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[In an excellent response to my original review of his article "Tuning,
Tonality, and Twenty-Two-Tone Temperament" in _Xenharmonikon_ 17
(Spring 1998:12-40), Paul Erlich has pointed out that I have sometimes
mixed two different concepts of an "n-limit" in just intonation.

The first concept, a "prime limit," focuses on the largest _prime_
number used for ratios in a given tuning, while an alternative "odd
limit" concept focuses on the largest _odd_ number used.

Paul not only helpfully distinguishes these concepts, but calls to my
attention an interesting case where my use of "n-limit" terminology
for tunings not strictly fitting the expected JI model may produce
strange results.]

> But then you speak of a non-prime, 9, as a limit:

> > Since the time of Debussy, more complex sonorities have been
> > accepted as stable: for example, tetrads such as the "added sixth"
> > (12:15:18:20 in one possible 5-limit tuning) and "minor seventh"
> > (12:14:18:21 in a 9-limit tuning).

> The prime limit of the first chord is indeed 5, but that of the
> latter, 7. However, as some of us have discussed before, there can be
> two equally useful concepts, an odd limit (closer to Partch's usage
> and useful for describing sonorities) and a prime limit (useful for
> describing resources of JI systems). The chords 12:15:18:20 and
> 12:14:18:21 are what I call the two "saturated 9-limit tetrads," since
> all the intervals in each chord are ratios of 9 or less, and no notes
> can be added to either chord without increasing the odd limit, and
> there are no other chords with these properties.

As a medievalist, I find that this distinction raises an interesting
question: which 9-based sonorities can be viewed as "complex 3-limit
sonorities," and which should be classed as specific to "9-limit"
tunings in Partch's sense?

I would tend to say that 4:6:9 (an actual Gothic sonority) and
(theoretically) 1:3:9, for example, are extended 3-limit sonorities,
but that 6:7:9 is 9-limit, not 3-limit (in part because it includes a
ratio based on 7, a larger prime than 3, of course).

The next quote refers back to 12:15:18:20 and 12:14:18:21:

> (Incidentally, both chords can be called minor seventh chords, as
> the first is octave-equivalent to 10:12:15:18.) This type of chord
> seems to have been missed by Partch, who may not have realized that
> utonalities and otonalities are not the only possible saturated
> chords for odd limits 9 and up. So perhaps you were thinking of the
> odd limit of the second chord, and the prime limit of the first
> chord?

Indeed, my recollections of writing the original post seem to fit well
with the explanation that I mixed the "prime limit" and "odd limit"
concepts without realizing the inconsistencies. Thanks to your
query, I've maybe become able to articulate why. First, a quick
historical aside on octave equivalence and inversion.

Certainly in the immediate context of the discussion, "since the time
of Debussy," we could apply the octave equivalence and inversion
concepts to call both sonorities "minor seventh chords." For the sake
of completeness, I should mention that in other periods, different
views (and appropriate terminologies) may prevail.

While the usual "inversion" concept draws a connection between
sonorities which might share the same pitch classes (e.g. e-g-b-d' and
g-b-d'-e'), medieval and Renaissance theory tend to focuses on
sonorities sharing the same _intervals_. Thus in medieval terms, the
first combination might be described as a "major sixth sonority," and
the second as a "minor seventh sonority."

In classical 18th-century theory, based on Rameau, g-b-d'-e' might be
described as _either_ an "added-sixth chord" or a "minor seventh chord
in inversion," depending on the context. This might be an interesting
case where even in a stylistic setting where "inversional equivalence"
usually holds, it gets qualified a bit in at least some areas of
theory.

I'm fascinated that Partch apparently didn't consider these combinations..
This sounds like a significant point that I'd love to see further
developed.

Again, your suggestion that I may have been applying the "largest
prime" concept to the first ratio, but the odd-limit concept to the
second, nicely explains my inconsistencies here. However, there's
another complicating factor we're about to consider.

