Re : Interval vs interval

Hi Paul!

These quotes in #20863 and #20971

<< A dyad interval is composed of two tones >>

<< One needs two notes to evaluate dyadic sonance >>

<< the only conditions under which [...]
is if the music is in two voices >>

helped me to see more clearly the distance between our languages. We
separate in the same manner notes and intervals but you talk always
concretely about intervals while my concept of musical interval is much
more abstract. I could develop on that later. I will try here only to
distinguish our respective paradigms so the simple words questions could be
settled.

---------------

You wrote :

<< Pitch height doesn't have to be measured in Hertz. It can be measured in
cents, or as a ratio relative to an arbitrary tonic. It is in the latter
sense that I understand your "widths". >>

I understand better why you tend to identify your pitch heights and my
widths. The main difficulty is not here but the slight ambiguity here don't
help.

Pitch heights have strictly to be measured with a frequency standard even
if a pitch height is perfectly pointed out by dyads (width, pitch height
reference) and (ratio, pitch height reference). It makes no sense to ask
what is the cents value of the international "LA" whose measure in Hz is 440.

However I have to admit your use of the term "measure" in a weak sense
since it is a current usage in acoustics. So the term decibel is used to
"measure" (level of) intensities while strictly it is purely a log (ratio).
It is then assumed (but maybe forget by some of its users) that the decibel
value has to be refered to the standard threshold 10^-16 watt/cm^2 to
obtain a true measure.

As an help to avoid misinterpretation I would like underline here that I
don't tolerate in my works this type on ambiguities. What is admissible in
the acoustical paradigm would become a serious flaw in the relational
paradigm where all stands on definitions.

You may understand "widths" in the sense you did but you have not to
understand my use of "width" in that sense for I don't understand or use it
in that sense.

---------------

You wrote

<< If you're always measuring the interval upwards from the
tonic, then all you're doing is characterizing the pitch-height
(or pitch-class, which is pitch-height modulo 2) of a tone. >>

"Pitch-height modulo 2" is not a well-defined object. If P1 and P2 are
pitch heights, it makes no sense to write P1+P2 or P1*P2. A fortiori, it
makes no sense to write P1 or P2 modulo 2 since the modulo definition
implies a composition law. That shows why I cannot tolerate ambiguities in
relational paradigm.

Forgetting that remark you're absolutely right : if I was always measuring
the interval upwards from the tonic . . . but I am never measuring the
interval upwards from the tonic . . . (thus simply characterizing
pitch-heights).

It is an interpretation to see scales composed with tones in sense of pitch
heights where I show modes composed with "tons" in sense of intervals
modulo 2.

There exist an enormous difference between your scales and my modes. Since
the difference is the same with your chords and my chords I begin with that
to show the difference between the objects within the acoustical paradigm
and the similar objects within the relational paradigm.

In your sense, if you have 3 tones <t1 t2 t3> such that

t1:t2:t3 = 4:5:6

then you have a major chord. The ratio 4:5:6 is also synonynous with the
major quality of a chord in relational paradigm. However the existence
itself of a such chord requires much more than 3 elements having the ratio
4:5:6.

In the relational paradigm an object like a chord exist within a finite set
S of intervals having a partial composition law. If 3 elements <a b c>
exist in S such that a:b:c = 4:5:6 then <a b c> is a chord if and only if
the 7 intervals

<1 6/5 5/4 4/3 3/2 8/5 5/3>

exist in S, otherwise <a b c> is called a discordance relatively to S.

A mode is a much more complex object than a chord. That requires many
axioms while only the first axiom is implied in the chord definition.
However, as with a chord, a mode exists within a finite set of intervals S
(gammoid) and its existence as a mode in S implies much more than the
existence of the intervals figuring in the sequence. For instance, the
blues set

<1 7/6 4/3 7/5 3/2 7/4>

exists in the set generated by the chord <1 3 5 7>. However
that's absolutely not a mode in that set for some intervals
and relations implied by this mode are missing. I note that
all intervals are not required as in the chord case and that
the main relation implied is always the partition in degrees.

The blues set is a mode in the minimal gammier <3 5 7 15 21>
and both a mode and a chord in the complete <3 5 7 15 21 35>.

If you decide to use concretely gammier modes as scales it remains possible
to add slight static or dynamic corrections depending of your goals, for
there may exist constraints with fixed tuning and also a space of
possibilities outside the constitutive harmonic frame described by the
gammier. It's there something regarding musicians rather than an explorator
of the articulation between the sensible and intelligible levels in kind of
music where intervals are intensively used.

Regards,

Pierre

Pierre wrote,

>Forgetting that remark you're absolutely right : if I was always >measuring
>the interval upwards from the tonic . . . but I am never measuring >the
>interval upwards from the tonic . . .

I thought you said you were. I could find the relevant quote, but . . . do
you agree with how I understand your definition of "interval", as I
characterized with the Zarlino JI scale example?

>In your sense, if you have 3 tones <t1 t2 t3> such that

> t1:t2:t3 = 4:5:6

>then you have a major chord.

Yes.

>The ratio 4:5:6 is also synonynous with >the
>major quality of a chord in relational paradigm. However the >existence
>itself of a such chord requires much more than 3 elements having the >ratio
>4:5:6.

>In the relational paradigm an object like a chord exist within a >finite
set
>S of intervals having a partial composition law. If 3 elements <a b >c>
>exist in S such that a:b:c = 4:5:6 then <a b c> is a chord if and >only if
>the 7 intervals

> <1 6/5 5/4 4/3 3/2 8/5 5/3>

>exist in S, otherwise <a b c> is called a discordance relatively to >S.

I don't understand this. Can you give a musical example? For instance, in
the key of C major, is the G major chord a discordance?

Anyhow, thanks for taking the time to talk about your ideas -- your system
is one of the most fascinating things to ever come up on this list. And
please bear with me as I, a musician who knows a little about
psychoacoustics, try to make sense out of it and goad you toward better
explaining your ideas to others like me . . .