Yahoo Tuning Groups Ultimate Backup tuning Re: 2-3-7 optimizations (for Graham Breed)

Hello, there, Graham Breed and everyone.

Thank you for sharing some observations based on experience with a
range of tunings, keyboard layouts, and intonational approaches which
add lots of interesting perspective to this dialogue.

One theme of recent threads which our dialogue may illustrate is the
way in which the same or similar terms often carry quite different
meanings and associations in different musical contexts.

For example, to me the term "enharmonic" as used in my accustomed
stylistic contexts typically means either the ancient Greek genus
dividing a fourth into a major third and two intervals each equal to
half a semitone, or the similar 16th-century "enharmonic" manner of
Vicentino, with its "fifthtones" equal to a meantone diesis, for
example the 128:125 (~41.06 cents) in 1/4-comma.

To express the idea of two conceptually different notes or intervals
getting mapped in a given tuning scheme to a single location, e.g. a
large major sixth and small minor seventh in 29-tET or 24-tET (both
mapped to 23/29 octave or 19/24 octave, roughly 26:15), I'd use some
other term such as "equivalent" or "identically mapped." However, I'm
aware that many people nowadays refer to this relationship as
"enharmonic."

The solution seems to me to recognize that terms can quite properly
have multiple meanings, with the context often clarifying the meaning,
although at times it can be (as you suggest in a thread on the term
"diaschismic") prudent to consider whether these alternative meanings
are likely to overlap in inconvenient ways.

> What are these fifthtone progressions, then?

In a Renaissance or Xeno-Renaissance setting, they are progressions
involving a meantone diesis, equal to about 1/5-tone. Nicola Vicentino
(1511-1576) made such progressions a hallmark of his style, presented
as a kind of "adaption" of the Greek enharmonic genus to modern
16th-century music with its use of temperament and of complex
polyphony.

Maybe a few progressions, with an asterisk (*) showing a note raised
by a diesis, may show some of the possibilities. First, here's a kind
of cadential ornament:

Eb4-Eb*4-E4
C4
G3
C3

Here, above the stationary lower parts, the upper voice moves through
two fifthtone steps from a usual minor third to Vicentino's "proximate
minor third" for which he gives the approximate ratio 5-1/2:4-1/2 (or
in other words 11:9) to a concluding regular major third (pure in the
1/4-comma tuning which seems a likely choice for Vicentino's
instruments).

On my 24-note subset of Vicentino's 36-note keyboard (in its complete
version featuring a circulating 31-note meantone cycle plus a few
extra notes to give pure fifths to some common diatonic notes on the
lower 19-note manual), the manuals are a diesis apart, so notes with
asterisks are played on the upper keyboard. Since Vicentino's 36-note
or 38-note design is based on two 19-note manuals, the note locations
are often somewhat different, but he likewise uses a dot above a note
(my ASCII asterisk) to show a note raised by a diesis.

Also, there are usual 16th-century progressions with altered
intervals, e.g.

G*4 A4 D5 C#*5
D*4 F4 G4 A*4
B*3 C4 D4 E*4
G*3 F3 or Bb3 A*3

Here we have the usual vertical meantone intervals (approximating the
ideal of 5-limit JI), but the altered melodic step of a minor semitone
B*3-C4 in the second-to-lowest voice of the first example, or D5-C#*5
and Bb3-A3 in the outer voices of the second example, normally the
chromatic semitone (~76.05 cents in 1/4-comma). This is a semitone of
about 2/5-tone rather than the usual 3/5-tone (~117.1 cents in
1/4-comma).

Also, in these progressions, we have motions by a large whole-tone of
about 6/5-tone or ~234.2 cents in 1/4-comma, e.g. G*3-F3 or D4-E*4.

While Vicentino speaks of a diesis as a fifth of a tone, in 1/4-comma
this is an approximate model, since we actually get two slightly
unequal fifthtone steps within a chromatic semitone, e.g. C-C*-C#,
roughly 41.06 cents (the 128:125 diesis) and 34.99 cents (the
chromatic semitone less this diesis, a difference which it happens is
very close to 50:49). This is close enough that counting fifthtones is
a very practical guide to the intervals, making a 31-note tuning in
1/4-comma a circulating system despite the slight theoretical
asymmetry.

In 31-tET, the fifthtone measurements would precisely apply, and by
the middle of the 17th century, Lemme Rossi (1666) associates
Vicentino's instrument with this temperament.

> I'd forgotten about 22. It would be the simplest on the list,
> because it'd only need 22 notes. So if you used all 22 notes, you'd
> get each chord at 22 different pitches.

This is another benefit of 22, maybe especially for people who focus
on circulation (something of a special effect in my outlook on
neo-Gothic, but possibly a central feature of some style in this
general area which someone might devise).

