Yahoo Tuning Groups Ultimate Backup tuning RMP definition of gentle temperament: 29-EDO to 17-EDO

Hello, all.

This is a quick summary of my new thoughts on the gentle
temperament as an RMP concept, which may make things at once
simpler and more intuitive.

Having read some of Gene Ward Smith's helpful writings on gentle,
and also benefited from an excellent offgroup discussion with Mike
Battagilia, I would like to propose the following approach to
defining gentle in clear terms based on characteristics present
in a simple rank-2 system with 17 or more notes.

In its basic rank-2 form, gentle may be identified by its mapping
of prime 7 to +15 fifths, and by its use of -8 and +9 generators
respectively to produce larger and smaller neutral thirds. The
64/63 and 144/143 are both observed, and mapped to the 17-note
comma. Thus +15 fifths represents 7/4, while -2 fifths represents
16/9 (or 39/22). Likewise, -8 fifths gives the better
representation of 16/13, and +9 fifths, the better representation
of 11/9.

The range of the gentle mapping thus extends from 29-EDO as the
lower bound to 17-EDO as the upper bound. Note that for septimal
intervals in gentle, +29 fifths represents 49/48 -- given that
-14 fifths represent 12/7, while +15 fifths represent 7/4. Below
29-EDO, in the Pythagorean region, -29 fifths likewise represents
49/48; precisely at 29-EDO, the 29-note comma is tempered out,
and along with it 49/48.

At the upper bound of gentle, 17-EDO, the situation changes
dramatically in two ways. First, +15 fifths and -2 fifths become
momentarily equivalent, with -2 fifths as the best mapping of 7
in the region beyond 17-EDO -- in other words, 64/63 is tempered
out, in contrast to its observance in gentle.

Also, in 17-EDO, -8 and +9 fifths momentarily become equivalent,
identically producing a single size of neutral third equal to
half the size of the fifth -- so that 144/143, along with the
17-note comma, is tempered out. Beyond 17-EDO, 144/143 is again
observed in such tunings as 56-EDO and 39-EDO -- but with the
direction of the 17-note comma now positive, so that +9 fifths is
a larger neutral interval than -8 fifths.

Of course, it is the mappings of 14/11 and 13/11 to +4/-3 fifths,
and the tempering out of 364/363, 352/351, 896/891, and
10648/10647 that are the most obvious feature of the system. I
happily join with Gene Ward Smith and others in calling 364/363
the gentle comma.

However, the observance of 64/63 and 144/143, together with the
mapping of 7 to +15 fifths and of larger and smaller neutral
thirds to -8 and +9 intervals, sets 29-EDO and 17-EDO as logical
lower and upper bounds.

In my view, the region around 704 cents -- especially when used
in rank-3 parapyth systems -- is the most characteristic for
gentle, with its potential through these systems for near-just
accuracy with four different sizes of neutral intervals.

However, someone wanting to explore and enjoy gentle in a simple
and accessible form might prefer 17 or 24 notes of 63-EDO or
80-EDO, for example, or maybe even a temperament a bit beyond one
of my previously proposed upper boundaries, 705 cents -- say with
a just 11/9 from +9 fifths. An upper bound of 17-EDO is logical,
rather than a matter of highly subjective taste or opinion as to
how much one can "gently" temper a fifth.

My conclusion is also that the region beyond 17-EDO clearly
belongs in a different category, both because of the tempering
out of 64/63, and because of the different behavior of neutral
intervals (with the region around 707 cents very attractive, as
George Secor demonstrates in his 17-WT system).

From the viewpoint of rank-2 prime mappings, the leapfrog region
around 63-EDO may be especially attractive; it is a kind of
"rank-2 equivalent of parapyth," with primes 2-3-7-11-13 all
reasonably close to just using only a single chain of fifths --
plus a virtually just prime 23 as a bonus! The two sizes of
neutral intervals are fine for some great Near Eastern music,
with the regularity of rank-2 making things likely a bit more
accessible than the complexities of parapyth. The minimax
solution of a virtually just 7/6 for other ratios of 7 is also a
big attraction of 17 or 24 of 63-EDO -- not to exclude the full
system.

While a rank-3 approach will indeed prefer the "704-cent
neighborhood" where the temperament of the fifth is indeed
"gentle," and comparable to that of 12n-EDO, a rank-2 approach
might well take 63-EDO (or the almost identical tuning with 7/6
as an eigenmonzo) as the ideal choice, or at least a fine
starting point.

Thinking in terms of "29-EDO to 17-EDO" makes the definition of
gentle clear, and leaves room for lots of preferences in terms of
more or less tempering, rank-2 or rank-3, etc.

Most appreciatively,

Margo Schulter
mschulter@...

🔗gdsecor <gdsecor@...>

Margo, how could you not mention 46-EDO in discussing this range of fifths?

--George

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Hello, all.
>
> This is a quick summary of my new thoughts on the gentle
> temperament as an RMP concept, which may make things at once
> simpler and more intuitive.
>
> Having read some of Gene Ward Smith's helpful writings on gentle,
> and also benefited from an excellent offgroup discussion with Mike
> Battagilia, I would like to propose the following approach to
> defining gentle in clear terms based on characteristics present
> in a simple rank-2 system with 17 or more notes.
>
> In its basic rank-2 form, gentle may be identified by its mapping
> of prime 7 to +15 fifths, and by its use of -8 and +9 generators
> respectively to produce larger and smaller neutral thirds. The
> 64/63 and 144/143 are both observed, and mapped to the 17-note
> comma. Thus +15 fifths represents 7/4, while -2 fifths represents
> 16/9 (or 39/22). Likewise, -8 fifths gives the better
> representation of 16/13, and +9 fifths, the better representation
> of 11/9.
>
> The range of the gentle mapping thus extends from 29-EDO as the
> lower bound to 17-EDO as the upper bound. Note that for septimal
> intervals in gentle, +29 fifths represents 49/48 -- given that
> -14 fifths represent 12/7, while +15 fifths represent 7/4. Below
> 29-EDO, in the Pythagorean region, -29 fifths likewise represents
> 49/48; precisely at 29-EDO, the 29-note comma is tempered out,
> and along with it 49/48.
>
> At the upper bound of gentle, 17-EDO, the situation changes
> dramatically in two ways. First, +15 fifths and -2 fifths become
> momentarily equivalent, with -2 fifths as the best mapping of 7
> in the region beyond 17-EDO -- in other words, 64/63 is tempered
> out, in contrast to its observance in gentle.
>
> Also, in 17-EDO, -8 and +9 fifths momentarily become equivalent,
> identically producing a single size of neutral third equal to
> half the size of the fifth -- so that 144/143, along with the
> 17-note comma, is tempered out. Beyond 17-EDO, 144/143 is again
> observed in such tunings as 56-EDO and 39-EDO -- but with the
> direction of the 17-note comma now positive, so that +9 fifths is
> a larger neutral interval than -8 fifths.
>
> Of course, it is the mappings of 14/11 and 13/11 to +4/-3 fifths,
> and the tempering out of 364/363, 352/351, 896/891, and
> 10648/10647 that are the most obvious feature of the system. I
> happily join with Gene Ward Smith and others in calling 364/363
> the gentle comma.
>
> However, the observance of 64/63 and 144/143, together with the
> mapping of 7 to +15 fifths and of larger and smaller neutral
> thirds to -8 and +9 intervals, sets 29-EDO and 17-EDO as logical
> lower and upper bounds.
>
> In my view, the region around 704 cents -- especially when used
> in rank-3 parapyth systems -- is the most characteristic for
> gentle, with its potential through these systems for near-just
> accuracy with four different sizes of neutral intervals.
>
> However, someone wanting to explore and enjoy gentle in a simple
> and accessible form might prefer 17 or 24 notes of 63-EDO or
> 80-EDO, for example, or maybe even a temperament a bit beyond one
> of my previously proposed upper boundaries, 705 cents -- say with
> a just 11/9 from +9 fifths. An upper bound of 17-EDO is logical,
> rather than a matter of highly subjective taste or opinion as to
> how much one can "gently" temper a fifth.
>
> My conclusion is also that the region beyond 17-EDO clearly
> belongs in a different category, both because of the tempering
> out of 64/63, and because of the different behavior of neutral
> intervals (with the region around 707 cents very attractive, as
> George Secor demonstrates in his 17-WT system).
>
> From the viewpoint of rank-2 prime mappings, the leapfrog region
> around 63-EDO may be especially attractive; it is a kind of
> "rank-2 equivalent of parapyth," with primes 2-3-7-11-13 all
> reasonably close to just using only a single chain of fifths --
> plus a virtually just prime 23 as a bonus! The two sizes of
> neutral intervals are fine for some great Near Eastern music,
> with the regularity of rank-2 making things likely a bit more
> accessible than the complexities of parapyth. The minimax
> solution of a virtually just 7/6 for other ratios of 7 is also a
> big attraction of 17 or 24 of 63-EDO -- not to exclude the full
> system.
>
> While a rank-3 approach will indeed prefer the "704-cent
> neighborhood" where the temperament of the fifth is indeed
> "gentle," and comparable to that of 12n-EDO, a rank-2 approach
> might well take 63-EDO (or the almost identical tuning with 7/6
> as an eigenmonzo) as the ideal choice, or at least a fine
> starting point.
>
> Thinking in terms of "29-EDO to 17-EDO" makes the definition of
> gentle clear, and leaves room for lots of preferences in terms of
> more or less tempering, rank-2 or rank-3, etc.
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@...
>

