Yahoo Tuning Groups Ultimate Backup tuning Re: The e-based tuning and metachromatic progressions (Part I)

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The e-based tuning and metachromatic progressions:
A feast of neo-Gothic flavors
(Part 1: Overview and vital statistics)
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In the early summer of the year 2000, the idea occurred to me of a
tuning based on Leonhard Euler's famous quantity _e_, the rate of
exponential growth, approximately 2.71828182845904523536029, with this
number defining Easley Blackwood's ratio "R" between the sizes of the
whole-tone and diatonic semitone.[1]

The resulting 12-note tuning (Eb-G#), with fifths around 704.61 cents
(~2.65 cents wide), provided a delightful initiation into what I would
come to describe as the most characteristic range of neo-Gothic
regular temperaments located in the general neighborhood of 704 cents.
Some cardinal features include rather gently tempered fifths and
fourths, the primary neo-Gothic consonances; complex and active major
and minor thirds around 14:11 and 13:11; and "supraminor/submajor"
thirds (augmented seconds and diminished fourths) not too far from
17:14 and 21:17, with convivially complex and active qualities.[2]

This convivial region, extending roughly from 29-tone equal
temperament or 29-tET (fifths ~703.45 cents, ~1.49 cents wide) to the
e-based temperament, is situated in the middle of the quintessential
neo-Gothic range between Pythagorean and 17-tET. While the e-based
tuning in its 12-note form typifies many charming qualities of the
region, we might say the same for any of the other tunings sharing
this neighborhood, each with its own unique hues and shadings.

Only some months later did I look into a 24-note version of this
temperament, and discover its secret giving it a very distinctive
place in the neighborhood: chains of 13, 14, or 15 fifths up or down
form near-pure intervals of the neo-Gothic "7-flavor" such as 9:7,
7:6, 12:7, 7:4, and 8:7. Most remarkably, 15 fifths up yield an
interval of around 969.10 cents, only about 0.28 cents larger than a
pure 7:4 (~968.83 cents).[3]

Along with these 7-flavor intervals inviting "streamlined" resolutions
in cadential progressions comes a new melodic interval with alluring
qualities for these progressions. In addition to the usual compact
e-based diatonic semitone of around 76.97 cents, and the large
chromatic semitone of about 132.25 cents, we have a diesis of around
55.28 cents (the difference between these two contrasting semitones)
very nicely serving as a small cadential semitone.

In this paper, I would like especially to focus on "metachromatic"
progressions using this 55-cent diesis as a melodic semitone, at the
same time documenting some general features of the tuning.

While striving to make what follows readable in itself, I would refer
readers who may be unfamiliar with neo-Gothic styles, cadences, and
intonational "flavors," or who might seek more background or detail,
to some related article. For a "Gentle Introduction to neo-Gothic
progressions," please see

/tuning/topicId_15038.html#15038 (1/Pt 1)
/tuning/topicId_15630.html#15630 (1/Pt 2A)
/tuning/topicId_15685.html#15685 (1/Pt 2B)
/tuning/topicId_16134.html#16134 (1/Pt 2C)

Part 2C of that series includes a discussion of some e-based cadences
and flavors set in a more general framework.

As I there remark, It seems especially felicitous that a tuning
designed to honor Euler should include near-pure versions of the
ratios 7:4 (~0.28 cents wide) and 7:6 (~2.37 cents narrow), both of
which he described and advocated for use in practical music in a
treatise of 1764.[4] While neo-Gothic music provides a radically
different musical context for their use than the 18th-century setting
addressed by Euler, this pleasant encounter between mathematics and
music is one much to be relished.

--------------------------------------------
1. Overview: keyboard diagram and Scala file
--------------------------------------------

We may define the e-based tuning as the regular temperament in which
the whole-tone (T) and diatonic semitone (S) have a ratio of T/S,
Easley Blackwood's R, equal to Euler's e. To find the size of the
diatonic semitone in cents for this tuning, we can use the following
formula based on the definition of a diatonic octave as including five
whole-tones and two diatonic semitones:

1200 1200 1200
S = ------ = ------ = ---------- = ~76.965461 cents
5R + 2 5e + 2 ~15.591409

From this semitone, we may calculate any other interval of the tuning:
for example, a regular fifth at ~704.606908 cents; a whole-tone at
~209.213815 cents; and a chromatic semitone at ~132.248354 cents.
The difference between the diatonic and chromatic semitones defines a
diesis of around 55.28 cents, also the amount by which three regular
major thirds of ~418.43 cents exceed a pure 1200-cent octave.

