Yahoo Tuning Groups Ultimate Backup tuning Re: "Miracle Scale" -- comparing notes on an exciting week

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The "Miracle Scale": Consensus and/or Pluralism?
Comparing notes on an interesting week
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Hello, there, everyone, and I'd like to applaud the "Miracle Scale" as
a brilliant development in theory and emerging practice while adding a
bit of perspective which might explain some of the mixed feelings I've
also seen expressed here.

As you'll see below, if you're inclined to follow this rather long
saga, the main effect of the "Miracle" thread was to impel me over the
last few days to explore two "new" 24-note tunings, one at least of
which turned out to be in fact a "rediscovery."

This is a special opportunity to honor Manuel op de Coul. After I had
"discovered" the first tuning by playing around with some generators
in Scala, I wisely searched the Scala archives with 'grep' for
'233.985' (the generator in cents) and learned that he had documented
_precisely_ the same tuning in a 31-note version.

This is a near-subset of the "Miracle Scale," as I discuss below, and
I hope that Manuel will get due recognition for this generator of
3:2^1/3 in any discussion of such subsets (see temp31g3.scl, which may
have also been mentioned in earlier lists of Scala files -- I'm not
sure).

The second scale took a different approach, also inspired by all the
excitement here: start with the familiar, and make "one basic change"
leading to some radically new experience. This is a science fiction
genre used in the 1970's in a workshop with Ursula K. LeGuin, and also
in some "alternate history novels" where one "decisive event" comes
out differently, and history along with it.

Now I find myself in a kind of alternative music history -- not that
this is a totally novel situation for me, only that yesterday it got a
lot more "alternative" in a curiously 21st-century way.

Please forgive me if a preface my odyssey with a bit of an "editorial"
on the topic of diversity -- something that's helpful to reaffirm upon
the discovery of _any_ monumentally new tuning system or music.

To allude to a favorite quote of Paul Erlich from Max Meyer, "_the
invention of scales_" can and should be a contagious experience.
While speaking only for myself, I hope that I may articulate some
important concerns and suggest, in part through my own experience of
this last exciting week, how one awe-inspiring discovery can stimulate
others.

- - - -

This is more than the issue of a scale: it is in good part a question
of what "optimization" means in a xenharmonic community with so many
different musical and intonational ideals. As one might ask, "Who
will optimize the optimizers?"

Is an "optimized" system the same thing for me, a follower of the
medieval-Manneristic European tradition, as for Jon Szantos, a devoted
musician in the tradition of Harry Partch; for Haresh Bakshi, a
musician skilled in the practice and theory of the raga tradition of
India, as for Judith Conrad, a harpsichordist and advocate of the
tradition of Baroque Europe; for Paul Erlich, an exponent of small
integer ratios as the basis of "ideal" temperaments, as for Jacky
Ligon, often a lover of large integer ratios?

As a medievalist, I'm unlikely to say that any new tuning makes
Pythagorean "obsolete," and I'd strongly suspect that Jon would say
the same about the 43-note system of Harry Partch. Please note that
we're not promoting these scales as total solutions either, of course.

Emphasizing with great circumspection that I don't want to imply that
people are promoting the "Miracle Scale" as a "total solution" either,
only to focus on why it might best be cherished and promoted as an
astounding breakthrough under one theoretical approach for certain
styles, I'd like to show how basic musical assumptions do differ
here.

If we have different assumptions and "rules" for making music -- and
thus for designing and "optimizing" tunings -- then we can make this a
virtue, presenting to the "non-xenharmonic" world the thriving face of
pluralism.

We can also present the "Miracle Scale" as _one_ example of what
people are doing -- for example, Mary's beautiful pieces I've heard
about here -- along with Jon's Partchian or other JI adventures, the
Monz's Aristoxenian 75:64 adventure, Paul's 22-tET decatonic music,
Jacky's Rational Intonation (RI), maybe in something of the tradition
of Kathleen Schlesinger, and so forth.

Dan Stearns has been focusing on the idea of "nontraditional
generators" for a long time here, and I'd see the "Miracle Tuning" as
one variety of this kind of solution.

