Yahoo Tuning Groups Ultimate Backup tuning Re:Two questions

Hi Robert,

For yet one or two weeks, I will not have time to write. So I post
immediately what is already written : the answer at the first question.

I will try to show why the D(X) function permits to find the degree of each
interval in a scale. I neglected, in part I, to distinguish the D(X)
function and the D(2;X) function using octave modularity.

In a future post then I will comment about general methods to obtain, in
our diatonic example, the trivalent set. In dimension N, one of them uses a
matrix which is a N + 1 root of the identity and the other uses what I name
the matrix S which permits to specify the same trivalent set but also, in
our diatonic example, a coherent quadrivalent set.

----------
QUESTION 1

I recall our diatonic example in <2,3,5> with D(X) = 7x + 11y + 16z and the
two statements : "harmonic-3 corresponds to degree 4" and "harmonic-5
corresponds to degree 2" (in the octave). I give first some details about
the "how", after what I will comment about the "why".

Let us calculate the value D(X) of the harmonic-3 image in all octaves :

{... 3/8, 3/4, 3/2, 3/1, 6/1, ...} --> {... -10, -3, 4, 11, 18, ...}

The first set corresponds to the harmonic-3 class in octave modularity and
the second set to the class 4 in Z/7 modularity. We have to see why it is
far from tricky to write

D(X mod 2) = D(X) mod 7

D(3 mod 2) = 11 mod 7 = 4
D(5 mod 2) = 16 mod 7 = 2

whose formalism has to be also explicitated.

---

Beyond the geometric interpretation, let us look at the properties of the
D(X) function. We know abstractly (Cf. Hellegouarch) that

G = <2,3,5> is an infinite abelian group

R = <81/80, 25/24> = <r,s> is an infinite subgroup of G

G/R is an infinite class group we name quotient-group.

Let us see gradually the properties implied.

a) The D(X) = 7x + 11y + 16z function applying G on Z specifies
the classes. Indeed, two intervals X and Y in G are in
the same class modulo R if D(X) = D(Y). The partition of G
simply follows from this relation of equivalence and the set
of its classes is designated by the notation G/R.

e.g. The intervals 8/5 == (3,0,-1) and 5/3 == (0,-1,1) are in
the same class for D(3,0,-1) = D(0,-1,1) = 5.

b) The D(X) function permits to label each class. There is an
infinite number of classes, each one containing an infinite
number of intervals. There is a bijection G/R --> Z between
the classes G/R and the numbers Z. For each number in Z,
e.g. 12723, we could specify generically the corresponding
intervals.

c) There is much more about D(X). A relation of equivalence is a
congruence if the algebraic structure is preserved. What
corresponds to the property (where * means coordinates adding
or interval multiplying)

D(X * Y) = D(X) + D(Y)

which indicates that the following diagram is commutative

G x G -------> Z x Z
| |
| * + |
| |
G -----------> Z
D

or (D)(*) = (+)(D x D).

Indeed, applying to (X,Y) successively we have

(D)(*)(X,Y) = D(X * Y)

(+)(D x D)(X,Y) = (+)(D(X),D(Y)) = D(X) + D(Y)

We say then that D is an abelian group homomorphism G --> Z.
The composition (table) of the numbers in Z, as images of the
intervals in G, is superposable on the composition (table) of
the intervals in G.

As illustration, the degrees of 5/32 and 12/5 are respectively
-19 and 9, and the degree of 3/8 is -10. Since 3/8 = (5/32)*(12/5)
the composition of the intervals is reflected in the composition
of the corresponding degrees : -10 = -19 + 9.

d) We name isomorphism an homomorphic bijection and epimorphism an
homomorphic surjection. Thus the surjection D is defined as an
epimorphism G --> Z which is composed

G ------> G/R ------> Z
D' D"

of the canonical epimorphism D' of intervals on classes and the
isomorphism D" of classes on numbers. We note (D")(D') the
composition of the applications meaning

(D")(D')(X) = D"(D'(X)) = D(X)

---

Now let us see how the equivalence modulo <2> or <3> or <5> may be
manipulated.

e) In our notation, <p> is the group of all intervals of the form
p^k where k is in Z. The quotient-group G/<2> or G/<3> or G/<5>
of the group G by the subgroup <2> or <3> or <5> is the group
of intervals modulo <2> or <3> or <5>.

