Yahoo Tuning Groups Ultimate Backup tuning Re: A neo-Gothic approach to 20-tET (Part 1)

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A neo-Gothic approach to 20-tET:
Essay in honor of Brian McLaren and Dan Stearns
Part 1
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Sometime maybe around the end of last summer, the renowned microtonal
composer and theorist Brian McLaren recommended 20-tone equal
temperament (20-tET) as a tuning for neo-Gothic music I hadn't yet
explored, but should.

In fact, this was a brilliant suggestion, most happily enriching my
musical practice and theory alike in all kinds of ways, and opening
the path to other related (or maybe not-so-obviously-related) scales.
Therefore this essay is most warmly dedicated to Brian McLaren, and
also to his revered colleague Ivor Darreg for the insight that all
tunings have musical value, offering a rainbow spectrum of "moods."

Also, this essay is dedicated to mathematician, musician, and virtuoso
scale designer Dan Stearns, who has made the Stern-Brocot Tree
flourish with new and rich ramifications for our art.

Early this year, in addition most patiently to guiding me through a
few of the mysteries of scale generation, Dan compared notes with me,
a newcomer to 20-tET, showing at once his creativity and his eagerness
to exchange viewpoints about a tuning system lending itself to a range
of approaches.

To honor Dan, and also a famous arboreal construction of the perennial
artist acclaimed for his high and branching concepts, Ervin Wilson, I
would call this essay simply a view of 20-tET from one particular
treehouse.

-------------------------------
1. Close encounters with 20-tET
-------------------------------

When Brian McLaren called 20-tET to my attention, pointing out the
great neo-Gothic major third at 420 cents (7/20 octave), my first
reaction was, "Yes, that's an excellent third -- but what about those
720-cent fifths?"

Would fifths at 12/20 octave, a full 18.04 cents wider than pure,
really be ideal for music where fifths and fourths are the main stable
concords?

Curiously, before getting around to tuning 20-tET, I tried an
experiment which took me a step further in confronting that particular
barrier: tuning 23-tET, where fifths and fourths are considerably
further from a pure 3:2.

Having found that yes, in a metallophone-like texture 23-tET fifths
and fourths can sound like fifths and fourths, I was ready to consider
tuning them at 720 cents and 480 cents almost a "mainstream" choice.

This January, I got down to the actual tuning, and had to decide on a
keyboard mapping. That involved questions like: "How do I construct a
diatonic scale, and find the cadences I'm looking for?"

----------------------------------
1.1. Keyboard mapping and spelling
----------------------------------

In my kind of neo-Gothic approach to 20-tET, the two 12-note manuals
look like this, with a "+" sign showing a note raised by 120 cents or
2/20 octave, an interval which I term an apotome or chromatic
semitone, arbitrarily taking a C-C octave as a point of reference,
with C4 as middle C:

300 420 780 900 1080
C#+4/ Eb+4/ F#+4/ Ab+4/ Bb+4
Eb4 E4 Ab4 A4
C+4 D+4 E+4 F+4 G+4 A+4 B+4 C+5
120 360 540 600 840 1020 1260 1380
240 180 60 240 180 240 60
-----------------------------------------------------------------
180 300 660 780 960
C#4 Eb4 F#4 Ab4 Bb4
C4 D4 E4 F4 G4 A4 B4 C5
0 240 420 480 720 900 1140 1200
240 180 60 240 180 240 60

In this "regularized keyboard" layout, each keyboard has identical
steps and intervals, arranged in a pattern which might be described as
a kind of "quasi-syntonic diatonic."

This patterns features a limma or diatonic semitone at 60 cents (1/20
octave), and alternating large and small whole-tones at 240 cents
(4/20 octave) and 180 cents (3/20 octave).

Accidentals are added to each keyboard so as to provide diatonic
semitones of 60 cents (e.g. C#-D, Eb-D, F#-G, Ab-G). This divides a
large whole-tone (e.g. C-D) into steps of 180-60 cents (e.g. F-F#-G),
and a small whole-tone into steps of 60-120 cents (e.g. D-Eb-E).