I must admit that often I simply tune 12 of 22, since a 12-note range
covers most Gothic music; however, there are some neat effects which
the full gamut makes available apart from the circulation question,
for example a cadence like

D4 E4 F4
A#3 B3 C4
G3 F4

Here the first sonority has the "schisma-like" augmented second of
22-tET, 7/22 octave, analogous to a Pythagorean diminished fourth,
close to 5:4, in a "split fifth" sonority (outer fifth, two thirds);
the voices then move to a regular major third and sixth (8/22 octave,
17/22 octave) near 9:7 and 12:7, expanding as usual to fifth and
octave.

Here, in neo-Gothic terms, the shift of a 22-tET step A#3-B3 moves us
from a 5-flavor to a usual 7-flavor major third; as in a typical
meantone, we don't have intermediate flavors like Pythagorean (81:64)
or 14:11 ("11-flavor").

> Meantone is simpler than schismic because you get 15 6:7:9 chords
> for your 24 notes. With positive tuning, you only get 8 6:7:9
> chords.

In part, this depends upon the musical style: I find the eight
locations in a Pythagorean-region tuning to be the ones I'm typically
interested in, although more don't hurt. Again, as someone accustomed
to a "normal" range of Eb-G# for accidentals, my view might be
different than someone trained in musical styles where more remote
accidentals are quite the norm.

Also, positive tunings vary somewhat: with the e-based tuning at
around 704.61 cents, I get 10 sonorities approximating 6:7:9, because
~9:7 is 13 fourths up and ~7:6 is 14 fifths up.

However, it's important to emphasize that typically I tune up a
positive tuning for neo-Gothic because the regular intervals fit the
style, just as the meantone ones fit a Renaissance style. It's
probably this basic convenience that leads to me use positive tunings
for Gothic/neo-Gothic and meantones for Renaissance, although you're
very right as well as creative to show that the reverse arrangements
are not merely possible but have their own advantages also.

> So that's conceptual simplicity on the 2-keyboard setup. You're
> simplifying meantone in that context to make it structurally
> identical to:

[I'm not sure if there might be missing text here -- Margo]

Yes, there's no special historical precedent for a 12x2 meantone
keyboard setup, it's just one solution of convenience. Maybe this kind
of approach has something in common with the very different
"generalized keyboard" concept: the one common element is the idea
that the manuals should have equivalent intervals mapped to the same
keys. This idea can be expressed by either 12x2 or some generalized
keyboard in the usual sense; maybe Fabio Colonna's _Sambuca Lincea_ of
1618 (the "Lynxian sambuca," named in honor of the "Academy of Lynxes"
in which Galileo was a member), with five seven-note manuals tuned at
intervals of 1/5-tone, is an early example.

>> Curiously, I consider a 24-note tuning in many ways less intricate
>> than a 19-note tuning, since the symmetry of the two "generalized"
>> 12-note keyboards can help in finding intervals and cadences.

> A 19-note tuning would really need a 19-note keyboard. If you had
> such a keyboard, the tuning would be very simple.

True, and such a keyboard, unlike my 24-note version, has a firm
historical basis. Vicentino's archicembalo (1555) could be described
as two 19-note manuals with the usual split keys, with E# and B# keys
on the lower manual of his practical 36-note version, or on both
manuals in his ideal 38-note version (which ran into technical
problems).

Costeley (1570) describes a similar 19-note keyboard for 19-tET, and
19-note keyboards or _cembali chromatici_ ("chromatic harpsichords)
had some vogue in Naples around 1600, the environment of the famed
composer Gesualdo, where keyboard composers such as Trabaci wrote
pieces especially for this instrument.

> You get some near-7 ratios with 12-note meantone as well. But
> 22-equal with a chain-of-fifths mapping gives the majority of thirds
> as 9-limit without 5. Usually, this wouldn't be my preferred
> mapping, but the neo-Gothic lens makes everything look different.

That's a really important point: n-limit terminology may not really
fit neo-Gothic, and from a neo-Gothic point of view regular major
thirds in 22-tET (chain-of-fifths mapping) at around 9:7 and 7:6 are
the main attraction, with the others more of a "special effect."

Incidentally, in neo-Gothic terms, both 9:7 and 7:6 are "7-flavor,"
because they are based on factors of 7; it may be natural in this
setting to group them together, because they or their octave
complements typically arise in a division of the fifth (6:7:9 or
14:18:21) or major sixth (7:9:12) where one implies the other.

However, in a music where a 7-limit sonority such as 4:5:6:7 is
common, the contrast between "7-limit" and "9-limit" makes great
sense.

>> Among listed systems, the next most accurate is 36-tET, with two
>> manuals at 1/6-tone (~33.33 cents) apart. This tempers the fifths
>> by about 1.95 cents, here in the narrow direction, and gives us,
>> overall, more accurate 7-based approximations than any of the
>> more heavily tempered alternatives.

> Yes, that comes from 12-equal being a good approximation to
> Pythagorean tuning. So whatever chords "work" will work in at least
> 12 positions.