An excellent approximation of this tuning is provided by 109-tET
(fifth ~704.587 cents, or 64/109 octave), where a whole-tone of 19/109
octave (~209.17 cents) is divided into a diatonic semitone of
7/19-tone (~77.06 cents) and a chromatic semitone of 12/109-tone
(~132.11 cents). Here T/S or Blackwood's R is equal to 19/7 or
~2.71429, quite close to Euler's e. In 109-tET, 88 steps yield an
interval of ~968.807 cents, less than 0.02 cents narrow of 7:4.[5]

In its 24-note form, the e-based tuning may be mapped to two standard
12-note keyboards a diesis apart, an arrangement we might also
describe as a regular chain of 23 fifths. Here an ASCII asterisk (*)
shows a note raised by a diesis. Either keyboard in itself provides a
usual 12-note tuning (Eb-G#).

187.53 341.462 682.924 892.138 1046.069
C#*3 Eb*3/D#3 F#*3 G#*3 Bb*3/A#3
_132.2|77.0_77.0|132.2_ _132.2|77.0_132.2|77.0_77.0|132.2_
C*3 D*3 E*3 F*3 G*3 A*3 B*3 C*4
55.283 264.497 473.145 550.676 759.890 969.104 1178.317 1255.283
209.214 209.214 76.965 209.214 209.214 209.214 76.965
-------------------------------------------------------------------------
132.248 286.179 627.641 836.855 990.786
C#3 Eb3 F#3 G#3 Bb3
_132.2|77.0_77.0|132.2_ _132.2|77.0_132.2|77.0_77.0|132.2_
C3 D3 E3 F3 G3 A3 B3 C4
0 209.214 418.428 495.393 704.607 913.821 1123.035 1200
209.214 209.214 76.965 209.214 209.214 209.214 76.965

Here is a Scala file[6] of the tuning, with notes alternating between
the two manuals, proceeding in terms of the above diagram from C*3 on
the upper manual (~55.283 cents) to C4 on the lower manual (1200 cents).

! neogeb24.scl
!
Neo-Gothic e-based lineotuning (T/S or Blackwood's R=e, ~2.71828), 24 notes
24
!
55.28289
132.24835
187.53125
209.21382
264.49671
286.17928
341.46217
418.42763
473.71052
495.39309
550.67598
627.64145
682.92434
704.60691
759.88980
836.85526
892.13815
913.82072
969.10362
990.78618
1046.06908
1123.03454
1178.31743
2/1

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2. Some vital statistics: Intervals and approximations
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One way to get acquainted with the e-based tuning is to survey the
most common intervals and some rational approximations. The following
table includes all intervals occurring within a usual 12-note tuning;
neo-Gothic "7-flavor" intervals formed from chains of 13, 14, or 15
fifths or fourths; and some other miscellaneous intervals.