Any scale, it seems to me, can be miraculous in the right time, place,
and stylistic context. If we attempt to take any one scale as
"optimal" in a style-independent context, however, it seems to me we
are in danger of reinventing the 12-tET hegemony. I'm not saying that
anyone is proposing to do this, only that when an "ideal" scale comes
along, it's a good time to remember that there are _lots_ of such
scales, as varied as the musical practices they support and foster.

For example, one effect this thread has had on me is to encourage me
to find my own _mirabile dictu_ scales (translation: "wonderful to
speak of"), sometimes a process of "rediscovery."

Dan Stearns and I were discussing his formula for generators a few
months ago, and what I recommend is that everyone with an interest in
xenharmonics try at least one of these "nontraditionally generated"
scales, whatever may fit one's own musical style or sense of
mathematical elegance.

For me, this thread plus a pleasant revisit to Dave Keenan's
"chain-of-fifths" regions (meantone, Pythagorean, 22-tET, Wolf fifths,
etc.) led me to look around in Scala for generators equal to or near
some precise fraction of a pure 3:2 fifth -- a Pythagorean touch, of
course. Dan Stearn, as I recall, last year did a 3:2^1/10 or
"decififth" scale somewhat like 17-tET, for which it took a while
before I caught on <grin> -- and Gary Morrison's 88-CET is close to
3:2^1/8.

Before telling the story rather briefly here, I'd like to call
attention to the role that Manuel op de Coul and Dave Finnamore will
play in it -- when I learn that my "discovery" is a rediscovery, as
well as a near subset of the "Miracle Scale."

Anyway, coming from a neo-Gothic outlook on optimization, I explored a
few areas in Scala like 683 cents and 718 cents -- intriguing from a
"chain-of-fifths" perspective focusing on 7-tET to 5-tET, with the
second generator near the 5-tET border and the first actually below the
7-tET border -- and also 3:2^1/4 or something like that.

Then I found what seemed to be _my_ "mirabile dictu" tuning of this
variety: 3:2^1/3, with a generator at ~233.985 cents (close enough!).

In Scala, I tried a 24-note version using this generator, and found
that it would yield 21 pure trines (2:3:4), 18 pure "quintal/quartal"
combinations like 6:8:9 or 9:12:16 (another Pythagorean favorite for
neo-Gothic music), and 17 near-pure "7-flavor" quads or tetrads
(12:14:18:21 or 14:21:24:28) and their subsets.

At the same time, I'd also have lots of friendly and familiar
Pythagorean "3-flavor" quads (54:64:81:96 or 64:81:96:108) -- in fact,
as Scala showed, 12 of them.

More than that, this was a _conceptually_ fascinating scale because it
crowded all these neat trines, triples, and quads in a compact
"geometry" outside of usual Pythagorean conventions. At least on a
12-note keyboard, I would find navigating it "new and different," to
say the least -- a challenge maybe a bit like classic multi-prime JI
systems.

The most obvious thing _not_ covered in this scale was a neo-Gothic
17-flavor (e.g. 14:17:21), but then I don't expect any one scale to
cover everything. As it happened, a tempered version of 3:2^1/3 in
36-tET (sound familiar?), or an Erlich-style least sum of squares
optmization for 12:14:18:21 at ~233.53 cents, would provide a
17-flavor -- although not in as many effortlessly accessible positions
as a simple 24-out-of-36-tET with the keyboards 1/6-tone apart.

What intrigued me especially about the pure 3:2^1/3 tuning was that it
might be called both "Just" and "quasi-just_ at the same time. All the
basic Pythagorean intervals were pure -- 2:1, 3:2, 4:3, 9:8 -- and
also the more complex thirds and sixths, etc.

By my usual definition, a "Just" tuning has all _stable_ consonances
as pure integer ratios -- and 3:2^1/3 meets this test, although it
doesn't meet my other usual test that "all intervals are derived from
integer ratios." Is it JI/RI? To borrow the title of Peter Abelard's
famous and sometimes very controversial book, _Sic et non_ -- "Yes,
and no."