True, 12-tET (or the "usual" 12-tET intervals on each keyboard of a
24-out-of-36 scheme) is a compromise between Pythagorean and meantone,
but weighted more toward the Pythagorean side. In certain kinder and
gentler timbres, this _might_ be acceptable for mixing Renaissance and
neo-Gothic styles, but with a slant toward neo-Gothic rather than
toward meantone thirds for the Renaissance sonorities.

In addition to the 2-3-7 sonorities, I like the "17-flavor" ones --
submajor and supraminor thirds at 366.67 and 333.33 cents, providing a
division of the fifth close to 14:17:21 (0-333-700 cents). These
sonorities, also characteristic of regular tunings around 704 cents,
make some neat neo-Gothic cadences.

> Another one I hadn't considered. I wonder how it'd work with each
> manual tuned to a traditional well temperament.

In 24 notes, there could be interesting possibilities here that I
hadn't considered. Once, for fun, I proposed the idea of a
Vallotti-Young with the diatonic and chromatic fifths reversed to make
the most commonly used thirds near-Pythagorean. Since then, I've
concluded that a 17-note well-temperament (with 17-tET as a point of
departure) may be typical for neo-Gothic -- but the idea of applying
the 24-note concept to 12-note well-temperaments is a whole new
question.

> The minimax tuning for these intervals is where 7:6 is just. That's
> a fifth of 696.319 cents. 9:7, 7:4 and 3:2 are all 5.6 cents out.

That's close to Kornerup's Golden Meantone, and not too far from
1/4-comma. It's a fascinating point of symmetry, maybe a bit like
Zarlino's 2/7-comma for the major and minor thirds (both varying from
pure by 1/7 comma).

> For the full 9-limit excluding 5, 9:8 would be the furthest interval
> equally mistuned, and 7:4 just. That's a lot closer to 31-equal.

Yes, for 10 fifths up or an augmented sixth to equal a 7:4, we'd have
a fifth of around 696.88 cents, or about 5.07 cents narrow, slightly
wider than 31-tET (~5.18 cents).

One fine point: in a neo-Gothic setting, I'd tend to associate 9:8
with a "3-limit," so to speak, since it's defined as the "difference
of the consonances" between a 3:2 fifth and a 4:3 fourth; it's present
or approximated a 12-note Pythagorean or near-Pythagorean tuning,
while something like 9:7 involves ratios of 7 (the "7-flavor"), not
present in a basic 12-note Pythagorean scheme.

However, each setting has its own logical associations and groupings
of intervals. An interesting point is that meantone neutralizes the
distinction between the 9:8 and 10:9 (so that neither can be very
closely approximated), while 22-tET does the same with the 9:8 and
8:7.

> 22-equal would be a lot worse if you included 9:8. It happens that
> the 9-limit intervals it's good at tend to be the ones that are
> important here. And it also has good 5-limit intervals.

From my point of view, I might say that 22-tET is not especially
accurate with the basic "Pythagorean" intervals: 3:2, 4:3, and 9:8,
which along with the 2:1 octave are the ones attributed to Pythagoras
(all included in a "quadrichord" with strings of 6:8:9:12). This is
the price of a regular tuning approximating "1/4-septimal-comma"
temperament of the fifth, with 9:8 and 8:7 mapped to the same interval
of 4/22 octave.

Fortunately, as Paul Erlich and I agree, 22-tET does have acceptable
approximations of sonorities such as 6:8:9 or 4:6:9, quite important
in neo-Gothic -- curiously, in the right timbre, I can also hear such
approximations in 20-tET (0-480-960 cents for 9:12:16), where the
"minor seventh" is actually smaller than a 7:4!

> Eb4 E4
> Bb3 B3
> F#3 E3

> The left-hand chord is the alternative version of Eb minor. So with
> a schismic tuning it would approximate 12:15:20.

A theorist in the 18th-19th century tradition might call the first
sonority "Eb minor," but I'd call it, in a neo-Gothic setting, a
"meantone version of the major sixth sonority F#3-A#3-D#4," with the
major third inviting expansion to a fifth and the major sixth to an
octave, as happens here. From a medieval perspective, the lowest note
is the _fundamentum_ or "foundation" of this sonority. Here the usual
Pythagorean-style spelling with sharps indicates an expectation of
_upward_ semitonal motion in the resolution -- A#3-B3 and D#4-E4.

As you remark, this would indeed be a close approximation of a
12:15:20 in a Pythagorean or similar setting, and is exactly the kind
of sonority often featured in early 15th-century keyboard pieces,

Typically such sonorities occur "implicitly" despite usual spellings,
because a popular 12-note tuning of the time (say 1400-1450) is Gb-B,
with the sharps tuned as Pythagorean flats. Thus we get

F#4 G4 Gb4 G4
C#4 D4 Db4 D4
A3 G3 A3 G3

written often sounded

The second version has a major sixth sonority on A3 very close to
12:15:20, just as you describe.

> The roles are reversed because 5-limit becomes 9-limit -- the
> alternative minor chord in meantone approximates 6:7:9. Or 7:9:12
> in this voicing, which doesn't look like the optimum meantone one as
> it exposes the 7:9.