We have to distinguish here the class group which contains an
infinite number of intervals in each class and a representing
system of these classes which corresponds to the choice of one
and only one interval for each class in such manner that this
chosen set be a group (with a modular operation) isomorph to
the class group.

The group <3,5> is a good system representing G/<2>. And so the
group <2,5> may represent adequately G/<3> like the group <2,3>
may be used to represent G/<5>.

However, we use generally the intersection of G with the first
octave [1,2[ as system representing G/<2> since it contains only
one interval in each class of G/<2> and it constitutes (with the
composition law modified for modular reentrance) a group isomorph
to G/<2>.

Naturally we could apply similar comments to G/<3> and G/<5>.

f) To keep clarity, we have to restrict the distinctions. So I note
simply

G/2 : G/<2> = <2,3,5>/<2>
or its representing groups (in octave or <3,5>)

G/2R : (G/2)/R = (G/R)/2 = G/<2,r,s>
or its representing groups

Z/n : Z/[n] = Z/nZ (further explanations below)

We could distinguish the class group and its representing group
in the first octave in adding the homomorphism G/<2> --> G/2.

With this simplification we obtain this commutative diagram of
all retained homomorphism permitting to manipulate as easily
as rigourously the formalism :

G x G ---------> G/R x G/R ---------> Z x Z
|\ |\ |\
| \ | \ | \
| \ | \ | \
| \ | \ | \
| \ | \ | \
| G/2 x G/2 ------> G/2R x G/2R ------> Z/7 x Z/7
| | | | | |
| | | | | |
G ----|---------> G/R ---|----------> Z |
\ | \ | \ |
\ | \ | \ |
\ ! \ | \ |
\ | \ | \ |
\| \| \|
G/2 -------------> G/2R ------------> Z/7

g) To complete the formalism we have to define some "modulo
equivalence" begining with the multiplicative and additive ones.

"x = y mod <n>" = "there exist k in Z such that x = y * n^k"

"x = y mod [n]" = "there exist k in Z such that x = y + n*k"

The notation [n] indicate here the ideal nZ which is a subgroup
of Z. When the context permits to distinguish them I use only
the form "x = y mod n".

e.g. (5/3 = 20/3 = 5/48) mod 2 (meaning mod <2>)

(18 = -10 = 4 = 11) mod 7 (meaning mod [7])

A such formalism may be used with all form of equivalence or
congruence relation. With X and Y in the group G and the
subgroup R = <r,s>, the symbol R indicate also a congruence :

"X = Y mod R" = "there exist K in R such that X = Y * K"

e.g. Using our diatonic example, with X and Y in G = <2,3,5>,
R = <r,s> = <81/80, 25/24>, and 2R = <2,r,s> we may write

(9/8 = 16/15 = 256/243) mod <r,s> (or mod R)

(9/64 = 1/30 = 8/243) mod <2,r,s> (or mod 2R)

(4,-1,-1) = (8,-5,0) mod R

By anticipation, I note also that I will use the content of a
matrix, which I call the S matrix, whose elements correspond to
a centered convex body in G to define the following congruence

"X = Y mod S" = "there exist K in S such that X = Y * K"

h) Now we can define uniformly the canonical epimorphism (mod m)
applying a group G on the classes modulo m noted G/m. For the
sake of clarity I may use the same symbol, like "mod 2", with
different definition domains when the effect is similar.

mod 2 : G --> G/2 : X :--> X mod 2

mod 2 : G/R --> G/2R : X :--> X mod 2

mod R : G --> G/R : X :--> X mod R

mod 2R : G --> G/2R : X :--> X mod 2R

mod 7 : Z --> Z/7 : x :--> x mod 7

So we see that mod R correspond to D' in the decomposition of
D = (D")(D') = (D")(mod R).

