Yahoo Tuning Groups Ultimate Backup tuning EDO alternatives, Leprechaun comma (4225/4224)

Hello, all,

For a long time, George Secor has been advocating and exploring
the process of approximating EDO tunings with just ratios. Here
is his charming and germinal parable on "Blarney," and a recent
post presenting a fine just alternative to 10-EDO.

</tuning/files/secor/blarney.txt>
</tuning/topicId_102422.html#102795>

Here, I'll go a step further by giving some EDO alternatives not
only in the form of JI approximations, but as "near-just"
approximations of those approximations in the MET-24 tuning.
We'll move from 5-EDO through 12-EDO, with another post to
address the interesting question George raised of 15-EDO.

Also, with the alternatives for 8-EDO and 11-EDO, we'll encounter
the Leprechaun Comma, 4225/4224 or 0.409808 cents, which I named
as a tribute both to George's "Blarney" and to Eire, the Emerald
Isle. It's appropriately a small comma that I discovered or
rediscovered in the process of approximating small EDO's, and
thus not unfitting to honor the wee people -- assuming that
this comma, mentioned as unnamed in some previous posts, has not
yet received a name.

Please be warned some of these small-EDO alternatives have what
might be considered a serious fault: nonparallel fifths!
"Nonparallel fifths" are intervals at or near 3/2 in an emulation
of an EDO without such "conventional fifths" -- say, with all
intervals outside Blackwood's regular diatonic range from 7-EDO
(685.71 cents) to 5-EDO (720 cents), or from 4/7 to 3/5 of a 2/1
octave. The presence of nonparallel fifths is a good warning that
these tunings are at least variations as "emulations" of the EDO
systems from which they draw their inspiration.

For 5-EDO, it's hard to top George Secor's choice, also available
in the Scala archive as slendro5_2.scl:

1/1 7/6 4/3 3/2 7/4 2/1

Here's the tempered version:

<http://www.bestII.com/~mschulter/met24-quasi_5-EDO_F.scl>

! met24-quasi_5-EDO_F.scl
!
One of George Secor's alternatives for 5-EDO (blarney.txt)
5
!
264.84375
495.70313
703.12500
967.96875
2/1

Next comes 6-EDO, a tuning associated with the "whole-tone scale"
of Debussy, and one with an air of impressionistic mystery. Why
not a variation with what may be the most vague and
impressionistic interval of all, and one I can easily imagine
Debussy loving: Phi, or a close approximation, here 21/13.
My basic idea was:

1/1 9/8 9/7 63/44 21/13 39/22 2/1

To follow more closely what was happening in MET-24, and also to
get two locations for a 441/352 major third quite close to 5/4, I
modified this slightly:

<http://www.bestII.com/~mschulter/quasi_6-EDO.scl>

! quasi_6-EDO.scl
!
Emulation of 6-EDO
6
!
8/7
352/273
63/44
21/13
39/22
2/1

Here's the tempered version:

<http://www.bestII.com/~mschulter/met24-quasi_6-EDO.scl>

! met24-quasi_6-EDO.scl
!
Emulation of 6-EDO
6
!
232.03125
439.45312
622.26562
829.68750
992.57812
2/1

For 7-EDO, George Secor again had already presented a solution
tough to outdo: Ptolemy's Equable Diatonic, which George gave
with tetrachords in a harmonic division of 9:10:11:12. To do
something just a bit different, I went back to Ptolemy's own
arithmetic division of 12:11:10:9 (descending numbers here show
string lengths -- or, if you like, wavelengths).

1/1 12/11 6/5 4/3 3/2 18/11 9/5 2/1

See the Scala archive, ptolemy_hom.scl. Here's my tempered
version:

<http://www.bestII.com/~mschulter/met24-ptolemy_hom_Bup.scl>

! met24-ptolemy_hom_Bup.scl
!
Tempering of Ptolemy's Homalon or Equable Diatonic
7
!
150.00000
312.89063
495.70313
703.12500
853.12500
1016.01563
2/1

For 8-EDO, I turned to a harmonic division -- 11:12:13.

