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Transcript
Hybrid Thinking in Meloharmony
Manfred Stahnke,
Hochschule für Musik und Theater Hamburg, Germany
[email protected]
www.manfred-stahnke.de
Proceedings of the International Conference « Composer au XXIe Siècle – Processus et
Philosophies »
Montréal (Québec) Canada, 28 février – 3 mars 2007
Abstract
This essay will concentrate on different approaches to thinking in terms of "meloharmony", a word
that I came up with to denote the interrelationship of vertical and horizontal pitch organization within
an open microtonal field. By definition, this field is open to every aspect of pitch organization;
interval relationships may exist within this field regardless of whether or not they are also related to
older forms of melodic-harmonic relationships. The only restriction lies in the avoidance of addressing
"anonymous" fields, where neither horizontal nor vertical pitch relationships play a distinct role.
In the first section I will discuss some positions that I consider important in reference to my term
"meloharmony":
1) Harry Partch and his "just intonation".
2) Michel Gonneville and his use of sixteenth-tones on a Julián Carrillo piano.
3) Gérard Grisey and his microtonal musical language using just or distorted spectra.
In the 2nd section I will analyze two works of my own, Partch Harp and Partota 8, as examples of a
hybridization of the first three positions.
My discussion here is related to my personal preferences and does not provide a broad basis for
microtonal thought as a whole.
Introduction
At some point we have to concede that the post-Gruppen styles (consider the music of Karlheinz
Stockhausen from the late 50s), as have been disseminated throughout the world in part due to
the influence of Darmstadt, represent but a few of the many possibilities for constructing a musical
language. Today post-Gruppen styles (such as post-serialism, New Complexity) are a well-known
landscape. Alternative routes have been shown and must still be found. It is now a necessity, and
will be more so in the future, to deal with questions of straightforward harmonicity and pulsation,
aspects of music that the Darmstadt styles have avoided. In Germany, the Darmstadt styles stand
for the aftermath of serialism and a post-Adorno way of thinking. With regard to harmonicity, we
have composers like Julián Carrillo, Ivan Wyschnegradsky, Harry Partch, or Gérard Grisey, who
showed us new paths. Regarding pulsation, we have the minimalism of Steve Reich or the
polymetric music of Conlon Nancarrow, as well as the many composers who look to the richness of
pulsation in the worlds of ethnic or jazz musics. The result is a spectrum of musical styles today
that is extremely rich, perhaps even chaotic.
Many composers comment on forms of past music, from medieval music to music of the
Renaissance, to the music of Richard Strauss. There is an immense richness in the music of the
world today, and a composer of the 21st Century has a good chance of finding his or her way in
this rich field. We have a Neo-Neo-Classicism, as well as a Neo-Avant-Garde (New Complexity) at
the same time. But if these strands were the only definitions of composed music in the 21st
Century, I would have to deplore this Century deeply. Retro has never been fresh. I would like to
have a music that is searching and inventive, one that is at risk of being misunderstood.
There are many possible ways to conceive an organization of pitches. Two of the most prolific
approaches go back to the ancient Greeks: 1) Archytas, who thought in whole-number ratios,
going beyond the approach of the older Pythagoreans by also using the numbers 5 and 7 in his
2
description of the enharmonic mode, and 2) Aristoxenos, who referred to the specific
characteristics of the way we hear. Aristoxenos may be considered the forefather of equal
temperament (ET). Western harmonic thought was determined by these two approaches. But we
must also consider developments of the 20th century: The instrumental sounds of trombones,
metallophones, and virtually all other instruments were measured in their time domain as well as in
their harmonic domain, and the possibilities of recreating such sounds via FM or additive synthesis
or physical modeling were devised. So we now have a broader awareness of what harmony may
consist of. There is no harmony without the time domain. There is no pitch without harmony. (Even
a sine wave will be distorted by the way we hear to give us a perception of harmony!)
The microstructure of harmony became a playground for many composers and, of course,
musicologists. Composers like Harry Partch, James Tenney, and LaMonte Young went back to base
their work on whole-number ratios, whereas others like Ivan Wyschnegradsky, Julián Carrillo or
Alois Hába pursued the path of equal temperament. A third approach is that of the Parisian
"spectralists", leaving simple whole numbers in their harmonic concepts (though basing their
harmonies on spectra of pitches but focusing on the details of complex spectra), and avoided
concepts of ET scales (though often notating deviations from 12-tone ET as quartertones for
practical reasons).
