Yahoo Tuning Groups Ultimate Backup tuning Re: Adaptive JI/RI tuning and neo-Gothic valleys

-------------------------------------------------------
Adaptive rational intonation and neo-Gothic valleys
A 24-note scheme a la Vicentino
(Essay in honor of John deLaubenfels)
Part I: Description and musical overview
-------------------------------------------------------

Over the past months, diverse contributors here including John
deLaubenfels, Paul Erlich, and Dave Keenan have been discussing such
concepts as adaptive tuning systems, harmonic entropy or complexity,
and possible distinctions between styles of tuning and music relying
exclusively on integer ratios. Here I would like to present a 24-note
tuning new to me -- however many times it may have been previously
discovered or rediscovered -- which may bring together many of these
themes.

While warmly thanking all the people who have contributed to these
discussions, I would like to dedicate this article especially to John
deLaubenfels, both for his courage as well as inventiveness in daring
to practice and incrasingly to perfect innovative adaptive tuning
techniques, and for a discussion which gives the "new" tuning
discussed here a special significance for me.

Some weeks ago, John and I shared views about the adaptive just
intonation (JI) system of Nicola Vicentino (1555), discussed in
Section 3 below. Vicentino's 38-note system, evidently featuring two
19-note manuals in meantone (Gb-B#) tuned 1/4 syntonic comma apart
(~5.38 cents), permits 16th-century sonorities with pure fifths and
minor thirds as well as major thirds (e.g. 4:5:6, 10:12:15).

Vicentino himself reported the exquisite qualities of this system as
realized in a 36-note version on his archicembalo ("superharpsichord")
and arciorgano ("superorgan"), but now as then it seems a daunting
task for even the most skilled human keyboardist to negotiate the
concourse of pure sonorities fluently in real time. Happily, the
beauties of Vicentino's system for 16th-century music can be
demonstrated today by automated technologies ranging from "player
harpsichords" (a la Colin Nancarrow) to MIDI sequencers.

One effect of this dialogue was to make me yearn for an adaptive
tuning system which I could experience with my own eager but very
modestly agile fingers. Now it is my delight, inspired by the examples
of Nicola Vicentino and John deLaubenfels, to explain how this
yearning has been fulfilled.

The tuning I am about to present may have a logic analogous in some
ways to Vicentino's, and features pure versions of some of John's
favorite intervals and sonorities (e.g. 12:14:18:21, 6:8:9), albeit in
a neo-Gothic musical context differing radically from that of either
Vicentino's JI system or John's adaptive tunings.

Even while writing this article, my viewpoint has been altered and
transformed by two germinal statements leading me into a realm of
neo-Gothic music based on small-integer just intonation (JI), a realm
where Easley Blackwood's "infatuation with sheer sound quality"
harmoniously coexists with incisive cadential progressions. One of
these inspiring statements is a description of David Doty's opus
_Uncommon Practice_, a kind of alternative music history in action;
the other is an article by Dan Stearns quoting Blackwood's passage[1]
on the possible future of music with "pure intervals":

http://www.syntonic-rec.com/ucp.html (David Doty, _Uncommon Practice_)
http://www.egroups.com/message/tuning/16408 (Dan Stearns on Blackwood)

Sharing this intonational adventure provides me with an opportunity to
honor another contributor to this List, John Chalmers. Back in 1998,
John remarked to me that a Pythagorean tuning if sufficiently extended
would produce intervals approximating ratios of 7. Out of this remark
grew my 24-note Pythagorean or "Xeno-Gothic" scheme with two 12-note
keyboards tuned a Pythagorean comma apart, from which it is but a
small step to the present rational adaptive variation featuring pure
fifths and 7-based intervals.