=====================================================================
Interval Cents Example ~Ratio Cents Variation
---------------------------------------------------------------------
1 0.00 C3-C3 1:1 (exact) 0.00 0.00
---------------------------------------------------------------------
limma 76.97 C3-Db3 23:22 76.96 + 0.01
or min2
---------------------------------------------------------------------
apotome 132.25 C3-C#3 14:13 128.30 + 3.95
or Aug1 27:25 133.24 - 0.99
13:12 138.57 - 6.32
---------------------------------------------------------------------
dim3 153.93 C#3-Eb3 12:11 150.64 + 3.29
---------------------------------------------------------------------
Maj2 209.21 C3-D3 9:8 203.91 + 5.30
26:23 212.25 - 3.04
---------------------------------------------------------------------
min3 286.18 C3-Eb3 33:28 284.45 + 1.73
13:11 289.21 - 3.03
---------------------------------------------------------------------
supram3 341.46 Eb3-F#3 17:14 336.13 + 5.33
or Aug2 C3-Eb*3/D#3 28:23 340.55 + 0.91
39:32 342.48 - 1.02
11:9 347.41 - 5.95
---------------------------------------------------------------------
subM3 363.14 C#3-F3 16:13 359.47 + 3.67
or dim4 Eb*3-G3 69:56 361.40 + 1.74
21:17 365.83 - 2.68
---------------------------------------------------------------------
Maj3 418.43 C3-E3 14:11 417.51 + 0.92
---------------------------------------------------------------------
4 495.39 C3-F3 4:3 498.04 - 2.65
---------------------------------------------------------------------
Aug3 550.68 Eb3-G#3 11:8 551.32 - 0.64
B3-E*4
---------------------------------------------------------------------
dim5 572.36 B3-F4 18:13 563.38 + 8.98
25:18 568.72 + 3.64
32:23 571.73 + 0.63
7:5 582.51 -10.15
---------------------------------------------------------------------
Aug4 627.64 F3-B3 10:7 617.49 +10.15
23:16 628.27 - 0.63
36:25 631.28 - 3.64
13:9 636.62 - 8.98
---------------------------------------------------------------------
dim6 649.32 G#3-Eb4 16:11 648.68 + 0.64
E*3-B3
---------------------------------------------------------------------
5 704.61 C3-G3 3:2 701.96 + 2.65
---------------------------------------------------------------------
min6 781.57 A3-F4 11:7 782.49 - 0.92
---------------------------------------------------------------------
supram6 836.86 C3-G#3 Phi (~1.618) 833.09 + 3.76
or Aug5 A3-F*4 34:21 834.17 + 2.68
112:69 838.60 - 1.74
13:8 840.53 - 3.67
---------------------------------------------------------------------
subM6 858.54 C#3-Bb3 18:11 852.59 + 5.95
or dim7 F*3-D4 23:14 859.45 - 0.91
28:17 863.87 - 5.33
---------------------------------------------------------------------
Maj6 913.82 C3-A3 22:13 910.79 + 3.03
56:33 915.55 - 1.73
---------------------------------------------------------------------
min7 990.79 C3-Bb3 23:13 987.75 + 3.04
16:9 996.09 - 5.30
---------------------------------------------------------------------
Aug6 1046.07 Eb3-C#4 11:6 1049.36 - 3.29
C3-Bb*3/A#3
---------------------------------------------------------------------
dim8 1067.75 C#3-C4 24:13 1061.43 + 4.32
F*3-E4 50:27 1066.76 + 0.99
13:7 1071.70 - 3.95
---------------------------------------------------------------------
Maj7 1123.03 C3-B3 44:23 1123.04 - 0.01
---------------------------------------------------------------------
8 1200.00 C3-C4 2:1 (exact) 1200.00 0.00
---------------------------------------------------------------------
diesis 55.28 Eb3-Eb*3/D#3 32:31 54.96 + 0.32
or comma F3-F*3 31:30 56.76 - 1.48
---------------------------------------------------------------------
Cadential "7-flavor" intervals in 24-note tuning
---------------------------------------------------------------------
large M2 230.90 D*4-F4 8:7 231.17 - 0.28
m3 - diesis
---------------------------------------------------------------------
small m3 264.50 G3-A*3 7:6 266.87 - 2.37
M2 + diesis
---------------------------------------------------------------------
large M3 440.11 G*3-C4 9:7 435.08 + 5.03
4 - diesis
---------------------------------------------------------------------
small m6 759.89 C4-G*4 14:9 764.92 - 5.03
5 + diesis
---------------------------------------------------------------------
large M6 935.50 G*3-F4 12:7 933.13 + 2.37
m7 - diesis
---------------------------------------------------------------------
small m7 969.10 G3-E*4 7:4 968.83 + 0.28
M6 + diesis
---------------------------------------------------------------------
Some miscellaneous intervals in 24-note tuning
---------------------------------------------------------------------
small 4 473.71 G3-B*3 21:16 470.78 + 2.93
M3 + diesis
---------------------------------------------------------------------
large 5 726.29 B*3-G4 32:21 729.22 - 2.93
m6 - diesis
---------------------------------------------------------------------
large M7 1144.72 F*3-F3 60:31 1143.23 + 1.48
8 - diesis 31:16 1145.04 - 0.32
---------------------------------------------------------------------
subdiesis 21.68 E*3-F3 81:80 21.51 + 0.18
or 17-comma 80:79 21.78 - 0.09
m2 - diesis
---------------------------------------------------------------------
---------------------------------------------------------------------