1/1 12/11 13/11 352/273 63/44 264/169 22/13 24/13 2/1

<http://www.bestII.com/~mschulter/quasi_8-EDO.scl>

! quasi_8-EDO.scl
!
Emulation of 8-EDO
8
!
12/11
13/11
352/273
63/44
264/169
22/13
24/13
2/1

Here's the tempered version:

<http://www.bestII.com/~mschulter/met24-quasi_8-EDO.scl>

! met24-quasi_8-EDO_Cup.scl
!
Emulation of 8-EDO
8
!
150.00000
289.45312
439.45312
622.26562
772.26562
911.71875
1061.71875
2/1

While 8-EDO gets no closer to 3/2 than 750 cents -- famous for
its use as a very wide fifth in the album _Galunlati_ by Jacky
Ligon and colleagues -- these tunings have one fifth as small as
717 cents, which could illustrate a neomedieval weakness for
nonparallel fifths.

Now for the Leprechaun Comma: it's the difference between a just
264/169 (772.218 cents) and 25/16 (772.627 cents), or 4225/4224.
Likewise, and maybe an easier place to start, it's the difference
between 169/132 (13/11 plus 13/12) and 32/25.

For 9-EDO, the 0-133.3-266.7 cent division not so surprisingly
suggested to me 14:13:12, and in fact Ibn Sina's full tetrachord
of 16:14:13:12 puts in an appearance -- and with it, nonparallel
fifths.

1/1 14/13 7/6 441/352 11/8 3/2 21/13 7/4 1323/704 2/1

<http://www.bestII.com/~mschulter/quasi_9-EDO.scl>

! quasi_9-EDO.scl
!
Emulation of 9-EDO
9
!
14/13
7/6
441/352
11/8
3/2
21/13
7/4
1323/704
2/1

<http://www.bestII.com/~mschulter/met24-secorian_9-like_Bb.scl>

! met24-secorian_9-like_Bb.scl
!
Emulation of 9-EDO
9
!
126.56250
264.84375
391.40625
554.29687
704.29687
829.68750
969.14062
1094.53125
2/1

For 10-EDO, I chose a tuning which combines familiar elements to
form a decatonic scale called Buzurg al-Erin, or Irish Buzurg --
a realization of the Near Eastern Buzurg mode around 1300
(here 14/13 16/13 4/3 56/39 3/2 8/7 24/13 2/1) along with lots of
support generally for tetrachords of 16:14:13:12 or 104:91:84:78
(1/1-8/7-26/21-4/3), both tunings favored by Ibn Sina. These
tetrachords with a tone, largish neutral third, and fourth might
be called "septimal Rast," although some might argue that Rast
implies a lower tone at or close to 9/8 rather than 8/7.

All steps except one are within 20 cents of the 120-cent interval
in 10-EDO, the exception or "wolf thirdtone" occurring between
56/39 and 3/2, 117:112 or 75.6 cents in the just version.

1/1 14/13 8/7 16/13 4/3 56/39 3/2 21/13 12/7 24/13 2/1

<http://www.bestII.com/~mschulter/buzurg_al-erin10.scl>

! buzurg_al-erin10.scl
!
Decatonic with septimal Buzurg, Rastlike modes (cf. Secor, blarney.txt)
10
!
14/13
8/7
16/13
4/3
56/39
3/2
21/13
12/7
24/13
2/1

<http://www.bestII.com/~mschulter/met24-buzurg_al-erin10_Cup.scl>

! met24-buzurg_al-erin10_Cup.scl
!
Decatonic with septimal Buzurg & Rastlike modes
10
!
126.56250
232.03125
357.42187
496.87500
622.26562
704.29687
829.68750
935.15625
1061.71875
2/1

An incidental effect of having steps at 14/13 and 8/7, or 7/4 and
13/7, is the production of steps at a just 52/49 (102.9 cents),
and of some more or less accurate approximations of 5/4, actually
closer in the tempered version. Mainly, however, this is a
wonderful tuning for Buzurg and septimal Rast.