🔗Marcel de Velde <marcel@...>

Hello Margo and list,

Thank you for your post.
I find this area of temperament very interesting coming from a Pythagorean standpoint.

I have been experimenting myself with different functionality between enharmonics in Pythagorean.
The difference between for instance a major third and a diminished fourth, or the difference between an augmented prime and a minor second.
I have also been trying to make sense of Arab and Turkish maqam music this way.
In which a neutral second would translate to for instance D-Gbbb. This however has been a longstanding point of doubt to me.
Whether to see the neutral second steps as D-Gbbb and Gbbb-F for instance, or D-Fb and Fb-F.
I have been working on techniques for harmonizing both versions, but everytime I think I've found a way to distinguish between for instance D-Gbbb and D-Fb I find out I failed and the D-Gbbb harmonization works in D-Fb as well (and often better).
I'm still in doubt on what the functional representation of the neutral seconds (and neutral thirds etc as well) are, but right now if I have to make a guess I'd say that the neutral seconds are infact augmented primes and diminished thirds who's intonation is exaggerated to distinguish them from the minor second and major second.
This leads to precisely the temperaments you discuss here. Something between 3/2 and 705 cents as a fifth (I personally consider something above 705 cents as too much of an exaggeration).
46edo as mentioned is very good here as well.

If we play for instance C-D-E-F-G-A-B-C melodically we will functionally hear it as such within a range of tuning differences. Whether it be Pythagorean or 1/4 comma meantone etc.
And when we play C-D-Fb-F-G-A-Cb-C in Pythagorean, which is functionally different, it is very close to C-D-E-F-G-A-B-C. In fact so close it's very hard to hear the functional difference when only playing it as a solo melody (as maqam music does). If we harmonize them the 2 are completely different and the distinction is made clear, but for solo melodies not.
So when we then exaggerate the D-Fb and Fb-F we enter the area of the neutral seconds. Make a lot of sense to me.

I've made 2 quick demonstrations of a few of the concepts mentioned here for the Xenharmonic alliance facebook group and uploaded these to soundcloud:

https://soundcloud.com/justintonation/tuning-demo-enharmonic

This one compares the melodies C-D-E-F-G-A-B-Cvs C-D-Fb-F-G-A-Cb-Cvs C-D-Gbbb-F-G-A-Dbbb-C
I can tell the tuning difference between al 3 of course, but the first 2 tend to both sound like the major scale to me, while the third one is no longer the major scale to me functionally but something different (maqam Rast is tuned about this way as you probably know).
Btw at times I do hear the Fb as functionally different also in the above example, and I know of several other modes where the difference is more easily clear to me, perhaps something for a future demo.

Then I harmonize all 3 versions the same as if all 3 are C-D-E-F-G-A-B-C with a standard I-ii-iii-IV-V-vi-viidim-I progression.
Here you can hear that the second and third version no longer represent a different functionality and simply sound out of tune. The progression clearly indicates an E and B there so simply tuning it differently up to a quarter tone does not change how we perceive it's function (mistuning more than a quarter tone wil flip it to become an Eb and Bb changing the progression)

Then I harmonize all 3 versions with a special (but very ugly haha) chromatic progression which does not have a preference for either E or Fb (and Gbbb could be possible to though less likely) where all 3 versions can be heard in the same context without comma shifts or small step sizes in any of the voices so we hear only the tuning difference.
I get the same result as with the solo melody version, only upon close listening I found that I tend to hear more likely the Fb as functionally different from the E, and the Fb and Gbbb as functionally the same.

The second demonstration:
https://soundcloud.com/justintonation/tuning-demo-improv-showing

First part is a quick random harmonization of C-D-Fb F (also going down to Cb in the end) which is played in the highest voice.
As such its a much better demo of harmonization of augmented primes and diminished third than the first demo. Though I did it quickly and one can make much prettier and coherent harmonizations of these chromatic steps. You can already hear an Arabic / Turkish vibe in it though (only now harmonically :) )

Second part is a demonstration I made for Mike about an E-G-Bb-Eb chord (which Mike tended to hear as having A# and D# instead) where I demonstrate that by resting on it is becomes clear to interpret it as E-G-Bb-Eb.
Not very relevant to this discussion though, but I do play an augmented prime up to E-G-B-E and down again etc.

Then third and fourth part are not relevant at all but were to demonstrate that the "Hendrix chord" has a minor third and not an augmented second (as it sais on wiki etc).

Sorry for the horrible playing! But I made it quickly on a cheap 1980s synth keyboard with a badly setup sampled piano on the computer and perhaps my playing isn't that great either :)

I plan on making a better demo demonstrating the use of augmented primes and diminished thirds as melodic step sizes.
Both solo melodies of different modes which are in Pythagorean and the exaggerated Pythagorean you describe (I'll probably pick 46edo for this).
And proper harmonizations of this.

Hope this is interesting to you.

