Re: A Scale of 2 notes

What I suppose I am trying to do is start with a scale that contains only two notes, and derive a tuning for it. So I begin with Ls and then transpose it to arrive at extra notes used as 'black notes' to modulate to new keys. I was asking the question, can it be done abstractly, only knowing that L is bigger than s? The extra notes created through transposing were not to be incorporated as white notes in the scale, if you'll pardon the analogy with a keyboard.

So, starting with just two notes, I transposed upward and downward by the L step, to arrive at the first sharp and the first flat, so to speak. I then transposed again by this interval to arrive at the second sharp and flat. Just as we transpose the diatonic scale by 5ths to arrive at different keys, but this time, without the notion of a temperament or EDO.

By investigating the three possible ratios of L to S, as rational numbers, it is possible to substitute 'graphical' positions of the notes by actual pitch-classes from an EDO. If L/s is modelled as irrational, we arrive at an endless spiral of notes. For either case, the scale consists of two adjacent elements from the circle or the spiral.

It should then be possible to generalise this to provide a means for determining tunings for different scales. I tend to call those where L > s > L-s as being 'meantone'-like, and those where L > L-s >s as being pythagorean. Looking for a moment at two nominals (white notes), the former inserts the black keys in the order sharp-flat (ascending), and the latter as flat-sharp (ascending). The case where L-s = s inserts only one black note, enharmonically sharp and flat.

What interests me is that we have a scale, viz: LsLsLsLLs, say, and we are looking to deriving a tuning for it. The simple approach is to assign steps of an EDO to it, like L= 2 and s=1. But what about determining a tuning that does not result in an EDO? We can choose to find the generator, in terms of L and s steps (in this case LsLsLsL), but how could one arrive at the narrowest and largest possible values for this interval without destroying the scale?

The sharps and flats way of looking at a scale gives rise to a stave, and clefs and key signatures, which are independent of the tuning. Just as we can write diatonic music and render it with 12 or 19 or 17 or 31 EDO (subject to some interesting effects of this translation), so we must be able to do with other scales. To write the music chained to a specific EDO locks out the possibility that it could be rendered with a different tuning. We should be thankful that western stave notation permits this possibility: what would we have done by adopting a 12-note notation sometime in the past?

The above scale is the 9 from 14 that Carl put on the list again. We can envisage a keyboard for it, and derive a (simple) harmonic structure. But how else may we tune it? L=3 s=2, this sets the scale as belonging to 23EDO. Setting L=3 and s=1, we get 19EDO. Other EDOS are possible for different values of L and s, but the 14EDO representation is the simplest. The possibility of writing a work on a stave devised for the LsLsLsLLs scale and then realising it in 14, 23 or 19 could give new insights into microtonal tunings of this scale. We could choose a scale that belongs to an EDO that contains multiple scales, and head off in different directions, merely by reinterpreting the tuning. 34EDO is particularly interesting in this respect.

This may be obvious to many on the list, but perhaps we should try to invoke microtones without the requirement for complexity that baffles the nontheoretical musician. What the scale with two notes is meant to do is act as a starting point for the exploration of tuning, merely through nominals, sharps and flats.

Mark

Mark,

What you're talking about is very familiar to many of us through the
work of Erv Wilson.

It appears that my decatonic scales don't fit into this scheme of
yours at all. Yet I believe it's founded on the very same principles
that make the diatonic scale work so well, is equally derivable from
JI, etc. Here's the original paper on them:

lumma.org/tuning/erlich/erlich-decatonic.pdf

And the 'Middle Path' paper I've been mailing out gives 54 systems
that derive from JI, all of which would be notatable with a single
pair of accidentals, and yet a good number don't have a period of one
octave (which appears necessary in order to fit into your scheme).

Just so you know my point of view.

Nevertheless, deeper discussion of your idea would certainly be of
interest to me, regardless of how much of the material we've gone
over before.

Best,
Paul

--- In tuning@yahoogroups.com, Mark Gould <mark@e...> wrote:
> What I suppose I am trying to do is start with a scale that
contains
> only two notes, and derive a tuning for it. So I begin with Ls and
then
> transpose it to arrive at extra notes used as 'black notes' to
modulate
> to new keys. I was asking the question, can it be done abstractly,
only
> knowing that L is bigger than s? The extra notes created through
> transposing were not to be incorporated as white notes in the
scale, if
> you'll pardon the analogy with a keyboard.
>
> So, starting with just two notes, I transposed upward and downward
by
> the L step, to arrive at the first sharp and the first flat, so to
> speak. I then transposed again by this interval to arrive at the
second
> sharp and flat. Just as we transpose the diatonic scale by 5ths to
> arrive at different keys, but this time, without the notion of a
> temperament or EDO.
>
> By investigating the three possible ratios of L to S, as rational
> numbers, it is possible to substitute 'graphical' positions of the
> notes by actual pitch-classes from an EDO. If L/s is modelled as
> irrational, we arrive at an endless spiral of notes. For either
case,
> the scale consists of two adjacent elements from the circle or the
> spiral.
>
> It should then be possible to generalise this to provide a means
for
> determining tunings for different scales. I tend to call those
where L
> > s > L-s as being 'meantone'-like, and those where L > L-s >s as
being
> pythagorean. Looking for a moment at two nominals (white notes),
the
> former inserts the black keys in the order sharp-flat (ascending),
and
> the latter as flat-sharp (ascending). The case where L-s = s
inserts
> only one black note, enharmonically sharp and flat.
>
> What interests me is that we have a scale, viz: LsLsLsLLs, say, and
we
> are looking to deriving a tuning for it. The simple approach is to
> assign steps of an EDO to it, like L= 2 and s=1. But what about
> determining a tuning that does not result in an EDO? We can choose
to
> find the generator, in terms of L and s steps (in this case
LsLsLsL),
> but how could one arrive at the narrowest and largest possible
values
> for this interval without destroying the scale?
>
> The sharps and flats way of looking at a scale gives rise to a
stave,
> and clefs and key signatures, which are independent of the tuning.
Just
> as we can write diatonic music and render it with 12 or 19 or 17 or
31
> EDO (subject to some interesting effects of this translation), so
we
> must be able to do with other scales. To write the music chained to
a
> specific EDO locks out the possibility that it could be rendered
with a
> different tuning. We should be thankful that western stave notation
> permits this possibility: what would we have done by adopting a 12-
note
> notation sometime in the past?
>
> The above scale is the 9 from 14 that Carl put on the list again.
We
> can envisage a keyboard for it, and derive a (simple) harmonic
> structure. But how else may we tune it? L=3 s=2, this sets the
scale as
> belonging to 23EDO. Setting L=3 and s=1, we get 19EDO. Other EDOS
are
> possible for different values of L and s, but the 14EDO
representation
> is the simplest. The possibility of writing a work on a stave
devised
> for the LsLsLsLLs scale and then realising it in 14, 23 or 19 could
> give new insights into microtonal tunings of this scale. We could
> choose a scale that belongs to an EDO that contains multiple
scales,
> and head off in different directions, merely by reinterpreting the
> tuning. 34EDO is particularly interesting in this respect.
>
> This may be obvious to many on the list, but perhaps we should try
to
> invoke microtones without the requirement for complexity that
baffles
> the nontheoretical musician. What the scale with two notes is meant
to
> do is act as a starting point for the exploration of tuning, merely
> through nominals, sharps and flats.
>
> Mark