---

i) Finally we can simply say that the following relation

D(X mod 2) = D(X) mod 7

corresponds to the fact that (D), (mod 2) and (mod 7) are
homomorphisms, and then the following diagram is commutative

D
G ---------> Z
| |
mod 2 | | mod 7
| |
G/2 -------> Z/7
D

(mod 7)(D) = (D)(mod 2)

For applying successively to X we have

(mod 7)(D)(X) = (mod 7)(D(X)) = D(X) mod 7

(D)(mod 2)(X) = D(X mod 2)

Thus it is why we can use so simply

D(X mod 2) = D(X) mod 7

j) Ultimately let us define the D(X) function applied to three
modularities, leaving only the formalism to talk.

D(2;X) : G --> G/2R --> Z/7
D(3;X) : G --> G/3R --> Z/11
D(5;X) : G --> G/5R --> Z/16

D(2;X) = D(X mod 2) = D(X) mod 7
D(3;X) = D(X mod 3) = D(X) mod 11
D(5;X) = D(X mod 5) = D(X) mod 16

D(2;X) = (7x + 11y + 16z) mod 7 = (4y + 2z) mod 7
D(3;X) = (7x + 11y + 16z) mod 11 = (7x + 5z) mod 11
D(5;X) = (7x + 11y + 16z) mod 16 = (7x + 11y) mod 16

Regards,

Pierre Lamothe

🔗Haresh BAKSHI <hareshbakshi@hotmail.com>

🔗genewardsmith@juno.com

🔗BobWendell@technet-inc.com

🔗monz <joemonz@yahoo.com>

> From: <BobWendell@technet-inc.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, October 10, 2001 9:10 AM
> Subject: [tuning] Re: Two questions
>
>
> They were all formidable improvisers.
> <snip>
> Early in his teens, Beethoven impressed Haydn and others as a
> prodigious improviser.
>
>
> --- In tuning@y..., "Haresh BAKSHI" <hareshbakshi@h...> wrote:
>
> > Hello everyone, I have two queries:
> >
> > [1] Do we know whether Bach, Mozart, or Beethoven ... any composer
> > did improvisation?

There are incredible stories of Beethoven's improvisatory feats.

For example, one time someone (I think it was a cellist) gave him
a part from which to improvise, and Beethoven supposedly put it
on the music stand upside-down and proceeded to elaborate a
magnificent improvisation from it.

There are numerous stories of his listeners in tears after
listening to him improvise.

> [Haresh again]
>
> > [2] Can we not call written scores in Western music, "frozen"
> > improvisation?
>
> From: <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, October 10, 2001 12:39 AM
> Subject: [tuning] Re: Two questions
>
> Not really, but a cadenza written out might qualify, for instance.

I would beg to differ with you on this, Gene, in the case of Beethoven
... and Harry Partch, too.

It sounds to me like many of Beethoven's innovative musical ideas stem
from improvisations, particularly in his piano sonatas, and most
particularly in the later ones.

I attended a recital by a pianist (don't remember his name) at
The Curtis Institute of Music (in Philadelphia) a couple of years
ago, which featured a performance of Beethoven's 4th Piano Concerto,
with the recitalist on the solo part and a faculty member or grad
student playing the orchestral part on another piano. The soloist
was magnificent, and I was mesmerized by his performance, as at
many points during the solo sections it seemed that I was watching
Beethoven himself make up the music as he played.

I've had similar experiences watching "our own" John Schneider
perform his reconstruction of the original version of Partch's
_Barstow_ with only voice and solo guitar. John has become
so comfortable with this piece that when I watch him perform
it, I swear it seems like I'm watching Partch himself walk along
the highway railing and improvise the music as he reads the grafitti.

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

🔗Paul Erlich <paul@stretch-music.com>

🔗mschulter <MSCHULTER@VALUE.NET>

Dear Haresh:

Thank you for your two questions on improvisation in Western European
music, both very important, and often having rather complicated
answers. These answers may vary depending on the historical period,
and also the style of music and the place.

First of all, it is easy to reply that a musician such as Bach was
indeed famed for his keyboard improvisations as well as his
compositions, and excelled at this form of performance.