From a 7-flavor perspective, it's "quasi-just," with the 7-based
ratios of 12:14:18:21 all ~2.81 cents impure. Not only the fifth, but
the Pythagorean limma or diatonic semitone (256:243, ~90.22 cents),
gets divided into three equal parts -- and also the 1029:1024 schisma
or the like, also known as a "gamelan residue" I would soon learn,
defined as the difference between the 64:63 and 49:48 commas. The
7-based intervals, as I would not be the first theorist to discover,
are impure by 1/3 of this schisma or residue.

Having described "my" tuning -- some earlier discoverers are coming up
soon, and in any event 3:2^1/3 isn't so surprising an idea when the
leading topic on this list has been 3:2^1/6 -- maybe I should say a
word about my implicit musical tastes and prejudices, some
(neo-)Gothic, but some also likely characteristically 20th-century.

While I've often described my use of sonorities like 6:8:9 or
12:14:18:21 in a neo-Gothic setting, I might suggest that in loving
these sonorities I am also a child of the era of Debussy and Bartok.
They feature fifths, fourths, and major seconds or minor sevenths --
but not the "strong discords" of either 13th-century or 20th-century
theory: minor seconds, major sevenths, or tritones or diminished
fifths.

In 20th-century terms, 6:8:9 and its kin are "fourth chords" or "fifth
chords," while as Dave Keenan has excellently explained, 12:14:18:21
is a 7-based variant on the minor seventh chord, a "subminor seventh
chord" to borrow his ideal nomenclature for this kind of context.
Likewise, following suit, we might describe 14:18:21:24 as a
"supermajor added sixth chord," a 7-flavor version of a favorite
closing sonority in some 20th-century genres.

Following either Jacobus of Liege in the early 14th century, or
various authors of books on "20th-century harmony" and the like such
as Ludmila Ulehla in her excellent _Contemporary Harmony_, I might
describe these sonorities as "pancompatible" -- they are very rich,
but include no strong or acute discords.

In an equitonal system such as Pythagorean tuning, meantone, 12-tET,
or 22-tET, by the way, more complex "pancompatible" sonorities are
possible -- pentads such as E-G-A-B-D (e.g. 0-5-9-13-18 in 22-tET),
which one might describe as "pentatonic clusters" from a chain of four
fifths, e.g. G-D-A-E-B. ("Anhemitonic pentatonic pentads" would be
more precise, since there are also pentatonic scales with semitones
and major thirds.) This is as far as we can go without introducing a
semitone or tritone.

Interestingly, if some of what I've read is correct, Japanese gagaku
music (very intriguing to my ears) uses complex clusters of this kind,
with forms including semitones or tritones deemed tenser than those
without, although at least one text cautions that the sense of
"concord/discord" or the like is different from the cadential dynamic
of much European music.

The only "anomaly" I associate with 12:14:18:21 or 14:18:21:24 is a
curious complication of multi-prime JI: we cannot construct a
pancompatible pentad from such a tetrad using pure fifths and fourths
because in systems where three fourths do _not_ equal a minor third.
Paul, you're written on this general point, and the 22-tET
pancompatible pentad (in a "Pythagorean" arrangement) might serve as
one example of your theme of "the advantages of tempering."

Fortunately 12:14:18:21 or 14:18:21:24 itself presents no problem in a
pure or quasi-pure 7-flavor, so I was ready to tune, once I had
figured out how to map this scale onto two 12-note Halberstadt
keyboards (also known as MIDI controller keybaords).

Somewhere around here, while getting ready both to tune and to post
here, I did a very wise thing -- using a UNIX-style 'grep' utility
to search the Scala archive for any scale with '233.985' in it. Lo and
behold, I promptly discovered temp31g3.scl -- and learned that my
precise 3:2^1/3 generator was a _re_discovery, not just the general
idea, but the exact tuning (in 31 notes rather my 24).

Manuel Op de Coul, the author of this scale as well as the version
of Scala I so happily use in MS-DOS 6.22, identified this scale as
"Cycle of 31 minor sevenths tempered by 1/3 gamelan residue."

Turning to intnam.par, I confirmed that this "gamelan residue" was
1029:1024, the same as the 49:48-64:63 comma or schisma I had
recognized was "trisected" by this tuning. I'm not sure whether
7-based intervals are involved in gamelan, or if maybe this is the
interval by which an octave is typically stretched or the like -- and
I'd love to learn the answer.