Generally I'd say that the exposed 9:7 of 7:9:12 is quite in keeping
with a neo-Gothic style, at a cadence anyway, where that major third
"strives to expand" to the fifth; so the extra tension of a 427-cent
as opposed to pure 9:7 tuning isn't really a problem, although a pure
7:9:12 does have its own allure. More generally, that major third
could be anything from Pythagorean to 9:7 in a typical neo-Gothic
style.

Your observation also raises a noteworthy point: different
arrangements like 6:7:9 or 7:9:12 are not precisely "equivalent,"
although in some theories they are inversions of each other. From a
14th-century or neo-14th-century viewpoint, both the major third and
major sixth are "expansive" intervals seeking the fifth and octave
respectively, while 6:7:9 with its "contractive" 7:6 minor third
seeking the unison has somewhat different qualities.

> This is a different role reversal to the usual one where Pythagorean
> intervals become 5-limit, and so cleaner than the 9-limit intervals
> that stay the same.

May I ask for a clarification on the last part of this sentence "and
so cleaner than the 9-limit intervals that stay the same"?

>> Also, as we see in this cadence, the meantone chromatic semitone,
>> the _smaller_ semitone of this tuning at around 76.05 cents,
>> becomes our neo-Gothic diatonic semitone: here Bb3-B3 and Eb4-E4 in
>> the two upper voices.

> That removes the melodic deficiency of meantone: that the semitones
> are too large. If you play with 9-limit chords, you can get those
> chromatic semitones to come out.

Curiously, while my "respelling" has this effect for neo-Gothic,
Vicentino also seems to lean toward cadences with his "enharmonic" or
fifthtone system where the usual meantone thirds and sixths are
maintained vertically, but the large diatonic semitone becomes a small
chromatic semitone. Maybe Paul Erlich or John deLaubenfels might
consider this a 16th-century variation on the "adaptive tuning" theme
(Vicentino's adaptive JI system is a different thing), and it could
also tie in with your point that a leading tone now and then of 21:20
or the like (~84.47 cents, maybe not so far from the 76-cent semitone
of 1/4-comma meantone) could do typical 18th-19th century harmony "a
world of good."

>> C#4 B3
>> Bb3 B3
>> Eb3 E3

> That's a 4:6:7 chord. The schismic equivalent would be 10:15:18.
> In this case, the meantone approximation makes a lot more sense.

Yes, in neo-Gothic terms, the minor seventh seeks a "closest approach"
to the fifth, so the narrower it is, the more efficiently it can
contract to its cadential goal -- and likewise the 7:6 minor third in
relation to its goal of the unison. At the same time, it's not only
more efficient but more "cool" or "mellow," so to speak.

In contrast, a minor seventh at or near 9:5 (e.g. a Pythagorean
augmented sixth, to continue our "role reversal" theme) seems to me
often more tense than either a 16:9 or a 7:4, although the 1020-cent
interval of 20-tET is quite fine, and provides a bit of "modal color"
in contrast to the 960-cent interval (in effect a sub-7:4 which also
serves as a 16:9).

> Yes, it happens that the same notation or fingering will work for
> either system. The 9-limit intervals from schismic are still
> 9-limit, the Pythagorean intervals become 5-limit, and the 5-limit
> schismic intervals become 9-limit. So the logic is the same, but
> the sound is different.

Here I wonder to what degree the keyboard arrangement itself has a
certain psychological effect on me: the neo-Gothic intervals are
"usual" in a typical neo-Gothic tuning, and the regular meantone or
5-limit ones in a meantone tuning. You nicely sum up the musical
symmetry.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗graham@microtonal.co.uk

Margo Schulter wrote:

> Maybe a few progressions, with an asterisk (*) showing a note raised
> by a diesis, may show some of the possibilities. First, here's a
kind
> of cadential ornament:
>
> Eb4-Eb*4-E4
> C4
> G3
> C3
>
> Here, above the stationary lower parts, the upper voice moves
through
> two fifthtone steps from a usual minor third to Vicentino's
"proximate
> minor third" for which he gives the approximate ratio 5-1/2:4-1/2
(or
> in other words 11:9) to a concluding regular major third (pure in
the
> 1/4-comma tuning which seems a likely choice for Vicentino's
> instruments).

I see you call it an ornament. It does look more like an ormanent
than a progression to me. Also, in this case the "fifthtones" could
be quartertones in 24-equal.

> Also, there are usual 16th-century progressions with altered
> intervals, e.g.
>
> G*4 A4 D5 C#*5
> D*4 F4 G4 A*4
> B*3 C4 D4 E*4
> G*3 F3 or Bb3 A*3
>
> Here we have the usual vertical meantone intervals (approximating
the
> ideal of 5-limit JI), but the altered melodic step of a minor
semitone
> B*3-C4 in the second-to-lowest voice of the first example, or
D5-C#*5
> and Bb3-A3 in the outer voices of the second example, normally the
> chromatic semitone (~76.05 cents in 1/4-comma). This is a semitone
of
> about 2/5-tone rather than the usual 3/5-tone (~117.1 cents in
> 1/4-comma).