Your second question raises two important issues: to what extent might
a composition be described as an "improvisation," and to what extent
might it be described as "frozen." These answers can vary depending on
style and period.

Certain compositions for solo instruments in the 16th century such as
lute or keyboard, for example, do have the quality of "frozen
improvisations," sometimes suggesting how a player may have
interpreted a known vocal composition for four voices, say, on which
the piece is based.

These keyboard pieces seem to me very much to fit your idea of "frozen
improvisations." They record, in a way like a CD or tape recorder, how
one musician may have played a given piece originally written for
voices, including things such as the choices of ornaments and the
accidental inflections raising or lowering certain notes by a
semitone.

This last point is very important. In much medieval and Renaissance
European music through the 16th century, many choices about
accidentals (sharps or flats) were left to the performers. Thus one
instrumental version of a given vocal piece might in a certain place
favor E and C; another Eb and C; and a third E and C#.

The same instrumental version might make different choices of
accidentals when a passage is repeated, showing how instrumentalists
and singers might have done this in unrecorded performances of the
period.

Thus a given piece of this kind is simply a "frozen" version or
recording of one interpretation; a different musician, or the same
musician performing the same kind of interpretation again, might make
different choices.

Some other kinds of keyboard pieces also suggest an improvisatory
approach: the 16th-century _toccata_, or "touch piece," might suggest
this style.

Also, there is not necessarily a sharp distinction in medieval and
Renaissance theory between "improvising" and "composing." The art of
_discant_, for example, might consist either in singing an improvised
melody against another given melody, or in composing in this manner.

Likewise, the art of _fantasia_ discussed by Tomas de Santa Maria in
his treatise of 1565 can mean either improvisation of composition of
music on a keyboard or other instrument such as a lute or harp capable
of playing several voices at once. Many of the observations in this
book about four-voice writing, for example, could apply to either kind
of music.

Here we run into other complication: a composer may be influenced by
the improvisations of other people, sometimes in ways of which we
cannot be sure.

For example, the scholar Claude Palisca has suggested that the
composer Claudio Monteverdi, sometime a bit before 1600, may have
heard singers perform a style of improvised counterpoint where
dissonances of a kind not used in conventional written compositions
would occur. Palisca suggests that Monteverdi could have heard this,
liked the effect, and made it a part of his new style of composed
vocal music.

Also, a composer may "find" some new or pleasing progression on a
keyboard, but then make this progression part of a style for
compositions written without any further reference to a keyboard
instrument.

Sometimes to the degree to which a given musician or community of
musicians can "internalize" a given style or tuning system is an open
question. For example, among the circle of musicians following the
"enharmonic" style of Nicola Vicentino with its division of the octave
into 31 equal or near-equal parts, could some of them compose in this
style without reference to a keyboard like Vicentino's? We are told
that singers tended to depend on such a keyboard to find the new types
of intervals; could some composers write enharmonic music confidently
without such an instrument, as they might write conventional
counterpoint?

Interestingly, the first recorded European "compositions" for more
than one voice, around the late 9th century, could be described as
examples to guide people improvising this kind of music. They may also
record how improvisers did things like starting at a unison, moving to
a 4:3, singing in 4:3 concords, and then returning to a unison.

By around the year 1000, it seems that the idea of composition, of
writing down a specific two-voice piece, was taking shape. By the 13th
century, at least, we find treatises with specific advice on how to
compose in various forms.

What I would want to emphasize is that in medieval and Renaissance
times, much in a composition is left to the performers, or at least
not specified on paper: many of the likely accidentals, ornaments of
various kinds, and the choice of voices or instruments for many
ensemble pieces.

Sometimes composers have sought to exert some guidance on this
process. In the 14th century, Guillaume de Machaut urged that one of
his part-songs be performed "neither adding nor taking anything away";
around 1500, Josquin des Prez is reported to have protested when a
singer ornamented one of his pieces in a manner he found untasteful,
suggesting that the singer should write compositions of his own if he
wanted an effect of this kind.

