Yahoo Tuning Groups Ultimate Backup tuning Arnold Schlick's well-temperament of 1511

Hello, there, Herbert Anton Kellner and everyone.

Thank you for your question about the organ temperament published by
Arnold Schlick in 1511, which gives me the opportunity to address a
period of music with which I am more familiar. Your inquiry has moved
me to suggest one possible interpretation of this tuning as a kind of
well-temperament, reflecting some features of musical style in early
16th-century Europe (the era of such composers as Josquin in his later
years, Isaac, and Senfl).

Incidentally, I have much enjoyed your discussion of some mathematical
techniques for dividing commas in the era of Werckmeister, which
remind me of the medieval and Renaissance theory with which I am
familiar. Also, I would much agree that riddles and puzzles are part
of the musical tradition, going back to at least the 13th or 14th
century: Glareanus (1547), in discussing the style of his model
Josquin, mentions that he sometimes follows the custom of writing
canons with arcane clues and solutions.

As you very wisely remark, Schlick's approach to temperament has much
fascinating musical philosophy to offer on such topics as the balance
between fifths and thirds, and also the very important observations
about beats which you mentioned, but does not define the tuning
process with any mathematical precision. It is a matter of taste and
judgement, with room for a range of interpretations.

Thus James Murray Barbour[1] has taken Schlick as suggesting a kind of
well-temperament not too far from 12-tone equal temperament (12-tET),
Barbour's own standard measure for tunings. Mark Lindley, in contrast,
has asserted that the tuning has a definite "Wolf fifth" which should
not be considered playable, better approximating a usual 16th-century
meantone tuning than a "well-temperament" model.[2] In part, Lindley's
view may represent a kind of reaction to Barbour, possibly I would say
an overcorrective reaction.

My own interpretation, based in part on Schlick's eloquent and nicely
considered language, and in part on sheer musical intuition, is that
his tuning is indeed a kind of well-temperament, in which many
intervals approach a meantone ideal and a few are quite marginal,
"little esteemed and seldom used," to quote Lindley's translation.

This well-temperament is fascinating not only for showing one method
of arranging 12 notes in a tuning circle using fifths of various sizes
and thirds moving progressively further from pure as we shift from
common to remote transpositions, but in evidencing some modal and
vertical preferences as they influence the tuning practices of at
least one sensitive composer and theorist in the second decade of the
16th century.

---------------------------
1. Schlick's tuning process
---------------------------

Here I rely on Mark Lindley's excerpts giving relevant text passages
and translations of Schlick's treatise, but have felt free to make my
own interpretation of the tuning and of "well-temperament" (a term not
used by Schlick himself).

A reader comparing my interpretation with Lindley's will see how I am
much indebted to his basic mathematical approach and conclusions, with
our main difference one of musical evaluation and characterization.

As noted by Lindley, Schlick gives a very vivid and readable
explanation of what would in later theory be called the "syntonic
comma problem," emphasizing the need (in a Renaissance as opposed to
medieval style, we might add) to narrow the fifths lest the major
thirds be "altogether too high."

He also gives a most memorable account of the beating of a properly
tempered fifth: the fifth should not be precisely "correct," but
should somewhat "break up" or "hiccup," yet so that a listener would
not criticize the fifth as obviously wrong.

He also introduces what might be termed the principle of irregular
temperament when he notes that the fifths between diatonic notes
should "suffer" more for the sake of their thirds than the fifths
involving accidentals.

In a wonderfully poetic expression, he explains that the notes of a
fifth should be tuned so that the two note do not quite fully satisfy
in their blend, but "ever and more crave one another."

These two concepts, of distinctly but discreetly tempered fifths and
of greater temperament for diatonic fifths than for those involving
accidentals, apply for the ten fifths in the circle from Eb to C#.

Now we come to a question which Bartoleme Ramos (1482) had earlier
raised: what about the choice, in a 12-note tuning, between Ab and G#?
Ramos concluded that Ab was the more prudent choice, noting that some
people attempt to satisfy both sides of the argument by using split
keys to provide both notes.

Schlick also leans toward Ab, but with a deliberate compromise between
the two notes within a 12-note tuning, so that the same key can serve
as either a reasonably serviceable Ab, or as a marginal G# for use in
aptly ornamented cadences on A. As Lindley points out, this preference
reflects the style of Schlick's own organ music, which has Ab-C-Eb as
a prolonged concord, but a more gingerly and ornamental use of G# in
cadences to A.