And that doesn't show anything progressing by a fifthtone.

> Also, in these progressions, we have motions by a large whole-tone
of
> about 6/5-tone or ~234.2 cents in 1/4-comma, e.g. G*3-F3 or D4-E*4.
>
> While Vicentino speaks of a diesis as a fifth of a tone, in
1/4-comma
> this is an approximate model, since we actually get two slightly
> unequal fifthtone steps within a chromatic semitone, e.g. C-C*-C#,
> roughly 41.06 cents (the 128:125 diesis) and 34.99 cents (the
> chromatic semitone less this diesis, a difference which it happens
is
> very close to 50:49). This is close enough that counting fifthtones
is
> a very practical guide to the intervals, making a 31-note tuning in
> 1/4-comma a circulating system despite the slight theoretical
> asymmetry.

As the middle chord is a neutral triad, that would suggest an equal
tuning.

> > So that's conceptual simplicity on the 2-keyboard setup. You're
> > simplifying meantone in that context to make it structurally
> > identical to:
>
> [I'm not sure if there might be missing text here -- Margo]

Oh, I probably went to answer another question while I was thinking of
a word for "the same setup with positive temperament".

> > A 19-note tuning would really need a 19-note keyboard. If you had
> > such a keyboard, the tuning would be very simple.
>
> True, and such a keyboard, unlike my 24-note version, has a firm
> historical basis. Vicentino's archicembalo (1555) could be described
> as two 19-note manuals with the usual split keys, with E# and B#
keys
> on the lower manual of his practical 36-note version, or on both
> manuals in his ideal 38-note version (which ran into technical
> problems).

How do you pronounce "archicembalo"?

> Costeley (1570) describes a similar 19-note keyboard for 19-tET, and
> 19-note keyboards or _cembali chromatici_ ("chromatic harpsichords)
> had some vogue in Naples around 1600, the environment of the famed
> composer Gesualdo, where keyboard composers such as Trabaci wrote
> pieces especially for this instrument.

So do any Gesualdo pieces use the extended gamut? I'm interested in
finding examples of how the extra notes were used.

> Incidentally, in neo-Gothic terms, both 9:7 and 7:6 are "7-flavor,"
> because they are based on factors of 7; it may be natural in this
> setting to group them together, because they or their octave
> complements typically arise in a division of the fifth (6:7:9 or
> 14:18:21) or major sixth (7:9:12) where one implies the other.

And, in response to Paul Erlich's suggestion, they aren't both
"intervals of 9" or "intervals of 7", being one of each. I find that
terminology's rarely useful in this context. I describe 6:7:9,
14:18:21 and inversions as "9-limit triads" although there are plenty
of other triads in the 9-limit. Calling them "intervals of 9 triads"
wouldn't even be correct, as only 1 of the 3 constituent intervals is
an interval of 9. (Not that you can have a chord with only intervals
of 9.)

> However, in a music where a 7-limit sonority such as 4:5:6:7 is
> common, the contrast between "7-limit" and "9-limit" makes great
> sense.

In chords that use a tritone, I'd call 7-limit tuning the usual, or
"unflavored" version. The line would then be drawn between 11-limit
chords on the one hand, and the 7-limit subset on the other. In which
case 4:5:6:7 is the unflavored 7 chord, and 1/(5:6:7:9) the 9-flavor
one. There's also the extended 5-limit tuning, which could be called
"completely flavorless". Or maybe that would itself constitute a
flavoring.

> > Another one I hadn't considered. I wonder how it'd work with each
> > manual tuned to a traditional well temperament.
>
> In 24 notes, there could be interesting possibilities here that I
> hadn't considered. Once, for fun, I proposed the idea of a
> Vallotti-Young with the diatonic and chromatic fifths reversed to
make
> the most commonly used thirds near-Pythagorean. Since then, I've
> concluded that a 17-note well-temperament (with 17-tET as a point of
> departure) may be typical for neo-Gothic -- but the idea of applying
> the 24-note concept to 12-note well-temperaments is a whole new
> question.

You can displace Valotti-Young by a tritone to make the white notes
Pythagorean. Or you can leave it as it is, and play on the black
notes. I don't know how well temperament would fit 17 notes, because
it's an extreme rather than a compromise temperament. I haven't seen
a good well temperament for 19-equal for the same reason, or maybe
because I haven't been paying attention.

> One fine point: in a neo-Gothic setting, I'd tend to associate 9:8
> with a "3-limit," so to speak, since it's defined as the "difference
> of the consonances" between a 3:2 fifth and a 4:3 fourth; it's
present
> or approximated a 12-note Pythagorean or near-Pythagorean tuning,
> while something like 9:7 involves ratios of 7 (the "7-flavor"), not
> present in a basic 12-note Pythagorean scheme.

Whatever it's called, if it's being used as a consonance, it has to be
on the charts.