How wonderful that the gamelan tradition, another beautiful world
music, should somehow involve the same ratio I had encountered from a
neo-Gothic perspective. This kind of connection seems to me one of the
greatest miracles a scale can accomplish.

Roughly around that same time, I also read a post from David Finnamore
explaining how 8:7^3 quite closely approximated 9:6 (n+1:m-1) or 3:2.
In any discussion of the 3:2^1/3 scale, I would also want to give him
credit and recognition for describing this pattern (of which I wasn't
aware) and generalizing it to other ratios.

Anyway, I did tune it, and after talking with my Mom on Mother's Day
and sharing the excitement, I played it -- with lots of adventure in
navigating the not-so-symmetrically mapped keyboards. It was a new
geometry, and that in itself gives the scale a special place.

Over those same few days of following the "Miracle Tuning" thread and
(re-)discovering 3:2^1/3, another idea also occurred to me -- a much
simpler approach to getting more 12:14:18:21 or 14:18:21:24 sonorities
in a 24-note tuning, and getting them as pure JI.

It was so simple, why hadn't it occurred to me before: just tune two
standard 12-note Pythagorean manuals at the distance of a 7:6. That
gives a usual Eb-G# tuning on each manual, hardly an unusual approach
for me <grin>, plus no fewer than 11 of these pure 7-flavor quads or
tetrads.

Wherever there was a pure fifth on each keyboard -- everywhere except
for the "Wolf," Eb-G# -- I would have a 12:14:18:21 or 14:18:21:24, in
other words in 11 out of 12 locations.

One point that both Dave Keenan and Graham Breed had made in
discussions of 7-based optimizations is that with a usual Pythagorean
chain you need 14 fifths for a near-7:4. From my neo-Gothic
perspective, you need 16 to get a full 12:14:18:21 (the number needed
to generate a 9:7 major third, to me a "7-based" interval).

Tuning the two keyboards a septimal rather than Pythagorean comma
apart makes the 7-flavor sonorities pure, and "only" 8 positions is
enough for lots of music.

Now, however, I could get these sonorities in 11 positions, plus all
the usual Pythagorean sonorities on either keyboard.

Was this optimization too good to be true? There didn't seem to be any
obvious "unintended side effect," other than a few new patterns for
7-flavor sonorities and cadences to learn.

As I joked to myself, I had "broken the Keenan barrier" by happening
upon a 24-note Pythagorean tuning with 11 pure 12:14:18:21 sonorities.
A major point of Dave's "chain-of-fifths" papers is that the _length_
of the tuning chain required for a given interval can be very relevant
to the question of utility.

http://www.uq.net.au/~zzdkeena/Music/1ChainOfFifthsTunings.htm
http://www.uq.net.au/~zzdkeena/Music/2ChainOfFifthsTunings.htm

The unspoken qualification and compromise here, of course, is that we
no longer are dealing with a single chain of fifths, but with _two_
such chains at the distance of a pure 7:6 -- in contrast to a distance
defined by an even division of the octave.

This is no big problem for me, since an Eb-G# gamut is quite roomy for
lots of Gothic and neo-Gothic music, as a corresponding meantone gamut
is for much 16th-century music.

However Dave or Graham might very appropriately point out that we are
in fact favoring JI at the expense of the transposibility which comes
with a usual 24-note Pythagorean chain or something close to it such
as the septimal comma spacing.

Maybe even more important (and easy to leave unspoken within this
stylistic setting), this tuning represents for me what I might call
"2-3/9-7 JI/RI." In other words, there are three main types of
intervals included, some but not all of which may be shared with
related or not-so-related musics:

(1) Basic Pythagorean intervals and combinations involving
prime factors of 2 and 3, including the factor of 9
(2:1, 3:2, 4:3, 9:8), e.g. 2:3:4 or 6:8:9:12;

(2) More complex Pythagorean or 3-prime intervals and
combinations, especially usual diatonic ones such as
those involving regular thirds and sixths (e.g. 64:81:96
or 54:64:81);

(3) Intervals and combinations involving factors of 7 together
with Pythagorean factors of 2 and 3 or 9, especially
12:14:18:21 or 14:18:21:24.