Kind regards,
Marcel de Velde

> Hello, all.
>
> This is a quick summary of my new thoughts on the gentle
> temperament as an RMP concept, which may make things at once
> simpler and more intuitive.
>
> Having read some of Gene Ward Smith's helpful writings on gentle,
> and also benefited from an excellent offgroup discussion with Mike
> Battagilia, I would like to propose the following approach to
> defining gentle in clear terms based on characteristics present
> in a simple rank-2 system with 17 or more notes.
>
> In its basic rank-2 form, gentle may be identified by its mapping
> of prime 7 to +15 fifths, and by its use of -8 and +9 generators
> respectively to produce larger and smaller neutral thirds. The
> 64/63 and 144/143 are both observed, and mapped to the 17-note
> comma. Thus +15 fifths represents 7/4, while -2 fifths represents
> 16/9 (or 39/22). Likewise, -8 fifths gives the better
> representation of 16/13, and +9 fifths, the better representation
> of 11/9.
>
> The range of the gentle mapping thus extends from 29-EDO as the
> lower bound to 17-EDO as the upper bound. Note that for septimal
> intervals in gentle, +29 fifths represents 49/48 -- given that
> -14 fifths represent 12/7, while +15 fifths represent 7/4. Below
> 29-EDO, in the Pythagorean region, -29 fifths likewise represents
> 49/48; precisely at 29-EDO, the 29-note comma is tempered out,
> and along with it 49/48.
>
> At the upper bound of gentle, 17-EDO, the situation changes
> dramatically in two ways. First, +15 fifths and -2 fifths become
> momentarily equivalent, with -2 fifths as the best mapping of 7
> in the region beyond 17-EDO -- in other words, 64/63 is tempered
> out, in contrast to its observance in gentle.
>
> Also, in 17-EDO, -8 and +9 fifths momentarily become equivalent,
> identically producing a single size of neutral third equal to
> half the size of the fifth -- so that 144/143, along with the
> 17-note comma, is tempered out. Beyond 17-EDO, 144/143 is again
> observed in such tunings as 56-EDO and 39-EDO -- but with the
> direction of the 17-note comma now positive, so that +9 fifths is
> a larger neutral interval than -8 fifths.
>
> Of course, it is the mappings of 14/11 and 13/11 to +4/-3 fifths,
> and the tempering out of 364/363, 352/351, 896/891, and
> 10648/10647 that are the most obvious feature of the system. I
> happily join with Gene Ward Smith and others in calling 364/363
> the gentle comma.
>
> However, the observance of 64/63 and 144/143, together with the
> mapping of 7 to +15 fifths and of larger and smaller neutral
> thirds to -8 and +9 intervals, sets 29-EDO and 17-EDO as logical
> lower and upper bounds.
>
> In my view, the region around 704 cents -- especially when used
> in rank-3 parapyth systems -- is the most characteristic for
> gentle, with its potential through these systems for near-just
> accuracy with four different sizes of neutral intervals.
>
> However, someone wanting to explore and enjoy gentle in a simple
> and accessible form might prefer 17 or 24 notes of 63-EDO or
> 80-EDO, for example, or maybe even a temperament a bit beyond one
> of my previously proposed upper boundaries, 705 cents -- say with
> a just 11/9 from +9 fifths. An upper bound of 17-EDO is logical,
> rather than a matter of highly subjective taste or opinion as to
> how much one can "gently" temper a fifth.
>
> My conclusion is also that the region beyond 17-EDO clearly
> belongs in a different category, both because of the tempering
> out of 64/63, and because of the different behavior of neutral
> intervals (with the region around 707 cents very attractive, as
> George Secor demonstrates in his 17-WT system).
>
> From the viewpoint of rank-2 prime mappings, the leapfrog region
> around 63-EDO may be especially attractive; it is a kind of
> "rank-2 equivalent of parapyth," with primes 2-3-7-11-13 all
> reasonably close to just using only a single chain of fifths --
> plus a virtually just prime 23 as a bonus! The two sizes of
> neutral intervals are fine for some great Near Eastern music,
> with the regularity of rank-2 making things likely a bit more
> accessible than the complexities of parapyth. The minimax
> solution of a virtually just 7/6 for other ratios of 7 is also a
> big attraction of 17 or 24 of 63-EDO -- not to exclude the full
> system.
>
> While a rank-3 approach will indeed prefer the "704-cent
> neighborhood" where the temperament of the fifth is indeed
> "gentle," and comparable to that of 12n-EDO, a rank-2 approach
> might well take 63-EDO (or the almost identical tuning with 7/6
> as an eigenmonzo) as the ideal choice, or at least a fine
> starting point.
>
> Thinking in terms of "29-EDO to 17-EDO" makes the definition of
> gentle clear, and leaves room for lots of preferences in terms of
> more or less tempering, rank-2 or rank-3, etc.
>
> Most appreciatively,
>
> Margo Schulter
> mschulter@... <mailto:mschulter%40calweb.com>
>

🔗Margo Schulter <mschulter@...>

Dear George (and All),

What I suspect is that I didn't mention 46-EDO, or the almost
identical (56/11)^(1/4), because it's so obvious! The question I
was addressing for gentle was one of lower and especially upper
boundaries, rather than the incontestable heartland of this
temperament (or, I might say, temperament family).

However, your very fair question provides me an opportunity to
note your use of (56/11)^(1/4) or 704.377 cents in your 17-WT in
1978, 22 years before I became intrigued with this same
generator, not yet aware of your earlier precedent which would be
catalytic in so many ways once I learned of it in the summer of
2001, and we got in contact.

Of course, as a regular temperament, 704.377 cents or the
minutely smaller generator of 46-EDO (704.348 cents) is in a
sense the capital of gentle, just as 1/4-comma meantone or 31-EDO
is the capital of meantone, with just major thirds at 14/11 and
5/4 respectively.

What should really be emphasized is that in many ways, your two
temperaments of 1978 staked out part of the territory of gentle,
as well as the territory a bit above 17-EDO where the limma is
very close to the 28/27 of Archytas. While 29-HTT neatly
demonstrates a rank-3 parapyth tuning (2, 703.579, 58.090) as one
of its subsets, 17-WT features the 704.377-cent generator and
some just 14/11 thirds.

Of course, what both temperaments also demonstrate are the
virtues of irregular systems, an area where a "middle way" theory
is needed in order to catch up with the practice of 35 years ago!
If a given scheme can't handle irregularity, then let's either
supplement it or replace it with an approach which can.

While we're on the topic of 46-EDO or (56/11)^(1/4), why don't I
mention an interesting characteristic of this region of gentle.

Your 29-HTT and 17-WT demonstrate two strategies for representing
many primes with a limited complement of notes. My concept of
gentle is more in line with 17-WT: 2-3-7-9-11-13. Here I mention
the odd factor 9, although it isn't a prime, because it can often
be the most inaccurate 13-odd factor (off by twice the tempering
of the fifth). Of course, 29-HTT is much more ambitious,
including 2-3-5-7-9-11-13-15 with 4:5:6:7:9:11:13:15 ogdads!
But here I'm focusing mainly on 2-3-7-9-11-13, with 9 carrying a
special message: "Temper no more than you really think
appropriate to meet the goals of the system!"

