Re: retuning midi clips

Hi Paul,

> Trying to order them chronologically is a fishy game. Normal
> Pythagorean was entrenched from about 800-1400, with schismic getting
> some play in the 15th century. This 15th century use of schismic is
> about as close as strict JI ever got to being important in Western
> music -- before 800 we just don't have enough evidence. Ragas are
> often in JI now because of the drone -- but the drone is a relatively
> recent addition to Indian music. 43 was an ET (very close to 1/5-
> comma meantone) that got some mention in the 1700s; a few theorists
> even went to 74 since that's close to 2/9-comma and 3/14-comma
> meantone.

Yes, that makes sense. Some of this should be said in the help for
the retuning midi player. What btw is normal Pythagorean as compared
with schismic? Hunt of the tuning definitions gives Pythagorean
defined as schismic, because going up eight fifths gives a good
approc. to the major third.

For the drop list itself, I want an easy to see indication of which
temperaments are suitable for which periods for the tyro to use.
I.e. one that will also work well if someone knowing nothing about
temperaments downloads FTS and then uses the retuning midi player
to listen to some clips, e.g. from the web.

So, I'll just do the best I can, perhaps with a catch all section
at the end or beginning for ones that don't seem to fit in anywhere.

Perhaps I should say "modern ragas"?

My idea about the j.i. scale was that it is a systematisation
of the kinds of pitches a folk singer would use. AT least that was
what I assumed.

But come to think of it, many folk traditions use pitches not at
all in the j.i. scale.

I suppose the 1/1 9/8 5/4 3/2 8/5 2/1 pentatonic scale might
be kind of pre-historic, being a very natural result of the
harmonic series. Pentatonic scales are common world wide,
I wonder if they most often are based on the j.i. pentatonic?

I wonder when the idea of building up a diatonic scale from
just intonation triads arose - in Margo's faq article she
describes the Pythagorean diatonic scale's origins, so I
suppose that might have come first in actual everyday
music making?

I suppose diatonic based on j.i. triads relates to ideas
of modulation? But then it is sometimes referred to
as Ptolemy's scale and I see that it does indeed date
back to his work on tunings, so in theoretical work it is
ancient. Here one is interested in usage and I wonder
how ancient is the actual use of the j.i. diatonic?

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi Paul,
>
> > Trying to order them chronologically is a fishy game. Normal
> > Pythagorean was entrenched from about 800-1400, with schismic
getting
> > some play in the 15th century. This 15th century use of schismic
is
> > about as close as strict JI ever got to being important in
Western
> > music -- before 800 we just don't have enough evidence. Ragas are
> > often in JI now because of the drone -- but the drone is a
relatively
> > recent addition to Indian music. 43 was an ET (very close to 1/5-
> > comma meantone) that got some mention in the 1700s; a few
theorists
> > even went to 74 since that's close to 2/9-comma and 3/14-comma
> > meantone.
>
> Yes, that makes sense. Some of this should be said in the help for
> the retuning midi player. What btw is normal Pythagorean as compared
> with schismic? Hunt of the tuning definitions gives Pythagorean
> defined as schismic, because going up eight fifths gives a good
> approc. to the major third.

Normal Pythagorean usually meant a chain of just fifths from Eb to
G#, while schismic usually meant a chain of just fifths from Gb to B -
- the latter gives near-just major triads on D, A, and E, while in
the former they are in a position unlikely to be exploited by
medieval composers.
>
> For the drop list itself, I want an easy to see indication of which
> temperaments are suitable for which periods for the tyro to use.
> I.e. one that will also work well if someone knowing nothing about
> temperaments downloads FTS and then uses the retuning midi player
> to listen to some clips, e.g. from the web.
>
> So, I'll just do the best I can, perhaps with a catch all section
> at the end or beginning for ones that don't seem to fit in anywhere.
>
> Perhaps I should say "modern ragas"?

Sure . . . I just meant the antiquity of JI isn't demonstrated just
because it's used for modern ragas, where the drone is present. In
fact, the old Indian nomenclature mapped the names Sa to Sa to a
Dorian, not Major, scale, and a 22-part division was used to specify
the scales as 3244324 and 3243424 -- note that in the latter, even
the fifth must have been quite flat.
>
> My idea about the j.i. scale was that it is a systematisation
> of the kinds of pitches a folk singer would use. AT least that was
> what I assumed.

