---------------------------------------------
Peppermint 24: A HEP neo-medieval temperament
Part I: Medieval European styles
---------------------------------------------
While the term "neo-medieval" when used on rec.music.early is apt to
suggest more specifically a style patterned on or inspired by that
medieval Europe, there is also the rich musical and intonational
tradition of the medieval Near East.
Peppermint 24 is a keyboard temperament with 24 notes per octave which
seeks to combine elements from both musical words, with some modern
embellishments and "artful distortions": a kind of "accentuated
Pythagorean" sound for 13th-14th century European styles, plus some
rather accurate representations of certain Near Eastern intervals,
scales, and modes from around the same era -- to the extent that a
flexible intonational practice can be represented on a fixed-pitch
instrument with a relatively small number of notes per octave.
From a European perspective, Peppermint 24 might draw its inspiration
in part from the _Lucidarium_ of Marchettus of Padua (1318), a
treatise advocating the use of extra-narrow cadential semitones or
"dieses" and extra-large major thirds and sixths in directed
progressions by stepwise contrary motion to the fifth and octave, and
in part on a bit of an alternative history scenario: "What if
musicians around the 15th or 16th century had experimented with
tempering fifths in the _wide_ direction, possibly developing, for
example, forms of chromaticism or enharmonicism quite different from
those of 16th-17th century meantone?"
From a Near Eastern perspective, Peppermint 24 offers a set of neutral
interval such as seconds and thirds which nicely approximate the just
ratios given by some medieval theorists of these traditions for
intervals, tetrachords, and scales in the _maqam_ or modal system: for
example, 14:13, 13:12, 12:11, 8:7, 7:6, 11:9, and 27:22. As Todd
McComb has emphasized, the intonation of Near Eastern ensemble music
is quite flexible, so that fixed interval sizes whether specified as
integer ratios or as tuning steps in a 19th-20th century scheme
dividing the octave into 24 equal parts should be taken at best as
general indications or approximations.
While one might improvise in either style, a "fusion" style is also
possible, for example with three-voice Gothic progressions applied to
Zalzal's tuning for lute or Ibn Sina's scale dividing a ratio of 7:6
into unequal neutral second steps of 14:13 and 13:12.
The Peppermint 24 temperament is a HEP system, that is, one based on
the ideal of "Historically Educated Performance" taking into account
period practices but adding new technical or interpretative elements.
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1. Origins of Peppermint 24: Accentuating Pythagorean intonation
----------------------------------------------------------------
Curiously, the regular 12-note temperament which provides a basis for
Peppermint 24 was proposed on the Alternative Tuning List not by a
medievalist, but rather by a young theorist and mathematician named
Keenan Pepper, who suggested a tuning system where the whole-tone and
chromatic semitone would have a ratio of logarithmic sizes (for
example, as measured in cents) equal to the Golden Section or Phi,
approximately 1.61803398874989484820459. He proposed this temperament
as a logical counterpart to the "Golden meantone" of Thorwald
Kornerup, in which the ratio of the whole-tone and _diatonic_ semitone
is equal to Phi, producing a Renaissance-like tuning somewhere between
1/4-comma and Zarlino's 2/7-comma.
In the following months, Kraig Grady pointed out that this temperament
also appears in music theorist Ervin Wilson's Scale Tree. Thus I often
refer to the basic temperament as the "Wilson/Pepper tuning."
As I quickly realized late in the summer of 2000 when I learned of
Pepper's proposal, this system with fifths at around 704.10 cents (or
about 2.14 cents wide) would nicely fit into my own exploration of
"accentuated Pythagorean" or "neo-Gothic" temperaments with fifths
_larger_ than the pure 3:2 of Pythagorean (~701.955 cents).
More specifically, Pepper's temperament, as I soon discovered, offers
a neat optimization of four of my favorite "neo-Gothic" interval
ratios which would be hard to outdo if one sought to calculate a
temperament specifically for this purpose. Major and minor thirds at
around 416.38 cents and 287.71 cents are very close to ratios of 14:11
and 13:11 (~417.51 cents and ~289.21 cents). Likewise augmented
seconds and diminished fourths at around 336.86 cents and 367.235
cents are very close to "supraminor" and "submajor" thirds at ratios
of 17:14 and 21:17 (~336.13 cents and ~365.83 cents).