For 2-3-7-9-11-13, the relevant subset of 29-HTT uses a rank-3
parapyth strategy: two chains of generators at 703.579 cents
(yielding a just 63/52 from +9 fifths) at 58.090 cents apart
(yielding a pure 7/4). The generator defining the spacing between
the chains is almost identical to 91/88, the difference between
22/13 and 7/4, since 22/13 (and 13/11) are virtually just. A
humorous footnote is that the HTT (High Tolerance Temperament)
generator of 703.579 cents or (504/13)^(1/9) differs from the
generator for a pure 22/13 or (44/13)^(1/3) at 703.597 cents,
when both generators are rounded off in cents to three decimal
places, by a typo. This irrational quantity, the difference
between 703.579 cents and 703.597 cents, might be called the
typisma -- not a serious naming proposal!

For 2-3-7-9-11-13, the 17-WT system uses a strategy outside the
scope of gentle itself as a family of regular rank-2 and rank-3
tunings: a tuning circle with nine larger fifths at 705.220 cents
which temper out 64/63. The result is not so accurate as in
strictly keeping to gentle and observing the 64/63 -- but very
musically practical, and quite awesome in the range of
isoharmonic and other chords made available with only 17 notes,
an amazingly appealing solution in terms of Fokker's concept of
seeking to optimize the balance between simplicity, accuracy, and
diversity.

Returning to the narrower field of gentle as a family of rank-2
and rank-3 systems, I would make a few observations about 46-EDO
or the almost identical just 14/11 tuning (704.377 cents).

First, from a 2-3-7-9-11-13 perspective, 46-EDO seems to me just
at the point where using the natural diesis (2d46, or 52.174
cents) starts to permit reasonably accurate approximations of
2-3-7-9 intervals. In 46-EDO, we get 7/4 at 965.217 cents from
+15 fifths -- curiously very close to 965.784 cents from +10
fifths in 1/4-comma meantone! -- and 7/6 at 260.870 cents, narrow
by six cents. With 9/7, however, the diesis isn't quite large
enough so that subtracting it from the fourth (as happens with
-13 fifths) gets us into the zone within five cents or so of this
ratio: we get 443.478 cents, closer to 128/99, for example (that
is, 4/3 less 33/32, which the 46-EDO diesis nicely approximates).

However, this imprecision might have its uses. From an
isoharmonic point of view, 0-547.8-965.2-1304.3 cents could serve
as a quite reasonable approximation of 8:11:14:17, with 756.522
cents wide by less than 2.9 cents from 17/11.

If the goal is a rank-2 gentle system that really maximizes
accuracy for 2-3-7-9 intervals, then something around 63-EDO
would be the minimax region (at or very close to a just 7/6).
Of course, from a viewpoint of neutral intervals, this moves away
from the very nice approximations of 14:17:21 in 46 (and a bit
below), and toward 32:39:48 (or 26:32:39), where 16/13 is simpler
than 17/14, but, as you observed back around late 2001, 14:17:21
is the lowest odd-limit neutral triad.

The other consideration, not central to the concept of gentle but
often of interest, is prime 5. In 46, the mapping of 5 is +21
fifths, or a "triapotome." In other words, if the apotome or
chromatic semitone (+7 fifths) is around 14/13 or a bit larger,
then three times this apotome (+21 fifths) will approximate 5/4.
With sufficiently long chains of fifths, this happens regardless
of the designer's intention.

Curiously, to optimize 5/4, the sweet spot would be just above
704 cents, or more specifically just above the point where 14/13
(128.298 cents) is just. The Keenan Pepper Noble Fifth tuning at
704.096 cents, with +21 fifths at 386.008 cents, comes very
close. In fact, two 29-chains of Pepper at 58.680 cents apart
(yielding a just 7/6) would produce no fewer than 32 locations
for 5/4 at either 386.008 cents (from +21 fifths within each
chain) or 388.077 cents (from -13 fifths less the spacing between
the chains, or possibly 22/17 less 121/117 for 234/187). This
would be Peppermint-58, a large but usual rank-3 parapyth. There
is also a superb 11:14:17, found in any Pepper chain of 17 notes
or more (with 14:11, 17:14, and 17:11 from +4, +9, and +13 fifths).

However, 46 from a 2-3-5-7-9-11-13 perspective has the advantage
that while its 7 is not as accurate as around 63-EDO, or its 5
as accurate as around Pepper or 121-EDO, it gets both reasonably
right (especially if 9/7 is not of special interest).

From a 2-3-7-9-11-13 perspective, of course, 46 will be of
interest in any event because of the virtually just 14/11.
And from this perspective, 17-WT is a very important landmark
because it uses a 704.377-cent generator to obtain 14/11 and 11/7
(as part of a larger scheme with numerous isoharmonic chords, for
example 7:9:11:13) in a context where prime 5 is not relevant and
prime 7 comes from a different source.

Most appreciatively,

Margo
mschulter@...

🔗Margo Schulter <mschulter@...>

Dear Marcel,

The Pythagorean interpretation of a Rast tetrachord as D-E-Abbb-G
or 0-203.9-360.9-498.0 cents is indeed one fine shading, with
steps of 204-157-137 cents. The version D-E-Gb-G or 204-180-114
cents is a standard modern Turkish tuning of Rast, although in
practice the third may range from just a bit lower than the
theoretical 384 cents (8192:6561) down to around 16/13 -- or just
about the same as our first version.

In practice, anything from around 29-EDO (703.45 cents) to 46-EDO
or a bit higher -- one of my tunings is 704.61 cents -- should
produce interval sizes which are neutral, with the third and
seventh steps of Rast quite bright or "submajor" in quality.
The small neutral step used in place of a semitone will range
from 124.1 cents in 29-EDO to 132.2 cents at 704.61 cents, close
to the ratio of 14/13 (128.3 cents). At 705 cents, we have this
step at 135 cents, closer to 13/12 (138.6 cents), which 19 fifths
up in Pythagorean neatly approximates at 137.1 cents.

While I think of maqam as primarily pure melody, there is an
approach to counterpoint or harmony which I much like, although
it is really a fusion of different musical traditions rather than
a part of the Near Eastern tradition.

The great French composer Guillaume de Machaut, whose music fits
with the medieval European Pythagorean system, interestingly is
known -- along with 14th-century composers in general -- for a
three-voice cadence I find is very nice with Rast, although quite
different from either a known 14th-century European or a Near
Eastern style.

In a temperament with fifths around 703.5-705 cents, we spell a
Rast scale using the simpler Pythagorean notation you discuss,
which hear actually produces neutral intervals, here with a
transposition making it easy to find Rast within a standard
12-note tuning with accidentals from Eb to G#. Let's assume we're
in 46, since it's a fine tuning:

B C# Eb E F# G# Bb B
0 209 365 496 704 913 1070 1200

Now a standard 14th-century final cadence is most often like
this, and very effective either in Pythagorean or in a
temperament like 46-EDO:

E F
B C
G F

We have a major third expand to a fifth, and a major sixth to an
octave, with each upper voice ascending by a semitone.

For Rast, we can use the same pattern, but with neutral or
"submajor" rather than regular major thirds and sixths, although
for Rast, the neutral intervals are indeed the "regular" ones!
In addition to the standard notes of Rast, we need an added step,
F (a diminished fifth above B) to form a neutral of submajor
third above C# in the first sonority of the cadence. This F will
be 626.1 cents above B, and 365 cents above C#.

