Yahoo Tuning Groups Ultimate Backup tuning-math notating ratios [from tuning]

>> I see no defects whatever in the definitions I've given (above).
>> Are they clear (or not) yet?
>
>No, they aren't clear. Is 3:2 always to be interpreted as either a
>dyad or else the interval involved in a dyad,

It is always interpreted as a musical interval, which includes
both melodic and harmonic intervals. I refer you to the music
dictionary of your choice.

>or can it carry the usual meaning of ratio?

It is obviously a ratio in the most basic/general sense.

>Is 4:5:6 automatically and explicitly a triad?

Almost always a chord (harmony) is meant. Sometimes a melodic
structure (such as an arpeggiated chord) is also possible.

>When you say 3:2 is a dyad, does this mean it is a particular
>element of the set of all pairs of frequecies in a 3:2 ratio,
>or does it denote all such pairs, being coextensive with the
>set of such pairs?

I'm sorry, I thought you used dyad=simultaneity, so that's
what I've been doing.

It means all such pairs. 3/2, on the other hand, is a
particular element of a set of pitches, or in some cases the
particular chord formed between 1/1 and 3/2 in that set of
pitches.

>What sort of mathematical operations are valid on 3:2--can you
>multiply them? Divide them? Add or subtract them?

You can only multiply and divide them.

>Is it possible to invert them?

Yes, and 2:3 = 3:2. But 4:3 != 3:2. Dave likes lower numbers
always first. I like bigger numbers when there are two, lower
numbers first when there are more than 2. As in, 5:4 and 4:5:6.

>What is 1:1?

That loveable interval known as the unison.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Carl Lumma" <ekin@l...> wrote:
> >> I see no defects whatever in the definitions I've given (above).
> >> Are they clear (or not) yet?
> >
> >No, they aren't clear. Is 3:2 always to be interpreted as either a
> >dyad or else the interval involved in a dyad,
>
> It is always interpreted as a musical interval, which includes
> both melodic and harmonic intervals. I refer you to the music
> dictionary of your choice.

I doubt this is true, for any dictionary, musical or otherwise. Can
you cite something to back this up?

> >or can it carry the usual meaning of ratio?
>
> It is obviously a ratio in the most basic/general sense.

Hardly! You just claimed the opposite--that it is *not* a ratio, but
rather a musical interval. Below I learn that is interpreted
extensively, as an equivalence class. I don't know what dictionaries
you have been reading, but I've never seen this.

> It means all such pairs. 3/2, on the other hand, is a
> particular element of a set of pitches, or in some cases the
> particular chord formed between 1/1 and 3/2 in that set of
> pitches.

I really hope this is not the standard meaning people have adopted,
because this is horrible.

> >What sort of mathematical operations are valid on 3:2--can you
> >multiply them? Divide them? Add or subtract them?
>
> You can only multiply and divide them.

How do you divide them, given that 2:3 = 3:2?

> >Is it possible to invert them?
>
> Yes, and 2:3 = 3:2.

That's not an inversion in the algebraic sense.

But 4:3 != 3:2. Dave likes lower numbers
> always first. I like bigger numbers when there are two, lower
> numbers first when there are more than 2. As in, 5:4 and 4:5:6.
>
> >What is 1:1?
>
> That loveable interval known as the unison.

Two notes playing in unison? A note followed by the same note? Both?

🔗Carl Lumma <ekin@lumma.org>

>> >> I see no defects whatever in the definitions I've given (above).
>> >> Are they clear (or not) yet?
>> >
>> >No, they aren't clear. Is 3:2 always to be interpreted as either a
>> >dyad or else the interval involved in a dyad,
>>
>> It is always interpreted as a musical interval, which includes
>> both melodic and harmonic intervals. I refer you to the music
>> dictionary of your choice.
>
>I doubt this is true, for any dictionary, musical or otherwise. Can
>you cite something to back this up?

They do not cover colon notation. They do cover melodic and
harmonic intervals. I thought Paul had straightened you out
on this apparent hole in your musical vocabulary.

>> >or can it carry the usual meaning of ratio?
>>
>> It is obviously a ratio in the most basic/general sense.
>
>Hardly! You just claimed the opposite--that it is *not* a ratio, but
>rather a musical interval. Below I learn that is interpreted
>extensively, as an equivalence class. I don't know what dictionaries
>you have been reading, but I've never seen this.

It's an interval, but one must resort to thinking of a ratio between
two pitches to discover which interval it is.

>> It means all such pairs. 3/2, on the other hand, is a
>> particular element of a set of pitches, or in some cases the
>> particular chord formed between 1/1 and 3/2 in that set of
>> pitches.
>
>I really hope this is not the standard meaning people have adopted,
>because this is horrible.

3:2 does mean all such pairs in colon notation. 3/2, in the
strict JI sense, is less standard because they don't (to my
knowledge) sit around on lists like this and standardize their
language. Therefore, my remarks about strict JI notation are
descriptive rather than prescriptive. In Dave's scheme, 3/2
means only a particular element from a set of pitches.

