Organising/Classifying scales - Think like a musician - instead of a numerologist (hung up on integer ratios).

Yes, Yahya;

I can appreciate what you are attempting to achieve; yet I believe that you are looking through the wrong end of the tele/microscope.

I like Gene's "basins" term, as a way to visualise. It implies a vague region with elastic sides, and a centre around which each basin can "catch a falling pitch".

To list the size of the intervals in ascending order within a octave is of little use to a real live musician.

Think about it in very down to earth practical terms; for example I wish to find out where I should put frets for a single string on my defretted guitar to
play the scale which you wish to describe, and hear.

We know the first octave fret will be at the midpoint of the nut to bridge distance, so let's put one at the midpoint, and restrict ourselves to the
placing of the other frets to the first octave. i.e. they must all go somewhere between the nut and the octave fret.

So this gives us a range of say 325 mm along which to find a position for each of our frets; of which there we now need to add one less than the number of intervals in your list of LMNPQRST... now for example we may have 3 of L; 2 of M; 1 of N, and say 4 of P, and the sum of the distances between the frets must equal 325 mm.

But this starts to get very complicated and requires four simultaneous equations to solve if we are to get any sort of precision happening.
The problem is further compounded by the fact that the fretboard is set up in geometric space, for the same distance between frets near the nut produces a smaller interval than it would near the octave fret.

Let's dump that approach and try something which a musician would do.......

(S)he would listen and think "that interval sounds pretty close to a fifth from the lowest note" so would stick a piece of masking tape near where you would normally find the seventh fret on a 12edo guitar.

(S)he would then continue until all the intervals from the lowest (tonic?) were approximately accounted for.

So we now have the correct number of pieces of masking tape although only roughly placed. Although the placements now precisely follow the LMNPQR pattern; this still leaves us lotsa scope for improvement. So we start experimenting .......

Say we tweak the sixth fret i.e. diabolical bVth or #VIth; any significant movement will change the pattern into some other pattern of LMN etc. as you have correctly observed,

I suggest that what we need is some hierarchy of significant intervals. e.g. steps of fourth and fifths; thirds and sixths; or seconds and sevenths seem the obvious, for these type of patterns seem to be common in all octave based tuning systems from all cultures.

If you want the link it's here:-)

http://www.lucytune.com/new_to_lt/pitch_05.html

> 1a.
> Re: Organizing and classifying scales
>
> Posted by: "yahya_melb" yahya@melbpc.org.au yahya_melb
>
> Sun Jun 18, 2006 6:47 am (PST)
>
> Hi Herman,
>
> --- In tuning-math@yahoogroups.com, Herman Miller wrote:
> [snip]
> > ... Are the example scales (0 3 5 7 9 11 14 of
> > 15-ET, 0 4 7 10 13 16 20 of 22-ET) included in his list in
> > some form, and do the tables make it apparent that they have
> > a similar structure?
> > What about modes (rotations) of these scales, like
> > 0 2 5 6 9 11 13 of 15-ET?
> >
> [snip]
> >
> > Basically what I want is something to get a better handle on
> > a scale than a Scala file with tuning in cents for each note.
> > Otherwise it's going to be impossible to keep track of these.
> > If there's already a classification system for these, that's
> > great ...
> [snip]
>
> What you're looking for is a classification of
> scales as patterns of relative step sizes, right?
>
> I don't believe I've ever seen anything more
> pertinent than my suggestion to use the letters
> LMNPQRSTUVWXYZ for decreasing relative sizes,
> always starting with L for the largest and using
> just as many letters as you have distinct sizes;
> then grouping all rotations ("modes") of one scale
> (sequence of relative stepsize letters), and using
> the letter sequence for the mode which sorts first
> lexically to represent the "melodic code" of the
> scale.
>
> Eg The pattern LPMMLNMM is an eight-step pattern,
> with relative sizes L > M > N > P. Its eight
> rotations are:
> LPMMLNMM
> PMMLNMML
> MMLNMMLP
> MLNMMLPM
> LNMMLPMM
> NMMLPMML
> MMLPMMLN
> MLPMMLNM
>
> Sorted lexically (alphabetically), they are:
> LNMMLPMM
> LPMMLNMM
> MLNMMLPM
> MLPMMLNM
> MMLNMMLP
> MMLPMMLN
> NMMLPMML
> PMMLNMML
>
> The first of these is LNMMLPMM, which becomes
> the "melodic code" for the scale.
>
> By convention, these letters represent (the
> relatives sizes of) the ascending scale steps.
> For scales which have different ascending and
> descending forms, I guess it makes sense to
> list both sequences, separated by, say, a pipe
> character |.
>
> Thus the rotation LNMMLPMM could be more fully
> described as LNMMLPMM|MMPLMMNL, although the
> second half could be left of for conciseness
> when both ascending and descending forms are
> the same.
>
> I note that this "melodic code" convention
> implies nothing particular about harmonic uses
> of the scale.
>
> Is there a more useful way of classifying scales
> *melodically*? (Naming a scale, although more
> concise, is not as informative.)
>
> Regards,
> Yahya
>
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Hi Charles,