People may note that while I usually choose a range of Eb-G# for
accidentals, here I've preferred Ab-C# in this arrangement, for
reasons having to do with cadences (see Part 2, Section 3.1).

Of course, since the tuning includes all 20 notes, our "G#4," here
spelled G+4 (840 cents above C4), is available also. Likewise, if we
want a "G#+" at 840 cents above C+, that's available at Bb4.

For some of my favorite modes, I can proceed more or less as if 20-tET
were a regular diatonic tuning, especially in a purely melodic context
or a polyphonic style after a 13th-century Western European manner
with lots of cadences in modes such as Mixolydian where all voices
move by whole-tones (e.g. major thirds contracting to unisons or minor
sixths expanding to octaves).

Here are F Lydian and G Mixolydian, for example, with T for large
tone, t for small tone, and S for semitone:

T t T s T t s
F3 G3 A3 B3 C4 D4 E4 F4
0 240 420 660 720 960 1140 1200
240 180 240 60 240 180 60

MIDI example: http://value.net/~mschulter/20tly001.mid

t T s T t s T
G3 A3 B3 C4 D4 E4 F4 G4
0 180 420 480 720 900 960 1200
180 240 60 240 180 60 240

http://value.net/~mschulter/20tmx001.mid

In these neo-Gothic modes of 20-tET, a major third of 420 cents is
formed from a large tone of 240 cents and a small tone of 180 cents.

This arrangement results in a "quasi-syntonic comma" equal to 60
cents, also the size of the usual diatonic semitone: the difference
between our major third at 420 cents, and four 720-cent fifths up less
two octaves, (2880-2400) or 480 cents. This comma is also equal to the
difference between our two sizes of whole-tones at 240 and 180 cents.

As in the syntonic diatonic of Ptolemy as applied to 16th-century
European polyphony by Fogliano and Zarlino, so here, this type of
comma causes special complications for the fifth spelled D-A in a
regular tuning.

Maybe as a kind of homage to Ben Johnston, a 720-cent fifth here is
spelled C#-A or D-Bb! The keyboard spelling D-A yields an interval of
660 cents, also the size of a usual augmented fourth (e.g. F-B, C-F#).

This produces three variations on the Dorian mode at D-D -- or, for
one variation, actually C#-C#:

"Hard Dorian" (C#-C#)

T s T t T s t
C#4 E4 F4 G4 A4 B4 C5 C#5
0 240 300 540 720 960 1020 1200
240 60 240 180 240 60 180

http://value.net/~mschulter/20tdr001.mid

"Soft Dorian" (D-D)

T s T T t s T
D4 E4 F4 G4 Bb4 B4 C5 D5
0 180 240 480 720 900 960 1200
180 60 240 240 180 60 240

http://value.net/~mschulter/20tdr002.mid

"Natural Dorian" (D-D)

T s T t T s T
D4 E4 F4 G4 A4 B4 C5 D5
0 180 240 480 660 900 960 1200
180 60 240 180 240 60 240

http://value.net/~mschulter/20tdr003.mid

The "soft Dorian" version on D has a very curious effect at the point
where we have two successive large whole-tones, F-G-Bb, where I am
somewhat reminded of the slendro scale for gamelan (close to 5-tET).
This effect seems a kind of meeting between neo-Gothic and gamelan
mediated through 20-tET's connection with 5-tET (compare the remarks
of Dan Stearns at the conclusion of this section).

The "natural Dorian" interestingly sounds quite pleasant to me as a
melodic scale featuring a 660-cent "fifth" at D-A; in a polyphonic
setting, one might modify either note to get a 720-cent vertical
interval (C#-A or D-Bb).

The "hard Dorian" at C#-C# has an interval C#-G of 540 cents (a
diminished fifth in size and also here in spelling) in place of a
usual fourth above the final or note of repose C#, and permits
interesting versions of some standard 13th-century cadences in three
or four voices (Part II, Section 3.2).

Here I might add that both for Dan Stearns and me, Mixolydian is an
especially beloved mode in 20-tET, with our interpretations and styles
suggesting some of the versatility of this tuning. Something like my
"soft Dorian" might suggest words which Dan has used about other types
of 20-tET patterns:

"What's special about scales like these is that they bath[e]
the familiar in pungent hues of the new and the exotic.