I believe otherwise.

> But come to think of it, many folk traditions use pitches not at
> all in the j.i. scale.
>
> I suppose the 1/1 9/8 5/4 3/2 8/5 2/1 pentatonic scale might
> be kind of pre-historic, being a very natural result of the
> harmonic series.

That's an odd scale -- with a minor sixth?

> Pentatonic scales are common world wide,
> I wonder if they most often are based on the j.i. pentatonic?

In my opinion, no. See for example,

http://www.iohk.com/UserPages/thompson/03b8intn.htm

to understand the types of problems JI presents for the pentatonic
scale in Chinese music -- probably this comes up just as much in many
different cultures
>
> I wonder when the idea of building up a diatonic scale from
> just intonation triads arose

I think it arose rather late and was used, anachronistically, to
describe earlier music.

> - in Margo's faq article she
> describes the Pythagorean diatonic scale's origins, so I
> suppose that might have come first in actual everyday
> music making?

Pythagorean, I would be willing to believe that.
>
> I suppose diatonic based on j.i. triads relates to ideas
> of modulation?

Hmm . . . not really. Certainly JI makes modulation difficult, and
the "pumps" are well known. So I'm not sure what you're thinking here.

> But then it is sometimes referred to
> as Ptolemy's scale and I see that it does indeed date
> back to his work on tunings,

For him, the scale primarily went from E to E rather than C to C, and
the "modern" 5-limit JI tuning was only one of dozens of tunings he
presented.

> so in theoretical work it is
> ancient. Here one is interested in usage and I wonder
> how ancient is the actual use of the j.i. diatonic?

Since vertical harmonies were not used in ancient music, we can have
little confidence that measurements on monochords, pipes, and the
like provided anything like a good approximation to what we call JI
intervals today (since the relationship between these measurements
and the physical nature of sound was not known until Galileo's day,
and the effect of bending a string down to a soundboard, or end-
effects in tubes, could certainly not have been accounted for). I'll
let John Chalmers comment further . . . certainly it would be good to
provide some ancient enharmonic scales to play with using your
program . . . as well as non-Western tunings . . . and maybe even a
few "modern" 12-tone tunings.

Hi Paul,

thanks, I understand what schismic Pythagorean now is as
compared with normal Pythagorean.

> That's an odd scale -- with a minor sixth?

Yes, should be 5/3 instead of 8/5 of course.

> > I suppose diatonic based on j.i. triads relates to ideas
> > of modulation?

> Hmm . . . not really. Certainly JI makes modulation difficult, and
> the "pumps" are well known. So I'm not sure what you're thinking here.

I was thinking that the idea of putting a triad on one of the other roots
of the scale suggest getting the scale to work with a chord progression,
and that the idea of a chord progression as kind of related to modulation
rather than modal type music, perhaps because the triad suggests the beginnings
of a scale on the new note as a root. I'm not well versed with music theory
so not sure if that makes sense.

Of course one can just do j.i. type chord progressions while remaining
in the same diatonic scale all the time, but once you have those triads
there it kind of suggests at least possiblity of taking off on any of
the triads into a new scale based on its root.

Then of course one gets the modulation difficulties and the pumps arising,
but the j.i. scale could be what propels one into those difficulties in
the first place.

> Since vertical harmonies were not used in ancient music, we can have
> little confidence that measurements on monochords, pipes, and the
> like provided anything like a good approximation to what we call JI
> intervals today (since the relationship between these measurements
> and the physical nature of sound was not known until Galileo's day,
> and the effect of bending a string down to a soundboard, or end-
> effects in tubes, could certainly not have been accounted for). I'll
> let John Chalmers comment further . . . certainly it would be good to
> provide some ancient enharmonic scales to play with using your
> program . . . as well as non-Western tunings . . . and maybe even a
> few "modern" 12-tone tunings.

Here I suppose you are talking about the 5/4 interval, as the octave, 3/2 and
4/3 must surely be very ancient.