Here's one way of expanding it into a 12-EDO alternative. by
adding degrees at the minor third and sixth. A small change in
the following JI version is that the classic 56/39 is slightly
compressed to 63/44, very close to MET-24:

1/1 14/13 8/7 16/13 4/3 56/39 3/2 21/13 12/7 24/13 2/1

<http://www.bestII.com/~mschulter/alternative12.scl>

! alternative12.scl
!
Superset of Buzurg al-Erin with 13/11, 39/22
12
!
14/13
8/7
13/11
16/13
4/3
63/44
3/2
21/13
12/7
39/22
24/13
2/1

<http://www.bestII.com/~mschulter/met24-alternative12_Cup.scl>

! met24-alternative12_Cup.scl
!
Buzurg al-Erin 10 plus approx 13/11, 39/22
12
!
126.56250
232.03125
289.45312
357.42187
496.87500
622.26562
704.29687
829.68750
935.15625
992.57812
1061.71875
2/1

This addition gives us a variation on one oft-quoted version of
the Archytas Diatonic at 32:28:27:24 or 1/1-8/7-32/27-4/3, here
slightly modified to 104:91:88:78 or 1/1-8/7-13/11-4/3. This
tetrachord is one of my favorites.

Now, having jumped from 10-EDO to 12-EDO, we come back to 11-EDO:

14/13 44/39 63/52 14/11 231/169 16/11 264/169 18/11 252/143 24/13 2/1

<http://www.bestII.com/~mschulter/quasi_11-EDO.scl>

! quasi_11-EDO.scl
!
Emulation of 11-EDO
11
!
14/13
44/39
63/52
14/11
231/169
16/11
264/169
18/11
252/143
24/13
2/1

<http://www.bestII.com/~mschulter/met24-quasi_11-EDO_Ebup.scl>

! met24-quasi_11-EDO_Ebup.scl
!
Emulation of 11-EDO
11
!
125.39063
207.42188
332.81250
414.84375
540.23438
645.70313
772.26563
853.12500
979.68750
1060.54687
2/1

Here, curiously, the closest fifths to 3/2 are at 77/52 (679.6
cents), for example 11:8 plus 14:13; and 117/77 (724.3 cents),
for example 9:7 plus 13:11 -- identically impure by 78/77 or 22.3
cents. While these are outside the regular diatonic range from
7-EDO to 5-EDO, one might argue that in an 11-EDO emulation
they are quite close by comparison to 654.5 cents -- if not
nonparallel fifths, at least nonsimilar ones (by analogy to
restrictions in 16th-century European counterpoint against
certain fifths approached by similar motion which were regarded
as uncomfortably close to outright parallels).

There are some incredible 11-EDO-like modes, some of them
demonstrating what I term the Battaglia effect, where, in a
melodic mode or vertical sonority involving a large major third
around 9/7 and a small minor sixth around 14/9 or 264/169, for
example, the minor sixth is perceived as if it were a fifth.
This _quinta stravagante di Battaglia_ is a new landmark in
expressive art and applied psychoacoustics alike -- Bravo!
And _stravagante_ (plural _stravaganti_) seems the right word.

<http://www.christopherstembridge.org/stravaganti.htm>

Peace and love,

Margo Schulter
mschulter@...

🔗genewardsmith <genewardsmith@...>

🔗Margo Schulter <mschulter@...>

🔗genewardsmith <genewardsmith@...>

🔗Andy <a_sparschuh@...>

🔗genewardsmith <genewardsmith@...>

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Hello, all,
>
> For a long time, George Secor has been advocating and exploring
> the process of approximating EDO tunings with just ratios. Here
> is his charming and germinal parable on "Blarney," and a recent
> post presenting a fine just alternative to 10-EDO.
>
> </tuning/files/secor/blarney.txt>
> </tuning/topicId_102422.html#102795>
>
> Here, I'll go a step further by giving some EDO alternatives not
> only in the form of JI approximations, but as "near-just"
> approximations of those approximations in the MET-24 tuning.
> We'll move from 5-EDO through 12-EDO, with another post to
> address the interesting question George raised of 15-EDO.

Margo (and George Secor if you're reading), I'm interested in what you think of the following idea.