"Speaking very broadly, the sound of these modes is like that
of 7-limit diatonic scales bent through the unique mood of
20 equal with its ancestral genealogy in 5 equal."[1]

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2. Intervals and categories
---------------------------

With 20-tET, as with more "conventional" tunings such as 29-tET or
46-tEt, how one maps scale steps into intervals and categories may
depend a great deal upon the desired style.

Here is a table of 20-tET intervals which may provide some overview of
my own neo-Gothic approach, with explanatory comments following:

=====================================================================
Interval Cents Example ~Ratio Cents Variation
---------------------------------------------------------------------
1 0 C3-C3 1:1 (exact) 0.00 0.00
---------------------------------------------------------------------
limma 60 E3-F3 28:27 62.96 - 2.96
or min2 G3-Ab3
---------------------------------------------------------------------
apotome 120 Eb3-E3 2187:2048 113.69 + 6.31
or dim3 G3-G+3/G#3 15:14 119.44 + 0.56
C#3-Eb3 14:13 128.30 - 8.30
---------------------------------------------------------------------
small 180 D3-E3 65536:59049 180.45 - 0.45
Maj2 G3-A3 10:9 182.40 - 2.40
---------------------------------------------------------------------
Large Maj2 240 C3-D3 8:7 231.17 + 8.83
quasi-min3 D3-F3 15:13 247.74 - 7.74
G3-Bb3 7:6 266.87 -26.87
---------------------------------------------------------------------
min3 300 C3-Eb3 32:27 294.13 + 5.87
E3-G3 19:16 297.51 + 2.49
---------------------------------------------------------------------
Neutral3 360 E3-Ab3 16:13 359.47 + 0.53
C3-D+3/D#3 21:17 365.83 - 5.83
D3-F+3
---------------------------------------------------------------------
Maj3 420 C3-E3 14:11 417.51 + 2.49
E3-G+3/G#3 23:18 424.36 - 4.36
---------------------------------------------------------------------
4 480 D3-G3 4:3 498.04 -18.04
Bb3-D4
---------------------------------------------------------------------
dim5 540 B3-F4 15:11 536.95 + 3.05
Super4 A3-D4 11:8 551.32 -11.32
---------------------------------------------------------------------
demi8 600 C3-F3+ 24:17 597.00 + 3.00
12:17 603.00 - 3.00
---------------------------------------------------------------------
Aug4 660 C3-F#3 16:11 648.68 +11.32
sub5 D3-A3 22:15 663.05 - 3.05
---------------------------------------------------------------------
5 720 C3-G3 3:2 701.96 +18.04
D3-Bb3
C#3-A3
---------------------------------------------------------------------
min6 780 C3-Ab3 11:7 782.49 - 2.49
D3-A+3
---------------------------------------------------------------------
Neutral6 840 F#3-Eb4 34:21 834.17 + 5.83
D3-Bb+3 13:8 840.53 - 0.53
G+3-F4
---------------------------------------------------------------------
Maj6 900 G3-E4 27:16 905.87 - 5.87
A3-F+4
---------------------------------------------------------------------
small min7 960 G3-F4 12:7 933.13 +26.87
Quasi-Maj6 A3-F#4 26:15 952.26 - 7.74
E3-C#4 7:4 968.83 - 8.83
---------------------------------------------------------------------
large min7 1020 A3-G4 9:5 1017.60 - 2.40
E3-D4 59049:32768 1019.55 - 0.45
---------------------------------------------------------------------
Aug6 1080 Eb3-C#4 13:7 1071.70 - 8.30
28:15 1080.56 - 0.56
15:8 1088.27 - 8.27
---------------------------------------------------------------------
Maj7 1140 F3-E4 27:14 1137.04 + 2.96
A3-G#4
---------------------------------------------------------------------
8 1200 F3-F4 2:1 (exact) 1200.00 0.00
_____________________________________________________________________
---------------------------------------------------------------------

As Brian McLaren pointed out to me, 20-tET has an excellent neo-Gothic
major third at 420 cents, somewhere between 14:11 (~417.51 cents) and
a suggested region of maximum "entropy" or complexity around 422-423
cents.[2]

Regular minor thirds at 300 cents, and major sixths at 900 cents, are
not too far from their usual Pythagorean ratios, although a bit
"subdued" or "mild"; these intervals have the same size as in 24-tET
or 36-tET (or more generally 12n-tET). However, when combined as they
typically are with the ebullient 420-cent major third, they can have a
most pleasantly active and shimmering quality.