Drones were prob. ancient, at least in Celtic music, it says in the book about the
ancient Scottish music I'm reading. So that suggest idea of j.i. as a possiblity
for same reason as in modern Ragas. Maybe you are talking about an earlier period
than this, but I think idea of a drone is quite natural, e.g. singing, for some
singers to hold a note or sing a repeated note while others sing ornamentations
/ elaborations above it. If one starts to do that one might well fall into
5/4 fifths I'd have thought,...

Especially if the timbre of the drone happens to have prominent fifth harmonics.

Any particular ones in mind for the modern and ancient scales that would be
good to provide?

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi Paul,
>
> thanks, I understand what schismic Pythagorean now is as
> compared with normal Pythagorean.
>
> > That's an odd scale -- with a minor sixth?
>
> Yes, should be 5/3 instead of 8/5 of course.

That would sound odd every time you had 5/6 or 5/3 next to 9/8 or
9/4 -- very common in pentatonic melodies as these are the notes
closest to 1/1 or 2/1.
>
> > > I suppose diatonic based on j.i. triads relates to ideas
> > > of modulation?
>
> > Hmm . . . not really. Certainly JI makes modulation difficult,
and
> > the "pumps" are well known. So I'm not sure what you're thinking
here.
>
> I was thinking that the idea of putting a triad on one of the other
roots
> of the scale suggest getting the scale to work with a chord
progression,
> and that the idea of a chord progression as kind of related to
modulation
> rather than modal type music, perhaps because the triad suggests
the beginnings
> of a scale on the new note as a root. I'm not well versed with
music theory
> so not sure if that makes sense.
>
> Of course one can just do j.i. type chord progressions while
remaining
> in the same diatonic scale all the time, but once you have those
triads
> there it kind of suggests at least possiblity of taking off on any
of
> the triads into a new scale based on its root.

I think that latter has much more to do with modal music than you may
suspect -- even if it's not modulation in the modern sense.
>
> > Since vertical harmonies were not used in ancient music, we can
have
> > little confidence that measurements on monochords, pipes, and the
> > like provided anything like a good approximation to what we call
JI
> > intervals today (since the relationship between these
measurements
> > and the physical nature of sound was not known until Galileo's
day,
> > and the effect of bending a string down to a soundboard, or end-
> > effects in tubes, could certainly not have been accounted for).
I'll
> > let John Chalmers comment further . . . certainly it would be
good to
> > provide some ancient enharmonic scales to play with using your
> > program . . . as well as non-Western tunings . . . and maybe even
a
> > few "modern" 12-tone tunings.
>
> Here I suppose you are talking about the 5/4 interval, as the
octave, 3/2 and
> 4/3 must surely be very ancient.

Yes, I agree.
>
> Drones were prob. ancient, at least in Celtic music, it says in the
book about the
> ancient Scottish music I'm reading. So that suggest idea of j.i. as
a possiblity
> for same reason as in modern Ragas.

Sure. You could put a few bagpipe scales into your program.
>
> Any particular ones in mind for the modern and ancient scales that
would be
> good to provide?

Let me ask you this -- what is the program going to be used for? To
play pre-existing MIDI files? Perhaps you should build some
more "intelligence" into it . . . John deLaubenfels comes to mind.

Hi Paul,

I know you can't really have a j.i. scale that has the most
commonly used diatonic triads all pure.

E.g. can have I, IV, and V pure, makes II impure,
or I, V, II pure makes IV impure.

Might be an idea to label them like that actually.
just intonation with I, IV, V pure - some folk music + many ragas
1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2/1

just intonation with I, II, V pure - some folk music + many ragas
= 1/1 135/128 9/8 6/5 5/4 27/20 45/32 3/2 8/5 27/16 9/5 15/8 2/1

Or, could just rotate it and as it rotates, show which triads
are pure, but that is getting into the next section on
chord recognition and support of the new SCALA list of chords,
that I have in mind to start work on very soon - not even
started yet :-(.

In my root control retuning experiments (using one of the midi
channels as a root control channel) I find one can often do quite
well for simple diatonic melodies by using root at I most of
the time, with the I V IV j.i. scale, and then using
root at II whenever one encounters a II chord.