From my viewpoint, categories (1) and (2) are "usual," or "bread and
butter"; category (3) is "xenharmonic," the "strange" or "guest"
element from a medieval Pythagorean perspective.

For a typical "JI" advocate, however, (1) and (3) may be routine but
the usual complex Pythagorean intervals of (2) something quite other
than "audibly just." Thus my use of "JI/RI" to describe the system as
a whole: for me, "ratios of 3" means all of the usual diatonic
Pythagorean intervals, including the rational but complex ones.

However, for such an advocate, an even more important _omission_ is
very likely to stand out: where are ratios of 5?

Someone seeking these ratios may follow a very different optimization
strategy than I might -- in often seeking instead, for example, other
kinds of complex thirds with ratios at or near 14:17:21 (not a major
strength for this tuning, although it does have a few neutralish
thirds around 345 and 357 cents to liven things up a bit further).

When I tuned it up yesterday, I very quickly found that it did have
another aspect that would give me a _radically_ new musical experience
within a few minutes of setting hands to keyboards, something that I
had noted at some level in theory, but utterly overwhelmed me when it
became very tangible practice.

I had figured out the rules or patterns for usual 7-flavor cadences,
and immediately confirmed that these patterns worked just as
predicted.

Now, however, it was time for something else. With this tuning,
playing two notes at a visual "unison" on the two keyboards produces
an actual 7:6 minor third; playing a "fifth" with the higher note on
the upper keyboard likewise produces a 7:4 minor seventh. An "octave"
or "fifth" with the lower note on the upper keyboard produces a 12:7
major sixth or 9:7 major third -- a 7:6 _smaller_ than the visual
interval.

Here's a layout arbitrarily using the octave C-C as a reference point,
with the ampersand symbol "&" showing a note raised by a 7:6. My
intent isn't to document the tuning at length, only to provide a
possible map for understanding what was about to happen:

381 561 879 1083 1263
C#& Eb& F#& G#& Bb&
C& D& E& F& G& A& B& C&
267 471 675 765 969 1173 1377 1467
--------------------------------------------------------------
114 294 612 816 996
C# Eb F# G# Bb
C D E F G A B C
0 204 408 498 702 906 1110 1200

Here I'm not so much concerned with fine points of ratios and cents as
with the logic of the hands.

Starting with a 2:3:4 trine, say D3-A3-D4 on the lower keyboard, I
shifted my left hand to D&3 on the upper manual, and played a series
of what _visually_ looked like parallel trines: D&3-A3-D4, E&3-B3-E4,
etc.

Actually, I was playing in parallel 7:9:12 sonorities! -- and at least
in this timbre (Yamaha TX-802, voice A56 "puff pipes" on both manuals)
it sounded absolutely beautiful and transporting. To conclude a
phrase, I did a usual cadence to a trine, say E&3-G3-C4 to D&3-A3-D4
for a landing on the final of Dorian. Maybe I might enrich that
penultimate sonority in my usual manner to E&3-G3-B&3-C4 (14:18:21:24),
a sound so rich and vibrant that it reminds me of the words _electra ut
sol_ from the Canticle of Canticles, "as bright as the sun," famously
set for example by Josquin.

People familiar a bit with late medieval or very early Renaissance
music may recognize this as a 7-flavor analogue of the 14th-century
English _cantilena_ style or the early 15th-century _fauxbourdon_ it
served to inspire on the Continent (e.g. Dufay or Binchois). We have
three voices starting on a trine, moving through a phrase in parallel
thirds, sixths, and upper fourths, and concluding it with a usual
cadence expanding to a trine.

However, there's one radical difference apart from the 7-based ratios:
the voices are moving in _identical_ parallels -- a string of 7:9:12
sonorities with _major_ thirds and sixths, rather than the usual
alternation of major and minor forms. It is somewhat like an organ
mixture, and a very 21st-century one to my ears.

Then I tried parallel 12:14:18:21 sonorities in a similar fashion,
e.g. starting at the fifth D3-A3, moving to D3-D&3-A3-A&3 and then
E3-E&3-B3-B&3, F3-F&3-C3-C&3, etc., eventually cadencing with a
progression of E3-E&3-B3-B&3 to D&3-A&3 (outer minor seventh
contracting to fifth).