>> >What sort of mathematical operations are valid on 3:2--can you
>> >multiply them? Divide them? Add or subtract them?
>>
>> You can only multiply and divide them.
>
>How do you divide them, given that 2:3 = 3:2?

You have to pick which way you're going to write them,
and stick to it.

>> >Is it possible to invert them?
>>
>> Yes, and 2:3 = 3:2.
>
>That's not an inversion in the algebraic sense.

Ah, maybe I should look that up sometime.

>> >What is 1:1?
>>
>> That loveable interval known as the unison.
>
>Two notes playing in unison? A note followed by the same note? Both?

Harmonic and melodic unison, both.

-Carl

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

>> They do not cover colon notation. They do cover melodic and
>> harmonic intervals. I thought Paul had straightened you out
>> on this apparent hole in your musical vocabulary.
>
>Paul as I recall was trying to explain what people meant by the colon,
>not attempting to give a definition in precise terms, which I would
>want before adopting any such notation myself. Nor did he claim the
>world at large had adopted any such definition.

If definitions implied understanding, AI would work. Which one do
you want? Are you telling me you don't understand the difference
between harmony and melody, that the former is a simultaneity while
the latter is not?

>> 3:2 does mean all such pairs in colon notation.
>
>I presume you mean it's coextensive,

That one's not on mathworld or wikipedia.

>rather than literally is all such
>pairs. It would be synonymous with "JI fifth", in other words.

Yes.

You may take "3:2" to mean "702.92setuhasecuaseu cents", and
3/2 to mean "seauthsaeu Hz.", no more no less.

-Carl

🔗monz <monz@attglobal.net>

--- In tuning-math@yahoogroups.com, Carl Lumma <ekin@l...> wrote:

> >> They do not cover colon notation. They do cover melodic and
> >> harmonic intervals. I thought Paul had straightened you out
> >> on this apparent hole in your musical vocabulary.
> >
> > Paul as I recall was trying to explain what
> > people meant by the colon, not attempting to give
> > a definition in precise terms, which I would
> > want before adopting any such notation myself.
> > Nor did he claim the world at large had adopted
> > any such definition.
>
> If definitions implied understanding, AI would work.
> Which one do you want? Are you telling me you don't
> understand the difference between harmony and melody,
> that the former is a simultaneity while the latter is not?

this can be a slippery issue, and i think some of
the arguing going on right now about the colon-vs-slash
notation, can be traced to this.

whether or not it's the correct way to go about things,
it's true that music-theory almost always divorces
the pitch aspect of music from the time aspect,
presenting conglomerations of pitches in the abstract,
with time playing a role only in the ordering of events
-- i.e., a chord progression generally sets up a dissonance
which must resolve in the following chord; or there may
be a whole series of chords.

of course we know that real music does not work like this,
because there, the way a long and very complex series of
sonic events unfold in time, is an essential part of the
whole structure.

harmonic theory takes a snapshot of a certain moment of
the whole musical piece (that moment may range in size
anywhere from one pair of chords to a comparison of,say,
the opening and closing chords of the movements of a
whole symphony), and analyzes the pitch aspect in
minute detail.

my main point is that, while "harmony" generally
carries the connotation of simultaneity, it
need not necessarily be so.

that analysis which i mention above, of the movements
of a symphony, could, rather than show the actual opening
and closing chords of each movement, instead be an
abstract rendering of important harmonic events ocurring
thruout the each whole movement. i've seen modern
adaptations of Schenkerian analysis which do this
sort of thing.

in this kind of example, an entire movement of a
Mahler symphony (c. 25 minutes long) might be boiled
down into a 1-line chord progression.

that chord progression is surely a visual representation
of the harmony of that entire 25-minute piece, but
it may not really correspond directly with any actual
sonic simultaneity -- the chords shown on such a
Schenkerian "graph" (as its practitioners like to call it)
are usually some type of reduction of melodic lines.

in the highly polyphonic Mahler example, there would
be several melodic lines all occuring simultaneously,
and these are all boiled down into simple chords or
melodic scale progressions.

and it's significant that Schenkerian graphs generally
contain mostly plain note-heads, without any
stems / flags / beams / dots indicating rhythm.
certain more important notes are connected to
each other by beams, and ones that are even more
important might be open (whole-notes) instead of
filled (black). they abstract rhythmic notation
to represent structural aspects of the construction
of the movement as a whole.

so the point of all this rambling is that sometimes
theorists want to show things on a much more "big picture"
level, in a way which blurs the distinctions between
harmony-as-simultaneity and melody-as-succession.
indeed, the whole study of counterpoint is about
reconciling these poles.

(and his mastery in doing the reconciliation is one
of the things i find so compelling about Mahler's music.)

and Gene, i can confirm that the world-at-large has
*not* adopted this precision of colon-vs-slash for ratios.
it was something we worked out here, and the only
ones i've seen continuing to use it consistently are
me, Paul, and Dave.

-monz