In my own work I like to think in terms of whole-number ratios, spanning from more common
intervals such as 2/1, which represents our octave, to less common intervals with no conventional
names, such as the frequency ratio 21/16, which could be identified as a "narrow fourth". My
interest lies in the unique physiological reaction that our hearing apparatus has developed. It can
be shown that our hearing reacts in a very specific manner even to complex ratios, creating a
sensation of "Stimmigkeit" (which is more than "harmonic correctness", maybe a harmonic supersignal). But my preference for an approach based on whole numbers does not mean that I avoid
new temperaments. Our hearing is not restricted to a singular means of perception. In fact, our
hearing apparatus constantly tempers the perception of the sound we are receiving while trying to
simplify it; or more simply put: The signal is "interpreted" according to its characteristic traits. This
temperament is far more complex than the mathematical temperament of pitches. I would not
venture to describe it in this context. However, one aspect of it seems absolutely apparent: The
perception of harmonies is always bound to the time domain. Our sense of hearing will immediately
place the vertical (harmonic) attributes of every perceived auditory event into a horizontal
(melodic) context, and differences that arise within this context will form our sense of the overall
harmonicity.
Section I:
The threefold approach: Just Intonation - Equal Temperament Distortion
In this first section I will discuss three positions that I consider important for my term
"meloharmony". These positions are hybrid in themselves.
1) Harry Partch was so individualistic that his specific approach of "just intonation", with its
"Otonality" and "Utonality", is difficult to explain briefly. But it is lovely and strange enough to
make discussions worthwhile. I myself studied with Ben Johnston, a friend and colleague of Harry
Partch. However, since Partch himself very seldom used truly "just" harmony, we should question
both the use of "just" intervals and the circumstances in which they could be used.
2) Julián Carrillo is only one person we can mention with regard to the many options of dividing the
octave into microtonal steps that differ from our 12-tone temperament. He is the forefather of the
approach pursued by Michel Gonneville from Montréal, an approach that I will analyze in more
detail later. Gonneville mixes the idea of sixteenth-tone equal temperament with that of nearly just
intonation.
3) The music of Gérard Grisey is only slowly becoming accepted in Germany following a prolonged
period of perception by many of his contemporaries that his spectral music is mere decoration.
However, it would be outdated to continue down the path of the Parisian spectralists today,
especially since Grisey himself eventually developed new approaches in his later years. In his last
pieces Grisey moved beyond spectralism, not giving up any quality in his harmonic approach but
further developing the richness of his unique microtonal language of distorted spectra.
3
I would like to discuss all three of these positions here briefly and provide support for the idea of
taking the best of each of these three paths to form a hybrid approach towards a new
"meloharmony". I am very aware of the fact that my discussion here is related to my own personal
preferences and by no means able to provide a broad basis for microtonal thinking as a whole. I
consider the development of an encyclopedic approach to systems of thought to be out of date in
our 21st century.
1) Hybrid thinking in Just Intonation - Harry Partch
Harry Partch (1901-1974) was a very interesting composer and theorist. Partch's just intonation is
only truly "just" in some aspects. Those aspects include the passages where he uses the
chromelodeon (his adapted harmonium) in a soloistic manner. None of his percussive instruments
tend to produce a justness of pitch. All of the instruments in Partch's ensemble more or less create
a kind of pleasantly unconventional sound, since Partch often used all of his 43 unevenly
distributed steps per octave in sequence. In all of his lovely arpeggios or glissandi on the kithara or
canon, or on his adapted viola or his guitars, he leans more on vocally based linear formulations
than on harmonically based just intonation.
We see this hybrid concept even at the lowest level of Partch's pitch organization, namely in his
Tonality Diamond and its expansion to a 43tone scale. In order to understand this we must take a
more specific look at the details of Partch's musical world.
Partch's original Expanded Tonality Diamond provides a nice visual representation of just intervals:
I have transcribed the pitches from this Expanded Tonality Diamond in the following example,
using the indices 5, 7, and 11 to indicate the 5th, 7th, and 11th partials. The frequency ratios of
each given pitch are given in parentheses following the note name. I use the German nomenclature
in these examples, in which the letter H is used to denote the pitch that is referred to as B-natural
within the English traditions of notation.