Bb B
F F#
C# B

The submajor third and sixth are at 365 and 861 cents, with the
leading tones in the upper voices at 130 cents, close to 14/13.
This is quite different from 78 cents for the regular limma or
semitone in 46, and yet not so far in its impression from that of
a "semitone" of some type -- a supraminor second, as it were.

Here a regular major third like G-B is a virtually just 14/11,
while a submajor third like C#-F (i.e. a diminished fourth) is
around 21/17. Both are rather bright and complex intervals, which
lends momentum to a cadence where the third expands to a fifth.

While this form is expansive, third to fifth and sixth to octave,
there is another form which corresponds to 14th-century European
practice based on Pythagorean. Here is a usual example:

B C
G F
E F

We have in the first sonority an outer fifth "split" into a lower
minor third and upper major third. The minor third contracts to a
unison, and the major third expands to a fifth, with the two
outer voices moving in parallel fifths.

In Rast, we get this very appealing variation:

F F#
C# B
Bb B

The first sonority in 46-EDO is 0-339-704 cents, quite close to a
just 14:17:21 or 0-336-702 cents of the kind that George and I
have discussed. It resolves to a fifth, again with the 130-cent
steps of 46. It sounds rather like a standard 14th-century
cadence of Machaut and his colleagues, and yet different.

Given that Near Eastern tunings of Rast vary not only over the
region as a whole (e.g. Egyptian tunings tending to be lower than
Turkish ones, with some Syrian practices around the middle) but
sometimes within a single part of the region (e.g. Turkish
practices ranging from around 359 to 384 cents for the Rast
third), it's not surprising that tastes in the temperaments we
are discussing may vary also.

With many thanks,

Margo
mschulter@...

🔗Margo Schulter <mschulter@...>

Previously I wrote to George Secor, without much explanation:

> In fact, two 29-chains of Pepper at 58.680 cents apart (yielding a
> just 7/6) would produce no fewer than 32 locations for 5/4 at either
> 386.008 cents (from +21 fifths within each chain) or 388.077 cents
> (from -13 fifths less the spacing between the chains, or possibly
> 22/17 less 121/117 for 234/187). This would be Peppermint-58, a
> large but usual rank-3 parapyth. There is also a superb 11:14:17,
> found in any Pepper chain of 17 notes or more (with 14:11, 17:14,
> and 17:11 from +4, +9, and +13 fifths).

Maybe this style of writing, like the Unix operating system and its
different variations, has the advantage of being "expert friendly" --
but not necessarily so easily understandable if one doesn't happen to
be already familiar with the specified commas and what they stand for
in practical musical terms. That's where some "plain language"
explanations can be very reader friendly.

One of the main things to explain here is the 121/117. It comes up in
any version of Peppermint -- and let's explain that first. Keenan
Pepper, in September 2000, proposed a "Noble Fifth" tuning with a
fifth at 704.096 cents. Peppermint, which I announced in the summer of
2002, uses two chains of this tuning at 58.060 cents apart. Pepper's
regular major second or tone at 208.191 cents, plus this spacing of
the chains, produces a pure 7/6 minor third.

Now while I happened to tune Peppermint-24 -- two 12-note chains of
the Pepper tuning at 58.060 cents apart -- there's no reason one
couldn't tune 17-note or 29-note chains, etc. At least, if one has a
keyboard or other instrument that can handle these larger tuning
sizes, along with the skills necessary to use that instrument
effectively, then why not tune them?

To understand the 121/117, we can use a Peppermint of 24 notes (two
12-note chains) or larger. In fact, it would come up in a
Peppermint-14, with 7 notes in each Pepper chain (i.e. a basic
diatonic scale). To show that there's nothing special here about
12-note chains (as unlikely as that might same after reading some of
my posts!), we can actually demonstrate the 121/117 with a
Peppermint-14 system.

Here are the two chains, using the Sagittal notation symbol /|\ to
show that the upper chain of fifths is raised by an interval, here
58.680 cents, which represents the best approximation of 33:32, the
difference between C-F (496 cents, near 4/3) and C-F/|\ (555 cents,
near 11/8):

C/|\ D/|\ E/|\ F/|\ G/|\ A/|\ B/|\ C/|\
58 267 475 555 763 971 1179 1258

C D E F G A B C
0 208 416 496 704 912 1120 1200

To understand 121/117, let's start with the regular minor third D-F in
the lower chain, at 288 cents, close to a just 13/11 (289 cents). Now
if we add the spacing between chains, we get D-F/|\ at 346 cents, just
over a cent narrower than 11/9 (347 cents).

Now the difference between 13/11 and 11/9 is 121/117 or 58 cents. This
is an important comma observed in Peppermint-14 (7-note chains),
Peppermint-24 (12-note chains), Peppermint-34 (17-note chains),
Peppermint-58 (29-note chains), or any desired tuning size of
Peppermint. And it even comes up in Peppermint-10, with two 5-note or
pentatonic chains, say D-F-G-A-C-D and D/|\-F/|\-G/|\-A/|\-C/|\-D/|\,
where we have D-F at around 13/11, and D-F/|\ at around 11/9.

Now 13/11 and 11/9 are very useful, but how does all this tie in with
the Peppermint-58 approximation of 5/4 at 388.077 cents? Let's trace
out the connection in what I hope may approximate readable prose.

In Peppermint-58, we have two 29-note Pepper chains. Each of these
chains includes a very interesting and very large major third formed
from a regular fourth (496 cents) less the Pepper 12-note comma or
"enharmonic diesis," the amount by which Eb is lower than D#. This
diesis is equal to 49.147 cents, and the fourth more precisely to
495.904 cents, giving us a large major third at 446.757 cents, for
example A#-Eb within a single Pepper chain of 17 notes or larger.

We can also describe this interval as a double diminished fifth.
The fifth at 704.096 cents is reduced by a chromatic semitone A-A# (128.669 cents), and again by Eb-E at the same size -- or by a small
minor third of 257.339 cents in all, leaving a large major third at
446.757 cents.

As it happens, this large third is almost identical to 22/17 or
446.363 cents. In Peppermint-58, let's see what happens when this
large third occurs in the upper chain, with the lower chain located
downward by 58.680 cents, taking A#/|\ on the upper chain as our
reference point.

A#/|\ Eb/|\
0.0 446.757

A# Eb
1141.320 388.077

Here A#/|\-Eb/|\ represents 22/17, while the spacing between the
chains can represent 121/117 (e.g. 13/11 vs. 11/9), as we've seen. Now
if we take 22/17 and subtract 121/117 -- that is, to seek out a just
ratio which the interval A#/|\-Eb might approximate -- we have a just
ratio of 234/187, or 388.164 cents. This is very close to the actual
A#/|\-Eb at 388.077 cents!

The more familiar and even more accurate representation of 5/4 in the
Pepper tuning, found within any single chain at an MOS of 29 or
larger, is from thrice the apotome or chromatic semitone at 128.669
cents, or 21 fifths up -- which gives us 386.008 cents, very close to
the just 386.314 cents. It occurs, for example, at C-Cx#, or Eb-Ex.