I suppose also in pentatonic, of course one can't have all
the major thirds pure and the major fifths because making
the fifths pure gives you the pythagorean major third.

Any thoughts about an optimal j.i. tuning for a pentatonic scale?

I see what you are saying about modal - can start thinking of
one of the triads as a kind of rest position in the scale
without any need to modulate to a new scale in conventional
sense - i.e. no new sharps or flats. Is that what you mean?

I certainly have in mind the possibility that at some point
in the future the retuning midi player could become an
adaptive retuning midi player, linking in with John's
program, following on from the ideas he and I discussed
here a while back.

It would be nice to have mix of different types of
temperament, traditional and modern ones.

What are the temperaments of yours that Margo mentioned?
- would be nice to include those if okay.

Yes bagpipe scales are an idea, certainly should have
those for retuning bagpipe 12 tone music, if for
nothing else.

Yes, idea is it is for retuning existing midi files.

One could also use it to the midi clips of files tuned
according to other conventions, such as the ones
Matts Oljare makes, and the ones I do, to be interpreted
as midi note numbers = steps up in the scale, whatever
it is.

To make that possible one would have a link to
the other FTS drop lists at the top of the drop list.

However, the default drop list of scales for the midi relaying
view, the one one gets if one resets it, will be this one
of various twelve tone temperaments.

Robert

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi Paul,
>
> I know you can't really have a j.i. scale that has the most
> commonly used diatonic triads all pure.
>
> E.g. can have I, IV, and V pure, makes II impure,
> or I, V, II pure makes IV impure.
>
> Might be an idea to label them like that actually.
> just intonation with I, IV, V pure - some folk music + many ragas
> 1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2/1
>
> just intonation with I, II, V pure - some folk music + many ragas
> = 1/1 135/128 9/8 6/5 5/4 27/20 45/32 3/2 8/5 27/16 9/5 15/8 2/1

Robert, the "modern Indian gamut" is neither of these but rather
1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 27/16 9/5 15/8 2/1

> Or, could just rotate it and as it rotates, show which triads
> are pure, but that is getting into the next section on
> chord recognition and support of the new SCALA list of chords,
> that I have in mind to start work on very soon - not even
> started yet :-(.

Sounds cool!
>
> Any thoughts about an optimal j.i. tuning for a pentatonic scale?

Pythagorean. Besides that, one could use very complex ratios to
approximate meantone . . . would this be acceptable?
>
> I see what you are saying about modal - can start thinking of
> one of the triads as a kind of rest position in the scale
> without any need to modulate to a new scale in conventional
> sense - i.e. no new sharps or flats. Is that what you mean?

Partially -- though lots of modal (pre-tonal) music is rich in sharps
and flats too.

> What are the temperaments of yours that Margo mentioned?

Where? I do have a few 12-tone subsets of 22-tET and the like . . .
but what did you have in mind?

[Robert Walker wrote:]
>>Any thoughts about an optimal j.i. tuning for a pentatonic scale?

[Paul E wrote:]
>Pythagorean. Besides that, one could use very complex ratios to
>approximate meantone . . . would this be acceptable?

Pathagorean? Uggh. How about the following, very meantone-like:

C +6.83 cents deviation from 12-tET
D 0.00 cents deviation from 12-tET
E -6.83 cents deviation from 12-tET
G +5.18 cents deviation from 12-tET
A -5.18 cents deviation from 12-tET

I got these numbers from a COFT on all notes sounding together. C to
E is pure 5:4, so to that extent it's 1/4 comma meantone, but G and A
are not quite consistent with that. Here's the whole COFT table:

Ptch Tuning Ptch Tuning Strength Ideal Actual Force Pain
---- ------ ---- ------ -------- -------- -------- ---------- ----------
0 6.83 2 -0.00 0.164 200.000 193.174 -1.118 3.817
0 6.83 4 -6.83 16.384 386.314 386.348 0.559 0.010
0 6.83 7 5.18 16.384 701.955 698.353 -59.013 106.277
0 6.83 9 -5.18 16.384 884.359 887.995 59.572 108.301
2 -0.00 0 6.83 0.164 1000.000 1006.826 1.118 3.817
2 -0.00 4 -6.83 0.164 200.000 193.174 -1.118 3.817
2 -0.00 7 5.18 16.384 498.045 505.179 116.887 416.951
2 -0.00 9 -5.18 16.384 701.955 694.821 -116.887 416.951
4 -6.83 0 6.83 16.384 813.686 813.652 -0.559 0.010
4 -6.83 2 -0.00 0.164 1000.000 1006.826 1.118 3.817
4 -6.83 7 5.18 16.384 315.641 312.005 -59.572 108.301
4 -6.83 9 -5.18 16.384 498.045 501.647 59.013 106.277
7 5.18 0 6.83 16.384 498.045 501.647 59.013 106.277
7 5.18 2 -0.00 16.384 701.955 694.821 -116.887 416.951
7 5.18 4 -6.83 16.384 884.359 887.995 59.572 108.301
7 5.18 9 -5.18 0.164 200.000 189.642 -1.697 8.790
9 -5.18 0 6.83 16.384 315.641 312.005 -59.572 108.301
9 -5.18 2 -0.00 16.384 498.045 505.179 116.887 416.951
9 -5.18 4 -6.83 16.384 701.955 698.353 -59.013 106.277
9 -5.18 7 5.18 0.164 1000.000 1010.358 1.697 8.790
---- ------ ---- ------ -------- -------- -------- ---------- ----------
RMS 4.71 strnSum 115.180 painSum 1279.491

JdL

Hi Paul,

I'm thinking of Margo's recent post on the Middle path, about
well temperaments.

> George Secor has published a superb 17-note well-temperament[1], while
> Paul Erlich and others have devised new 12-note systems.

In the Scala mode list I see
0 3 1 1 3 1 3 1 3 1 1 3 1 Twelve-tone Chromatic (1/3-comma positive)
but it doesn't look much like a well temperament!

Perhaps I'll do it as:

"
just intonation with I, IV, V pure
(1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 5/3 9/5 15/8 2/1)
just intonation with I, II, V pure
(= 1/1 135/128 9/8 6/5 5/4 27/20 45/32 3/2 8/5 27/16 9/5 15/8 2/1)
Modern Indian Gamut - ragas
1/1 16/15 9/8 6/5 5/4 4/3 45/32 3/2 8/5 27/16 9/5 15/8 2/1
Modern bagpipe scale
Pythagorean (Middle ages)
Quarter comma mean-tone (renaissance / baroque)
Werckmeister III (Bach's time, late baroque / early classical)
[perhaps a couple of other well temperaments from this time]
Variable comma mean-tone, 0 to 1/4 (via modulation wheel or other controller)
Sixth comma mean-tone (Mozart's time, classical)
Variable comma mean-tone, 1/6 to 1/3 (via modulation wheel or other controller)
Vallotti and Young (late classical)
[some more well temperaments here]
12 tone equal temperament (twentieth century / early lute music).
[modern well temperaments go here]
Bagpipe scales...
Early n-tet meantones...
n-tet meantones...
"

So not strictly chronological: First a j.i. section, then historical,
then links to drop lists of scales.

I'll do something or other, any suggestions welcome. It's not a scholarly
work, just a nice list of scales suitable for anyone who wants to use the
retuning midi player to listen to midi files originally for 12-tet
and one would want to include the more common well temperaments etc. in the
main list to make this easier for the user, and save need to look for the desired
scale in the SCALA archive or copy / paste / type it in from some other
source.

Plus would include a few other nice and intriguing ones, and example modern ones.
I feel it's okay if it is a little eccentric in some way or another.

Users can always edit the list and customise it to add new scales to it.
I plan to make it easier to edit the scales and modes lists in near
future.

When I add new ones to it, I'd back date the file to just after the
date it was first made, and before the first upload, so
that the installer will only replace the file if it has been
left in its original state (I've set the installer to replace files only
if the replacement is newer than the file it replaces).

> > Or, could just rotate it and as it rotates, show which triads
> > are pure, but that is getting into the next section on
> > chord recognition and support of the new SCALA list of chords,
> > that I have in mind to start work on very soon - not even
> > started yet :-(.

> Sounds cool!