"Otonality" on G, to be read on the diamond from lower left to upper right:
4
"Utonality" under G, to be read on the diamond from lower right to upper left:
Partch transposes these ratios into one octave. Utonality would be more easily recognizable as an
inversion of Otonality if he had written the ratios as 1/1, 8/9, 4/5, 8/11, 2/3, 4/7.
Sound example 1: Harry Partch - Otonality and Utonality, with some modulations, given by
Partch in Genesis of a Music
The Expanded Tonality Diamond provides the "primary tonalities" (a term created by Partch) that
are shown in my next figure. These summarize the entirety of the diamond's content plus added
pitches. These additions ("secondary tonalities" etc.) were chosen by Partch in part to fill in the
gaps that arise between the diamond's primary pitches. Partch maintains his just thinking while
achieving a nearly even distribution of pitches that can be used, as I mentioned early, in arpeggios
or chains of linear microsteps.
5
The result is Partch's 43-tone scale:
6
I have had to sometimes use double indices in my transcription. The indices placed below the notes
indicate Otonality pitches; those placed above the notes indicate Utonality pitches. Combined
indices are necessary to indicate ratios of the 5th, 7th, or 11th partials that are built on pitches
already derived from such ratios. For example, a 7 below a note indicates an Otonality ratio of 7/4
(sixth-tone lower). In combination with a 5 above the note, we know that the fundamental tone is
a Utonality 4/5 (twelfth-tone higher). The 11th partial deviates by roughly a quartertone. The
precise values in cents are given in Harry Partch's Genesis of a Music, New York, N.Y. (Da Capo
Press) 1979, p.133.
2) Hybrid Thinking in Equal Temperaments
We are all familiar with our Western music history and the gradual development of temperaments
leading up to the development of equal temperament as a tuning system for keyboard instruments,
which is the predominant system of temperament in Western music today. The primary challenge
through the centuries was the superposition of fifths and thirds within a single system. But this was
only one part of the problem. The other issue was always the congruency of vertical, harmonic
aspects of a tuning system with melodic linearity. Prior to the development of the first meantone
temperaments, the fifths were just and the thirds were imperfect consonances, since they were
derived from superimposed fifths. (However, as early as the 1300s, Englishmen such as Walter
Odington addressed this weakness, and argued that thirds be considered consonant). During the
Renaissance (Pietro Aron is a prominent name in this context) the major third became a just 5/4
within meantone keyboard tunings, while the fifth was heavily tempered (down by 5.4 cents
against just 3/2). The unequal size of the intervallic steps (from 76 to 117 cents, with the famous
"wolf" between G-sharp and E-flat) led to the construction of keyboard instruments with split keys,
sometimes with 19, or even with as many as 36, such as on the archicembalo of Nicola Vicentino.
Our modern equal temperament is a pleasant compromise, balancing out the thirds and fifths and
evenly distributing the seconds. The degree to which this tuning system can be considered hybrid
is seen in the coexistence of both Chopin and Schoenberg on the same piano. Chopin constructed
chords in a manner much like composers of the Renaissance, except that he included the seventh,
and even the ninth, as crucial components of his harmonic language. Schoenberg, on the other
hand, considered the field of 12 non-hierarchical pitches per octave to be his point of departure,
with no reference to the old chords, which he strictly rejected for his Dodecaphony.
Let us take a look now at one modern approach to equal temperament. Montreal composer Michel
Gonneville (*1950) wrote a piece entitled Naturel tempéré for the sixteenth-tone piano, an
instrument that Julián Carrillo designed and had built by the German piano makers Sauter in the
mid 20th Century. The piece was sponsored by Bruce Mather for the composer's community in
Canada. There are two primary aspects to Gonneville's piece: The first refers to the nearly endless
intervallic possibilities of such a sixteenth-tone scale. The octave on the sixteenth-tone piano is
divided into 96 pitches, and there is only one octave on the instrument. Let us look at the first
example from Naturel tempéré. For me it is the central passage, the passage that opens my ears
and engages my listening:
7
Sound example 2: Michel Gonneville - Detail from Naturel tempéré (includes both score
examples)
This passage utilizes the possibilities of the instrument extremely well. It is as though the all too
familiar sound of the piano becomes fluid and fresh again. A new concept of interval is derived
from the unison, resulting in intervals for which we have no names. We experience a world of
sound that lies between a bent or blurred single tone and the emerging and vanishing of the
interval. This effect is of course due to the extreme temperament constructed of minute steps of
12.5 cents, which our ear barely perceives as steps.