This table is by no means exhaustive: it includes all intervals
involving chains of 16 or fewer fifths or fourths, and thus occurring
in at least eight positions of a 24-note tuning. Additionally, it
includes a structurally important interval we might call the
"subdiesis" or "17-comma."

This interval formed from 17 fourths up, for example E*3-F3 or B*3-C3,
has a size of ~21.68 cents. We may define it either as a "subdiesis"
equal to the difference between a 76.97-cent diatonic semitone
(e.g. E3-F3) and a 55.28-cent diesis (e.g. E3-E*3), or as a "17-comma"
by which 17 fifths up fall short of 10 pure octaves.

Melodically, while the diesis typically serves as a kind of small
cadential semitone in progressions which I term "metachromatic" (see
Sections 3.3 and 4), subdiesis shifts may also sometimes occur in
approaching these progressions. Two notes a subdiesis apart, for
example F3 and E*3, may at times be treated as versions of the "same"
modal or cadential center. Thus the e-based subdiesis can play a role
somewhat analogous to that of a comma in just or rational intonation
systems.

As a 17-comma, this interval serves as one vital statistic placing the
e-based temperament on the portion of the range between Pythagorean
and 17-tET somewhat closer to the latter tuning. As we move from
Pythagorean toward 17-tET, the Pythagorean comma or diesis by which 12
fifths exceed 7 pure octaves grows larger, while the 17-comma grows
smaller; together, these two intervals make up a usual limma or
diatonic semitone.

In the following graph, the Pythagorean comma or diesis or "12-comma"
is shown by "12," and the 17-comma by "17." The number appearing at
the top of each bar shows the size of the diatonic semitone, equal to
their sum, in a given tuning[7]:

90.22
_________ 82.76
| | _________ 76.97
| | | | _________ 70.59
| | 17 | | 17 | 21.68 | _________
| 66.76 | | 41.38 | |_______| | |
17 | | |_______| | | | |
|_______| | | 12 | 55.28 | 12 | 70.59 |
| | 12 | 41.38 | | | | |
12 | 23.46 | | | | | | |
|-------|---------|-------|-------|-------|-------|-------|
Pyth 29-tET e-based 17-tET
701.96 703.45 704.61 705.88
0.00 +1.49 +2.65 +3.93

In Pythagorean tuning, the limma at 256:243 (~90.22 cents) divides
into a Pythagorean comma at 531441:524288 (~23.46 cents) and a
17-comma at 134217728:129140163 or ~66.76 cents, a very useful small
cadential semitone.

In 29-tET, the limma at 2/29 octave or ~82.76 cents divides into two
identical intervals of 1/29 octave or ~41.38 cents: the diesis and
17-comma are precisely equal, either serving as a striking kind of
cadential step distinct from a "semitone" in the usual sense.

In the e-based tuning, the limma of ~76.97 cents divides into a diesis
of ~55.28 cents and 17-comma of ~21.68 cents, with their Pythagorean
roles reversed: it is now the diesis which serves as a small semitone.

In 17-tET, the limma of ~70.59 cents is identical to the diesis, and
17 fifths are equal to 10 pure octaves, so that the 17-comma is
neutralized or dispersed.

For a regular 24-note Pythagorean, 29-tET, or e-based tuning, the
12-comma or diesis measures a "natural" distance between two 12-note
manuals[8]. In the e-based tuning, this interval is an alluring
cadential semitone, opening a universe of progressions flowing between
the manuals in special patterns here termed metachromatic.