I'm looking forward to working on this.

For the pentatonic, I was actually thinking, what might be a good twelve
tone to add to the list for retuning pentatonic music, e.g. midi clips of
pentatonic folk melodies with no accompaniment, since this is a
quite likely use, and neither of the j.i. scales is optimal,
as you said, with 27/20 fourths in one place or another.

Prob. there is no solution to it really, if one wants a j.i. like one
with the 9/8, 10/9 steps. If one is happy with a meantone like one
and wants the 1/1 5/4 of the pentatonic scale pure, then quarter
comma meantone or some close temperament, as John's COFT pentatonic
scale is fairly close to qc meantone.

Really, even when optimising the pentatonic scale, I see
one is already setting off on the adventure of devising
various temperaments...

John's scale:
1/1 193.174 386.348 698.353 887.995 2/1

C.f. qc meantone
1/1 193.1569 386.3137 696.5784 889.7353 2/1

(Thanks, John)

Robert

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Robert Walker wrote:]
> >>Any thoughts about an optimal j.i. tuning for a pentatonic scale?
>
> [Paul E wrote:]
> >Pythagorean. Besides that, one could use very complex ratios to
> >approximate meantone . . . would this be acceptable?
>
> Pathagorean? Uggh. How about the following, very meantone-like:
>
> C +6.83 cents deviation from 12-tET
> D 0.00 cents deviation from 12-tET
> E -6.83 cents deviation from 12-tET
> G +5.18 cents deviation from 12-tET
> A -5.18 cents deviation from 12-tET

That's fine, but it's not JI! Robert said JI.

--- In tuning@y..., "Robert Walker" <robertwalker@n...> wrote:
> Hi Paul,
>
> I'm thinking of Margo's recent post on the Middle path, about
> well temperaments.
>
> > George Secor has published a superb 17-note well-temperament[1],
while
> > Paul Erlich and others have devised new 12-note systems.
>
> In the Scala mode list I see
> 0 3 1 1 3 1 3 1 3 1 1 3 1 Twelve-tone Chromatic (1/3-comma positive)
> but it doesn't look much like a well temperament!

It sure isn't! My 22-tone well-temperament has 22 tones per octave.
If you're looking for my 12-tone subsets of 22-tET, Manuel has coded
them as .kbd files in Scala (right, Manuel?)
>
> For the pentatonic, I was actually thinking, what might be a good
twelve
> tone to add to the list for retuning pentatonic music, e.g. midi
clips of
> pentatonic folk melodies with no accompaniment, since this is a
> quite likely use, and neither of the j.i. scales is optimal,
> as you said, with 27/20 fourths in one place or another.

The scales you already list come to mind. Pythagorean, meantones,
well-temperaments.

[Robert Walker wrote:]
>>>>Any thoughts about an optimal j.i. tuning for a pentatonic scale?

[Paul E wrote:]
>>>Pythagorean. Besides that, one could use very complex ratios to
>>>approximate meantone . . . would this be acceptable?

[I wrote:]
>>Pythagorean? Uggh. How about the following, very meantone-like:

>>C +6.83 cents deviation from 12-tET
>>D 0.00 cents deviation from 12-tET
>>E -6.83 cents deviation from 12-tET
>>G +5.18 cents deviation from 12-tET
>>A -5.18 cents deviation from 12-tET

[Paul E:]
>That's fine, but it's not JI! Robert said JI.

We're arguing over words. The above is the closest possible approach
to JI that this combination of notes permits (according to a fairly
reasonable definition of "closest" and "JI"). It seems to be the type
of solution Robert was grasping for, if I've understood his response
correctly.

The Pythagorean major third at 81:64 would be called JI by some, but
most list members, yourself included (?), would consider ratios this
complex to be well outside JI.