--- In tuning-math@yahoogroups.com, Charles Lucy wrote:
>
> Yes, Yahya;
>
> I can appreciate what you are attempting to achieve;
> yet I believe that you are looking through the wrong
> end of the tele/microscope.

Can't the bug look back at the scientist? ;-)
Seriously - I don't think what you wrote later
invalidates this approach at all.

> I like Gene's "basins" term, as a way to visualise.
> It implies a vague region with elastic sides, and a
> centre around which each basin can "catch a falling
> pitch".

Nice image!

> To list the size of the intervals in ascending order
> within a octave is of little use to a real live
> musician.

Get outta here! (No, no, don't leave ...!)

I dispute your implication. If there exists one "real
live musician" that finds it useful, your assertion
fails. I am that one real live musician (I promise I
am not merely a clever computer simulation of such a
wondrous beast). When thinking of modes, it's most
useful to me to know where to put the small, average
and large steps. This is of practical use, whether
I'm playing keyboard or strings (fretted or not).

> Think about it in very down to earth practical terms;
> for example I wish to find out where I should put
> frets for a single string on my defretted guitar to
> play the scale which you wish to describe, and hear.

How easy that is! It's simple maths, based on the
chosen tuning. And if you haven't chosen the tuning
yet, why are you setting frets? - it's simply
premature, since it makes more sense to try out your
tuning ideas on a fretless instrument, even on a
monochord or synth.

> We know the first octave fret will be at the midpoint
> of the nut to bridge distance, so let's put one at
> the midpoint, and restrict ourselves to the placing
> of the other frets to the first octave. i.e. they
> must all go somewhere between the nut and the octave
> fret.

I see; you're only interested in octave-equivalent
scales then? I'm not.

> So this gives us a range of say 325 mm along which
> to find a position for each of our frets; of which
> there we now need to add one less than the number of
> intervals in your list of LMNPQRST... now for example
> we may have 3 of L; 2 of M; 1 of N, and say 4 of P,
> and the sum of the distances between the frets must
> equal 325 mm.
>
> But this starts to get very complicated and requires
> four simultaneous equations to solve if we are to
> get any sort of precision happening.

No, it only requires a decision as to what sizes you
wish to make those intervals, which can be accomplished
sequentially, starting with L and proceeding in the
direction of Z. For each of such interval, I, say, the
choice is constrained by
1 < I * (count of I in scale)
<= equivalence-interval - intervals already chosen
(= applies to the last interval chosen only)
and
I < previous interval chosen.

You have plenty of choices in those ranges, but if you
respect the constraints, you can't go far wrong. It is
possible to end up with something "left over" at the
end, which you can then redistribute over all the
intervals. And if there isn't enough to go round, you
can think of this as a negative excess, which you can
redistribute in the same way.

Another way is to start with all steps equal to the
average step size (= equivalence-interval/step count),
then add small amounts to the "large" steps and
subtract equivalent amounts from the "small" steps,
until you're satisified with the sound.

All I'm really saying is there are heuristic methods
available for the "real live musician" to decide on
what tuning he will choose that conforms to a given
"melodic structure" LMN...YZ.

> The problem is further compounded by the fact that
> the fretboard is set up in geometric space, for the
> same distance between frets near the nut produces a
> smaller interval than it would near the octave fret.

Once you've decided how many cents in your step,
it's only simple arithmetic to find the fret
position, isn't it?

> Let's dump that approach and try something which a
> musician would do.......

I don't get this. Are you implying that I'm
*not* a musician?!