-----
Notes
-----

1. On Easley Blackwood's ratio "R," see his book _The Structure of
Recognizable Diatonic Tunings_ (Princeton: Princeton University Press,
1985). Curiously, in August 1998, the idea had occurred to me of a
different kind of regular e-based tuning, devised by analogy with John
"Longitude" Harrison's famous 18th-century temperament where the size
of the major third is equal to a pure 2:1 octave divided by pi
(~3.14159), 1200/pi or ~381.97 cents, with fifths at ~695.49 cents
(~6.46 cents narrow), now the basis of Charles Lucy's system of
LucyTuning. In this analogous tuning, a major third would be equal to
1200/e, or ~441.46 cents, with fifths at ~710.36 cents (~8.41 cents
wide). Although I did not realize it at the time, there is a certain
musical parallel between these tunings. Harrison/LucyTuning (roughly
3/10-comma meantone) leans toward pure 6:5 minor thirds (1/3-comma)
rather than pure 5:4 major thirds (1/4-comma); the 1200/e scheme,
roughly a "3/10-septimal-comma" temperament, likewise leans toward
pure 7:6 minor thirds rather than pure 9:7 major thirds.

2. For an early appreciation of these points, focusing on the e-based
tuning it its 12-note form, see "Neo-Gothic tunings and temperaments:
Meantone through a looking glass," Parts I and II (7-8 July 2000),
/tuning/topicId_11096.html#11096 (Part I)
/tuning/topicId_11108.html#11108 (Part II). For a
more detailed discussion of the "four convivial ratios" (14:11, 13:11,
21:17, 17:14), see "Optimizing the Four Convivial Ratios" (9 October
2000), /tuning/topicId_14180.html#14180.

3. For a first report on these 7-flavor ratios in a 24-note tuning,
see "Quick bulletin -- e-based tuning and 7-flavored valleys" (14
October 2000), /tuning/topicId_14361.html#14361.

4. See http://www.ixpres.com/interval/monzo/euler/euler-en.htm for Joe
Monzo's brilliant translation of paper in French by Patrice Bailhache
on Euler's music theory, with Monzo's alternative translations and
commentary on Euler's original Latin.

5. For a virtually identical approximation, we may define a "nonoctave"
tuning dividing a pure 7:4 minor seventh into 88 equal steps, or
7:4^1/88, producing a step size of ~11.009385 cents (by comparison
with ~11.009174 cents in 109-tET), and a "stretched" 109-step octave
of ~1200.023 cents.

6. This file is formatted for import into Manuel Op de Coul's
outstanding program Scala, available for free on the Internet, which
can provide a list of all intervals found in the tuning and show
temperings for any desired rational ratios, among many other
features.

7. With the e-based tuning, as it happens, rounding errors produce a
situation where the diesis at a rounded 55.28 cents and 17-comma at a
rounded 21.68 cents add up to diatonic semitone or limma at a rounded
76.97 cents -- a curious "anomaly" of decimal arithmetic which I allow
to stand in the bar graph and accompanying discussion.

8. That is, if a regular tuning based on a single chain of fifths is
carried to 24 notes (23 fifths), then the two manuals will "naturally"
be at the distance of a 12-comma or diesis (12 fifths up). Carrying
17-tET to 24 notes (or any number beyond 17), however, can be done
only by using two or more chains of fifths placed at some arbitrary or
"artificial" distance. If this distance is precisely half of a 17-tET
step, the result is 34-tET, a tuning noted for closely approximating
5:4 and 6:5 thirds. For many neo-Gothic styles, one alternative is a
second chain placed above the first at the distance by which a 17-tET
whole-tone (3/17 octave, ~211.76 cents) differs from a pure 7:6,
~55.106 cents, an interval almost identical as it happens to the
e-based diesis. In a 34-note version (two 17-note manuals), this
arrangement would make available all of the usual 17-tET intervals
plus such attractions as a 7-flavor, a 17-flavor (submajor/supraminor
thirds and sixths), and also a 3-flavor (thirds and sixths close to
Pythagorean values).

Most respectfully,

Margo Schulter
mschulter@value.net

Re: The e-based tuning and metachromatic progressions (Part I)

🔗M. Schulter <MSCHULTER@VALUE.NET>

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🔗David Clampitt <david.clampitt@yale.edu>

🔗jpehrson@rcn.com

🔗M. Schulter <MSCHULTER@VALUE.NET>