-------------------------------------------------------------------
1. Introduction: What is adaptive just/rational intonation (JI/RI)?
-------------------------------------------------------------------

This 24-note neo-Gothic "adaptive rational tuning," based on integer
ratios only, consists of two regular 12-note Pythagorean tunings
(Eb-G#) a septimal comma apart (64:63, ~27.26 cents). Using the carat
sign (^) to show a note raised by a septimal comma, we can notate the
tuning as follows:

C#^ Eb^ F#^ G#^ Bb^
C^ D^ E^ F^ G^ A^ B^ C^
-------------------------------------------------------
C# Eb F# G# Bb
C D E F G A B C

In addition to the usual Pythagorean intervals available on each
keyboard, this scheme makes available pure intervals combining factors
of 2, 3, and 7, including 7:6 minor thirds (e.g. E^3-G3); 7:4 minor
sevenths (e.g. E^3-D4); 8:7 major seconds (e.g. D3-E^3); 9:7 major
thirds (e.g. G3-B^3); and 12:7 major sixths (e.g. G3-E^4).

To reveal the one notable compromise involved, we may look at the two
manuals as 12-note segments of two Pythagorean chains of fifths
separated by the small rational interval known as a septimal schisma
or Eduardo Sa/bat's "Beta 2," 33554432:33480783 (~3.80 cents).[2] This
interval is equal to the difference between the septimal comma of
64:63 and the Pythagorean comma (e.g. Eb-D#) of 531441:524288 (~23.46
cents). In the following diagram, Vicentino's "comma sign" or
apostrophe ('), which could stand for various small intervals,
indicates that the upper chain of fifths is a septimal schisma higher:

(Eb^-Bb^-F^- C^ - G^ - D^ - A^ - E^)...
-D#'-A#'-E#'-B#'-F##'-C##'-G##'-D##'...
|
-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G#-

The fifth connecting the two manuals, G#-D#' (G#-Eb^), is therefore
stretched by a septimal schisma of ~3.80 cents, taking on the large
integer ratio of 16777216:11160261 (~705.76 cents), a size very close
to that of the fifth in 17-tone equal temperament or 17-tET at around
705.88 cents, or about 3.93 cents wide. All other fifths are pure.

In addition to being a rational intonation (RI) system based entirely
on integer ratios, the scheme is thus an "adaptive tuning" where notes
of a regular tuning are shifted by a small amount in order to obtain
more pure intervals.

Here the notes of the upper manual are raised by the 3.80-cent
septimal schisma from their positions in a regular 24-note Pythagorean
or Xeno-Gothic tuning, where they would be a Pythagorean comma
(e.g. Eb-D#) rather than a septimal comma (e.g. Eb-D#') above those of
the lower manual. This shift is comparable to the evident adjustment
of 1/4 syntonic comma (~5.38 cents) in Vicentino's adaptive scheme.

As a user of Xeno-Gothic for some two years, I might add that the idea
of adaptively nudging the two manuals apart by an additional septimal
schisma to obtain pure 7-based intervals occurred to me just within
the last few weeks, with a catalyzing role played by Graham Breed's
schismic temperaments, to which musical lineage this tuning may also
belong (see Section 5).[3]

Here is a Scala file for the tuning, with notes alternating between
the two manuals:

! xenoga24.scl
!
Xeno-Gothic rational adaptive tuning, 3-7 ratios (keyboards 64:63 apart)
24
!
64/63
2187/2048
243/224
9/8
8/7
32/27
2048/1701
81/64
9/7
4/3
256/189
729/512
81/56
3/2
32/21
6561/4096
729/448
27/16
12/7
16/9
1024/567
243/128
27/14
2/1

Here is a keyboard chart using a MIDI-style octave notation with C4 as
middle C. As in previous diagrams, a carat (^) shows a note raised by
a septimal comma, and an apostrophe (') a note raised from its regular
Pythagorean position by a septimal schisma. These signs are used to
show equivalent spellings for the accidentals Eb^4/D#'4 and Bb^4/A#'4:

243:224 2048:1701 81:56 729:448 1024:567
140.95 321.40 638.99 842.90 1023.35
C#^4 Eb^4/D#'4 F#^4 G#^4 Bb^4/A#'4
_113.7|90.2_90.2|113.7_ _113.7|90.2_113.7|90.2_90.2|113.7_
C^4 D^4 E^4 F^4 G^4 A^4 B^4' C^5
64:63 8:7 9:7 256:189 32:21 12:7 27:14 128:63
27.26 231.17 435.08 525.31 729.22 933.13 1137.04 1227.26
203.91 203.91 90.22 203.91 203.91 203.91 90.22
------------------------------------------------------------------------
2187:2048 32:27 729:512 6561:4096 16:9
113.7 294.13 611.73 815.64 996.09
C#4 Eb4 F#4 G#4 Bb4
_113.7|90.2_90.2|113.7_ _113.7|90.2_113.7|90.2_90.2|113.7_
C4 D4 E4 F4 G4 A4 B4 C5
1:1 9:8 81:64 4:3 3:2 27:16 243:128 2:1
0 203.91 407.82 498.04 701.96 905.87 1109.78 1200
203.91 203.91 90.22 203.91 203.91 203.91 90.22

--------------------------------------------------------
1.1. Musical context and qualities: neo-Gothic JI or RI?
--------------------------------------------------------

In a neo-Gothic setting, this tuning takes on lively musical qualities
inviting two intonational outlooks: a "just intonation" (JI) approach
striving consistently to achieve pure ratios for stable and unstable
sonorities alike; and a familiar neo-Gothic approach often favoring
complex Pythagorean ratios for unstable sonorities, which might more
communicatively be termed "rational intonation" (RI).

In exploring these approaches, we enter the neo-Gothic universe of
stable trines, unstable quads, and "3-flavor" and "7-flavor"
progressions. While seeking to explain relevant terms here as the
narrative unfolds, in good part by example, I would warmly refer
readers desiring a step-by-step presentation of these concepts in a
fuller musical context to the series "A gentle introduction to
neo-Gothic progressions," much welcoming questions or other feedback:

http://www.egroups.com/message/tuning/15038 (1/Pt 1)
http://www.egroups.com/message/tuning/15630 (1/Pt 2A)
http://www.egroups.com/message/tuning/15685 (1/Pt 2B)
http://www.egroups.com/message/tuning/16134 (1/Pt 2C)

A general musical overview of our adaptive JI/RI system might be as
follows:

(1) All fifths and fourths are pure, except for G#-D# or D#-G#,
"tempered by ratio" a septimal schisma (G#-D#' or D#'-G#).

(2) Stable three-voice sonorities or trines therefore have pure ratios
of 2:3:4 (e.g. D3-A3-D4) or 3:4:6 (e.g. D3-G3-D4), except that forms
such as G#3-D#4-G#4 or D#4-G#4-D#5 are about as impure as in 17-tET.

(3) Unstable four-voice cadential sonorities or quads are available in
pure "7-flavor" ratios such as 14:18:21:24 (e.g. G3-B^3-D4-E^4) or
12:14:18:21 (e.g. D^3-F3-A^3-C4); and likewise unstable three-voice
subsets or "triples," e.g. G3-B^3-E^4 (7:9:12); D^3-F3-A^3 (6:7:9);
D^3-A^3-C4 (4:6:7).

(4) Mildly unstable "quintal/quartal" triples combining fifths and/or
fourths with major seconds or ninths or minor sevenths are also pure:
e.g. G3-C4-D4 (6:8:9); G3-D4-A4 (4:6:9); G3-C4-F4 (9:12:16).

(5) Paul Erlich's 4:6:7:9, in neo-Gothic terms the unstable and
exotically engaging _nona fissa Erlichana_ or "Erlichan split ninth,"
is also available in a pure version (e.g. A^3-E^4-G4-B^4).