In Naturel tempéré Gonneville proceeds through many variations of seconds and thirds etc.,
intervals that have no proper names in our conventional nomenclature (except perhaps "neutral
third"). But in many places he approaches just intonation. In these locations in the music he
invents a "nearly just intonation", using a mode consisting of partials 8 through 15.
Let us take a look at the second example from the Gonneville piece Naturel tempéré:
8
Starting in bar 121 Gonneville approaches what Harry Partch would have described as the "C
Otonality 15-limit". In measure 125 he uses all of the pitches from this Otonality based on C, from
partial 8 to 15:
The chart I have put together below shows:
1st row: The partial number
2nd row: Gonneville's notation of sixteenth-tones (in his upper system), as deviation in sixteenthtone steps from ET.
3rd row: The cent deviations of the former.
4th row: The deviations of the just partials from ET.
Partial number:
8
9
10
11
12
13
14
15
Gonneville's labeling of sixteenthtone deviation from ET:
0
0
-1
-4
0
3
-2
-1
Deviation in cents of Gonneville's
pitches:
0
0
-12.5
-50
0
37.5
-25
-12.5
Deviations in cents of the just
partials from ET:
0
4
-13.6
-48.7
2
40.5
-31
-11.7
We can easily follow Gonneville's concept: The acoustic impression is that of the overtone series
above C. Our ear completely fails to notice the minor deviations. What Gonneville is addressing in
his piece is the congruency, the coming together of temperament and justness of the natural
intervals within a texture of minute steps.
3) Distorted spectra in quartertone notation: Gérard Grisey
9
Gérard Grisey (1946-1998) was one of the founders of the ensemble "L'Itinéraire" in Paris 1973,
together with Tristan Murail, Roger Tessier, and Michaël Levinas. Hugues Dufourt also later joined
the group. They began focusing on aural spectra and their microstructural construction with regard
to pitch and the time domain through computer-aided analysis. In fact, the connection and
coherence between these two domains was one of the group's primary topics.
In his late composition Vortex temporum for ensemble, Grisey seldom makes use of the natural
spectrum. Instead, he primarily distorts spectra by stretching or compressing them. Even in their
nearly natural form, Grisey writes out the spectra without any indices for the deviations of the 5th
or 7th partial tones from ET. Only the 11th and 13th partials are notated with quartertone signs, as
is the 14th partial on occasion (which is odd, since the 7th partial, the 14th's octave relation, is not
notated with any index). But these inconsistencies also appear within Grisey's work with distorted
spectra (see below). In his late pieces Grisey moved away from a systematic design of his various
spectra. Instead, he seems to have decided what was appropriate for the intended harmonic
picture on a solely local level. He chose quartertones as the smallest step for the majority of his
work, without composing "quartertone music" as such. His music still sounds "spectral".
Let us have a look at Grisey's view on spectra. Immediately at the beginning of Vortex temporum
Grisey chooses the partials 5, 6, 7, 8, 9, 11, 12, 13, 14, and 17 of a stretched spectrum. The curly
bracketed { } notes appear later in this passage. The "fundamental" pitch of this stretched
spectrum is C-sharp.
Notes 1, 2, 4, and 8 form non-octaves (1200+50 cents each)
Notes in parentheses ( ) appear later (figure 4 in the score).
Notes in curly brackets { } are used later, partly in transpositions of the mode
Sound example 3: Gérard Grisey - the first two chords from Vortex temporum, with the
"fundamental". Example includes the chord shifting at index 10
We hear a semi-tonal harmonicity, but without a fundamental since Grisey begins Vortex
temporum with the fifth tone of his series. This structure is extremely ambiguous. We may even
perceive a compressed spectrum from the F-sharp to its neutral third, the low A-sharp, then to the
C instead of the fifth C-sharp, progressing on to the low D-sharp as the seventh instead of E, and
ending on F instead of the octave F-sharp.