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Robert Walker wrote:]
> >>>>Any thoughts about an optimal j.i. tuning for a pentatonic
scale?
>
> [Paul E wrote:]
> >>>Pythagorean. Besides that, one could use very complex ratios to
> >>>approximate meantone . . . would this be acceptable?
>
> [I wrote:]
> >>Pythagorean? Uggh. How about the following, very meantone-like:
>
> >>C +6.83 cents deviation from 12-tET
> >>D 0.00 cents deviation from 12-tET
> >>E -6.83 cents deviation from 12-tET
> >>G +5.18 cents deviation from 12-tET
> >>A -5.18 cents deviation from 12-tET
>
> [Paul E:]
> >That's fine, but it's not JI! Robert said JI.
>
> We're arguing over words. The above is the closest possible
approach
> to JI that this combination of notes permits (according to a fairly
> reasonable definition of "closest" and "JI"). It seems to be the
type
> of solution Robert was grasping for, if I've understood his
response
> correctly.

On the contrary, I believe I had already convinced Robert that strict
JI was not optimal for a pentatonic scale, and he was then asking
(above), "if you had to use JI, i.e., ratios, what would you use"?
Robert has a very strong interest in rational tunings, as opposed to
tempered ones, at least that's my impression from his contributions
to the various tuning lists.
>
> The Pythagorean major third at 81:64 would be called JI by some, but
> most list members, yourself included (?), would consider ratios this
> complex to be well outside JI.

I think Dave Keenan is the only one who considers 81:64 not to be JI.
It comes up in the simplest JI tuning systems from a chain of four
3:2s.

Hi John,

Actually Paul is right about this, I was looking for a low limit
j.i. one for the j.i. section for the retuning midi player drop list.

In fact, thinking it over, if one was singing over a drone,
or using harmonics, one would surely have four of the five notes
as
1/1 9/8 5/4 3/2 2/1.

So then choice is over 27/16 or 5/3 for the remaining note,
i.e. 9/8 above the 3/2, or 10/9 above it

1/1 9/8 5/4 3/2 5/3 2/1
steps
9/8 10/9 6/5 10/9 6/5
or
1/1 9/8 5/4 3/2 27/16 2/1
steps
9/8 10/9 6/5 9/8 32/27

and I suppose singer sings one or the other depending on the
context. Could use the 27/16 if previous note was the 9/8,
choose the 5/3 if it was 5/4 or 1/1, and choose either if prev.
note was the 3/2.

Something to be said for a split key there.

Or, adaptive j.i. pentatonic? Because if one sings the 27/16 then
goes say to the 2/1 as a 6/5 above it instead of 32/27, one is off
into a whole new pentatonic scale.

Or, do split keys for entire pentatonic scale to get ten note scale,
with all the notes 81/80 above the simplest one, then add another one
81/80 below.

In fact could do a hexagonal type arrangement of keys with
1/1 9/8 5/4 3/2 5/3 2/1

for each row and 81/80 as the shift between rows.

You can play that one from p.c. keyboard in FTS by setting
1/1 9/8 5/4 3/2 5/3 2/1
as the scale then going to Seed | Play keyboard options
| Pitches for keyboard rows | Custom modulation (p.a. only)

A new field appears:
Modulate each row by (p.a. only)
and one then can set 81/80 as the interval between p.c. keyboard rows.

So now, first row of keyboard plays
1/1 9/8 5/4 3/2 5/3 2/1,...
second row plays
81/80 729/640 81/64 243/160 27/16 81/40,...
third row plays
6561/6400 59049/51200 6561/5120 19683/12800 2187/1280 6561/3200
and so on.

So one can play pentatonic melodies on p.c. keyboard, and whenever
you want the 3/2 from degree 1 to deg 4, drop down to the next
row of the p.c. keyboard, and keep doing that, getting pentatonic
melodies with comma drift, if that is the way one likes it.

If one had a drone and did this sort of thing, I suppose one might
feel a need to comma shift the drone at some point - I wonder
if any traditional drone based music has done that...

I expect for most of you this is all old hat, but I haven't quite
thought it through before (even though I had all the pieces before),
so maybe a few others may be helped by seeing this too.

It was nice to see your pentatonic scale too, and thanks for
that.

Robert

[Paul E wrote:]
>I think Dave Keenan is the only one who considers 81:64 not to be JI.
>It comes up in the simplest JI tuning systems from a chain of four
>3:2s.