As Lindley discusses, this suggests making the major third Ab-C rather
tense but "acceptable," possibly comparable to 12-tET at around 400
cents, and E-G# (or E-Ab) roughly Pythagorean (around 408 cents).

Schlick also notes that the fifth C#-G#/Ab will be too large, but says
that this should not trouble the musician, since it is "not used," and
advises that transpositions might best avoid this interval if
possible.

The major third B-D#/Eb is also "little esteemed and seldom used."

While Lindley takes Schlick's remarks about C#-G#/Ab to imply an
outright Wolf, I interpret them instead as comparable with a
"marginally playable" kind of fifth. In fact, Lindley's own suggestion
of a fifth about 8-10 cents wide would seem to fall in the latter
category, at least by many later standards, in contrast to the usual
"Wolf fifths" of regular meantones in the range of 1/3-1/6 syntonic
comma.

In such regular meantones, a 12-note tuning has a single "Wolf fifth"
about 15.97 cents wide in 1/6-comma, and yet wider in Renaissance
meantones such as 1/4-comma, 2/7-comma, or 1/3-comma. In these
tunings, the 11 regular fifths are each narrowed by the same amount so
as to achieve pure or near-pure thirds; the remaining diminished sixth
(typically G#-Eb, but sometimes C#-Ab or D#-Bb) absorbs the full brunt
of balancing the Pythagorean comma.[3]

In contrast, whether by design or as a side-effect of the calculated
compromise between Ab and G#, Schlick's tuning produces _two_ fifths
wider than pure, Ab/G#-Eb and C#-Ab/G#, which "bisect" the Wolf,
albeit in an unequal manner leaving the latter as a kind of
"near-Wolf."

While any mathematical interpretation is indeed that, a possible
translation into numbers of Schlick's own qualitative rather than
quantitive approach, I would like to suggest such an interpretation
close to Lindley's in its interval sizes, and then to consider a
couple of interesting musical implications.

---------------------------------------------------
2. An interpretation: a "marginal well-temperament"
---------------------------------------------------

In his interpretation of Schlick's tuning, Lindley suggests making the
six fifths between diatonic notes (F-C-G-D-A-E-B) each about 4 cents
narrow, and the four fifths involving accidentals other than G#/Ab
(Eb-Bb-F, B-F#-C#) about 3 cents narrow.

This arrangement involves about 36 cents of total narrowing for these
ten fifths, exceeding the Pythagorean comma of 543411:524288 (~23.46
cents) by roughly 12 cents.

He interprets Schlick's compromise regarding Ab/G# to call for a fifth
Ab-Eb about two cents wider than pure, leaving about 8-10 cents for
what he terms the "Wolf" fifth C#-Ab/G#, and I would term a marginally
playable fifth. This makes E-G# roughly Pythagorean, as described
above, and the more "serviceable" Ab-C about as in 12-tET.

Here I would like again to observe, as does Lindley, that any
mathematical interpretation of Schlick's temperament -- or tuning of
it by ear on a keyboard instrument -- represents an exercise in the
art of musical estimation. Just as a 16th-century vocal composition
leaving many decisions about accidentals to the judgment of the
performers might inspire instrumental versions resolving these
decisions in a variety of ways, so Schlick's vivid and musically
sensitive description leaves much room for taste and choice.

While Lindley uses cents as a convenient unit in his interpretation of
Schlick's tuning, I shall here use fractions of the Pythagorean comma,
or PC, with 1/24 PC just a tad smaller than 1 cent.

Since both cents and fractions of a Pythagorean comma are units
favored in later eras of theory, there is a certain anachronism here,
but an advantage of a focus on the Pythagorean comma is that it may
reflect how the "marginally well-tempered" tuning circle is being
balanced, by design or in part by accident.

My approach, generally similar to Lindley's, is to narrow each
diatonic fifth by an arbitrary amont of 1/6 PC (~3.91 cents); and each
fifth involving accidentals other than Ab/G# by 1/8 PC (~2.93 cents).

We can conveniently express these fractions in what I might term
"Pythagorean cents" (rather like "metric tons"?) of 1/24 PC, or around
0.9775 cents of the usual variety. Each of the six diatonic fifths is
narrowed by 4/24 PC, and each of the four accidental fifths we are now
considering by 3/24 PC.

