<html><head></head><body style="word-wrap: break-word; -webkit-nbsp-mode: space; -webkit-line-break: after-white-space; "><div>Dear Nicolas,</div><div><br class="Apple-interchange-newline"><blockquote type="cite"><div bgcolor="#FFFFFF" text="#000000">
<font face="Calibri">You may be making too much of Jacques
Handschin's ideas. </font></div></blockquote><div><br></div>Handschins had a very wide historical perspective and his ideas were driven by his sharp awareness of open theoretical problems. His act of upgrading the line of fifths to a central music-theoretical concept is the main concern of his book "Der Toncharakter". Remarkably, it starts with a departure from its "boring" role as a vehicle for harmony teachers and piano tuners. At any rate Handschin's ideas really deserve to be reviewed. It is a different question, of course, whether our algebraic approaches go in the right direction. </div><div><br><blockquote type="cite"><div bgcolor="#FFFFFF" text="#000000"><font face="Calibri">The fact is that a sharpward series of fifths
in Pythagorean intonation does raise in pitch, by one Pythagorean
comma after twelve steps.<br></font></div></blockquote><div><br></div><div>That's a sophisticated perspective. With the same line of argument one could say that five sharpward fifths lower the pitch by a semitone, and that seven sharpward fifths raise it by an augmented prime. But the pitch height direction of those "commata" is not directly concerned with the pitch height directions of the scale steps themselves. It only measures their difference. My concern about directionality is more elementary. </div><br><blockquote type="cite"><div bgcolor="#FFFFFF" text="#000000"><font face="Calibri">
In short, a mode is not a scale.<br></font></div></blockquote><div><br></div>A mode relates the perfect fifth and perfect fourth to a species of the fifth and a species of the fourth. It further relates the perfect octave to the concatenations of the species of the fifth and the fourth. In the dorian mode, for example, the species of the fourth is Y = TST (Tone, Semitone, Tone) and the species of the fifth is X = TSTT, wherein the species of the fourth is a prefix, i.e. X = YT, with T playing the role the major step as well as the role of the diazeuxis. </div><div>The point where I might be making too much <span class="Apple-style-span" style="font-family: Calibri; ">of Jacques Handschin's ideas </span>is the construction of a "species of the major step" T = Y^(-1) X and a "species of the minor step" S = T^(-1)YT^(-1) = X^(-1)YYX^(-1)Y. Handschin doesn't deliberately distinguish between an ascending fifth and a descending fourth. These concepts seem to be algebraically motivated neologisms. But is this historically true? Schenker's "ausgeworfener Grundton" is quite close to that construction and maybe other theorists have also considered the five species of the semitone?</div><div><br></div><div>Sincerely</div><div>Thomas Noll</div><div><br><blockquote type="cite"><div bgcolor="#FFFFFF" text="#000000"><font face="Calibri">
<br>
Yours,<br>
</font>Nicolas Meeùs<br>
Université Paris-Sorbonne<br>
<font face="Calibri"> <br>
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<div class="moz-cite-prefix">Le 1/08/2012 15:08, Thomas Noll a
écrit :<br>
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<blockquote cite="mid:7EAEB86B-3481-4552-A715-2F0683FA009E@cs.tu-berlin.de" type="cite">If we assume a directional markedness in the pitch
height dimension (e.g. downwards being unmarked) we might assume
an analogous markedness along the line of fifths (e.g. flatward
being unmarked). Such an assumption implies an interesting
question: How do the two kinds of markedness interrelate? Jacques
Handschin argues in favor of an affinity between ascending pitch
height and sharpward oriented fifths. That same type of affinity
would then hold between descending pitch and flatward oriented
fifths. This affinity is contrapuntally supported by the ultimate
progression between tenor and bass in the cadence (as well as in
the Ursatz). But for modal tone relations Handschin's assumption
might nevertheless be wrong. There are good mathematical reasons
to postulate that the combination between ascending pitch and
flatward oriented fifths is the unmarked one.