If we assume this is a stretched spectrum, we recognize the basic principle of the stretched octave
1200+50 cents in the lower register with a very pragmatic adjustment of the partials in between
based on quartertone steps.
The following chart shows:
1st row: Number of the partials within a just spectrum
2nd row: Deviation in cents between just spectrum and ET (cents rounded)
3rd row: Distance in cents between neighboring spectral pitches within a just spectrum
4th row: Distance in cents between pitches in Grisey's spectrum in the score (beginning of Vortex
temporum)
5th row: Tendency of Grisey's distorted spectrum as compared to a just spectrum
1
2
3
4
+2
5
6
7
-14
+2
-31
8
9
10
11
12
13
14
15
+4
-14
-49
+2
+40
-31
-12
1200 702
498
386
316
267
231
204
182
165
151
139
128
119
112
1250 750
500
400
350
250
250
250
150
200
150
150
150
100
150
+
=
+
+
- !!
+
+
- !!
+
=
+
+
- !!
+
+
16
10
On three occasions in the construction of his spectrum, Grisey does not stretch the intervals
between the neighboring spectral pitches but actually even compresses them. I have indicated this
in the table with "!!".
At figure 10 in Vortex temporum, Grisey changes the fundamental and accompanying spectrum for
the first time to a stretched G, a tritone transposition of the stretched C-sharp that we saw before.
In this instance Grisey starts with the 9th tone of the new spectrum. As listeners we perceive a
similar field of pitches that appear to be merely differently illuminated. An important element that
leads to this consists of the quartertone distances between F and low F-sharp, high F-sharp and G,
B-flat and low A-sharp, C-sharp and high C-sharp, and high D-sharp and D-sharp.
Grisey plays with our tendency to combine many individual elements together to hear wholes
rather than the individual parts; in this case, the combined wholes are overtone structures. He
liked to use the term "threshold" (limen; écriture liminale). Meloharmonic perception of sound is
just one aspect of threshold listening. Pulsation, or even formal processes are also components of
Grisey's focus on liminal composition. The whole of Vortex temporum is like one instrument that we
must discover. For example, we can think of the first section of the piece as one big accelerando,
formed by the shortening of structures that resemble the decay of bell-like sounds. The longest of
theses decays lasts 65 sixteenth notes; by the end, the shortest is only 3 sixteenth notes long. The
shortening occurs according to a segment of the Fibonacci series: 3, 5, 8, and 13.
The following graph shows the duration of the strokes of "bells", starting from lenghts of 64
sixteenth notes (each time shortened by 3), at the sixth "stroke" of 65 (shortened by 5), at the
20th stroke of 40 (shortened by 8), at the 31st of 39 sixteenth notes (shortened by 13). The four
layers are intermingled:
Another aspect of hybridization in this work (in this case, a hybridization of old and new), is a
reference to Maurice Ravel's Daphnis et Chloë that can easily be recognized at the beginning of
Vortex temporum:
11
This oscillating structure is present throughout the whole piece and provides a starting point for
other aspects of pitch, pulsation, and form. In its initial distortion of Ravel, this structure serves as
the playground for other extensive distortional thinking in the piece.
The question now arises as to whether a synthesis of these ideas can be achieved. Perhaps the
ideas themselves are not so distant from one another. Could we bring Partch, Carrillo/Gonneville,
and Grisey together, assuming these names to be prototypes for 1) ratio-based intervals, 2) equal
temperament, and 3) complex spectral constructions?
Section II:
My approach to hybridization: Partch Harp and Partota 8
I would like to describe some aspects of hybridization within a few microtonal works of my own
that have close links to all three of the previously discussed positions of just intonation, equal
temperament, and spectral distortion.
In the early 1980s I began to develop a special harp tuning for my chamber opera Der Untergang
des Hauses Usher (The Fall of the House of Usher, based on the short story by Edgar Allen Poe).
Not too long before beginning work on that piece I had studied with Ben Johnston in Urbana,
Illinois, examining Partch's just intonation and Johnston's own extension of Partch's microtonal
approach. My harp tuning uses the just interval of 5/4 (close to a major third) and the just interval
of 7/4 (close to a minor seventh). It uses no 3/2 intervals (just fifths). I used this tuning again in
later pieces, for example in the duo Partch Harp for harp and synthesizer (1987):
Sound Example 4: Manfred Stahnke - Partch Harp, Otonality and Utonality chords
The octaves are tuned normally in this piece. With the special harp pedal mechanism, I was able to
use transpositions of these chords at distances of 100 cents, such as B--E-flat--A, E-flat--G--D-flat,
or G--C-flat--F. "Utonalities" (inversions of the previous "Otonality" chords) were also possible: D-A-sharp--E, F-sharp--D--G-sharp, with transpositions.
