equal, so that one can validly compare
> distances between intervals using them.
>
> In 29-tet, an extra-wide or "maximal" major sixth at 23 steps is
> indeed precisely 6/5-tone (or 6/29 octave) from either the usual goal
> of the octave (29 steps) or the less usual goal by contrary motion of
> the fifth (17 steps). This doesn't mean that Marchettus necessarily
> intended vocal intonation precisely in 29-tet, or using this scale as
> the basis for an "adaptive JI" scheme to make fifths, fourths, and
> major seconds pure -- only that 29-tet seems to realize the idea of a
> cadential major sixth (or "maximal sixth") about equally far from the
> octave or the fifth.
>
> As his second answer to the problem of why the resolution to the
> octave is more favored, Marchettus analyzes the melodic motion of each
> voice in the cadence. Generally, he presents ideal cadential action as
> involving movement by _both_ voices in contrary motion: both voices
> share in the "dissonance," and both participate in its resolution.
>
> Comparing the two resolutions by contrary motion of the major sixth,
> he notes that in the usual resolution to the octave, the lower voice
> descends by a regular whole-tone or five dieses, while the upper voice
> ascends by one diesis, the narrowest possible interval of the system.
>
> In contrast, in the unusual resolution to the fifth, the lower voice
> ascends by the usual diatonic semitone or limma (to Marchettus, an
> "enharmonic semitone") of two dieses, while the upper voice descends
> by its wide "chromatic semitone" of four dieses.
>
> If we look at these progressions in 29-tet, using the maximal sixth in
> either case, these descriptions exactly and intuitively fit. Here
> numbers in parentheses show vertical intervals in scale steps, and
> signed numbers show melodic motion in scale steps or "dieses":
>
> C#*4 -- +1 -- D4 C#*4 -- -4 -- C4
> (23) (29) (23) (17)
> E3 -- -5 -- D3 E3 -- +2 -- F3
>
> Mx6 8 Mx6 5
>
> Marchettus explains that the usual resolution to the octave is more
> satisfying in part because one of the voices moves by the narrowest
> interval possible, the cadential diesis equal to "one of the five
> parts" of a tone -- here C#*-D in the upper voice. In contrast, the
> smallest interval involved in the unusual resolution to the fifth is
> the usual semitone of "two dieses," here E3-F3 in the lower voice.
>
> At the least, this comparison of the cadential M6-8 and M6-5
> progressions suggests that Marchettus indeed regards the cadential
> diesis as substantially smaller than the usual semitone or limma --
> with the 29-tet distinction between these melodic intervals at
> 2/5-tone and 1/5-tone nicely fitting the terms of his comparison.
>
> In a slightly different realization, one could tune the cadential
> major thirds and sixths (our "maximal" thirds and sixths) at precisely
> 13:10 and 26:15 in a JI system with a 3-odd-limit of stability but a
> 13-prime-limit or higher for generating unstable intervals. Many
> neo-Gothic temperaments including 29-tet tend to approximate such a
> state of affairs.
>
> Finally, I should note that the 29-tet interval of 23 steps can be
> used either as a Marchettus-like(?) "maximal sixth" or as a "minimal
> seventh" regularly resolving to the fifth by a progression featuring a
> cadential diesis, e.g.:
>
> D4 -- -5 -- C4
> (23) (17)
> E*3 -- +1 -- F3
>
> Although Marchettus focuses on thirds and sixths as unstable cadential
> intervals, the resolution from minor seventh to fifth occurs in
> various 13th-14th century styles. Like the progression from maximal
> sixth to octave, this intonational variation using 23/29 octave as a
> "minimal seventh" contracting to a fifth features motion by a
> whole-tone (five dieses) in one voice, and by a single cadential
> diesis in the other.
>
> In short, 29-tet is a temperament rich with many neo-medieval
> possibilities, ranging from the rather gently neo-Gothic to the
> adventurous ultra-Gothic. This isn't to exclude all the other uses of
> this tuning, only to offer a sketch of one very attractive side of its
> musical flavor.
>
> Most respectfully,
>
> Margo Schulter
> mschulter@value.net
>
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
> tuning-subscribe@egroups.com - join the tuning group.
> tuning-unsubscribe@egroups.com - unsubscribe from the tuning group.
> tuning-nomail@egroups.com - put your email message delivery on hold for the tuning group.
> tuning-digest@egroups.com - change your subscription to daily digest mode.
> tuning-normal@egroups.com - change your subscription to individual emails.

--
Hello,

I have recently discovered a new graphic representation for the
I Ching hexagrams that show that the I Ching was also intended to
be a hologram generator. By hologram, I mean something like the
traditional hieroglyphic style of Chinese writing except that it
has an internal sequence/logic. I accomplished this by representing
the changing lines differently. You can see the new graphic
arrangement at my webpage (it's about 2 paragraphs down):

http://home.earthlink.net/~troubledoor

You can also download the software for free. I use Norton's anti-virus with the
latest software upgrades so the download is virus free.
Please distribute it to your other I Ching friends. Thanks.

symmetric keyboard:
http://x31eq.com/instrum.htm