This means that the tempering of the diatonic fifths involves a total
of 24 Pythagorean cents, or a full comma (4/24 x 6). The four
accidental fifths involve an additional (3/24 x 4), or 12 Pythagorean
cents -- another 1/2 comma.

Thus these ten fifths involve a total of 36 Pythagorean cents, or 12
cents beyond the amount necessary to close the tuning circle, and the
remaining two fifths must divide this excess.

Here, in a result that seems to me rather like Lindley's, I propose
tempering Ab/G#-Eb at 1/8 PC or 3 Pythagorean cents wide, leaving 3/8
PC or 9 Pythagorean cents for the marginal C#-Ab/G#. The latter fifth,
at around 710.75 cents or ~8.80 cents wide, might well be honored
mainly in the avoidance, as Schlick suggests, yet could be playable in
a pinch, for example in unusual transpositions of certain modes.

As I will explain in Section 3, its marginal playability might be
improved in one situation where it would be likely to arise --
especially in music following a cadential style not much favored in
"modern" early 16th-century practice.

Here is a scale file for Manuel Op de Coul's excellent and free scale
definition and analysis software program Scala, which I again
emphasize is simply one interpretation of Schlick's temperament:

! as1511ms.scl
!
Possible well-tempered interpretation of Arnold Schlick's tuning of 1555
12
!
88.2700
196.0900
303.9100
392.1800
501.9550
589.2475
698.0450
799.0225
894.1350
1002.9325
1090.2250
2/1

Having suggested a possible tuning circle, I would like to discuss a
few musical implications of this "semi-well-temperament" from a
fascinating era in European music, pursuing my own perspective with
much indebtedness both to Lindley and to Barbour's earlier study.[4]

---------------------------------------------------
3. Schlick's artful balance and its musical context
---------------------------------------------------

While the term "modified meantone" is sometimes applied to the
well-temperaments of the late 17th-19th centuries, fitting a system of
major and minor keys, this description might even more aptly fit
Schlick's artful approach to interval aesthetics in an early
16th-century modal setting.

A very salient feature of Schlick's tuning, if an approach such as
Lindley's or mine is generally correct, is the relatively smooth or
"meantone-like" quality of all regular major thirds not involving the
equivocal Ab/G# (F-A, C-E, G-B, Bb-D, Eb-G, D-F#, A-C#). All of these
thirds are within about 7.82 cents of pure, or 1/3 Pythagorean comma,
with a gentle gradation:

F-A C-E G-B ~392.18 cents ~5.87 cents wide of 5:4
Bb-D D-F# ~393.16 cents ~6.84 cents wide of 5:4
Eb-G A-C# ~394.14 cents ~7.82 cents wide of 5:4

These "sweet" thirds throughout the most frequently used portion of
the modal gamut in an early 16th-century style such as Schlick's
reflect an aesthetic expressed in regular meantones also. One should
strive in a 12-note tuning to make all the usual thirds not very far
from pure, not being too concerned about compromising the more remote
intervals.

When we focus on this "meantone-like" portion of Schlick's tuning, the
obvious difference with later 16th-century practice and theory is the
treatment of E-G#.

By around 1520-1530, a fully concordant major third above E is
becoming an essential feature of style both for final cadences in the
mode of E Phyrgian favoring a major third in the closing sonority, and
at many other places in pieces in various modes. Pietro Aaron's
_Toscanello_ of 1523, and his treatise on the modes of 1525, discuss
and offer examples of this preference for the major third.

Thus the most common 16th-century meantone range for 12-note
instruments becomes Eb-G# rather than Ab-C#, and instruments with more
notes per octave such as Vicentino's (1555) tend to place G# as the
"usual" accidental and Ab as the "less usual" one.

In Schlick's era, however, there was evidently no such stylistic and
intonational consensus: Schlick treats G# quite gingerly as a
cadential semitone in ornamental approaches to A, and feels free to
tune his compromise Ab/G# accordingly, with the accent on a relatively
more concordant Ab-C, which he uses as a sustained consonance in his
own writing for organ.

In 1482, Ramos seems to observe that while some people favor G# as a
major third in E Phyrgian, the "incitative" nature of this mode also
admits of the minor third below and major third above (E-G-B), or the
simple fifth (E-B); his remark may well describe the stylistic
situation around 1500 also.