As far as I know, the reasons to use JI instead of equal temperament are:
* To have some purely intoned, beatless intervals, and
* To have a variety of subtly different intervals (rather than "homogenized" or "whitewashed" intervals)
I think those two broad categories cover all the reasons that have been put forward. If you know of a reason that falls completely outside those two, please let me know of it!

So, my idea is that both of those goals can be furthered by using an EDO approximation where each EDO step is approximated, not by a single JI ratio, but by a cluster of them that are spaced much closer together than the EDO steps. That way, the scale approximates the EDO just as well as a scale with single pitches for each note (as long as you're comfortable with the mental leap of treating it as a scale of clusters rather than a scale of individual pitches), and also, there can be more pure concordances as well as a larger variety of subtle shades of intervals.

> Also, with the alternatives for 8-EDO and 11-EDO, we'll encounter
> the Leprechaun Comma, 4225/4224 or 0.409808 cents, which I named
> as a tribute both to George's "Blarney" and to Eire, the Emerald
> Isle. It's appropriately a small comma that I discovered or
> rediscovered in the process of approximating small EDO's, and
> thus not unfitting to honor the wee people -- assuming that
> this comma, mentioned as unnamed in some previous posts, has not
> yet received a name.

Since it's a 13-limit superparticular it was listed on http://xenharmonic.wikispaces.com/List+of+Superparticular+Intervals ; it didn't yet have a name so I added yours.

> Please be warned some of these small-EDO alternatives have what
> might be considered a serious fault: nonparallel fifths!
> "Nonparallel fifths" are intervals at or near 3/2 in an emulation
> of an EDO without such "conventional fifths" -- say, with all
> intervals outside Blackwood's regular diatonic range from 7-EDO
> (685.71 cents) to 5-EDO (720 cents), or from 4/7 to 3/5 of a 2/1
> octave. The presence of nonparallel fifths is a good warning that
> these tunings are at least variations as "emulations" of the EDO
> systems from which they draw their inspiration.

With cluster scales we can have nonparallel fifths starting and ending on every step!

> For 5-EDO, it's hard to top George Secor's choice, also available
> in the Scala archive as slendro5_2.scl:
>
> 1/1 7/6 4/3 3/2 7/4 2/1

Here's my clusterization of this one, which serves as a good example to introduce the concept:

1/1 {8/7 7/6} {21/16 4/3} {3/2 32/21} 7/4 2/1

If we analyze this we see that:
* For each cluster, the exact 5edo step occurs between the pair of notes in the cluster. The clusters are thus quite evenly spaced if you consider their "average positions"; even more so than the original scale.
* From any cluster to the cluster 3 steps above it, there is some choice of pitches that results in an exact 3/2 (and the same goes for clusters 2 steps apart and 4/3).
* The addition of only 3 new pitches greatly increases the variety of intervals available.

(The astute reader will note how closely this scale is related to slendric temperament, but I'm deliberately avoiding posting the slendric-tempered version of it because that's a "whitewashed" version that would merely complicate matters. Do note, however, the occurrence of 147/128 (239.6 cents) and 512/441 (258.4 cents) intervals; if you treat those as equivalent to 8/7 or 7/6 respectively then you're effectively using slendric temperament.)

I plan to make a "clusterized" version of each of these scales, but that can wait for a later email (and perhaps be informed by replies to this one). For now, let me leave with this (preliminary) clusterized version of the 7edo approximation:

> For 7-EDO, George Secor again had already presented a solution
> tough to outdo: Ptolemy's Equable Diatonic, which George gave
> with tetrachords in a harmonic division of 9:10:11:12. To do
> something just a bit different, I went back to Ptolemy's own
> arithmetic division of 12:11:10:9 (descending numbers here show
> string lengths -- or, if you like, wavelengths).
>
> 1/1 12/11 6/5 4/3 3/2 18/11 9/5 2/1

1/1 {12/11 11/10 10/9} {6/5 121/100} {4/3 27/20} {40/27 3/2} {121/75 18/11} 9/5 2/1

Keenan

🔗Margo Schulter <mschulter@...>

> Margo (and George Secor if you're reading), I'm
> interested in what you think of the following idea.

