Re: Sliding tones (29-tET diesis -- for Orphon Soul)

Hello, there, Orphon Soul and everyone.

As someone who uses 29-tone equal temperament (29-tET) in a neo-Gothic
setting based on medieval European practice and theory of the
13th-14th centuries, I would say that the diesis of 1/5-tone or 1/29
octave (~41.38 cents) is a distinct melodic step, and a striking kind
of cadential step larger than a "comma" and yet smaller than a usual
"semitone."

Marchettus of Padua in his _Lucidarium_ (1318) describes a fivefold
division of the whole-tone, and advocates the use of directed cadences
involving a diesis step equal to only "one of the five parts of a
tone." If we take these five parts to be equal, then a model for
flexible vocal intonation quite close to 29-tET results.

(Here I would emphasize that I am not aware of any medieval Western
European theory of temperament or equal divisions of the octave, so
that 29-tET itself might best be considered a _neo_-Gothic tuning.)

Please let me agree that Pythagorean and 29-tET are in many ways
closely related tunings, with the most common regular intervals formed
by chains of up to 5 or 6 fifths or fourths generally quite similar.

However, I would say that the 41-cent diesis is a distinctive feature
of 29-tET, along with two varieties or "flavors" of unstable cadential
intervals complementing the regular near-Pythagorean ones: submajor
and supraminor thirds and sixths; and "maximal/minimal" intervals
resolving by the kind of cadential diesis steps or "fifthtones" which
Marchettus _may_ be describing.

Before developing these points, I might just add that by an
interesting coincidence, the 29-tET diesis of ~41.38 cents is almost
identical to the Renaissance diesis in 1/4-comma meantone or 5-based
JI of 128:125 (~41.06 cents). This latter diesis is equal to the
difference between three pure 5:4 major thirds up (125:64) and a pure
2:1 octave, and is also equal to the difference in 1/4-comma between
G# and Ab, for example.

The 29-tET diesis is likewise equal to the amount by which three
regular major thirds of 10/29 octave (~413.79 cents) exceed a 2:1
octave, here a single scale step of 1/29 octave.

The 29-tET diesis is _positive_, with G# _above_ Ab (as also in
Pythagorean tuning, where the Pythagorean comma is equivalent to this
diesis); the meantone diesis is _negative_, with G# _below_ Ab.[1]

In a 5-based JI system, the 25:24 ratio which you mentioned defines
the chromatic semitone (e.g. C-C#) or difference between the pure 5:4
major third and the 6:5 minor third (e.g. A-C#, A-C). The use of this
interval as a direct melodic step is one definition of 16th-century
"chromaticism," with direct use of the diesis or fifthtone is termed
"enharmonicism" (e.g. Nicola Vicentino, 1555), since the diesis
resembles in its size the dieses of the ancient Greek enharmonic
genus.

-----------------------------------------------------
1. Maximal/mininal intervals and the cadential diesis
-----------------------------------------------------

While the most common regular intervals of 29-tET are quite close to
those of Pythagorean tuning, with fifths and fourths tempered by only
about 1.49 cents in the wide and narrow directions respectively, the
41-cent diesis has a quality which seems distinct from that of either
a Pythagorean comma (531441:524288, ~23.46 cents) or a usual
"semitone" starting somewhere 50 cents, or possibly a bit smaller.

The distinctive quality of the diesis as direct neo-Gothic cadential
step is exemplified in progressions such as the following, very
typical of 24-out-of-29-tET with two keyboards tuned a diesis apart.
Here C4 is middle C, numbers in parentheses show rounded intervals in
cents between a given voice and any higher voices, and signed numbers
show ascending (+) or descending (-) melodic steps. An ASCII asterisk
(*) shows a note raised by a diesis:

E*4 -- +41 -- F4 F4 -- -41 -- E*4
(497) (497) (703) (703)
B*3 -- +41 -- C4 Bb3 -- -41 -- A*3
(952,455) (1200,703) (952,248) (703,0)
G3 -- -207 -- F3 G*3 -- +207 -- A*3

(M6-8 + M3-5) (m7-5 + m3-1)

In the first progression, an extra-wide major third expands to a fifth
(M3-5), and an extra-wide major sixth to an octave (M6-8), one voice
in either of these resolutions descending by a usual whole-tone and
the other ascending by a 41-cent diesis, taking the role as it were of
a cadential semitone.

In the second progression, an extra-narrow minor third contracts to a
unison (m3-1), and an extra-narrow minor seventh to a fifth (m7-5),
with the same melodic steps -- this time an ascending whole-tone and
descending diesis.