<div>Sincerely</div>
<div>Thomas Noll <br>
<div><br class="Apple-interchange-newline">
<blockquote type="cite">
<div bgcolor="#FFFFFF" text="#000000"> <font face="Calibri">Curt
Sachs writes, in one of his books, that descending
motion is more common, in all musics of the world, than
ascending one. He does not say, however, how common it
is to describe such motion as 'descending'. In general,
though, melodies end at the pitch at which they began:
they can but descend what they first climbed. the
'descending' effect probably results from the fact that
they go up by leaps and come down by conjunct steps.<br>
There is an interesting paper on the verticalization
of pitch in Western music by M. E. Duchez in Acta
musicologica 51/1, 1979. She indicates that the
verticalization by no means is universal and that it
appeared slowly and lately in the West (after the 9th
century). The verticalization of pitch may be the
consequence (rather than the cause) of the vertical
disposition in notation. It does not seem to have
existed in Latin (or Greek), where pitches were
described as acutus (oxus) and gravis (barus). For some
time, no clear distinction was made between pitch and
intensity ('musica alta' was loud, not high).<br>
In harmonic music, singing in just intonation tends
to shift pitch. With respect to the cycle of fifths
(Pythagorean tuning) taken as reference, the pitch
shifts down a comma for each ascending major or
descending minor third, and the reverse. Think of a
neo-Riemannian network and of the change of line
corresponding to 3d-relations: horizontal lines are a
comma apart in just intonation. Tonal harmonic
progressions tend to shift down – one of the reasons why
a capella choirs shift down: they sing too much in tune!<br>
<br>
</font>Nicolas Meeùs<br>
Université Paris-Sorbonne<br>
<br>
<br>
<font face="Calibri"><br>
</font>
<div class="moz-cite-prefix">Le 30/07/2012 03:23,
Christopher Bonds a écrit :<br>
</div>
<blockquote cite="mid:5015E223.5010504@willy.wsc.edu" type="cite"> A quick comment. Seems like success in
relating any kind of musical event to gravity depends on
the answers to a couple of questions. First, whether
descending intervals, stepwise lines, root progressions,
etc., generally always create a sense of closure or at
least a lessening of tension; and if so, are these style
and culture independent? Second, if so, could there be
other explanations for this phenomenon? Third, if some
sort of relationship could be established between the
physical law and gravity, what effect, if any, will
Einstein's general theory of relativity have on musical
perception, now or in some future time? Finally, is the
concept of "up and down" in music universal and innate,
or is it something we have learned by association?<br>
<br>
(For the record, my personal thinking is that the
musical brain has learned to associate higher and lower
pitches with up and down in space. Maybe because low
sounds are associated with heavier objects, which seem
to be tending downward more seriously than lighter
objects (although they accelerate at the same rate when
falling.))<br>
<br>
Christopher Bonds<br>
Wayne State College (retired)<br>
<blockquote class=" cite" id="mid_2ED0C71B_BA17_4B58_9832_DF5232F963BD_cs_tu_berlin_de" cite="mid:2ED0C71B-BA17-4B58-9832-DF5232F963BD@cs.tu-berlin.de" type="cite"> </blockquote>
<br>
<blockquote cite="mid:2ED0C71B-BA17-4B58-9832-DF5232F963BD@cs.tu-berlin.de" type="cite"> </blockquote>
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<div>*********************************************************</div>
<div>Thomas Noll</div>
<div><a moz-do-not-send="true" href="http://user.cs.tu-berlin.de/%7Enoll">http://user.cs.tu-berlin.de/~noll</a></div>
<div><a moz-do-not-send="true" href="mailto:noll@cs.tu-berlin.de">noll@cs.tu-berlin.de</a></div>
<div>Escola Superior de Musica de Catalunya,
Barcelona </div>
<div>Departament de Teoria i Composició </div>
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