Hi, Keenan, and while I can't speak for George, I think you
have a great idea! In a way, it's what happens in a tuning
like Peppermint or MET-24 for maqam -- and also, by
transposition, in something like George's 17-WT as well.

> As far as I know, the reasons to use JI instead of equal temperament are:

> * To have some purely intoned, beatless intervals, and

True, or "close to beatless" in a near-just unequal temperament.

> * To have a variety of subtly different intervals (rather
> than "homogenized" or "whitewashed" intervals)

Also true. And this wording can hold regardless of what
people think about questions such as whether superparticular
melodic steps may have any special attraction. Lots of maqam
music features varied steps -- and likewise gamelan music --
without any necessary focus on the superparticular.

> I think those two broad categories cover all the reasons
> that have been put forward. If you know of a reason that
> falls completely outside those two, please let me know of
> it!

Well, if in doubt, "variety of subtly different intervals"
covers lots of the relevant ground!

> So, my idea is that both of those goals can be furthered
> by using an EDO approximation where each EDO step is
> approximated, not by a single JI ratio, but by a cluster
> of them that are spaced much closer together than the EDO
> steps. That way, the scale approximates the EDO just as
> well as a scale with single pitches for each note (as
> long as you're comfortable with the mental leap of
> treating it as a scale of clusters rather than a scale of
> individual pitches), and also, there can be more pure
> concordances as well as a larger variety of subtle shades
> of intervals.

This is very fitting, and can apply not only to an EDO, but
to what might be called an "abstract maqam," or more
generally an "abstract modality."

Indeed Can Akkoc, as a scholar of Turkish music (note that
"c" in the modern Turkish alphabet is like English "j"),
suggested that often a _perde_ or abstract "step" in a maqam
is actually represented by a "cluster" of pitches covering
the area of a comma or considerably more. Subtle shadings,
primary melodic, are the main point. So your own insight and
the Akkoc research independently arrive at a similar point
with which I strongly agree.

In fact, sometimes EDO-like notations can actually represent
an "abstract maqam" (or "abstract jins," a jins, with the
plural form ajnas, being a genus -- a trichord, tetrachord,
or pentachord, usually). Especially with a notation in 17 or
24, the idea of each step as a "cluster" captures lots of
the Near Eastern viewpoint of traditional performers.

For example, Rast would in 17-notation be 3 2 2. Among some
Turkish or Syrian musicians, we might get something in
practice like 1/1-9/8-26/21-4/3 (0-204-370-498 cents). But
that high neutral third at 26/21, in a Turkish conception,
is really part of a "cluster." In descending to a final
cadence, one might use 16/13 or possibly 11/9 -- lower,
pulling toward the tonic or final. This is called _cazibe_
or "gravity," and your cluster concept allows for precisely
this sort of thing.

Some Turkish musicians go so far as to think of the whole
region between 6/5 and 5/4, say, as "a 3-comma glissando
zone," with a middle or neutral interval often free to
wander around that zone as the musical context or
performer's intuition might dictate.

At a very modest level, something like Peppermint or MET-24
does provide, essentially, a division of the octave into 17
thirdtones plus 7 additional comma pairs. Depending on where
we are in the system, we might have a choice between 26/21
and 11/9; or 7/6 and 13/11, etc.

[On 4225/4224 or Leprechaun comma]

> Since it's a 13-limit superparticular it was listed on
> <http://xenharmonic.wikispaces.com/List+of+Superparticular+Intervals>
> ; it didn't yet have a name so I added yours.

Thank you! This is one of things that I found by accident,
and the name is definitely a tribute to George's
blarney.txt.

> With cluster scales we can have nonparallel fifths
> starting and ending on every step!

As you show brilliantly!

> Here's my clusterization of this one, which serves as a good example to
> introduce the concept:
> 1/1 {8/7 7/6} {21/16 4/3} {3/2 32/21} 7/4 2/1

A neat set, and it makes sense that a JI or near-JI set
might be larger than the underlying EDO.

> If we analyze this we see that:

> * For each cluster, the exact 5edo step occurs between the pair of notes in the
> cluster. The clusters are thus quite evenly spaced if you consider their
> "average positions"; even more so than the original scale.

That "even more so than the original scale" catches my
attention.