When we consider the 21 possible pitches per octave (7x3, since the harp has 7 strings per octave,
and 3 pedal positions), the interval distances are quite irregular, as is shown in the next example.
The distances between the possible pitches are indicated in cents above the notes, which are
notated in ascending order. The cent deviations from ET are listed below the notes:
12
The synthesizer in Partch Harp uses the basic notes of the harp, organized in three superimposed
chords with the intervals 5/4 and 7/4, as seen in the first Partch Harp example. I saw that I could
easily use the 12 pitches per octave on the keyboard: Each step would have to be a little smaller
than 100 cents if I wanted to arrive at 5/4 (=386c instead of the tempered 400c) after 4 steps. I
was then able to arrive at a 7/4 after 10 steps. Of course, I would no longer arrive at a proper
octave after 12 steps. The synthesizer in Partch Harp thus has a very specific non-octaving
temperament: 12th root of 1.9560685 (=1.057504), which produces a 5/4 and a 7/4 up or down
from any note with deviations smaller than 1 cent.
The basic mathematical idea behind my approach lies in the use of the 4th root of 5/4
(=1.0573713), which is very close to 10th root of 7/4 (=1.0575571). I used the average value of
these two numbers to blend chains of 5/4 and 7/4, resulting in my non-octave of 1.9560685/1
(instead of 2/1). All major thirds are just 5/4 ratios, and all minor sevenths are just 7/4 (the tiny
deviation of 0.8c is negligible, since we cannot perceive it in any practical circumstances).
As can be seen from my graph, a side effect of my approach is the near 11/8 ratio (with a
deviation +9c) in pitch material of both the synthesizer and the harp. Partch Harp makes use of the
many just intervals and many nearly just intervals, as well as of "wrong" ones. The following
example is the beginning of the second movement of Partch Harp, which is a kind of blending of
Renaissance oriented chanson and jazz oriented song:
13
Sound Example 5: Stahnke - Partch Harp for Harp in scordatura and Synthesizer, detail
Partch Harp uses a kind of shimmering, watery set of pitches. The 21 pitches of the harp, which
still remain within an octaving environment, do not match the 12 pitches of the synthesizer, with
its non-octaving ambience. The meloharmonic situation changes with every (non-)octave.
To conclude this essay on hybrid forms of microtonal thinking, I would like to offer a final example
of my own work, Partota 8 for Disklavier and computer. Partota 8 served to continue and expand
14
my view on microtonality and leads to an extreme non-scale oriented meloharmony. All local
harmonies are calculated by the computer according to specific preset chord structures and the
sounds are generated by a sampler.
Partota 8 uses no definite set of pitches. Instead, the chord played on the Disklavier is analyzed
according to the so-called "cubic difference tone" perceived by the ear upon hearing a narrow
interval. As a basic example, we could imagine a primary interval, such as a just major third (a 5/4
ratio in just intonation), and a secondary pitch "K" that arises according to the cubic difference
tone generated by the formula K=2f2-f1 (where f1 is the higher pitch): 2x4-5 results in tone 3 in the
example below. I call this tone "K". The lowest tone, "Q", is the "quadratic difference tone" (Q=f1f2, result 5-4=1).
(In these examples, I use accidentals with arrows for sixth-tones, and "+" or "-" for twelfth-tones
to approximate the actual pitches of the just intervals.)
In reality, the cubic difference tone is a cascade of virtual pitches. In my next example, I consider
9/8 to be a primary interval. The secondary cubic tones are given to the right: 7, 6, 5, and 4 above
the fundamental A. The farther away from the primary tones these secondary, virtual pitches are,
the more quietly they will sound.