This is only one possible answer to some parts of your question, but I
hope it may give some background and invite further discussion.

Most appreciatively,

Margo Schulter
mschulter@value.net

🔗genewardsmith@juno.com

🔗Jon Szanto <JSZANTO@ADNC.COM>

🔗Jay Williams <jaywill@tscnet.com>

>> --- In tuning@y..., "Haresh BAKSHI" <hareshbakshi@h...> wrote:
>>
>> > [2] Can we not call written scores in Western music, "frozen"
>> > improvisation?
In some cases, most definitely yes. The famous 10th century organist,
Marcel Dupre (1886o-1971) was a stormer of an improviser and manhy of his
compositions such as the "Stations of the Cross" are recreations from
memory of his improvisations. In the worl,d of organists the art of
improvisation is very much alive and well. Got to
http://www.pipedreams.org and follow links to find cd's of such as Naji
Hakkim. This is as ear-stretching music as a pipe organ will allow.
And, if you're ever touristing in Salt Lake City, stop by the Tabernacle at
noon on any weekday. The organist must include as part of the recital, an
improv combining "Come, Come ye Saints" with any other tune (secular or
sacred) of his/her choice. These folks are pretty stellar performers!
>> To: <tuning@yahoogroups.com>
>> Sent: Wednesday, October 10, 2001 12:39 AM
>> Subject: [tuning] Re: Two questions
>>
>> Not really, but a cadenza written out might qualify, for instance.
>
>
>I would beg to differ with you on this, Gene, in the case of Beethoven
>... and Harry Partch, too.
>
>It sounds to me like many of Beethoven's innovative musical ideas stem
>from improvisations, particularly in his piano sonatas, and most
>particularly in the later ones.
>
>I attended a recital by a pianist (don't remember his name) at
>The Curtis Institute of Music (in Philadelphia) a couple of years
>ago, which featured a performance of Beethoven's 4th Piano Concerto,
>with the recitalist on the solo part and a faculty member or grad
>student playing the orchestral part on another piano. The soloist
>was magnificent, and I was mesmerized by his performance, as at
>many points during the solo sections it seemed that I was watching
>Beethoven himself make up the music as he played.
>
>I've had similar experiences watching "our own" John Schneider
>perform his reconstruction of the original version of Partch's
>_Barstow_ with only voice and solo guitar. John has become
>so comfortable with this piece that when I watch him perform
>it, I swear it seems like I'm watching Partch himself walk along
>the highway railing and improvise the music as he reads the grafitti.
>
>
>
>love / peace / harmony ...
>
>-monz
>http://www.monz.org
>"All roads lead to n^0"
>
>
>
>
>
>_________________________________________________________
>Do You Yahoo!?
>Get your free @yahoo.com address at http://mail.yahoo.com
>
>
>
>You do not need web access to participate. You may subscribe through
>email. Send an empty email to one of these addresses:
> tuning-subscribe@yahoogroups.com - join the tuning group.
> tuning-unsubscribe@yahoogroups.com - unsubscribe from the tuning group.
> tuning-nomail@yahoogroups.com - put your email message delivery on hold
for the tuning group.
> tuning-digest@yahoogroups.com - change your subscription to daily digest
mode.
> tuning-normal@yahoogroups.com - change your subscription to individual
emails.
> tuning-help@yahoogroups.com - receive general help information.
>
>
>Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/
>
>
>
>

Re:Two questions

🔗Robert Walker <robert_walker@rcwalker.freeserve.co.uk>

🔗Pierre Lamothe <plamothe@aei.ca>

🔗Haresh BAKSHI <hareshbakshi@hotmail.com>

🔗genewardsmith@juno.com

🔗BobWendell@technet-inc.com

🔗monz <joemonz@yahoo.com>

🔗Paul Erlich <paul@stretch-music.com>

🔗mschulter <MSCHULTER@VALUE.NET>

🔗genewardsmith@juno.com

🔗genewardsmith@juno.com

🔗Jon Szanto <JSZANTO@ADNC.COM>

🔗Jay Williams <jaywill@tscnet.com>