At the opening of the 16th century, composers such as Josquin and
Isaac are starting regularly to close pieces with sonorities including
a third, and such closes also occur in German keyboard music of the
period -- but the third is often minor.

Of course, the traditional close on what in a medieval setting could
be called a complete trine, e.g. E3-B3-E4 in Phrygian (here C4 shows
middle C), also remains very common in this era around 1500-1520 in
Germany and elsewhere.

Under these stylistic conditions, the main call for G# might indeed be
what Schlick describes, and caters for in his tuning: a cadence on A
requiring a marginally playable third E-G#, which an organist can
tastefully ornament to avoid making the impurity of this interval too
obvious.

Focusing on the more "usual" portion of Schlick's gamut, Eb-C#, we
find he has achieved the same basic ideal as a regular 16th-century
meantone: rather uniformly "sweet" thirds, although not quite as sweet
as in regular schemes such as 1/4-comma or 2/7-comma.

By way of compensation, if Lindley and I are correct in reading the
approximate temperings of fifths which may be intended, these
intervals are somewhat less impure -- around 3-4 cents -- than in such
regular meantones where they are compromised by about 5-6 cents.

For the more remote intervals, possibly in part by design and in part
by accident, Schlick's scheme suggests how it is possible not only to
strike a compromise between Ab-C and E-G#, especially in a style where
both may be somewhat less frequently used than other thirds, but also
to "bisect" the Wolf fifth of a regular meantone so as to make all
fifths (or diminished sixths) at least marginally "playable."

If we take the "tempering excess" in this tuning at about 1/2 PC or 12
of our slightly smaller "Pythagorean cents," then allocating 3 cents
to Ab/G#-Eb, and the remaining 9 cents to C#-Ab/G#, happens to avoid
any outright "Wolf" while strongly favoring Ab-Eb, Schlick's primary
intention.

This kind of engagingly "ragged-edged" well-temperament would offer in
a Renaissance kind of setting some strikingly plangent sonorities and
stylistic effects near the extremes of the tuning chain with an
intriguing "partial interchangeability," while retaining many of the
usual attractions of regular meantones.

As in these meantones, the emphasis is on the "sweetness" of the usual
gamut -- a higher priority in this kind of modal style than in the
tonally-oriented well-temperaments of Werckmeister and successors.
This "usual" gamut, as in Schlick's system, might be defined to
include tenth fifths (e.g. Eb-C#), and the seven major thirds formed
within their portion of the chain.

Two more thirds, for example Ab-C and E-G# in Schlick's system, are
tempered to be more or less "serviceable," somewhere (as Lindley and I
interpret it) between 12-tET and Pythagorean, more or less.

The three remaining diminished fourths are more decidedly marginal
(e.g. B-Eb, C#-F, F#-Bb in Schlick's system), generally "little
esteemed and seldom used" in a conventional Renaissance setting, as
Schlick aptly puts it, although available in a pinch.

In either Lindley's model or my generally similar one, these intervals
have a size of around 414 cents. In a 21st-century setting, these
thirds would be highly esteemed indeed in styles such as neo-Gothic,
where they very closely approximate the regular intervals of 29-tET;
in a 16th-century setting, they would have a plangent flavor as
idiomatically used diminished fourths, but possibly with more utility
as substitute major thirds than their counterparts in a usual meantone
temperament (e.g. 1/4-comma at around 427 cents).

The most compromised fifth, C#-Ab/G# in Schlick's system, would be
decidedly more beatful than a regular meantone fifth even in the
19-tET of Costeley or almost identical 1/3-comma temperament of
Salinas (a bit more than 7 cents impure), but yet not "unplayable" in
a pinch at something like 9 cents wide.

When Schlick remarks that the impurity of this fifth should not
trouble the reader because "it is not in use," he may be reflecting a
change in cadential aesthetics making the fifth C#-G# -- or most
typically the fourth G#-C# -- indeed less important by around 1500
than in styles of the early to middle 15th century.

In these earlier styles, a cadential resolution in which a major sixth
expands to an octave while a major third expands to a fifth is one of
the most popular progressions, e.g.