These four types of thirds, known in neo-medieval theory as the "Four
Convivial Intervals" (14:11, 13:11, 17:14, 21:17) are all within 1.5
cents of their rational ratios. While the regular 14:11 and 13:11
thirds could be taken as accentuated variations on the Pythagorean
81:64 and 32:27 (at ~407.82 cents and ~294.13 cents), the supraminor
and subminor or "semi-neutral" thirds are quite different from the
intervals of recorded medieval European practice, although some
Persian tuning systems seem to feature similar sizes.
From a Gothic -- or neo-Gothic -- aesthetic stance, these thirds at
around 17:14 and 21:17 have the quality, like Pythagorean thirds, of
being quite complex and active, lending themselves to new flavors of
cadences after a 13th-14th century style. Interestingly, they vary
from the simplest ratios for thirds of 5:4 and 6:5 by about the same
amount as Pythagorean thirds, but in the opposite directions. Thus a
three-voice progression like this, with supraminor third resolving to
unison and submajor third to fifth, can sound like a variation on
Machaut as interpreted in some alternative history or universe (here
C4 is middle C):
Bb3 B3
F#3 E3
Eb3 E3
Melodically, the wide chromatic semitones at Eb3-E3 and Bb3-B3 give
this progression a "different" flavor: their size is around 128.67
cents, almost identical to a just 14:13 step (~128.30 cents).
In contrast to these chromatic semitones or "2/3-tones" are the usual
diatonic semitones at around 79.52 cents, very close to the 22:21 step
appearing in some ancient Greek and medieval Near Eastern tunings
(~80.54 cents). In more or less conventional 13th-14th century
European styles, this semitone can serve as a yet narrower version of
the already quite incisive 256:243 diatonic semitone or limma of
Pythagorean tuning (~90.22 cents).
--------------------------------------------
2. Wilson/Pepper 12 and neo-Gothic mannerism
--------------------------------------------
On a single 12-note keyboard, Wilson/Pepper is a regular tuning based
on a chain of 11 fifths, typically Eb-G#, all of the same size; one
can play conventional 13th-14th century European music much as one
would on a usual Pythagorean keyboard in an Eb-G# tuning. The full
24-note Peppermint system for two keyboard manuals described below
(with keyboard diagrams and tuning data) retains this familiar
arrangement on each 12-note manual.
The obvious compromise of Wilson/Pepper is that the fifths are indeed
tempered, in contrast to the pure 3:2 fifths and 4:3 fourths of
medieval Pythagorean intonation. This compromise is significant,
although not severe: fifths are about 2.14 cents wide, and fourths
equally narrow, a degree of impurity very slightly greater than that
of 12-tone equal temperament (fifths 700 cents, ~1.96 cents narrow),
albeit in the opposite direction.
While this degree of impurity is quite mild by comparison to
Renaissance syntonic comma or meantone temperaments, for example,
where fifths are often tuned narrow on the order of 5 cents, it
nevertheless does represent what one might call a Manneristic
accentuation or distortion of the classic ideal of Pythagorean
tuning. From this viewpoint, it seems more characteristic of HEP than
of HAP, or Historically Approximated Performance, where the goal is to
adhere more closely to known or likely period practices.
As a 12-note chromatic tuning applied to Gothic or similar styles,
Wilson/Pepper can have many of the same qualities as Pythagorean
tuning, only exaggerated: wide major intervals, narrow minor
intervals, active thirds and sixths, and efficient "closest approach"
cadences where unstable intervals resolve by stepwise contrary motion,
one voice proceeding by a whole-tone and the other by a diatonic
semitone.