(6) Additionally, either keyboard provides usual Pythagorean versions
of unstable quads and triples featuring complex "3-flavor" ratios:
e.g. G3-B3-D4-E4 (64:81:96:108); D^3-F^3-A^3-C^4 (54:64:81:96);
G3-B3-E4 (64:81:108); D^3-F^3-A^3 (54:64:81); D3-A3-C4 (18:27:32).

To place some of these intonational features in a musical context, we
might well begin with basic quad-to-trine cadences in their "7-flavor"
and "3-flavor" versions.

In Gothic or neo-Gothic music, the complete three-voice unit of stable
concord is the _trine_ consisting of outer octave, lower fifth, and
upper fourth, e.g. D3-A3-D4 or F3-C4-F4.

The most efficient cadential progressions involve motion from an
unstable _quad_ with four voices and four unstable intervals to a
stable trine or its prime interval of the fifth, each unstable
interval resolving by stepwise contrary motion with one voice moving
by a whole-tone and the other by a semitone (M2-4, M3-5, M6-8, m3-1,
m7-5):

Expansive most proximate quads Contractive most proximate quads

Intensive Remissive Intensive Remissive

E4 F4 E4 F#4 D4 C4 D4 C#4
D4 C4 D4 C#4 B3 C4 B3 C#4
B3 C4 B3 C#4 G3 F3 G3 F#3
G3 F3 or G3 F#3 E3 F3 or E3 F#3

(M6-8 + M3-5 + m3-1 + M2-4) (m7-5 + m3-1 + M3-5 + m3-1)

An _expansive_ most proximate quad features an outer major sixth
expanding to the octave of a complete trine, while its _contractive_
counterpart features an outer minor seventh contracting to a fifth.
Intensive resolutions of these sonorities have ascending melodic
semitones, while remissive resolutions have descending semitones.

In our adaptive tunings, these cadences may occur in either "7-flavor"
forms with quads featuring pure 7-based intervals, or in "3-flavor"
forms featuring complex Pythagorean intervals based on powers of 3:2.
For example, here are 7-flavor and 3-flavor versions of the intensive
cadences we have just considered, with sonorities shown as frequency
ratios, and also in rounded cents with respect to the lowest voice:

7-flavor cadences

E^4 F4 D4 C4
D4 C4 B^3 C4
B^3 C4 G3 F3
G3 F3 E^3 F3

14:18:21:24 2:3:4 12:14:18:21 2:3:4
0-435-702-933 0-702-1200 0-267-702-969 0-702-1200

3-flavor cadences

E4 F4 D4 C4
D4 C4 B3 C4
B3 C4 G3 F3
G3 F3 E3 F3

64:81:96:108 2:3:4 54:64:81:96 2:3:4
0-408-702-906 0-702-1200 0-267-702-969 0-702-1200

Both flavors share pure 2:3:4 trines and incisive cadential semitones
at 256:243 or around 90.22 cents in the usual Pythagorean or 3-flavor
versions (e.g. E3-F3, B3-C4), and a yet more efficient 28:27 or around
62.96 cents in the 7-flavor versions (e.g. E^3-F3, B^3-C4).

As discussed at more length in Section 2, the streamlined and
"superefficient" 7-flavor cadences may often be favored in either a JI
style seeking pervasive small-integer-ratio sonorities, or an RI style
favoring complex 3-flavor Pythagorean sonorities at many points. For
example, the following cadence from an unstable 7-flavor triple at a
pure 7:9:12 to a complete trine (M6-8 + M3-5) might occur in either
style:

E^3 F4
B^3 C4
G3 F3
7:9:12 2:3:4
0-435-933 0-702-1200

What distinguishes the two styles is the overall context, as in these
RI and JI versions of a cadential idiom found in various compositions
of the 14th and early 15th centuries:

large-integer RI low-integer JI
(7-flavor cadence) (consistent 7-flavor)