In Peppermint-58, we'd get each type of approximation at 16 locations.
With each 29-note chain, we have 8 locations for the 21-fifth chain
producing the 386-cent version; plus 16 locations in the upper 29-note
chain for the near-just 22/17, which when reduced by the spacing
between the chains produce the 388-cent version. Thus there are 32
locations for a near-just 5/4 in all.

Note that in single-chain or rank-2 MOS systems for Peppermint up to
17 notes, prime 5 doesn't come up. In Peppermint-24, it's incidentally
approximated at three remote locations (395.540 cents, representing
49/39 or 14/13 plus 7/6), but not as a main theme of the 2-3-7-9-11-13
system.

However, with even one chain at a 29-MOS or larger, and yet more
dramatically with two or more such chains (with Peppermint-58 as an
illustration), near-just 5/4 locations are a main theme of the system,
whether or not they happen to be relevant to a given musical agenda.
Similarly, 43-EDO is an intriguing system for Persian music, although
steps near 13/12 and 13/8 were hardly a priority for advocates of this
tuning (around 1/5-comma meantone) in 18th-century Europe.

Getting back to 46-EDO, an important topic in this thread raised by
George, I might comment that its history also reflects the different
kinds of theoretical and practical leanings one can have in
approaching a tuning.

The 20th-century often takes an interest in 46-EDO mainly from the
viewpoint of canvassing every EDO up to a certain size for
approximations of 3/2, 5/4, 6/5, and possibly 7/4. By this standard,
46-EDO is not quite so accurate for 3/2 as 12-EDO (although quite
comparable), but far more accurate for the other ratios, which are
available in a radically non-meantone scheme.

George Secor's 17-WT took a different approach, using a generator of
704.377 cents for a just 14/11 third or 11/7 sixth in an irregular
tuning where 5 is not a factor.

In the year 2000, not yet having learned of George's 17-WT and 29-HTT
(both from 1978), I asked myself, "Why is it that 46-EDO and similar
tunings are not recognized for the virtually just 14/11, 13/11, 17/14,
21/17, and other ratios of interest, which are present even in a
similar 12-note chromatic tuning, and present new possibilities?"

This was what my encounters with George Secor, a wonderful mentor, in
2001-2002 showed me was basically a 2-3-7-9-11-13 outlook. However, as
George showed as early as 1978, one can also choose to combine these
worlds in a 2-3-7-9-11-13-15 outlook, for example, as in the family of
HTT or High Tolerance Temperament sets.

My experience is that a 24-note set of 2-3-7-9-11-13 is practical and
intriguing. But this shouldn't stop anyone from getting yet more
adventurous and diverse with 29-HTT, 41-HTT, Peppermint-58 (not quite
so accurate on many intervals including 3/2 as HTT), or any other
system!

Best,

Margo
mschulter@...

🔗Marcel de Velde <marcel@...>

Dear Margo,

> The Pythagorean interpretation of a Rast tetrachord as D-E-Abbb-G
> or 0-203.9-360.9-498.0 cents is indeed one fine shading, with
> steps of 204-157-137 cents. The version D-E-Gb-G or 204-180-114
> cents is a standard modern Turkish tuning of Rast, although in
> practice the third may range from just a bit lower than the
> theoretical 384 cents (8192:6561) down to around 16/13 -- or just
> about the same as our first version.
>

Oh wow thank you for this information. I had no idea that the standard modern Turkish tuning of Rast is D-E-Gb-G.
If you have any references / links to more information for this ready I'd love to read more about this.

> For Rast, we can use the same pattern, but with neutral or
> "submajor" rather than regular major thirds and sixths, although
> for Rast, the neutral intervals are indeed the "regular" ones!
> In addition to the standard notes of Rast, we need an added step,
> F (a diminished fifth above B) to form a neutral of submajor
> third above C# in the first sonority of the cadence. This F will
> be 626.1 cents above B, and 365 cents above C#.
>
> Bb B
> F F#
> C# B
>
> In Rast, we get this very appealing variation:
>
> F F#
> C# B
> Bb B
>

The 2 progressions you give, Bb-F-C# > B-F#-B and F-C#-Bb > F#-B-B are not easily interpreted as such I find.
As they are similar to A#-C#-E# > B-F#-B and E#-C#-A# > F#-B-B which are the simpler diatonic version (B-D-F# > C-G-C and F#-D-B > G-C-C when transposed up a minor second).

In order to use an interval like the augmented second instead of a minor third it must be clearly indicated as such by preparation and then resolved before we can hear it as such.
So to simply play C-D#-G instead of a C-Eb-G is not going to be heard as a D# but will still be interpreted as an Eb functionally.
I demonstrated this partly in the audio I linked to in my previous email (could you hear this btw? if I remember correctly you could hear audio on soundcloud right? otherwise I'd be happy to send it to you another way), where harmonizing in the classical way the C-D-E-F-G-A-B-C scale, then playing with the same harmonization C-D-Fb-F-G-A-Cb-C and then C-D-Gbbb-F-G-A-Dbbb-C and it is clear that we hear all 3 tunings as the same progression, no functional difference, only the Fb and Cb and Gbbb and Dbbb start sounding as out of tune E and B.

Similarly we cannot simply change the tuning in a Bach piece for instance to have augmented seconds instead of minor thirds and diminished fourths instead of major thirds and expect out brain to interpret these as new intervals. We will still hear them as minor thirds and major thirds regardless.

Therefore we cannot harmonize Rast in triads with only notes from the mode itself as then we will functionally indicate the western major mode instead.

I will make and upload a harmonization later today of makam Rast in which I use only major and minor chords and the simplest interpretation (according to Pythagorean thinking) is that of makam Rast and it will clearly be very distinctly different to our brain from a diatonic harmonization.

On the other hand, if one does not follow Pythagorean logic, but sees the neutral third as a stable interval in its own right (and with a unique higher limit ratio etc) then logic would say it can be harmonized as in your example.
However I have heard several such attempts and tested this myself and I've never heard this approach produce results which sound anything like harmonized Rast to me. The Rast expression / unique identity of the neutral intervals get completely lost in this approach as some other have found as well.

> Given that Near Eastern tunings of Rast vary not only over the
> region as a whole (e.g. Egyptian tunings tending to be lower than
> Turkish ones, with some Syrian practices around the middle) but
> sometimes within a single part of the region (e.g. Turkish
> practices ranging from around 359 to 384 cents for the Rast
> third), it's not surprising that tastes in the temperaments we
> are discussing may vary also.
>

Again very interesting information. And I'd love to read more about this.

Thank you!
And kindest regards,

Marcel de Velde

🔗Margo Schulter <mschulter@...>

Dear Marcel,

Please let me apologize for not clarifying some important points
which may make our dialogue easier, and help us in exploring
whether our perceptions are indeed quite different on the musical
questions we are discussing, or if differences in style may be a
factor along with the aspects of neutral intervals that we are
addressing.

Before getting to these, I should give links to a few of the
sources you asked about. For Turkish music, searching on Google
for "Turkish makam tuning" yields many webpages.

As another resource addressing some of the things I discuss below
about 14th-century European music and neutral intervals, may I
suggest George Secor's paper of his 17-tone well-temperament, and
a companion paper of mine to accompany his. Regrettably, what I
am linking to for my paper is not precisely what was published in
Xenharmonikon 18, and there some errors and inconsistencies --
but George's paper is perfect, and the two together may provide
more background.