For example, consider this standard four-voice cadence from the 14th
century, and a 21st-century variation:
14th-century 21st-century
G#4 A4 G#4 A4
C#4 D4 C#4 D4
G#3 A3 B3 A3
E3 D3 E3 D3
(Maj10-12 + Maj6-8 + Maj3-5) (Maj10-12 + Maj6-8 + Maj6-8 + Maj2-4)
In either formula, the major thirds and tenths at around 14:11 and
28:11 are rather wider than in Pythagorean intonation, and so "closer"
to their goal of a stable fifth or twelfth; the complexity of the
outer major tenths can lend a special intensity and excitement to
these cadences. As mentioned above, the narrow diatonic semitones near
22:21 can also contribute to the overall incisive effect.
By experimenting with unusual transpositions, we can also derive
variations of usual medieval European octave species or modes with
supraminor or submajor intervals. Here, for example, is a version of
Dorian with a supraminor third and seventh above the final:
Bb3 C4 C#4 Eb4 F4 G4 G#4 Bb4
cents: 0 208 337 496 704 912 1041 1200
208 129 159 208 208 129 159
In addition to the 337-cent supraminor third and 1041-cent supraminor
seventh, which can have a pleasant effect above a drone, we have
melodically a mixture of regular whole-tones at around 208 cents (a
bit larger than the pure Pythagorean 9:8 at ~203.91 cents) and two
varieties of smaller steps: 129-cent chromatic semitones (e.g. C4-C#4)
and 159-cent diminished thirds (e.g. C#4-Eb4). While the chromatic
semitone can sound like a large semitone, the diminished third can
have the effect of a small whole-tone, producing a charming variation
on the usual medieval European Dorian pattern of T-S-T-T-T-S-T.
----------------------------------------
3. From 12 to 24: Enter the quasi-diesis
----------------------------------------
The Peppermint 24 tuning features two 12-note keyboards, each with a
regular chain of fifths (Eb-G#) as described above, with a distance
between the manuals of about 58.68 cents, the difference between the
regular major second at ~208.19 cents and a pure 7:6 minor third at
~266.87 cents. Thus the system is defined in part by the just 7:6
third or 12:7 major sixth, with all other intervals tempered except
the 2:1 octave.
This 58.68-cent interval is called a "quasi-diesis," or QD for short,
since it is considerably smaller than the usual 80-cent diatonic
semitone and often serves as a special cadential step like the diesis
of Marchettus. It could be taken as a narrower version of the 28:27
"thirdtone" (~62.96 cents) favored by the Greek theorist Archytas.
The resulting 24-note keyboard layout looks like this, with the
raising by a quasi-diesis of a note on the upper keyboard indicated by
an asterisk (*) symbol:
187.349 346.393 683.253 891.445 1050.488
C#* Eb* F#* G#* Bb*
C* D* E* F* G* A* B* C*
58.680 266.871 475.062 554.584 762.775 970.967 1179.158 1258.680
7/6
-------------------------------------------------------------------------
128.669 287.713 624.574 832.765 991.809
C# Eb F# G# Bb
C D E F G A B C
0 208.191 416.382 495.904 704.096 912.287 1120.478 1200
As an overall system, Peppermint 24 approximates just intonation
intervals based on prime factors of 2-3-7-11-13, and also certain
ratios of prime factor 17 (e.g. 17:14, 21:17, 17:12, 24:17), in
contrast to 16th-19th century European systems of just intonation (JI)
based on factors of 2-3-5, with a primary role for 5:4 and 6:5 thirds.
The results are often conducive to a medieval European or Near Eastern
style, as well as to some avant-garde 20th-21st century styles, while
quite different from European styles of the Renaissance-Romantic eras.
Here is a tuning file for Manuel Op de Coul's Scala, a tuning program
freely available on the Internet which can analyze tunings and also
generate MIDI tuning data for some popular synthesizers or sound
cards:
---------------- Scala file starts on next line of text -----------
! peprmint.scl
!
Peppermint 24: Wilson/Pepper apotome/limma=Phi, 2 chains spaced for pure 7:6
24
!