This use of the 41-cent diesis as a direct cadential step in moving
from an unstable interval to a stable one by stepwise contrary motion
seems to me strongly to distinguish this step in 29-tET from the
Pythagorean or septimal comma. At least, while I've used comma shifts
in various contexts, I haven't attempted to substitute such a comma
for a usual diatonic cadential semitone; in contrast, this is maybe
the most common use of the 29-tET diesis.

A charming feature of 29-tET is that the extra-large or "maximal"
sixth is identical to the extra-narrow or "minimal" seventh -- both
23/29 octave, or ~951.72 cents (very close to 26:15, ~952.26 cents),
making possible "puns" like the following example on a 24-note
keyboard:

F4 -- -41 -- E*4 F4 -- +41 -- F*4
(248) (0) (248) (497)
D*4 -- +207 -- E*4 D*4 -- -207 -- C*4
(952,703) (703,703) (952,703) (1200,703)
G*3 -- +207 -- A*3 G*3 -- -207 -- F*3

(m7-5 + m3-1) (M6-8 + M2-4)

In these examples, the unstable sonority G*3-D*4-F4 (0-703-952 cents)
is identical, but invites two different interpretations.

In the first interpretation, the 952-cent interval between the outer
voices represents a small minor seventh contracting to a fifth, and
the 248-cent interval between the upper voices (more precisely ~248.28
cents) a small minor third contracting to a unison.

In the second interpretation, however, the 952-cent interval instead
represents a large major sixth expanding to an octave, and the
248-cent interval a large major second between the upper voices
expanding to the upper fourth of the resolving 2:3:4 trine
F*3-C*4-F*4. (In Gothic or neo-Gothic music, a trine is a complete or
saturated stable sonority with outer octave, lower fifth, and upper
fourth).

-----------------------------
2. Supraminor/submajor thirds
-----------------------------

In Pythagorean tuning, diminished fourths and augmented seconds have
ratios of 8192:6561 (~384.36 cents) and 19683:16384 (~317.60 cents),
only a 3-5-schisma from a pure 5:4 (~386.31 cents) or 6:5 (~315.64
cents). Because of this small difference of a 3-5-schisma
(32805:32768), they are often known as "schisma thirds."

In contrast, at least in a neo-Gothic context, corresponding 29-tET
intervals have a certain affinity to submajor and supraminor thirds at
around 21:17 (~365.83 cents) and 17:14 (~336.13 cents). In 29-tET,
these alternative thirds have sizes of 9/29 octave or ~372.41 cents,
and 8/29 octave or ~331.03 cents.

Here is a typical 29-tET cadence featuring these supraminor/submajor
thirds:

Bb3 -- +124 -- B3
(372) (703)
F#3 -- -207 -- E3
(703,331) (703)
Eb3 -- +124 -- E3

Note that in this type of resolution, the supraminor third contracts
to a unison while the submajor third expands to a fifth -- both
resolutions involving melodic motion by an apotome or chromatic
semitone of 3/29 octave or ~124.14 cents, here Eb3-E3 and Bb3-B3.
The contrast between this cadential semitone and the usual one of 2/29
octave (~82.76 cents) gives this type of resolution a distinctive
melodic as well as vertical quality.

Incidentally, the border regions between this supraminor/submajor
flavor of thirds or sixths and the near-5-based flavor of the
immediate Pythagorean neighborhood or the more neutral flavor of the
17-tET neighborhood can be quite fuzzy.

One rough and ready rule is that supraminor/submajor thirds should
differ from ratios of 5:4 and 6:5 at least as much as the regular
thirds of 12-tET do in the opposite direction: this places the border
area at just below 29-tET along the spectrum of regular tunings.[2]

To distinguish supraminor/submajor from more "neutral" thirds, we
might in one approach say that these thirds should differ in size by
at least a syntonic comma of 81:80 (~21.51 cents), or possibly by the
85:84 comma (~20.49 cents) distinguishing ratios of 5:4 and 6:5 from
the simplest submajor/supraminor ratios of 21:17 and 17:14.

By either test, the submajor/supraminor thirds of the e-based tuning
(fifths ~704.61 cents, ratio of whole-tone to diatonic semitone equal
to Euler's e at ~2.71828) are getting close to a more neutral flavor:
these thirds differ by about 21.68 cents.

----------
3. Summary
----------

The similarities between Pythagorean tuning and 29-tET may illustrate
how a rather small amount of tempering (here ~1.49 cents for 29-tET
fifths at ~703.45 cents) can make a dramatic musical difference for
intervals derived from longer chains of fifths or fourths. At the same
time, regular intervals built from shorter chains are indeed quite
similar in the two tunings.