With more complex primary intervals, such as 19/16, the results will be values such as 13, 10, 7,
etc. I calculated 65 of these chords for Partota 8. Some of them sound very similar to familiar
chords; some of them sound very strange. The three following examples are chord passages from
Partota 8 in which I morph from one chord to the next in microsteps. Again, I use accidentals with
arrows for sixth-tones, and "+" or "-" for twelfth-tones. The boxes at the very right of the
examples indicate the chord that is actually physically played. The bracketed tones on the left are
the pitches calculated by the computer.
The first chord has as its fundamental an E that is a sixth-tone high. We could imagine the partials
14 and 13 from this high E as the primary pitches. The cascade of the cubic difference tone would
then give rise to the partials 12, 11, etc.
The second chord uses the partials 37 and 34 over a fundamental of B-natural ("H" in the German
nomenclature). The cubic cascade then results in 31, 28, 25 etc. (the distance between the partials
here is 3).
The third chord is derived in the same manner described for the first two.
15
Sound example 6: Morphing of three chords from Stahnke - Partota 8
Since Partota 8 is to be played on a normally tuned, computerized piano in ET, we do not have just
fifths or multiples of that interval. Instead I use tempered forms of the fifth. I chose the interval of
16.7 cents (from 72nd root of 2 in equal temperament) as the smallest common step for the
pitches that are calculated by the computer and produced by a sampler. Partota 8 is an example of
a microtonal concept, mixing concepts of just intonation with practical elements of equal
temperament. It utilizes harmony based on simple, nearly just intonation (as appears in my third
chord) as one pole in a rich field of meloharmony, with seemingly distorted harmony at the other
pole (as seen in my second chord). We must always be aware of the fact that the listener will
recalculate all these chords, as well as the melodic structures arising from them, according to his or
her previous knowledge or preferences. The interesting fact is that we only perceive such
microtonal differences in the context of changes over time. We need the individual, specific
differences within the detail in order to perceive the whole. That is the essence of meloharmony.
References
Julián Carrillo:
Carrillo, Julián s.a. Musical composition in third, quarter, eighth and sixteenth-tones: Musical
critic's comments, México: Sonido 13, 39 pp.
Carrillo, Julián ed. 1927. The Thirteenth Sound: The Herald of America's Musical Culture. New York.
(Revista mensual bilingüe español-inglés/Monthly Bilingual Spanish-English Magazine)
Michel Gonneville:
Gonneville, Michel 1997. "Chute/Parachute" pour piano et bande magnetique, et, "PetitTchaikovski", drame de musique et de theatre: Deux moments d'une demarche, Université de
Montréal (Canada), 504 pp.; Dissertation, ISBN 9780612653122
Gérard Grisey:
16
Stahnke. Manfred 1999. Die Schwelle des Hörens: "Liminales" Denken in "Vortex temporum" von
Gérard Grisey. In: ÖMZ 6/1999, 21-30
with modifications also in:
——— 2001. Zwei Blumen der reinen Stimmung im 20. Jahrhundert: Harry Partch und Gérard
Grisey. In: Komposition als Kommunikation. Festschr. P. Petersen, ed. Constantin Floros et al.,
Peter Lang-Verlag, Frankfurt/M, 369-390
Hervé, Jean-Luc 2001. Dans le vertige de la durée. Vortex Temporum de Gérard Grisey.
L'Harmattan, Collection L'Itineraire, Paris, ISBN-10: 2-7475-1275-4, 68 pp.
More on "Vortex temporum":
http://www.beckmesser.de/komponisten/grisey/periodes-vortex.html
Interview 1996 on spectralism:
http://www.angelfire.com/music2/davidbundler/grisey.html
Harry Partch:
Partch, Harry 1974. Genesis of a Music, New York: Da Capo Press. ISBN 0-306-80106-X
Partch, Harry 1991. Bitter Music: Collected Journals, Essays, Introductions and Librettos,
Champaign: University of Illinois Press.