A4 G4 F4 E^4 F4 A^4 G^4 F4 E^4 F4
E4 D4 C4 B^3 C4 E^4 D^4 C4 B^3 C4
C4 Bb3 A3 G3 F3 C4 Bb3 A^3 G3 F3

In the RI version, the descending sixth sonorities leading up to the
cadence have 3-flavor or Pythagorean ratios, a complexity nicely
fitting their musical role as "forerunners and handmaidens" of the
coming stable concord, to borrow the evocative words of Johannes Boen
in 1357. Against this "inertial frame" of Pythagorean intonation, the
sonority G3-B^3-E^4 stands out as a point of culminating cadential
tension. Its "stretched" 7-flavor intervals (9:7, 12:7) and the
extra-narrow melodic semitones of approach and resolution (C4-B^3-C4,
F4-E^4-F4) make the final resolution stand out in a special way.

In the JI version, however, pure 7-flavor intonation is itself the
pervasive norm: we have a stream of 7:9:12 and 18:21:28 sonorities, of
which the penultimate G3-B^3-E^4 is an integral part rather than a
special cadential amplification. Melodically, 63-cent semitones at
28:27 likewise become the diatonic norm (e.g. Bb3-A^3 in the lowest
voice, as well as the cadential B^3-C4 or E^4-F4). The alternating 9:8
and 8:7 whole-tones in the manner of Archytus or al-Farabi (see
Section 2) also lend this pure form of JI a special melodic quality.

If the RI version represents a usual 3-flavor Gothic rendition
adorned by the accentuated 7-flavor cadence, the JI version might
represent a kind of "neo-Gothic impressionism" recalling Debussy's
love of sheer sonority. Both renditions share the same cadential
grammar of a series of unstable sixth sonorities leading up to a
stable trine, while varying in vertical and melodic color alike.

------------------------------------------
1.2. A 24-quad system: mapping the lattice
------------------------------------------

In addition to 24 notes, our adaptive tuning includes 24 complete
"most proximal quads" of the usual 3-flavor and 7-flavor varieties we
have met in some of the above examples. Expansive versions of these
quads have a major third, fifth, and major sixth above the lowest
voice (e.g. G3-B3-D4-E4), while contractive versions have a minor
third, fifth, and minor seventh (e.g. E3-G3-B3-D4).

In locating these quads, the following lattice diagram may be
helpful. As explained near the opening of Section 1, a carat (^) shows
a note raised by a septimal comma, and an apostrophe (') in an
alternative spelling (e.g. D#') shows a note raised by a septimal
schisma from its usual Pythagorean position:

D#' A#' E#' B#'
Eb^-Bb^-F^- C^- G^ - D^ - A^ - E^ - B^ - F#^ - C#^ -G#^
/ / / / / / / / /
/ / / / / / / / /
Eb - Bb - F - C - G - D - A - E - B - F# - C# - G#

In this diagram, the eight slanted rectangles joining the two
keyboards indicate pure 7-flavor quads at 14:18:21:24 (expansive) or
12:14:18:21 (contractive). Starting at the left, we begin with the
quad Eb-G^-Bb-C^ (expansive form) or C^-Eb-G^-Bb (contractive form),
concluding at the right with E-G#^-B-C#^ or C#^-E-G#^-B.

For each of these eight quads, the lowest note of the expansive form
is located at the lower left (e.g. Eb of Eb-G^-Bb-C^); the lowest note
of the contractive form at the upper left (e.g. C^ of C^-Eb-G^-Bb).