> However, each setting has its own logical associations and groupings
> of intervals. An interesting point is that meantone neutralizes the
> distinction between the 9:8 and 10:9 (so that neither can be very
> closely approximated), while 22-tET does the same with the 9:8 and
> 8:7.

Not so much that neither can be closely approximated, but they can't
both be so approximated at the same time.

> > Eb4 E4
> > Bb3 B3
> > F#3 E3
>
> > The left-hand chord is the alternative version of Eb minor. So
with
> > a schismic tuning it would approximate 12:15:20.
>
> A theorist in the 18th-19th century tradition might call the first
> sonority "Eb minor," but I'd call it, in a neo-Gothic setting, a
> "meantone version of the major sixth sonority F#3-A#3-D#4," with the
> major third inviting expansion to a fifth and the major sixth to an
> octave, as happens here. From a medieval perspective, the lowest
note
> is the _fundamentum_ or "foundation" of this sonority. Here the
usual
> Pythagorean-style spelling with sharps indicates an expectation of
> _upward_ semitonal motion in the resolution -- A#3-B3 and D#4-E4.

Ah, so in modern notation that would be called "F# 6 no5".

Actually, that's the same kind of the idea as the Neapolitan sixth,
where the third is in the base, which is also the fourth of the key.
So although it's always described as a bII chord, because that's
simplest, it still looks a bit like a 6th chord on IV. All of which
ties in with my thesis that chromatic roots are compositionally
irrelevant in Common Practice music ;)

> As you remark, this would indeed be a close approximation of a
> 12:15:20 in a Pythagorean or similar setting, and is exactly the
kind
> of sonority often featured in early 15th-century keyboard pieces,

Do you have any examples of such pieces? Are the schismic 5-limit
chords used as the rule or the exception?

> Typically such sonorities occur "implicitly" despite usual
spellings,
> because a popular 12-note tuning of the time (say 1400-1450) is
Gb-B,
> with the sharps tuned as Pythagorean flats. Thus we get
>
> F#4 G4 Gb4 G4
> C#4 D4 Db4 D4
> A3 G3 A3 G3
>
> written often sounded
>
> The second version has a major sixth sonority on A3 very close to
> 12:15:20, just as you describe.

That also relates to the Neapolitan 6th. In C major, it'd be written
as F-Ab-Db. But in an Eb-G# meantone, it'd sound as F-G#-C#, which is
a 9-limit major triad. It also means that C# and G# are chromatic
semitones from C and G, which may have melodic advantages. So I
wonder how often the cadence would have been heard "wrongly" in this
way.

> > The roles are reversed because 5-limit becomes 9-limit -- the
> > alternative minor chord in meantone approximates 6:7:9. Or 7:9:12
> > in this voicing, which doesn't look like the optimum meantone one
as
> > it exposes the 7:9.
>
> Generally I'd say that the exposed 9:7 of 7:9:12 is quite in keeping
> with a neo-Gothic style, at a cadence anyway, where that major third
> "strives to expand" to the fifth; so the extra tension of a 427-cent
> as opposed to pure 9:7 tuning isn't really a problem, although a
pure
> 7:9:12 does have its own allure. More generally, that major third
> could be anything from Pythagorean to 9:7 in a typical neo-Gothic
> style.

Hmm, well. The existence of 9-limit harmony when you're expecting
5-limit harmony should be powerful enough without the chords going out
of tune as well. Indeed, the more poorly tuned the 9-limit chord is,
the less likely it is to be recognized as such, rendering it
arbitrary. As the 9:7 is poorly tuned around 31-equal, I'd avoid
placing it at the bottom of the chord, as I find intervals there have
a strong influence on the overall consonance. So the prevalence of
7:9:12 chords would count against a 31-like meantone tuning.

In general, you can't tell from the inversion whether or not a minor
triad should be tuned 5-limit or wider 9-limit. In both cases, the
simplest ratio is root position (either 6:7:9 or 10:12:15). With
major triads, the 5-limit version is best tuned as 3:4:5. If you find
the fifth in the bass that would argue for a 5-limit tuning.

All of which is fine in theory, but if experiment shows 7:9:12 chords
to be no problem here in meantone, so be it.

> Your observation also raises a noteworthy point: different
> arrangements like 6:7:9 or 7:9:12 are not precisely "equivalent,"
> although in some theories they are inversions of each other. From a
> 14th-century or neo-14th-century viewpoint, both the major third and
> major sixth are "expansive" intervals seeking the fifth and octave
> respectively, while 6:7:9 with its "contractive" 7:6 minor third
> seeking the unison has somewhat different qualities.

With octave-equivalent harmony, that'd still come to the same thing.

> > This is a different role reversal to the usual one where
Pythagorean
> > intervals become 5-limit, and so cleaner than the 9-limit
intervals
> > that stay the same.
>
> May I ask for a clarification on the last part of this sentence "and
> so cleaner than the 9-limit intervals that stay the same"?

9-limit is 9-limit in both systems. In meantone, the 9-limit
harmonies will be more complex than the usual 5-limit ones. But in
schismic, the usual intervals are the Pythagorean ones that are more
complex than the 9-limit. Hence the dynamic of tension/resolution is
reversed.