<http://www.anaphoria.com/Secor17puzzle.pdf>
<http://www.bestII.com/~mschulter/Secor_17-WT_draft.zip>

First, one of the biggest things I left unsaid but should have
clarified is that 14th-century European music in Pythagorean
intonation is outside the major/minor system, and has very
different although equally compelling methods for creating a
sense of motion, tension, and resolution in textures for two,
three, or sometimes four voices. The rules of 16th-century
counterpoint, let alone those of 18th-century tonality, simply do
not apply.

One view might hold that Pythagorean intonation or the tempered
variations of it we are discussing are especially fitting for
14th-century music, where major or minor thirds and sixths are
not stable, but act as dynamic intervals "striving" to resolve to
stable sonorities, with 2:3:4, 1:2:3, or sometimes in four-voice
writing 2:3:4:6 as the ideal stable harmonies. In this view, some
kind of meantone or irregular temperament is appropriate some
music starting around 1450, and for most during the era of
1500-1900 or so.

In a different view, Pythagorean might be applied to these later
styles, where thirds and sixths are stable intervals, as well to
medieval music where they are unstable and the complex
Pythagorean ratios (or tempered equivalents) seem acoustically to
reinforce this perception. This I understand to be mostly your
perspective, although you do not exclude the use of meantone
temperament for historical styles where it was clearly standard.

With 14th-century music, before we get into the possibility of
neutral progressions based on some tuning of Maqam Rast, we need
to understand that the usual Pythagorean intonation, although
without these neutral intervals, presents something quite other
than 18th-century practice. In fact, augmented and diminished
intervals may have arisen rather often in ornamented cadences,
for example, and may in practice sometimes have been tuned as
neutral steps. However, it is the usual intervals which are the
starting point for clarifying some basic progressions.

As for as some usual progressions, I have written of these in two
presentations which may give a summary or overview:

<http://www.medieval.org/emfaq/harmony/13c.html>
<http://www.medieval.org/emfaq/harmony/pyth.html>

One caution is that back in 1997-1998 when I prepared these pages
for Early Music FAQ Editor Todd McComb, I was not so familiar
with the practice of using fractions such as 22/21 for tuning
ratios, being then more familiar with the 22:21 form. Thus the
symbols in these articles such as 5/4 mean not the ratio 5:4, but
rather a type of music theory notation somewhat like basso
continuo showing the intervals of a fifth and a fourth with
relation to the bass -- in terms of ratios, 6:8:9 or the like.
Obviously, if I had spent some time on this list before starting
the project, I might have used a different notation, or at least
given a caution about the possible ambiguity.

Now let us return to ordinary progressions in Machaut and other
14th-century composers. We start with the 2:3:4 sonority, which
is the ideal conclusion for a piece, and is more generally used
when complete and stable harmony is desired. We may write this
D-A-D, for example.

In a sense, we can infer an ideal stepwise cadence -- since most
of the strongest 14th-century progressions tend to involve motion
of all voices by step -- by starting at D-A-D, and seeking the
most powerful and compelling way to approach this goal. We have
two ideal solutions: E-G#-C# to D-A-D or Eb-G-C to D-A-D.

In both progressions, an unstable major third expands to a fifth,
and an unstable sixth to an octave. While both of these two-voice
progressions remain very common in the 16th century, and indeed
are used by Monteverdi and others in the early 17th-century or
late Manneristic era, they take on a different context because
thirds and sixths are in these styles of counterpoint stable.
In the 14th century, however, thirds and sixths signal that the
music will continue until a resolution is reached, with 2:3:4
ideal.

Now E-G#-C# to D-A-D is in a 14th-century style especially
compelling and conclusive because of the ascending semitonal
motion of the two upper parts, coupled with a descending
whole-tone motion in the lower part. In Pythagorean, the tone
will be at 9:8, and the semitones at 256:243. In the temperaments
we are considering, the tone will be slightly larger, around
206-209 cents, while the semitone will be yet smaller and more
incisive, at around 80 cents or so, or 22/21, for example.

The cadence Eb-G-C to D-A-D also has a major third expanding to a
fifth and a major sixth to an octave, but with descending
semitonal motion in the lowest voice, and ascending motion in the
upper voices. Typically this is used as a half-cadence, less
conclusive, and often signalling completion of one section of a
form, with another to follow.

This cadences are "expansive," since the major third and sixth
tend to resolve outward to the fifth and octave, in each case
with one voice moving by a tone and the other by a semitone or
limma.

There is also a "contractive" type of cadence, most often with a
minor third contracting to a unison, and sometimes with a minor
seventh contracting to fifth (notable in Machaut). One form would
be C#-E-G# to D-D-A, with the lower two voices resolving from
minor third to unison, and the upper voices from major third to
fifth. The outer voices move in parallel fifths -- which would,
of course, be contrary to the rules against parallel or
consecutive perfect concords in the era of around 1450-1900.
We may also have C-Eb-G to D-D-A, for example, typically less
conclusive.

Resolutions involving sevenths, with the minor seventh considered
milder and often used by composers such as Machaut, although they
are treated more cautiously by others such as Landini, are also
effective. For example, we may have E-B-D to F-C-C, with the
outer minor seventh contracting to a fifth, and the upper minor
third to a unison. Another possibility is E-G-D to F-F-C.

These contractive resolutions often cadence to a simple 3:2
rather than a complete 2:3:4, but either is acceptable. Endings
on a bare unison or octave sometimes also occur, but the full
2:3:4 (or 1:2:3 or 2:3:4:6), or next best a simple 3:2, are
generally preferred.

A preference for "closest approach" generally applies, although
there is room for varying tastes among composers and performers
(who may often add accidentals based on context and judgment), so
that this is not an invariable rule. Generally major intervals
tend to expand (e.g. Maj2-4, Maj3-5, Maj6-8, Maj10-12), while
minor intervals tend to contract (e.g. min3-1, min7-5, min10-8).

* * *

What George Secor and I found was that, to a certain point, a
large neutral second step may sound rather like a tone, and a
small neutral second step rather like a semitone. Thus 11:10
can have the effect of a small tone, or 14:13 of a large semitone
-- although, in certain contexts, their "middle" or neutral
character becomes clear.

However, at some point middle or neutral intervals sound quite
different from either minor or major ones -- the progression
takes a new and unmistakably different quality. I would say that
by around 13/12 (139 cents), a neutral step tends to sound
distinctly different than a semitone. George Secor suggested that
around 12/11 (150 cents) is the smallest size where a larger
neutral second might give the convincing impression of a tone.

To judge these progressions in a setting where 14th-century
European patterns of counterpoint are being used and altered, two
things are helpful.

The first is familiarity with standard 14th-century
progressions. The second is some familiarity with different sizes
of middle or neutral intervals.

Your remarks are very correct on one point especially: a third at
around 26/21 or 370 cents can indeed give the impression of a
"subditone" or small ditone, expanding to the fifth much as a
regular major third at 81:64 (408 cents), or in these
temperaments a bit larger at somewhere between 33:26 (413 cents)
and 14:11 (418 cents), might do. The color is different, and the
"semitone" steps much larger (typically around 125-130 cents, as
compared to 90 cents for the Pythagorean limma and 80 cents or so
for the tempered limma of these tunings), but the usual
progression remains quite recognizable.