C#4 D4
G#3 A3
E3 D3

Interestingly, as Lindley has elsewhere commented, in this type of
progression an upper fourth (here G#3-C#4) impure even by a whole
Pythagorean comma -- and thus a "Wolf" by any usual standard -- may be
quite tolerable both because of the cadential context, and of the more
or less pure quality of the other intervals.

Thus a player of Schlick's keyboard playing more "old-fashioned" music
might not find the impurity by around 9 cents of the fourth G#3-C#4
that much of a problem in the main situation where it might occur.

Could Schlick's 12-note tuning, or this possible interpretation of it,
be adapted to later 16th-century or similar styles where one desires
to tune E-G# as a usual near-pure third?

One possible solution would be to regard the "usual" gamut of ten
fifths as Bb-G#, thus including E-G# as the usual major third above
the final in Phyrgian, while regarding Eb-G and Ab-C as the two
thirds to be compromised in Schlick's manner.

Thus E-G# would be rather close to pure, along with the other "usual"
fifths; Eb-G might be comparable to 12-tET, and Ab-C to Pythagorean.

We would have Eb-Bb at around 3 Pythagorean cents wide, and Ab-Eb as
the "marginally playable" fifth around 9 cents wide.

The pragmatic assumption here is of a style where E-G# occurs more
often than either Ab-Eb or Eb-Bb, an assumption that seems to me to
hold for many 16th-century pieces in what is termed a _cantus durus_
signature with B-natural (German H) rather than Bb (German B).

The distinct qualities of such a 12-note temperament, somewhere
between those of a regular Renaissance meantone and a conventional
18th-century well-temperament, could have a special appeal --
especially in the kind of 21st-century setting where diminished fourths
at around 414 cents could also invite digressions from a Renaissance
to a neo-Gothic style (where these intervals would make excellent
usual major thirds).

Comparing this type of circulating "modified meantone" tuning with a
well-temperament in the usual 18th-century fashion may illustrate a
main point of distinction from the viewpoint of playing music in a
Renaissance or related style.

In the Renaissance-style circulating temperament, seven major thirds
will be within 1/3 Pythagorean comma or 8 Pythagorean cents of just.
In a typical well-temperament of the Bach era or later, in contrast,
we may find three or four major thirds of this "sweet" variety as
judged by 16th-century standards.

A final caution I would certainly want to add, having presented
Schlick's tuning within its framework of a 12-note system, is that in
my view larger meantone tuning sets permit yet more uniformly smooth
or "sweet" thirds in a Renaissance setting while opening up various
kinds of special effects, including artful uses of the "odd" intervals
also available in these tunings.

-----
Notes
-----

1. J, Murray Barbour, _Tuning and Temperament: A Historical
Survey_ (East Lansing: Michigan State College Press), pp. 137-139.

2. "Early Sixteenth-century Keyboard Temperaments," _Musica
Disciplina_ 28:129-151 (1974), pp. 129-139 on Schlick.

3. For the kinds of historical European styles we are considering in
characteristic harmonic timbres, a "Wolf" fifth is described by some
recent scholars as an interval impure by about one half Pythagorean
comma (~11.73 cents) or more, this comma also being known as the
ditonic comma.

4. Barbour's model, see n. 1 above, p. 138, is also of interest.
Rightly correcting earlier notions in the historical literature that
Schlick was describing 1/4-comma meantone, he describes the system as
"somewhere between meantone and equal temperament." Offering one
possible interpretations, he suggests narrowing each of the six
diatonic fifths by 4 cents, and each of the four accidental fifths not
involving Ab/G# by 2 cents. Both Lindley and I follow his first
estimate, but suggest a somewhat greater temperament of the latter
fifths by about 3 cents, roughly as in 1/7-comma meantone rather than
12-tET. In Barbour's more "balanced" tuning circle, Ab-C# and E-G# are
both slightly less impure than in Pythagorean tuning, while Ab-Eb and
C#-G# are each about 4 cents wide. Here I would agree with Lindley
that Schlick calls for Ab-C to be decidedly more serviceable as a
sustained 16th-century concord than E-G#, and for C#-Ab/G# to be
much more marginal than Ab-Eb. At the same time, even while accepting
Lindley's parameters as appealing, I would agree with Barbour's
interpretation of a kind of well-temperament, albeit more tentative
and "ragged at the edges" than the one he postulates.

Most appreciatively,

Margo Schulter
mschulter@value.net

Arnold Schlick's well-temperament of 1511

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