Note that fauxbourdon in parallel minor sevenths is _not_ a late
medieval or early Renaissance practice, just my curious offshoot,
again with _identical_ parallel intervals.

As I was playing a chantlike melody, I was stunned and encompassed by
the sheer beauty of the sound, as if it distilled the sonorousness and
sweetness of Debussy, Vaughan Williams, and some of the music of my
own youth favoring minor seventh chords or the like. It was a
fauxbourdon for the 21st century, summing up some of the musical
moments of the previous century also.

Debussy especially occurred to me. I asked myself, why does so much of
the 20th century seem to be a struggle either to "fight tonality" or
to extend it? Why not just flow around it in what I might call a
Debussyan manner, taking a melody and enriching it with sheer
sonority?

In my new technique of chant harmonization or fauxbourdon, was I
hearing something analogous to what Dufay and his listeners heard in
the early 15th century -- a new and captivating texture? It seemed so
"modern," so fitting in the year 2001 -- as Dufay's music can also
sound to me at the right moments.

The old and the new are all there: regular Pythagorean keyboards for
lots of intricate discant and contrapuntal weaving, plus amplified
7-flavor cadences, plus this radically new kind of parallelism in
which the 7-flavor sonorities seem like "supersaturated concords," at
once independent moments of incredibly rich sweetness in themselves
and still very agreeably progressing to the usual trines.

(Prudence and due humility bid me add that if this "new" tuning and
kind of 7-flavor parallelism have been discovered before, something
not unlikely given the popularity of various kinds of parallelism from
Impressionism to today, it will make my own experience no less
exciting and transporting, but maybe even more meaningful in
initiating me into some community of travellers who have come to know
this inviting region.)

For me, this is indeed a "wonder tuning" -- but very happily, it
doesn't make any of the others obsolete.

If we can keep the same perspective on the "Miracle Tuning" and other
tunings discussed here, enthusiasm and pluralism may reinforce one
another both on this List and in the larger arena of world musics.

Most respectfully,

Margo Schulter
mschulter@value.net

> A quick ASCII-Monzo lattice of Margo's Sesquisexta tuning,
> using ASCII 72-EDO to notate the pitches:

Hello, there, Monz, and please let me thank you for your lattice
diagram, which also gives me a chance to look at 72-tET/72-EDO
notation for a neo-Gothic scale.

May I add that this detailed a response, including the lattices, is
especially moving given your other commitments -- I really appreciate
the perspective you can bring to something like this.

Here the 72-EDO "<" sign and my conventional neo-Gothic "@" sign are
closely analogous, both representing in this tuning a note lowered by
a septimal comma of 64:63 (~27.26). In 72-EDO, this comma is
represented by 2/72 octave, or 33.333... cents.

Comparing your lattice of the ratios of 3 and 7 with a considerably
less elegant one I had drawn for my own reference, I notice one
difference which I suspect might represent some difference in our
theoretical approaches.

In your lattice, the notes lined up on the 7-axis are a 7:4 or 7-based
minor seventh apart (e.g. C-Bb< in 72-EDO notation, or C-Bb@ in my
neo-Gothic notation).

In my lattice, the paired notes on the 7-axis are a 7:6 minor third
apart -- the distance between the two keyboards -- so that they form
quadrilaterals showing the 11 full 7-flavor "quads" or tetrads with
pure ratios of 12:14:18:21 or 14:18:21:24. Here's my version, with
more elegant renditions warmly invited:

Gb@ - Db@ - Ab@ - Eb@ - Bb@ - F@ - C@ - G@ - D@ - A@ - E@ - B@
/ / / / / / / / / / / /
/ / / / / / / / / / / /
Eb - Bb - - F - - C - - G - - D - A - E - B - F# - C# - G#

Each quadrilateral in this diagram represents a complete quad, with
the 12:14:18:21 version starting from the lower left note
(e.g. Eb-Gb@-Bb-Db@), and the 14:18:21:24 version starting from the
upper left note (e.g. Gb@-Bb-Db@-Eb).