> > Yes, it happens that the same notation or fingering will work for
> > either system. The 9-limit intervals from schismic are still
> > 9-limit, the Pythagorean intervals become 5-limit, and the 5-limit
> > schismic intervals become 9-limit. So the logic is the same, but
> > the sound is different.
>
> Here I wonder to what degree the keyboard arrangement itself has a
> certain psychological effect on me: the neo-Gothic intervals are
> "usual" in a typical neo-Gothic tuning, and the regular meantone or
> 5-limit ones in a meantone tuning. You nicely sum up the musical
> symmetry.

I find that the neutral triads in a neutral third MOS tuning also come
to be "usual". This may also be for the psychological reason that
they look like consonances, but also because they get used often
enough to become familiar. Both reasons also apply to Pythagorean
triads.

Graham

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗M. Schulter <MSCHULTER@VALUE.NET>

Hello, there, Graham Breed, Paul Erlich, and everyone.

To the dialogue on 2-3-7 optimizations, I might just at this point add
a few odd remarks.

First of all, while I tend to use the term "fifthtone progression" in
a 16th-century or similar meantone setting to mean various types of
altered progressions in the "enharmonic" manner involving a shift by a
diesis or fifthtone in the usual destination, maybe an example of a
direct fifthtone progression involving motion by several voices might
be helpful.

Here the notation is like Vicentino's, his dot above a note to show
its raising by a diesis being represented by an ASCII asterisk (*):

G*4 G#4
E*4 E4
C*4 B3
C*3 E3

While a 1/4-comma meantone tuning actually makes our dieses or
fifthtones slightly unequal, for most purposes we can use the model of
a division of the whole-tone into five quasi-equal parts.

The two upper parts each moves by a diesis up or down: G*4-G#4 (up) or
E*4-E4 (down). The lowest part moves up by 9/5-tone, a neutral third
C*3-E3, while the next to lowest part moves down by a small whole-tone
of 4/5-tone, C*4-B3.

Remove the enharmonic inflections from the notes of the first
sonority, and we get a characteristic 16th-century chromatic
progression, but without the special nuances of the enharmonic
version.

My own pronunciation of _archicembalo_ follows more or less the
Italian, with _ch_ before _e_ (or _i_) as in English _church_.

Gesualdo's use of the more remote accidentals in a 19-note Gb-B# range
is discussed in Glenn Watkins, _Gesualdo: The Man and His Music_
(University of North Carolina Press, Chapel Hill, NC, 1973, ISBN
0-8078-1201-3), pp. 194-201, under the heading "Accidentals and
Chromaticism."

Interestingly, Gesualdo's range of accidentals in his madrigals
coincides almost exactly with the Gb-B# gamut of the 19-note _cembalo
chromatico_ or "chromatic harpsichords" popular in his own milieu
around Naples; one madrigal additionally uses Cb.

Watkins concludes that Gesualdo's motivation for using the four most
remote sharps (D#, A#, E#, B#) is to obtain the major third of a
sonority, the norm both for a more conclusive ending of a phrase, and
for various cadences calling for a major third or sixth to proceed to
an octave.

The situation in Vicentino's known enharmonic compositions -- one
complete secular Latin motet in honor of his patron, plus some
sections and excerpts from madrigals -- may be somewhat different:
these examples can each fit in some 24-note meantone range with two
manuals a diesis apart (Bb-D# plus Bb*-Eb; Ab-C# plus Ab*-Db).

As noted by the early 17th-century theorist Doni, Gesualdo sometimes
practices a form of modal transposition, for example in a passage
where all sharps might be ignored and the same melodic and vertical
intervals result (i.e. transposition by a chromatic semitone up).

For a piece in a 31-note tuning with cadences following a circle of
fifths and visiting all 31 steps in turn, we must wait until Fabio
Colonna's treatise on his _Sambuca Lincea_ of 1618, which includes an
"Example of Circulation" to demonstrate the division of the whole-tone
into five parts without any complications such as commas which would
impede such a journey.

It may be a moot point whether a hypothetical xenharmonic researcher
with a frequency counter would find Colonna's instrument in 1618 tuned
to something more closely resembling 1/4-comma meantone or 31-note
equal temperament (31-tET), and the variances in a tuning by ear could
likely overshadow the theoretical difference between these models.

Both Vicentino and Colonna tell us that the first 12 or 19 notes of
their instruments are tuned in the common manner, making 1/4-comma a
more likely scenario than 31-tET, and that a whole-tone is divided
into five equal parts, a model literally realized in 31-tET. Late
17th-century theory (Rossi, Huygens) draws a mathematical distinction
between these two slightly different models -- but it is interesting
to ask whether such a line would have been drawn in 1555 or 1618.