Here I'm tempted to ask, "Isn't 0-240-480-720-960-1200 about
as evenly spaced as you can get?" -- knowing, however, that
"average positions" could have various meanings.

> * From any cluster to the cluster 3 steps above it, there
> is some choice of pitches that results in an exact 3/2
> (and the same goes for clusters 2 steps apart and 4/3).

So, in terms of slendro, we have different flavors of
_kempyung_ available.

> * The addition of only 3 new pitches greatly increases
> the variety of intervals available.

Yes, it does. And as Jacques Dudon has discussed in his
articles on JI gamelan, we also get the important element of
intervals like 512/441 (21/16 to 32/21) that could serve as
hemifourths of a kind. One of the weakness of MET-24, by the
way, is the absence of anything between 8/7 and 7/6, etc.
But your 8-note scale has it!

> (The astute reader will note how closely this scale is
> related to slendric temperament, but I'm deliberately
> avoiding posting the slendric-tempered version of it
> because that's a "whitewashed" version that would merely
> complicate matters. Do note, however, the occurrence of
> 147/128 (239.6 cents) and 512/441 (258.4 cents)
> intervals; if you treat those as equivalent to 8/7 or 7/6
> respectively then you're effectively using slendric
> temperament.)

Here the 1029/1024 contrast has its advantages, one of which
might be clearest in polyphony. Jacques Dudon and I have
both independently noticed that 147/128 (or a tempered 240
cents in 20-EDO, say) can serve as a "quasi-third"
contracting to a unison. I just tried it with 8/7, and get
the sense that it doesn't quite give this "thirdlike"
impression -- somehow things are just a bit too close. To
get the contrapuntal effect, it might help to have an
additional step at 49/36 say, so that we could resolve a
dyad 21/16-32/21 to 49/36 by stepwise contrary motion.
It would be bit like, in 20-EDO, having 0-240 cents
resolving to a unison on 60 cents.

But with your 8-note system or more, we get both some pure
7/6 and 7/4 intervals, and that 147/128. As you point out,
in slendric temperament we would get less variety.

In a Peppermint-34, by the way, we'd have a few locations of
238.566 and 257.338 cents, as well as the usual just 7/6
thirds and 7/4 approximations at 970.967 cents. Similarly
for MET-24 -- the limiting factor is only having 12-note
chains of fifths, where 16 generators gives the hemifourths
of interest.

> I plan to make a "clusterized" version of each of these
> scales, but that can wait for a later email (and perhaps
> be informed by replies to this one). For now, let me
> leave with this (preliminary) clusterized version of the
> 7edo approximation:

[Here I'm arbitrarily breaking the first line at 3/2 -- M.S.]

> 1/1 {12/11 11/10 10/9} {6/5 121/100} {4/3 27/20} {40/27 3/2}
> {121/75 18/11} 9/5 2/1

Having three notes to a cluster is neat, and I notice
intervals at 121/100 (330 cents, square of 11/10) and 121/75
(828 cents) right near the line between "5-based minor" and
"supraminor" intervals (e.g. 63/52, 21/13).

But the simultaneous permutations from the same location are
striking. I'm wondering about a 363/200, which would be
quite close to 6 steps of 7-EDO (just as 147/128 is very
close to 1 step of 5-EDO).

With many thanks,

Margo

🔗Keenan Pepper <keenanpepper@...>

I'm working on figuring out what's going on with all of the EDO-approximating JI scales from the first email; here are brief responses to some particular points for the time being.

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
> That "even more so than the original scale" catches my
> attention.
>
> Here I'm tempted to ask, "Isn't 0-240-480-720-960-1200 about
> as evenly spaced as you can get?" -- knowing, however, that
> "average positions" could have various meanings.

By "original scale" I meant the original JI scale, 1/1 7/6 4/3 3/2 7/4 2/1. The average positions of the clusters are more evenly spaced than these 5 JI pitches, but not, of course, more evenly spaced than 5edo (which is, as you say, as even as possible since the steps are exactly equal).