58.679693 cents
128.669246 cents
187.348938 cents
208.191213 cents
7/6
287.713180 cents
346.392873 cents
416.382426 cents
475.062119 cents
495.904393 cents
554.584086 cents
624.573639 cents
683.253332 cents
704.095607 cents
762.775299 cents
832.764852 cents
891.444545 cents
912.286820 cents
970.966512 cents
991.808787 cents
1050.488479 cents
1120.478033 cents
1179.157725 cents
2/1
--------- Scala file ended on newline after last line of text --------
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4. The quasi-diesis in action: Marchettan progressions
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As already mentioned, one important application of the 58.68-cent
quasi-diesis is as a narrow semitone in directed progressions such as
those described by Marchettus of Padua in 1318. Prudently I would
emphasize that Marchettus is discussing the flexible intonation of
singers, so that Peppermint 24 simply offers one possible
interpretation or approximation on a fixed-pitch instrument.
Consider, for example, this trecento figure cited by Christopher
Page[1], with the metrical indications here counting crotchets or
quarters in Page's example, and all divisions binary, so that the
pattern at the third crotchet or quarter in the upper voice is two
semiquavers or sixteenths followed by a quaver or eighth::
1 + 2 + 3 + + 4 + | 1...
E4 F4 G4 F4 G4 F4 E4 F4 E4 D4
A3 G3 A3 B3 C#4 D4
Page aptly terms this figure a "convergence cadence," since the lowest
voice "rises through a fifth to converge on a unison with the voice
above" -- that is, G3-D4. He notes that this voice ascends by three
successive whole-tones G3-A3-B3-C#4 leading up to the cadential
semitone C#4-D4. The melodic progression calls for a special
adjustment of the kind advocated by Marchettus "so that the c'# before
the convergence may be almost startlingly `colourful,' emphasizing the
transparency of the unison which is to follow."[2]
To achieve this kind of Marchettan effect, we place the regular or
_musica recta_ gamut for this passage on the _upper_ keyboard, with
the cadence converging on D*4. This permits us to substitute, in place
of the usual ascending 80-cent cadential semitone C#*4-D4 in the
lowest voice, a narrow 59-cent quasi-diesis step D4-D*4:
1 + 2 + 3 + + 4 + | 1...
E*4 F*4 G*4 F*4 G*4 F*4 E*4 F*4 E*4 D*4
A*3 G*4 A*3 B*3 D4 D*4
Vertically, this special inflection makes the narrow cadential minor
third D4-E*4 at a just 7:6 or around 267 cents -- actually a major
second plus quasi-diesis -- "more closely approach" the unison than
would the regular Peppermint minor third C#*4-E*4 at around 288 cents.
Also, the written diminished fourth preceding this cadential third
(C#*4-F*4 if played in a usual spelling on the upper keyboard) takes
on a special color as the neutral third D4-F*4 at around 346.39 cents,
very close to a just 11:9 (~347.41 cents).
Among other things, this passage shows how neutral thirds of a kind
routine in medieval and later Near Eastern styles _might_ have arisen
in some performances of 14th-century Italian music. However, another
significant if less dramatic point it illustrates is how one can make
these Marchettan inflections while maintaining pitch stability for the
regular notes of the gamut.
In his presentation, Marchettus himself advocates using extra-narrow
cadential dieses specifically in directed closest approach
progressions involving the sign of _musica ficta_ -- that is,
progressions involving sharps. In Peppermint 24, this means taking the
upper keyboard as our base for such a style, and using notes on the
lower keyboard only as special inflections for written or unwritten
sharps. Thus the usual diatonic notes of a piece remain unaltered,
with ascending semitone steps in closest approach progressions
involving such sharps realized as quasi-diesis steps moving from the
lower to the upper keyboard (e.g. C#*4-D*4 realized as D4-D*4; F#*4-G*4
realized as G4-G*4).
Some standard 14th-century cadences for three voices may also
illustrate this technique:
Usual notation Peppermint realization
C#4 D4 D4 D*4
G#3 A3 A3 A*3
E3 D3 E*3 D*3
(Maj6-8 + Maj3-5)
Here the Peppermint version uses a wide cadential major third E*3-A3,
a fourth-less-quasi-diesis (or "4-QD" for short) at around 437.225
cents, and a wide cadential major sixth E*3-D4, a minor seventh less
quasi-diesis or min7-QD at a just 12:7 (~933.13 cents). The wide major
third is very close to a just 9:7 (~435.08 cents), so that the
cadential sixth sonority nicely approximates a pure 7:9:12. The lowest
voice descends by a regular whole-tone (E*3-D*3) while each upper
voice ascends by a quasi-diesis (A3-A*3, D4-D*4).