In a 1/4-comma meantone tuning of from 24 to 31 notes, some of the
more remote intervals have two versions differing in size by about
6.07 cents, the "comma" by which 31 such meantone fifths fall short of
18 pure octaves. For example, in my 24-note tuning with the two
keyboards a diesis of 128:125 (~41.06 cents) apart, the neutral third
C*-E is about 345 cents (C-E less C-C*), while C-Eb* is about 351
cents (C-Eb plus Eb-Eb*).

As you suggest in a recent post, Graham, a 31-tET tuning would
actually give a more accurate approximation of 11:9 than either of
these 1/4-comma sizes: 9/31 octave or ~348.39 cents, within one cent
of the pure interval at ~347.41 cents.

Anyway, I hope that these remarks may give a bit more information
about Gesualdo's use of remote accidentals and some of the other
meantone-related topics of this thread.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗graham@microtonal.co.uk

Margo Schulter wrote on the 10th:

> First of all, while I tend to use the term "fifthtone progression" in
> a 16th-century or similar meantone setting to mean various types of
> altered progressions in the "enharmonic" manner involving a shift by a
> diesis or fifthtone in the usual destination, maybe an example of a
> direct fifthtone progression involving motion by several voices might
> be helpful.
>
> Here the notation is like Vicentino's, his dot above a note to show
> its raising by a diesis being represented by an ASCII asterisk (*):
>
> G*4 G#4
> E*4 E4
> C*4 B3
> C*3 E3

This looks mighty strange, so I had to find out what it sounds like. It
is, in fact, strangely beautiful. It's great to see that such avant-garde
stuff was being tried out back in the C16h.

> While a 1/4-comma meantone tuning actually makes our dieses or
> fifthtones slightly unequal, for most purposes we can use the model of
> a division of the whole-tone into five quasi-equal parts.
>
> The two upper parts each moves by a diesis up or down: G*4-G#4 (up) or
> E*4-E4 (down). The lowest part moves up by 9/5-tone, a neutral third
> C*3-E3, while the next to lowest part moves down by a small whole-tone
> of 4/5-tone, C*4-B3.

Actually, I left out the lowest part. The two manuals of my archicembalo
are opposite each other, so I had to play each chord with one hand. So I
still have to imagine that neutral second.

> Remove the enharmonic inflections from the notes of the first
> sonority, and we get a characteristic 16th-century chromatic
> progression, but without the special nuances of the enharmonic
> version.

That also sounded strange to start with, but I quickly got used to it.

> My own pronunciation of _archicembalo_ follows more or less the
> Italian, with _ch_ before _e_ (or _i_) as in English _church_.
>
> Gesualdo's use of the more remote accidentals in a 19-note Gb-B# range
> is discussed in Glenn Watkins, _Gesualdo: The Man and His Music_
> (University of North Carolina Press, Chapel Hill, NC, 1973, ISBN
> 0-8078-1201-3), pp. 194-201, under the heading "Accidentals and
> Chromaticism."

I might go into Bristol on Saturday, so I'll see if there's anything in
the library.

> Interestingly, Gesualdo's range of accidentals in his madrigals
> coincides almost exactly with the Gb-B# gamut of the 19-note _cembalo
> chromatico_ or "chromatic harpsichords" popular in his own milieu
> around Naples; one madrigal additionally uses Cb.

"Almost exactly" isn't such a coincidence. Cb-E# is equal numbers of
sharps and flats. Gb-B# is an equal number of fifths either side of A,
which may be important as A=440Hz is our reference pitch, and similar
ideas may have been around earlier.

> Watkins concludes that Gesualdo's motivation for using the four most
> remote sharps (D#, A#, E#, B#) is to obtain the major third of a
> sonority, the norm both for a more conclusive ending of a phrase, and
> for various cadences calling for a major third or sixth to proceed to
> an octave.

That's disappointing. No playing with the 9-limit, then.

> As noted by the early 17th-century theorist Doni, Gesualdo sometimes
> practices a form of modal transposition, for example in a passage
> where all sharps might be ignored and the same melodic and vertical
> intervals result (i.e. transposition by a chromatic semitone up).

That's simple enough.

> It may be a moot point whether a hypothetical xenharmonic researcher
> with a frequency counter would find Colonna's instrument in 1618 tuned
> to something more closely resembling 1/4-comma meantone or 31-note
> equal temperament (31-tET), and the variances in a tuning by ear could
> likely overshadow the theoretical difference between these models.

Oh yes, the two are practically equivalent. I treat my guitar as a subset
of 31-equal, but I actually fretted it even further away then 1/4-comma.
Some 11-limit intervals are a bit further out, but they still work.

> Both Vicentino and Colonna tell us that the first 12 or 19 notes of
> their instruments are tuned in the common manner, making 1/4-comma a
> more likely scenario than 31-tET, and that a whole-tone is divided
> into five equal parts, a model literally realized in 31-tET. Late
> 17th-century theory (Rossi, Huygens) draws a mathematical distinction
> between these two slightly different models -- but it is interesting
> to ask whether such a line would have been drawn in 1555 or 1618.

It would likely have been easier to tune with pure thirds.

Graham