Yes, I'm aware of where 81:64 comes from. So, is it your position that
all intervals which can be connected by a chain of pure fifths, however
long, automatically qualify as "JI intervals"? For example, 3^12:2^19
(531441:524288): is that "JI"? I thought that the term "RI" had been
agreed upon for very complex, yet integer, ratios.

Of course, no fixed scale with very many notes, though they may be
connected by low-number (JI) ratios, will have _all_ intervals connected
by low numbers. I had thought that the convention was to describe the
_scale_ as JI even though many of its intervals (for practical purposes)
are not JI.

[Robert wrote:]
>Actually Paul is right about this, I was looking for a low limit
>j.i. one for the j.i. section for the retuning midi player drop list.

Oh, ok. Kyool.

>Or, adaptive j.i. pentatonic?

That'd be my favorite approach (no surprise!).

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

<I thought that the term "RI" had been
> agreed upon for very complex, yet integer, ratios.

The defintions I've seen of "JI" as anything other than "RI" have
been essentially meaningless, so I'm sticking with JI=RI until that
changes.

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:
> [Paul E wrote:]
> >I think Dave Keenan is the only one who considers 81:64 not to be JI.
> >It comes up in the simplest JI tuning systems from a chain of four
> >3:2s.
>
> Yes, I'm aware of where 81:64 comes from. So, is it your position that
> all intervals which can be connected by a chain of pure fifths, however
> long, automatically qualify as "JI intervals"? For example, 3^12:2^19
> (531441:524288): is that "JI"? I thought that the term "RI" had been
> agreed upon for very complex, yet integer, ratios.

This debate was a very heated one and I opted to stay out of it. There is way too much
supporting literature on both sides of the debate to make a clear decision on the definition of JI
when it comes to isolated intervals.
>
> Of course, no fixed scale with very many notes, though they may be
> connected by low-number (JI) ratios, will have _all_ intervals connected
> by low numbers. I had thought that the convention was to describe the
> _scale_ as JI even though many of its intervals (for practical purposes)
> are not JI.

In the writings of composers and theorists who use JI systems of many notes (Helmholtz-Ellis,
Partch and Johnston come to mind), I've never seen any claim that any of their intervals are not
JI. Not consonant, sure -- but still JI. Of course, proponents of temperament (such as myself)
are always quick to point out that there's no qualitative difference between a JI dissonance and a
tempered dissonance -- and that there's nothing audibly special about, for example, the
2/7-comma meantone chromatic semitone or the 1/6-comma meantone tritone, just because
they happen to coincide with JI ratios.

wrong address this is not John take me off your mailing address!!

[I wrote:]
>>Yes, I'm aware of where 81:64 comes from. So, is it your position
>>that all intervals which can be connected by a chain of pure fifths,
>>however long, automatically qualify as "JI intervals"? For example,
>>3^12:2^19 (531441:524288): is that "JI"? I thought that the term "RI"
>>had been agreed upon for very complex, yet integer, ratios.

[Paul E:]
>This debate was a very heated one and I opted to stay out of it. There
>is way too much supporting literature on both sides of the debate to
>make a clear decision on the definition of JI when it comes to isolated
>intervals.

Ok.

>In the writings of composers and theorists who use JI systems of many
>notes (Helmholtz-Ellis, Partch and Johnston come to mind), I've never
>seen any claim that any of their intervals are not JI. Not consonant,
>sure -- but still JI. Of course, proponents of temperament (such as
>myself) are always quick to point out that there's no qualitative
>difference between a JI dissonance and a tempered dissonance -- and
that there's nothing audibly special about, for example, the 2/7-comma
meantone chromatic semitone or the 1/6-comma meantone tritone, just
because they happen to coincide with JI ratios.

So, what I should've written is: the above [suggested pentatonic tuning]
is the closest possible approach to consonant intervals that this
combination of notes permits (according to a fairly reasonable
definition of "closest" and "consonant").

JdL

--- In tuning@y..., "John A. deLaubenfels" <jdl@a...> wrote:

> So, what I should've written is: the above [suggested pentatonic
tuning]
> is the closest possible approach to consonant intervals that this
> combination of notes permits (according to a fairly reasonable
> definition of "closest" and "consonant").

And I would have agreed 100%. It should be in the Scala archive
as "optimal pentatonic" or something.