Another common 14th-century cadence has a lower minor third contract
to a unison while an upper major third expands to a fifth:
Usual notation Peppermint realization
C#5 D5 D4 D*4
A4 G4 A*4 G*4
F#4 G4 G4 G*4
(min3-1 + Maj3-5)
Here the middle voice with its _musica recta_ notes remains on the
upper keyboard throughout, while the two outer voices realize their
ascending cadential semitones as quasi-dieses (G4-G*4, D4-D*4). As in
Page's two-voice example, the lower pair of voices form a narrow 7:6
minor third -- a major second plus quasi-diesis, or Maj2+QD --
converging on a unison. The upper voices form a wide, near-9:7 major
third (4-QD), expanding to a fifth. The unstable cadential sonority
approximates a just 6:7:9, at once offering a pleasant touch of color
and inviting a very efficient directed resolution.
A common medieval progression illustrating a minor complication of the
system evidently advocated by Marchettus, with two alternative
solutions on the Peppermint keyboard, is the following:
Usual notation Peppermint realizations
F#4 G4 F#*4 G*4 G4 G*4
D4 C4 D*4 C*4 D*4 C*4
B3 C4 B*3 C*4 or C4 C*4
(min3-1 + Maj3-5)
Our minor dilemma involves the outer fifth of the first sonority,
B3-F#4. If we literally follow the advice of Marchettus, then the
_musica ficta_ step of F#4-G4 should be realized as a narrow diesis
(Peppermint G4-G*4) considerably smaller than the regular limma or
semitone B3-C4 (Peppermint B*3-C*4), resulting in this progression:
G4 G*4
D*4 C*4
B*3 C*4
The problem is that the vertical interval B*3-G4 in the first sonority
has a size of around 724.94 cents, almost 23 cents larger than a just
3:2, and in usual harmonic timbres quite distinct from a stable fifth!
While this interval, close to a complex 7-based ratio of 32:21
(~729.22 cents), can be of interest in situations where we want _very_
special effects, here we would likely want to keep vertical fifths at
their usual sizes (a just 3:2 in Pythagorean; about 2 cents wide in
Peppermint).
Thus we might choose between the two Peppermint realizations suggested
above. The first simply uses the usual notes on the upper keyboard,
sticking with regular Peppermint intervals:
F#*4 G*4
D*4 C*4
B*3 C*4
Note that this is exactly the same progression that one would play in
a simple 12-note tuning. However, we can also obtain an accentuated
version with narrow cadential quasi-dieses while maintaining
the concord of the first vertical fifth -- if we are willing to
compromise the rule of leaving _musica recta_ steps unaltered:
G4 G*4
D*4 C*4
C4 C*4
Here we have the same melodic and vertical intervals as in our
previous (min3-1 + Maj3-5) cadence on G, with ascending quasi-diesis
motions in both outer voices, which move smoothly in fifths. In order
to achieve this result, the usual _musica recta_ semitone B*3-C*4 must
be altered to the quasi-diesis C4-C*4.
Thus we may sometimes have to use two slightly different "versions" of
the same note -- Peppermint F#*4 and G4, or possibly B*3 and C4 -- in
order to handle the vertical fifth B-F# (realized as B*3-F#*4, or
possibly sometimes as C4-G4).
In practice, this minor complication should not seriously affect pitch
stability, as long as one otherwise sticks with the usual Pythagorean
(or Peppermint) degrees except for sharps. Here it is not my purpose
to urge against more experimental practices, only to make the point
that generally Marchettus calls for the alteration in directed
progressions of _musica ficta_ rather than _musica recta_ notes.
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4.1. Marchettan chromaticism: Subdividing the whole-tone
--------------------------------------------------------
While the above "Marchettan" nuances involve typical 14th-century
European progressions, Marchettus also describes and advocates some
direct chromatic progressions bringing into play another special
interval: the _chroma_ or extra-wide chromatic semitone (e.g. G3-G#3),
larger than the regular apotome (e.g. Bb3-B3).