Stahnke, Manfred 1981/82. Gedanken zu Harry Partch. In: Neuland. Ansätze zur Musik der
Gegenwart, Jahrbuch Band 2, ed. Herbert Henck, Bergisch Gladbach, 243-251
Blackburn, Philip 1998. Harry Partch: Enclosure III, Saint Paul: Innova. ISBN 0-9656569-0-X[8]
Gilmore, Bob 1998. Harry Partch, A Biography, New Haven: Yale University Press
Stahnke, Manfred 2001. Partchenogenese - wie ein Komponist sich selbst erfand. Hörstück für den
WDR. In: Stahnke, Manfred 1998ff. Den Ton finden - Schriften zur Musik. Hamburg, section II, 5465
Manfred Stahnke:
Books and essays by Manfred Stahnke on the topic of this paper:
Stahnke, Manfred 1998ff. Den Ton finden - Schriften zur Musik. Hamburg, 284 pp., Stahnke-Verlag
http://www.manfred-stahnke.de
(The following articles are found together in this book)
——— 1981/82. Gedanken zu Harry Partch. In: Neuland. Ansätze zur Musik der Gegenwart,
Jahrbuch Band 2, ed. Herbert Henck, Bergisch Gladbach, 243-251
——— 1988. Neue Skalen. Ein Essay über neue Denkweisen um György Ligeti in Hamburg und
außerhalb. In: Für Ligeti. Die Referate des Ligeti-Kongresses 1988, ed. Constantin Floros et al.,
Laaber-Verlag, Hamburg 1991, 259-276
——— 1990. Partch Harp. (Er)Findung einer nicht-oktavierenden Musik. In: Musikkulturgeschichte.
Festschrift für Constantin Floros, ed. Peter Petersen, Breitkopf & Härtel, Wiesbaden, 11-26
——— 1991. Ordnung und "Chaos" in komplexen musikalischen Stimmsystemen auf der Basis
naturreiner Intervalle. In: Studien zur Wertungsforschung Band 23: Musikalische Gestaltung im
Spannungsfeld von Chaos und Ordnung, ed. Otto Kolleritsch, Universal Edition, Wien-Graz, 136148
——— 1998. Wanderung voller Abenteuer. Zu meinem Orchesterstück "Trace des sorciers". In:
Das Andere, ed. Kreutziger-Herr, Annette, Peter Lang-Verlag, Frankfurt/M., 397-411. Else in:
Musik-Texte Heft 80, 8/1999, 47-53
——— 1999. Die Schwelle des Hörens: "Liminales" Denken in "Vortex temporum" von Gérard
Grisey. In: ÖMZ 6/1999, 21-30
——— 2000. Mein Weg zu Mikrotönen. In: Musik als Lebensprogramm. Festschr. für C. Floros, ed.
Gottfried Krieger and Matthias Spindler, Peter Lang-Verlag, Frankfurt/M, 249-263
17
——— 2001. Zwei Blumen der reinen Stimmung im 20. Jahrhundert: Harry Partch und Gérard
Grisey. In: Komposition als Kommunikation. Festschr. P. Petersen, ed. Constantin Floros et al.,
Peter Lang-Verlag, Frankfurt/M, 369-390
——— 2001a. Partchenogenese - wie ein Komponist sich selbst erfand. Hörstück für den WDR. In:
Stahnke, Manfred 1998ff. Den Ton finden - Schriften zur Musik. Hamburg, section II, 54-65
Other publications on the topic of this paper that are not included in: Stahnke, Manfred 1998ff. Den
Ton finden - Schriften zur Musik. Hamburg:
——— 2001b: Infinite Meloharmonik. Vermutungen über den Wind. In: Neue Musik 2000. Fünf
Texte von Komponisten, ed. Stahmer, Klaus H., Würzburg, ISBN 3-8260-2056-1. CD with Stahnke,
Manfred: "Trace des sorciers" (SWR-Sinfonieorchester, Olav Henzold)
——— 2005. MeloHarmonik. In: Mikrotöne und mehr. Auf György Ligetis Hamburger Pfaden, ed.
Stahnke, Manfred. ISBN 3-932696-62-X; 392 pp., CD with sound examples. Hamburg, 207-224
——— 2005b: (Dis-)Harmonie, in: Science and Music. Leopoldina Symposium 2005, Nova Acta
Leopoldina NF 92, Nr. 341, Wien, 49-61
Scores:
Gonneville, Michel. Naturel tempéré.
Private publishing Gonneville, Michel <[email protected]>
Grisey, Gérard. Vortex temporum. Ricordi NR 13944400 http://www.ricordi.it
Stahnke, Manfred. Partch Harp. Stahnke-Verlag http://www.manfred-stahnke.de
Stahnke, Manfred. Partota 8. Stahnke-Verlag http://www.manfred-stahnke.de