In fact, it had occurred to me that in this tuning the two keyboards
themselves form a kind of visual lattice, with each manual as a
horizontal 3-axis (11 pure fifths), and the vertical 7-axis between
the two keyboards a pure 7:6 apart.

The fit between the conceptual and visual lattices can be shown by
including the "Sesquisexta-style" keyboard notation where an ampersand
or "&" sign shows a note on the upper keyboard raised by a 7:6, as
well as the equivalent "@" notation (or "<" in the 72-EDO system):

Eb& Bb& C& G& G& D& A& E& B& F#& C#& G#&
Gb@ - Db@ - Ab@ - Eb@ - Bb@ - F@ - C@ - G@ - D@ - A@ - E@ - B@
/ / / / / / / / / / / /
/ / / / / / / / / / / /
Eb - Bb - - F - - C - - G - - D - A - E - B - F# - C# - G#

By the way, for people interested in the musical significance of these
quads in a neo-Gothic setting, my "Gentle Introduction to neo-Gothic
progressions" might provide some context for this lattice-mapping
approach:

/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)

Here it may be worth emphasizing that the neo-Gothic 7-flavor quad
reflects a different kind of intonational logic than the "7-limit" or
"n-limit" concept often discussed here: it doesn't include any ratios
of 5, and does include 9:7, regarded as "9-limit" by many people
taking the n-limit approach.

Thank you for your very interesting comments and lattice showing the
approximations of 5. Maybe I should take some of my mathematical
musings in response to this and make them a separate reply, to keep
the length of this one manageable, especially in view of your Miracle
and related activities.

For now, I might mainly remark that your discussion of Pythagorean
approximations of 7:6 at once brings back lots of history we've shared
in on this List, and calls a new possibility to my attention which I
hadn't considered, although it makes beautiful sense the moment I look
at it:

> 3^-15 = ~270.674987 cents = ~3.804081415 cents wider than 7:6
> 3^-68 = ~267.0599412 cents = ~0.18903555 cent wider than 7:6

The first approximation is an important part of my "Xeno-Gothic"
tuning first posted here in 1998, with two 12-note Pythagorean manuals
at a Pythagorean comma apart, forming a chain of 23 pure fifths. As
you note, 7-prime intervals in this regular 24-note Pythagorean tuning
are impure by about 3.80 cents, the 3-7 schisma.

We've had some very interesting discussions about this kind of 7-based
approximation as a "Xenharmonic bridge," with a great entry on your
Web site chronicling some of this dialogue.

Then, late last year, I proposed a "modified Xeno-Gothic 3/7 JI"
tuning where the two keyboards are a _septimal comma_ apart, making
the 7-flavor intervals and quads pure; the first post of this series
gives a little background on Xeno-Gothic and the role of John Chalmers
in inspiring the original version:

/tuning/topicId_16640.html#16640 (Part I)
/tuning/topicId_17034.html#17034 (Part II)
/tuning/topicId_17339.html#17339 (Part III)
/tuning/topicId_17389.html#17389 (Part IV)
/tuning/topicId_17672.html#17672 (Part V)

While approximating a 7:6 almost perfectly by 68 fourths up hadn't
occurred to me, it makes perfect sense, because this interval is
impure only by the difference between the 3-7 schisma and the comma of
Mercator (or Kirchner?) by which 53 pure fifths exceed 31 pure
octaves, or about (3.80 - 3.62) or 0.18 cents -- actually closer to
0.19 cents with more accurate rounding, as you show.

Although I had described this very small interval in Part V of my "3/7
JI" series and proposed the name "nanisma," I didn't realize what a
neat application it could have: a virtually pure 7:6 using only pure
fifths as generators!

Most appreciatively,

Margo Schulter
mschulter@value.net

Re: "Miracle Scale" -- comparing notes on an exciting week

🔗mschulter <MSCHULTER@VALUE.NET>

🔗monz <joemonz@yahoo.com>

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🔗Dave Keenan <D.KEENAN@UQ.NET.AU>

🔗manuel.op.de.coul@eon-benelux.com

🔗Graham Breed <graham@microtonal.co.uk>

🔗mschulter <MSCHULTER@VALUE.NET>