Consider, for example, this two-voice progression:
Usual notation Peppermint realization
G4 G#4 A4 G*4 A4 A*4
C4 E4 D4 C*4 E*4 D*4
Here the first step G4-G#4 in the upper voice, involving a _musica
ficta_ inflection or sharp, is an extra-wide chroma -- realized in
Peppermint as G*4-A4, a major-second-less-quasi-diesis (Maj2-QD) at
around 149.51 cents, very close to the 12:11 neutral second (~150.64
cents) of some ancient Greek and medieval Near Eastern tuning
systems. In contrast, the narrow Marchettan cadential semitone or
diesis G#4-A4 is realized as the 59-cent quasi-diesis A4-A*4. These
two intervals together add up to a regular whole-tone: G*4-A4-A*4.
While the melodic element of direct chromaticism is itself a striking
effect in an early 14th-century setting, this effect is here made yet
more colorful by the disparity in size between these steps -- in
Peppermint, respectively about 149 and 59 cents, adding up to a usual
208-cent tone.
Vertically, the first note of the upper voice (G4, Peppermint G*4)
forms a fifth with the lower voice; the second note forms a wide major
third (E4-G#4, Peppermint E*4-A4), resolving to a fifth (D4-A4,
Peppermint D*4-A*4) with the lower voice descending by a regular tone
and the upper voice ascending by a narrow diesis G#4-A4 (or Peppermint
quasi-diesis, A*4-A4).
Another Marchettan chromatic progression also illustrates the
chroma/diesis division of the tone and its Peppermint realization:
Usual notation Peppermint realization
C5 C#5 D5 C*5 D5 D*5
F4 E4 D4 F*4 E*4 D*4
Here the first note of the upper voice forms a fifth with the lower
voice, with this voice then ascending by a wide chroma C5-C#5
(Peppermint C*5-D5) to form a large cadential major sixth with the
next note of the lower voice (E4-C#5, Peppermint E*4-D5) which
resolves to the octave. Again, the Peppermint version features a
contrast between the 150-cent chroma (C*5-D5) and the 59-cent
quasi-diesis (D5-D*5).
As Marchettus observes, this last progression also shows how the
cadential diesis in the upper voice is smaller than the regular limma
or semitone appearing as the first melodic interval in the lower voice
(F4-E4, Peppermint F*4-E*4) -- respectively around 59 cents and 80
cents in Peppermint.
He also describes what might be called a time-reversed version of this
last formula, with the same intervals in the opposite order:
Usual notation Peppermint realization
D5 C#5 C5 D*5 D5 C*5
D4 E4 F4 D*4 E*4 F*4
As Marchettus explains, the first two vertical intervals should be
intoned as if the performers were going to make a cadence returning
from the wide major sixth E4-C#5 (Peppermint E*4-D5) to the octave
D4-D5 (Peppermint D*4-D*5) --
Usual notation Peppermint realization
D5 C#5 D5 D*5 D5 D*5
D4 E4 D4 D*4 E*4 D*4
but with this sixth instead contracting by contrary motion to a fifth,
F4-C5 (Peppermint F*4-C*5)! In this surprising resolution, the lower
voice ascends by a usual limma or semitone E4-F4 (Peppermint E*4-F*4),
and the upper voice by the wide chroma (Peppermint D5-C*5).
Marchettus describes this as a _color ficticius_ or "feigned color" --
a deceptive inflection or cadence, we might say.
Realizing these Marchettan chromatic progressions in Peppermint has
introduced an interval, the 150-cent or near-12:11 neutral second,
very useful also for medieval or modern Near Eastern styles.
Part II looks at Peppermint mappings and nuances for Near Eastern
scales, and also some modern idioms and variations drawing on medieval
European or Near Eastern elements.
-----
Notes
-----
1. Christopher Page, "Polyphony before 1400," in Howard Mayer Brown
and Stanley Sadie, eds., _Performance Practice: Music before 1600_
(New York: W. W. Norton, 1990), p. 80 and Ex. 3.
2. Ibid.
Most appreciatively,
Margo Schulter
mschulter@calweb.com