> (S)he would listen and think "that interval sounds
> pretty close to a fifth from the lowest note" so
> would stick a piece of masking tape near where you
> would normally find the seventh fret on a 12edo guitar.

Depends what this parti-gendered beast *wants*.
Suppose it doesn't want fifths?

> (S)he would then continue until all the intervals
> from the lowest (tonic?) were approximately
> accounted for.
>
> So we now have the correct number of pieces of
> masking tape although only roughly placed. Although
> the placements now precisely follow the LMNPQR
> pattern; this still leaves us lotsa scope for
> improvement. So we start experimenting .......
>
> Say we tweak the sixth fret i.e. diabolical bVth
> or #VIth; any significant movement will change the
> pattern into some other pattern of LMN etc. as you
> have correctly observed,

All this tweaking and experimenting comes about
*before* you place your frets, doesn't it?

> I suggest that what we need is some hierarchy of
> significant intervals. e.g. steps of fourth and
> fifths; thirds and sixths; or seconds and sevenths
> seem the obvious, for these type of patterns seem
> to be common in all octave based tuning systems
> from all cultures.

What can I say, except to note that not everyone on
this list is interested in doing what's been done
before - no matter how common it may be. A real
live musician may not desire recognisable fifths,
thirds or seconds; a serialist in particular may
avoid these as rigorously as he already avoids the
octave.

> If you want the link it's here:-)
>
> http://www.lucytune.com/new_to_lt/pitch_05.html

Link to what???

> > Re: Organizing and classifying scales
> >
> > Posted by: "yahya_melb"
> >
> > Sun Jun 18, 2006 6:47 am (PST)
> >
> > Hi Herman,
> >
> > --- In tuning-math@yahoogroups.com, Herman Miller wrote:
> > [snip]
> > > ... Are the example scales (0 3 5 7 9 11 14 of
> > > 15-ET, 0 4 7 10 13 16 20 of 22-ET) included in his list in
> > > some form, and do the tables make it apparent that they have
> > > a similar structure?
> > > What about modes (rotations) of these scales, like
> > > 0 2 5 6 9 11 13 of 15-ET?
> > >
> > [snip]
> > >
> > > Basically what I want is something to get a better handle on
> > > a scale than a Scala file with tuning in cents for each note.
> > > Otherwise it's going to be impossible to keep track of these.
> > > If there's already a classification system for these, that's
> > > great ...
> > [snip]
> >
> > What you're looking for is a classification of
> > scales as patterns of relative step sizes, right?
> >
> > I don't believe I've ever seen anything more
> > pertinent than my suggestion to use the letters
> > LMNPQRSTUVWXYZ for decreasing relative sizes,
> > always starting with L for the largest and using
> > just as many letters as you have distinct sizes;
> > then grouping all rotations ("modes") of one scale
> > (sequence of relative stepsize letters), and using
> > the letter sequence for the mode which sorts first
> > lexically to represent the "melodic code" of the
> > scale.
> >
> > Eg The pattern LPMMLNMM is an eight-step pattern,
> > with relative sizes L > M > N > P. Its eight
> > rotations are:
> > LPMMLNMM
> > PMMLNMML
> > MMLNMMLP
> > MLNMMLPM
> > LNMMLPMM
> > NMMLPMML
> > MMLPMMLN
> > MLPMMLNM
> >
> > Sorted lexically (alphabetically), they are:
> > LNMMLPMM
> > LPMMLNMM
> > MLNMMLPM
> > MLPMMLNM
> > MMLNMMLP
> > MMLPMMLN
> > NMMLPMML
> > PMMLNMML
> >
> > The first of these is LNMMLPMM, which becomes
> > the "melodic code" for the scale.
> >
> > By convention, these letters represent (the
> > relatives sizes of) the ascending scale steps.
> > For scales which have different ascending and
> > descending forms, I guess it makes sense to
> > list both sequences, separated by, say, a pipe
> > character |.
> >
> > Thus the rotation LNMMLPMM could be more fully
> > described as LNMMLPMM|MMPLMMNL, although the
> > second half could be left of for conciseness
> > when both ascending and descending forms are
> > the same.
> >
> > I note that this "melodic code" convention
> > implies nothing particular about harmonic uses
> > of the scale.
> >
> > Is there a more useful way of classifying scales
> > *melodically*? (Naming a scale, although more
> > concise, is not as informative.)

Regards,
Yahya