A delightful ramification of our two sizes of whole-tones in
quasi-diatonic modes (240 cents and 180 cents) is the use of
"quasi-thirds" at 240 cents spelled and treated vertically like minor
thirds contracting to unisons, and likewise "quasi-sixths" at 960
cents spelled and treated like major sixths expanding to octave.
In other contexts, these same interval sizes also serve as usual major
seconds or minor sevenths.

To introduce the quasi-minor-third (q3 for short) at 240 cents, let us
first consider a routine two-voice progression with a "regular" minor
third at 300 cents. Here numbers in parentheses show vertical
intervals, and signed numbers show ascending (+) or descending (-)
melodic intervals in each voice:

D4 -- -240 -- C4 -- -60 -- B3
(720) (300) (0)
G3 -- +180 -- A3 -- +240 -- B3

5 - m3 - 1

http://value.net/~mschulter/20tq3000.mid

This typical 13th-century progression from fifth to minor third to
unison takes on an engagingly different "modal color" at positions
where the 240-cent quasi-third is the natural minor third:

C4 -- -240 -- Bb3 -- -60 -- A3
(720) (240) (0)
F3 -- +240 -- G3 -- +180 -- A3

5 - q3 - 1

http://value.net/~mschulter/20tq3001.mid

Here my impression is that the 240-cent quasi-third _acts_ like a
usual minor third, and so I tend to hear it in this way.

Similarly, a major sixth may be realized by either the usual interval
of 900 cents, or a "quasi-major-sixth" (Q6 for short) at 960 cents.
Here is a routine 13th-14th century progression from fifth to major
sixth to octave, given first with the usual 900-cent major sixth:

D4 -- +180 -- E4 -- +60 -- F4
(720) (900) (1200)
G3 --------------- -- -240 -- F3

5 - M6 - 8

http://value.net/~mschulter/20tq6000.mid

We might observe that while the _vertical_ intervals are routine, the
upper voice moves melodically through a quasi-third (D4-E4-F4) made up
of a small 180-cent major second or whole-tone plus a usual 60-cent
semitone.

Here is a version of the same basic progression with a resolution of
quasi-sixth to octave (Q6-8):

E4 -- +240 -- F#4 -- +60 -- G4
(720) (960) (1200)
A3 --------------- -- -180 -- G3

5 - Q6 - 8

http://value.net/~mschulter/20tq6001.mid

The same type of Q6-8 progression, this time featuring a descending
semitonal motion, occurs naturally in the mode of E Phrygian:

C4 -- +240 -- D4 -- +180 -- E4
(720) (960) (1200)
F3 --------------- -- -60 -- E3

5 - Q6 - 8

http://value.net/~mschulter/20tq6002.mid

Thus the frequent interchangeability of quasi-thirds and quasi-sixths
with usual minor third and major sixths is one charming feature of
neo-Gothic "diatonicity" in 20-tET.

------------------------
2.1. A bit of philosophy
------------------------

Neo-Gothic tuning systems such as 29-tET, and also 24-tET when used in
a neo-Gothic setting, feature intervals about midway between a regular
whole-tone and minor third, or between a major sixth and minor
seventh. In 29-tET, intervals of 6/29 octave (~248.28 cents) and 23/29
octave (~951.72 cents) admirably fill this role, closely approximating
15:13 (~247.74 cents) and 26:15 (~952.26 cents). In 24-tET, intervals
of 250 cents and 950 cents lend themselves to the same treatment.[3]

In two ways, we might say that 20-tET goes these systems one better.