A very important lesson of intonational practice and theory is that
mathematical parameters take on artistic meaning in specific musical
contexts. This article has addressed specifically a Gothic/neo-Gothic
type of style, where intervals such as thirds and sixths are unstable
and typically quite active.

In such a setting, 29-tET augmented seconds and diminished fourths can
serve as "gentle" variations on supraminor/submajor thirds most
characteristically around 17:14 and 21:17. These intervals complement
the regular 29-tET thirds of 10/29 octave (~413.79 cents or ~127:100)
and 7/29 octave (~289.66 cents, ~13:11).

Especially dramatic and distinctive are progressions using the 41-cent
diesis in place of a usual cadential semitone in resolving "maximal"
seconds, thirds, and sixths (M2-4, M3-5, M6-8) a diesis wider than
their regular counterparts, and "minimal" thirds and sevenths a diesis
narrower (m3-1, m7-5).

This use of the diesis in a neo-Gothic setting seems to distinguish it
from smaller intervals such as the Pythagorean or septimal comma; at
the same time, it also has a rather different quality from more
typical semitones of around 50-55 cents or larger, as with the regular
diatonic semitone of 22-tET (~54.55 cents) or the diesis of the
e-based tuning featured in "metachromatic" progressions (~55.28
cents).

Having discussed some mathematical parameters of 29-tET, I would
emphasize that such parameters take artistic shape in specific musical
contexts.

For example, the 29-tET augmented second at around 331 cents might
represent either a "gentle" 17:14 or a rather inaccurate 6:5; the same
interval or tuning may often lend itself to such alternative
interpretations.[3]

----
Note
----

1. We can also define the Pythagorean comma or the diesis in 29-tET or
meantone as the difference between 12 fifths up in a given tuning and
7 pure octaves. For this measurement, a fifth of 700 cents (12n-tET)
is the "neutral" reference point: regular tunings the same distance
from this point in opposite directions will have dieses of the same
magnitude. For example, 29-tET (~703.45 cents, +~3.45 cents) and
1/4-comma meantone (~696.58 cents, -~3.42 cents) have dieses of almost
identical sizes, but respectively positive and negative.

2. We can get a similar result by saying that thirds of this flavor
should differ from each other by no more than the ratio of 459:448
(~41.99 cents) between a 17:14 supraminor third and a usual
Pythagorean minor third at 32:27 (~294.13 cents).

3. Similarly, the 400-cent third of 12n-tET might suggest either a
somewhat "subdued" variation on the usual Pythagorean 81:64 (~407.82
cents), or a rather "beatful" version of a 5:4.

Most appreciatively,

Margo Schulter
mschulter@value.net

On 5/30/01 8:34 PM, "mschulter" <MSCHULTER@VALUE.NET> wrote:

> [a whol' lotta things an' stuff...]

WOW ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !

So much for one of the first questions I ever asked here,
"does *anyone* ever use 29?"
I give you 29 exclamation points.
A simple "uh huh..."
Neo-gothic eh?
I know some people who would like that.

Margo, I thank you profusely for that most eloquent reply.
It was very well organized and annotated.
I'll have to study it at length.

Marc

--- In tuning@y..., "Orphon Soul, Inc." <tuning@o...> wrote:
> On 5/30/01 8:34 PM, "mschulter" <MSCHULTER@V...> wrote:
>
> > [a whol' lotta things an' stuff...]
>
> WOW ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !
>
> So much for one of the first questions I ever asked here,
> "does *anyone* ever use 29?"
> I give you 29 exclamation points.
> A simple "uh huh..."
> Neo-gothic eh?
> I know some people who would like that.
>
> Margo, I thank you profusely for that most eloquent reply.
> It was very well organized and annotated.
> I'll have to study it at length.
>
> Marc

Margo is great, isn't she? A model of how discourse should be conducted on this list.

Anyway, 29 . . . 29 is the first ET consistent in the 15-limit. A related fact is that two 29-tET
keyboards or guitars tuned a few cents apart give you 29 otonal and 29 utonal complete 15-limit
ogdoads (those are 1:3:5:7:9:11:13:15 and 1/(15:13:11:9:7:5:3:1) chords, with factors of 2
"free", of course). This guy Graham, he's brilliant.

On 5/30/01 11:44 PM, "Paul Erlich" <paul@stretch-music.com> wrote:

> Margo is great, isn't she? A model of how discourse should be conducted on
> this list.

I think I actually sat up straight when I typed my response to her, yeah.

Most refreshing. Makes me want to be a tech writer when I grow up.