Taking a "close up" look at such a 7-flavor quad may make its
intonational structure clearer:

Expansive form Contractive form

C^4 G^3 C^3 G^3
933 435 0 702
12:7 9:7 1:1 3:2
24 _ _ _ 4:3_ _ _ _18 12 _ _ _ _ _ _ _ 18
/ . 498 / / . /
/ . / / . /
/ . / / . /
/ 8:7 / 7:6 / 9:7 / 7:6
/ 231 / 267 / . 435 / 267
/ . / / . /
/ . / / . /
/_ _ _ _ _ _ _ _ ./ /_ _ _ _3:2_ _ __/
14 21 14 702 21
1:1 3:2 7:6 7:4
0 702 267 969
Eb3 Bb3 Eb3 Bb3

Here the numbers at the corners of a rectangle identify each of the
four notes with a frequency ratio number (14-18-21-24 or 12-14-18-21),
and give the tuning ratio and interval in cents of each note in
reference to the lowest voice (1:1-9:7-3:2-12:7 or 1:1-7:6-3:2-7:4;
0-435-702-933 or 0-267-702-969 cents).[4]

Numbers near the midpoint of a side or inner diagonal of the
rectangle show intervals between upper voices. In the expansive form
Eb3-G^3-B3-C^4, for example, the diagonal connecting Bb3-C^4 shows an
interval of 8:7 or 231 cents.

A 7-flavor quad includes six intervals: four unstable intervals always
occurring between notes on different manuals, and shown by slanted
lines or diagonals; and two stable intervals of the fifth/fourth class
always occurring between notes on the same manual, and shown by
straight lines. Here are these intervals for our two quads above:

-----------------------------------------------------------------
Expansive | Contractive
notes interval ratio size | notes interval ratio size
-----------------------------------------------------------------
4 unstable intervals
-----------------------------------------------------------------
Eb3-C^4 M6 12:7 933 | C^3-Eb3 m3 7:6 267
Eb3-G^3 M3 9:7 435 | Eb3-G^3 M3 9:7 435
G^3-Bb3 m3 7:6 267 | G^3-Bb3 m3 7:6 267
Bb3-C^4 M2 8:7 231 | C^3-Bb3 m7 7:4 969
-----------------------------------------------------------------
2 stable intervals
-----------------------------------------------------------------
Eb3-Bb3 5 3:2 702 | Eb3-Bb3 5 3:2 702
G^3-C^4 4 4:3 498 | C^4-G^4 5 3:2 702
-----------------------------------------------------------------

To keep intervals and ratios in musical perspective, let us observe
our pure 7-flavor quads in cadential action:

Expansive/remissive Contractive/remissive

C^4 D^4 Bb3 A^3
Bb3 A^3 G^3 A^3
G^3 A^3 Eb3 D^3
Eb3 D^3 C^3 D^3

(M6-8 + M3-5 + m3-1 + M2-4) (m7-5 + m3-1 + M3-5 + m3-1)

In these standard resolutions, the four unstable intervals resolve by
stepwise contrary motion, while stable fifths or fourths progress by
parallel motion (e.g. Eb3-Bb3 to D^3-A^3 and G^3-C^4 to A^3-D^4 in the
expansive cadence).

Here we have shown only remissive resolutions with descending melodic
semitones and ascending whole-tones. As it happens, we are in the far
flat region of our tuning system where standard intensive resolutions
with ascending semitones and descending whole-tones would take us
outside our 24-note gamut, requiring the additional notes Db and Ab:

Expansive/intensive Contractive/intensive

C^4 Db4 Bb3 Ab3
Bb3 Ab3 G^3 Ab3
G^3 Ab3 Eb3 Db3
Eb3 Db3 C^3 Db3

(M6-8 + M3-5 + m3-1 + M2-4) (m7-5 + m3-1 + M3-5 + m3-1)

More generally, while any expansive or contractive most proximal quad
has in principle both an intensive and a remissive resolution, if we
wish to maintain usual note spellings and melodic step sizes then only
one of these resolutions may be available in regions near the flat or
sharp edge of our 24-note gamut. From another viewpoint, these regions
offer "nonstandard" variations on the other resolution, for example a
progression with melodic diminished thirds substituted for usual major
seconds[5].