> But then you go on to say,

> > Similarly, in Gothic style, a combination of three superimposed
> > thirds (e.g. e-g-b-d') may resolve to a stable 3-limit sonority
> > (e.g. f-c'); in triadic harmony, it may resolve to a 5-limit
> > sonority; in 7-limit harmony, it may serve as a stable tetrad.

> Now any chord that could be notated as e-g-b-d' cannot be a stable
> tetrad if the odd limit is 7, so you must be back to the prime limit
> definition and thinking of 12:14:18:21, or you should have said 9
> instead of 7, in which case you could have meant either 12:14:18:21 or
> 10:12:15:18.

First, of course, you're absolutely right that I should have said
"9-limit" instead of "7-limit" in reference to a stable 12:14:18:21
tetrad (odd-limit sense).

Now we come to a critical point which may be of interest in many
discussions of "harmonic evolution" and JI theory, and which you have
very helpfully brought into focus.

If these ratios really mean what they say, a not-unfair assumption,
then indeed they are impossible without a 9:7 interval, calling for a
9-(odd)-limit system in order for the tetrad to be stable.

However, what I left vaguely floating somewhere out there in my
conceptual space is the awareness that in a medieval 3-limit
(Pythagorean) context, or the 12-tone equal temperament (12-tet)
context assumed by much 20th-century keyboard music, the tuning does
_not_ actually treat minor sevenths as 7-based intervals, and nor does
it have anything very close to a 9:7.

In medieval 3-limit (Pythagorean) just tuning, for example, we have
54:64:81:96, or notes at about 0-294-702-996 cents; in a 20th-century
keyboard setting premised on 12-tet, we have notes at 0-300-700-1000
cents. In both cases, the m7 is at or close to 16:9, not 7:1, and the
thirds at or closer to 3-limit than to either 5-limit or 7-limit/9-limit
(6:7, 9:7).

Note that your possible 5-(prime)-limit interpretation of e-g-b-d' as
10:12:15:18, or about 0-316-702-1018 cents, involves the ratio 18:10
(9:5) for the minor seventh, but does _not_ involve any 9:7 large
major third, arguably an interval involving a distinctive level of
tension not found except in the version involving ratios with _both_ 7
and 9 as factors. Similarly, the 3-limit tuning involves 16:9 for this
minor seventh, but not a 9:7 third, which seems to me a definitive
interval for "9-(odd)-limit" in a certain idiomatic sense of "a system
of stable sonorities including the ratio between the odd-limit 9 and
the nearest smaller prime of 7."

In other words, what I suspect I may have been doing without realizing
it was to focus on the prime limit of 3 or 5 where the sonority has a
16:9 or 9:5 interval but no 9:7 interval, but to focus on the odd
limit when 9:7 was present.

This raises an interesting question: how _should_ we describe e-g-b-d'
as a stable 20th-century sonority on a 12-tet organ or piano, say?
Should we use "n-limit" terminology at all when the actual tuning in
question doesn't really fit such a model, and more closely fits
3-limit than 5-limit or 7-limit/9-limit?

While writing as imprecisely as I did may not necessarily be cause for
great celebration, having a reader such as yourself to point out my
inconsistencies, thus inviting more careful consideration of some
underlying issues, may be beneficial not only to me but to the
progress of xenharmonics.

> An easy way to get a feel for these and other possibilities on almost
> any retunable synth is to use a 12-out-of-22 tuning where E-F and B-C
> are 1/22 oct intervals and all the rest are 2/22 oct. Then:

[ ... ]

> 7) The interval B-f approximates 11:8 with an error of 6 cents and
> sounds good in the chord B-a-b-eb'-f, which approximates
> 4:7:8:10:11. It is interesting to resolve this interval by contrary
> motion through the very incisive 1/22 oct intervals to c-e. The latter
> interval is approximates a 9:7 with an error of 1 cent and is
> therefore more stable than B-f. 9:7 sounds like a traffic noise to
> most people unless supplied with a context, such as D-Gb-A-c-e, which
> approximates 4:5:6:7:9.