First, in 29-tET or 24-tET, the applicable intervals are about equally
poised between a large major second and a small minor third, or
between a large major sixth and a small minor seventh. In 20-tET,
however, 240 cents is decidedly closer to a major second, and 960
cents to a minor seventh -- yet they can also serve as minor thirds or
major sixths.[4]

Secondly, in 20-tET these quasi-thirds and quasi-sixths are an
integral part of "diatonic" scale structure, in contrast to a system
like 29-tET where they play more of a supplementary role to the
regular intervals and cadences.[5]

Both as vertical intervals, and as routine melodic intervals in
various modes, quasi-thirds and quasi-sixths may illustrate how usual
diatonic patterns are "bent," as Dan Stearns puts it, when mapped into
a system such as 20-tET -- definitely not a regular diatonic tuning,
but a most intriguing one for neo-Gothic music.

-----
Notes
-----

1. See </tuning/topicId_24348.html#24348>.

2. See Margo Schulter and David Keenan, "The Golden Mediant: Complex
ratios and metastable intervals," TD 810:3 (17 September 2000):
</tuning/topicId_12915.html#12915> -- with the term
"Noble Mediant" now preferred for the Phi-based mediant described in
that paper. As estimated by the Noble Mediant, the region of maximum
complexity between 5:4 (~386.81 cents) and 9:7 (~435.08 cents) may be
located around 422.48 cents; Paul Erlich's "harmonic entropy" approach
similarly suggests a region around 423 cents.

3. In neo-Gothic theory, these intervals around 15:13 or 26:15 --
along with wide major thirds around 13:10 (~454.21 cents), e.g. 11/29
octave in 29-tET (~455.17 cents) or 9/24 octave in 24-tET (450 cents)
-- are known as "13-flavor" intervals. In 1318, Marchettus of Padua
provides a _possible_ precedent for their use when he describes a
cadential major sixth (typically expanding to an octave) equally
distant in size from the 3:2 fifth and 2:1 octave, differing by "six
diesis" from either interval. If we take his division of the 9:8
whole-tone into five dieses to be an equal division, then he may be
describing a system of adaptive tuning for singers approximately
modelled by 29-tET on a fixed-pitch instrument, with the cadential
major sixth close to 23/29 octave. However, while this interpretation
very nicely fits his description of this sixth, it is only of the
various readings considered by various scholars. See, for example, my
recent paper
http://value.net/~mschulter/marchetmf.txt (ASCII text);
http://value.net/~mschulter/marchetmf.zip (zip, text and PostScript)
See also Joseph L. Monzo, _Speculations on Marchetto of Padua's
"Fifth-Tones"_ (1998),
<http://www.ixpres.com/interval/monzo/marchet/marchet.htm>;
and Jay Rahn, "Practical Aspects of Marchetto's Tuning," _Music Theory
Online_ 4.6 (1998),
<http://boethius.music.ucsb.edu/mto/issues/mto.98.4.6/mto.98.4.6.rahn.html>.

4. At 240 cents, the large major second and quasi-minor-third is
considerably closer to 8:7 (~231.17 cents) than to 7:6 (~266.87
cents); at 960 cents, the small minor seventh and quasi-major-sixth is
likewise closer to 7:4 (~968.83 cents) than to 12:7 (~933.13 cents).
See also the table of intervals in Section 2.

5. Possibly 24-tET is something of an intermediate case: while the
usual 12-tET intervals offer a reasonable approximation of Pythagorean
intonation, I would say that the intervals of 250 cents and 950 cents
(and also the large major third at 450 cents) are a main attraction of
this tuning, at least for me. The dramatic contrast between these
"ultra-Gothic" intervals and the regular "subdued Pythagorean" ones
gives 24-tET, at least for me, a certain 20th-century modernistic
flavor to which I take a special affection.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: A neo-Gothic approach to 20-tET (Part 1)

🔗mschulter <MSCHULTER@VALUE.NET>

🔗David J. Finnamore <daeron@bellsouth.net>

🔗D.Stearns <STEARNS@CAPECOD.NET>

🔗Paul Erlich <paul@stretch-music.com>