> Here the 1029/1024 contrast has its advantages, one of which
> might be clearest in polyphony. Jacques Dudon and I have
> both independently noticed that 147/128 (or a tempered 240
> cents in 20-EDO, say) can serve as a "quasi-third"
> contracting to a unison. I just tried it with 8/7, and get
> the sense that it doesn't quite give this "thirdlike"
> impression -- somehow things are just a bit too close. To
> get the contrapuntal effect, it might help to have an
> additional step at 49/36 say, so that we could resolve a
> dyad 21/16-32/21 to 49/36 by stepwise contrary motion.
> It would be bit like, in 20-EDO, having 0-240 cents
> resolving to a unison on 60 cents.

That's an interesting perspective on the difference in impressions between 8/7 and a slightly wider interval. But I must say, the resolution you describe is quite different from the way I imagined these kind of cluster scales being used. I conceive of the pitches in each cluster as intonational variants of the same "note", exactly the way you described things in makam music. Different pitches of the same cluster would be used in different contexts to achieve diversity of expression (or simply pure consonant intervals), but the contrast between them would never be exploited in a cadential resolution such as the one you mention.

In other words, resolving from one note of a cluster to another is treating it as an ordinary scale with >5 notes, in contrast to what I had in mind which was treating it as a 5-note scale with intonational variants, and using it more similarly to 5edo itself. A resolution has to resolve from one note to another; if the two pitches are intonational variants of the same note then it can't be a "resolution" without a clash of concepts happening.

Keenan

🔗Margo Schulter <mschulter@...>

>> Here I'm tempted to ask, "Isn't 0-240-480-720-960-1200
>> about as evenly spaced as you can get?" -- knowing,
>> however, that "average positions" could have various
>> meanings.

> By "original scale" I meant the original JI scale, 1/1
> 7/6 4/3 3/2 7/4 2/1. The average positions of the
> clusters are more evenly spaced than these 5 JI pitches,
> but not, of course, more evenly spaced than 5edo (which
> is, as you say, as even as possible since the steps are
> exactly equal).

Dear Keenan,

Thank you for this clarification! When I asked my question,
I felt a bit like the student in one of those dialogues on
counterpoint where the teacher clears up a beginner's
misunderstanding in a humorous way -- feeling confident that
you would likely clear things up, as you have.

And your explanation really brings home an interesting
concept of "untempering" -- not just putting in a more
accurate integer ratio for each step of the original, but
seeking out alternative rational pitches, in your example
two or three, which the tempered step might represent.
So the "average" concept really is relevant!

> That's an interesting perspective on the difference in
> impressions between 8/7 and a slightly wider
> interval. But I must say, the resolution you describe is
> quite different from the way I imagined these kind of
> cluster scales being used. I conceive of the pitches in
> each cluster as intonational variants of the same "note",
> exactly the way you described things in makam
> music. Different pitches of the same cluster would be
> used in different contexts to achieve diversity of
> expression (or simply pure consonant intervals), but the
> contrast between them would never be exploited in a
> cadential resolution such as the one you mention.

I agree! Your conception is the relevant and correct one for
this project or approach, while my little digression was
actually out of place in this context for two reasons. The
first is exactly as you state: the idea is to have
intonational variations of the same "note," not a "contrast"
between them. And the second ties in with what you say next.

> In other words, resolving from one note of a cluster to
> another is treating it as an ordinary scale with >5
> notes, in contrast to what I had in mind which was
> treating it as a 5-note scale with intonational variants,
> and using it more similarly to 5edo itself. A resolution
> has to resolve from one note to another; if the two
> pitches are intonational variants of the same note then
> it can't be a "resolution" without a clash of concepts
> happening.

My second reason is a rather simple corollary of your
statement here. A basic implication of 5-EDO or your JI
cluster version is that 240 cents or 147/128, say, is a
"one-step" interval which therefore is not expected to
resolve to a unison by contrary motion, which my
"quasi-third" resolution would require! Indeed, the very
idea of a "third" here has a contrapuntal sense that there
is a "step between" the two notes to which they can both
resolve. And that, indeed, would violate the premise of a
cluster tuning by "treating it as an ordinary scale with >5
notes."

In short, you're obviously right!

Wishing you a Happy New Year,

Margo