Curiously, I've tried this kind of Renaissance-Romantic era resolution
of a diminished fifth to a large major third using two voices, e.g.

f'-e'
b -c'

in 17-tet, where the resolution isn't _quite_ so incisive (b-c'
and f'-e' being 1/17 octave, or about 70.6 cents, rather than the 54.5
cents of 22-tet). In 17-tet, the large major third is a bit smaller,
being 6/17 octave or about 423.5 cents.

It seemed to work nicely, and maybe demonstrates Ludmila Ulehla's
concept of a "dual-purpose" sonority: the 17-tet M3 can be quite
urgently "discordant," and yet relatively "restful" when it follows a
diminished fifth (8/17 octave, about 564.7 cents).

Typically I might use 17-tet in a more Gothic manner as a
"hypermodern" variation on Pythagorean, but it's interesting that the
Renaissance and later d5-M3 resolution (which Zarlino recognizes as
very pleasing in 1558) also works well in this tuning.

[The following message by Paul refers to this "12-out-of-22" scheme
for tuning 22-tet with e-f and b-c' equal to 1/22 octave and all other
intervals to 2/22 octave, which he points out could nicely fit the
"neo-late-Gothic" applications I mentioned in my review]

> I wrote,
>
> > 5) A major, A-flat major, C minor, and C-sharp minor give
> > approximate 5-limit just intonation, in a fashion analogous to the
> > late Gothic keyboard tuning.

> > I meant the _keys_ that are conventionally called A major, A-flat
> > major, C minor, and C-sharp minor, not (just) the _chords_ that go
> > by those names. Each of the keys has a consonant I, IV, and V chord
> > which are mutually exclusive from those of the other keys; hence
> > there are 12 consonant 5-limit triads in this 12-out-of-22 keyboard
> > mapping.

This looks like a very interesting system, and for many typical
applications, speaking in terms of major/minor triads and keys is the
natural approach to communicate this information in an efficient way.

May I just add a caution directed mainly to some more historically
specific discussions of Gothic and early Renaissance tunings? Authors
sometimes attempt to apply 18th-century key concepts in explaining the
development of these tunings, as with Owen Jorgensen in _Tuning the
Historical Temperaments by Ear_ (Northern Michigan University Press,
Marquette, 1977), e.g. pp. 48-56.

My caution would be that major/minor key systems are really specific
to European music of the era around 1680-1900, and to subsequent works
following such styles.

In the early 15th-century context of the Pythagorean tunings with
prominent schisma thirds and sixths, for example, we can best
understand the role of these intervals by focusing on Gothic concepts
of harmonic action and color.

This situation may be very analogous to the 22-tet nomenclature issues
you discussed in your initial reply, emphasizing that either decatonic
or more conventional notations might be appropriate depending on the
musical context.

To take one quick example, consider a sonority with smooth schisma
thirds and sixths in the tuning with a Wolf at F#-B (actually Gb-B) of
around 1400: e-g#-c#'. To understand the harmonic role of this
sonority, we need to be familiar with one of the most popular final
cadences of the epoch, and indeed of the preceding century:

c#'-d'
g# -a
e -d

Here the major third expands to a stable fifth, and the major sixth to
a stable octave. The schismatic tuning doesn't alter this basic
vertical logic, but gives the cadence a different color both by making
the M3 and M6 somewhat smoother than in a usual Pythagorean tuning,
and by making the melodic semitones g#-a and c#'-d' somewhat less
keen and efficient.

This point quite aside, I'm fascinated and delighted to see a
discussion of 22-tet as a "neo-late-Gothic" tuning lead to possible
5-limit applications in later stylistic contexts. This is also a
striking illustration of how the same combinations of notes might be
useful in _either_ a 15th-century or 18th-century context, although
the progressions and the expectations behind them might be very
different.

Most appreciatively,

Margo Schulter
mschulter@value.net

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End of TUNING Digest 1520
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🔗John Starrett <jstarret@...>

🔗"M. Schulter" <mschulter@...>