In addition to the eight most proximal quads in the 7-flavor marked by
the slanted rectangles of our diagrams, each keyboard offers eight
such quads in the usual Pythagorean 3-flavor, adding up to a total of
24 regular most proximal quads in these two flavors.

Here I have focused mainly on the pure 7-flavor quads, because usual
Pythagorean or 3-flavor quads are discussed at length in "A gentle
introduction to neo-Gothic progressions," for which URL links appear
at the beginning of Section 1.1. Indeed, each manual is a complete
12-note Pythagorean tuning in itself, and one of the attractions of
this JI/RI system is that the new 7-flavor sonorities and traditional
Pythagorean ones are both there to be used however a given style may
desire.

-----
Notes
-----

1. Easley Blackwood, _The Structure of Recognizable Diatonic Tunings_
(Princeton: Princeton University Press, 1985), p. 153, suggesting that
in a tertian 7-limit setting, JI may gratify an "infatuation with
sheer sound quality" but weaken "harmonic progressions."

2. Eduardo Sa/bat, _Principios de la Gama Dina/mica_ (Montevideo:
Arca, 1994), Section 6.3, pp. 208-209, defines Beta 2 (Beta sub-dos)
as the difference between a pure 8:7 major second (~231.17 cents) and
the Pythagorean approximation formed from 14 fifths up minus eight
octaves at 4782969:4194304 (~227.37 cents), a quantity he approximates
by the decimal ratio of 1.00220 (~3.80 cents).

3. We might consider this a special kind of schisma tuning in which
the full (not-so-onerous) burden of the 3.80-cent septimal schisma is
placed on a single fifth (G#-E^, i.e. G#-D#') thus "virtually
tempered" or "tempered by ratio." Since this schisma is itself an
integer ratio, the tuning remains entirely rational-based. Possibly
the category "Rational Adaptive Schisma Tuning" (RAST) might express
both the Vicentino-like and schismic aspects.

4. Note that the tuning ratios simply represent vertical intervals
with reference to the lowest voice, as opposed to scale steps or the
like. A cadence from the expansive quad Eb3-G^3-Bb3-C^4 to D3-A3-D4,
or from the contractive quad C^3-Eb3-G^3-Bb3 to D3-A3, might occur in
pieces centered on various medieval modes or octave species.

5. Within the limitations of our 24-note system, we might choose the
variant intensive resolutions of Eb3-G^3-Bb3-C^4 to C#3-G#3-C#4 and
C^3-Eb3-G^3-Bb3 to C#3-G#3. In contrast to usual 7-flavor resolutions
with melodic whole-tones at 9:8 (~203.91 cents) and semitones at 28:27
(~62.96 cents), this variation involves Pythagorean diminished thirds
or small whole-tones (Eb3-C#3, Bb3-G#3) at 65536:59049 (~180.45 cents)
and semitones (e.g. G^3-G#3, C^4-C#4) a septimal schisma smaller than
the usual Pythagorean 256:243 or ~90.22 cents, at the rather imposing
ratio of 137781:131072 (~86.42 cents). Vertical interval sizes remain
unaltered. At the sharp end of the system, intonational variations are
by comparison rather minor, involving the "septimal schisma fifth"
G#-D#' and certain "3-flavor" intervals with this fifth in their
tuning lineages. For example, the remissive cadence F#3-A#'3-C#4-D#'4
to E#'3-B#'3-E#'3 (F#3-Bb^3-C#4-Eb^4 to F^3-C^4-F^4) involves a quad
with major third and sixth a septimal schisma (~3.80 cents) wider than
the usual Pythagorean sizes, and semitone motions (F#3-E#'3, C#4-B#'3)
narrower by the same amount.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Adaptive JI/RI tuning and neo-Gothic valleys

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗John A. deLaubenfels <jdl@adaptune.com>

🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗John A. deLaubenfels <jdl@adaptune.com>

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗John A. deLaubenfels <jdl@adaptune.com>