Likewise, a progression like Bb-C#-F to B-B-F# with Bb-C# at
somewhere around 63:52 (332 cents) or 17:14 (336 cents), for
example, will sound not too different from B-D-F# to C-C-G, but
different enough to be subtly distinct. Here the large minor
third is a kind of "suprasemiditone" which may substitute for a
usual 32:27 (294 cents), or in these tunings often 13:11 (289
cents) or so, without obscuring the nature of the progression,
but giving it a different color.

However, as George Secor and I found almost 12 years ago, when
intervals move more into the central neutral region, then the
effect may be immediately and radically different. For example, a
regular F-A-D to E-B-E cadence is recognizable for its familiar
descending semitone step in the lowest voice F-E, which in some
medieval and Renaissance European sources is called "remissive,"
which is to say "relaxed" or, in musical terms, proceeding
downward by a semitone. This descent would be 256:243 in
Pythagorean, or about 80 cents in the types of temperaments we
are discussing -- and in George Secor's 17-WT, as little as 64
cents, close to the just 28/27 step of Archytas.

What becomes immediately clear is that this is something quite
different from a usual cadence with ascending or descending
semitones. I would emphasize that this effect is by no means
unique to neutral variations on 14th-century European music, or
to temperaments with wide fifths.

For example, in a full set or subset with the needed neutral
intervals of a 31-note circle of 1/4-comma meantone, or 31-EDO,
progressions involving a 4/5-tone (4d31) can be quite
astonishing. Consider something like this, analogous to our last
example, but with thirds as stable intervals in a 16th-century
fashion: F*-A*-D*-A* to E-B-E-G#. In these meantones, usual F-E
would be a large semitone down, 117 cents in 1/4-comma or 116
cents in 31-EDO, Here, however, it is a 41-cent diesis or
fifthtone larger in 1/4-comma (158 cents), or 155 cents in
31-EDO. Moreover, all four voices are moving by this 4/5-tone
step!

In these circumstances, as George and I found, a neutral interval
immediately stands out from any familiar tone or semitone step.

Again, I deeply appreciate your interest in these questions of
musical perception and taste, and warmly invite further
dialogue.

Most appreciatively,

Margo Schulter
mschulter@...

🔗Marcel de Velde <marcel@...>

Dear Margo,

> Dear Marcel,
>
> Please let me apologize for not clarifying some important points
> which may make our dialogue easier, and help us in exploring
> whether our perceptions are indeed quite different on the musical
> questions we are discussing, or if differences in style may be a
> factor along with the aspects of neutral intervals that we are
> addressing.
>

No need to apologize at all.
I'm aware of our different perceptions in tuning and style.
We had a talk about meantone tuning for medieval period music before where we differed in end conclusion on the role of Pythagorean vs meantone :)

But I think me may have some miscommunication going on due to me not explaining well what I mean.
I'm not trying to discuss tuning "color" differences or differences in style, historic tuning preferences, or taste etc.
I'm trying to discuss functional difference between tones.
As the Eb is a functionally different tone from the E, and the Ab is a functionally different tone from the G#.
And I'm interested in how we make these functional differences clear to the listener.
Regardless of which tuning we use to achieve this be it 12tet or Pythagorean or meantone or superpythagorean etc.

I've uploaded a simple harmonization of the western major scale vs makam Rast:
https://soundcloud.com/justintonation/demo-makam-rast-harmonization
Here you can hear that regardless of tuning they are 2 functionally different scales.
The second time I've added a viola that plays the scale clearly C-D-E-F-G-A-B-C vs C-D-Fb-F-G-A-Cb-C and one can hear that the E and Fb, and Cb and B are different tones.

Yet, in the other example I posted earlier:
https://soundcloud.com/justintonation/tuning-demo-enharmonic
The first set of harmonizations based on the major scale harmonization it does not matter one bit whether we tune the E and B to Fb and Cb, or even 2 Pythagorean commas lower to Gbbb and Dbbb we still hear them as the same tone as the E and B only as out of tune.

I'll try uploading a musical example tomorrow. A harmonization of a Turkish melody.
The way they use the augmented second and diminished third really brings out that exotic eastern sound that simply walking the steps of the scale as I did above doesn't demonstrate well.

Btw, can you let me know if you can indeed hear the audio examples? (as a few month ago you had some trouble with that and I'm not sure if that was about soundcloud or youtube)
I do not think this discussion will be as productive when you cannot hear them.

Kindest regards,
Marcel de Velde

🔗Margo Schulter <mschulter@...>

Dear Marcel,

Thank you for your links to your two examples, which I have now
heard. My conclusion is that they are very interesting musically,
rather like more traditional and avant-garde styles of
20th-century European orchestration, but quite different than the
kind of maqam melody I am addressing, with or without any element
of polyphony. And also, needless to say, they are quite different
from the medieval European melody and polyphony which doubtless
influences my approach to Near Eastern style.

However, I have not yet, of course, heard your Turkish Rast
example that you mention, and so what I say might not apply
to that -- I am only describing what I heard in the two
examples to which you linked.

An interesting point is that what you are addressing was an
aspiration of some 20th-century Near Eastern musicians and
theorists such as Vaziri in Iran, who wished to arrive at a
tuning system which would permit combining traditional maqam or
dastgah with European harmony and orchestration. That was one
argument in favor of tuning systems such as 24-EDO, although
interestingly, in Turkey, it became an argument for Pythagorean
(which can be mathematically simplified into the 53-EDO model
very common there today).

What I hear in your examples is a Western kind of tonal
progression -- or maybe sometimes a bit modal -- which eventually
gets more complex or dissonant in a 20th-century kind of "jazzy"
way, not unpleasant, just a change of style. All I can say is
that whatever norms or patterns of tuning hold there, pure maqam
melody or for that matter 13th-14th century European polyphony
may present patterns that are quite different -- at least to me.

To me, a neutral interval is not just a major or minor interval
that is recognizable but out of tune -- it is something quite
else, and exactly what I want when playing Rast or Huseyni or
whatever. Likewise, progressions involving steps of 12/11 and
13/12, or 13/12 and 14/13, often sound to me (and also George
Secor) quite different from progressions with regular tones and
semitones.

What I have found is that pure melody is the best test of maqam,
even though polyphony can add interest, and some forms of
polyphony (such as drones, or parallel fourths or fifths at
times) are an authentic element of Near Eastern practice even
though it is not wrong to say that maqam and the related Persian
dastgah system are in principle monophonic. That is, it is
melody, not vertical events, which defines the path of a piece,
with drones or parallel consonances as adornments. That is also
an oversimplification, since there are pieces where a melodic
pattern may realize in effect a melody-plus-drone pattern. But it
is a helpful oversimplification, especially in the West where the
primacy of melody may not be so well known.

This brings me back to my point that melodic discrimination may
operate differently in different styles, so your observations and
mine may both be perfectly accurate as accounts of what we hear.

These issues became familiar to me what I first began to
experiment with neomedieval tunings involving wide fifths,
essentially the kinds of systems we are now discussing. A friend,
a piano technician admirably skilled in tuning meantone and
various historical well-temperaments and in advocating for these,
asked about Werckmeister's precept that a major third should be
no more than a comma wide of 5/4 -- in other words, not wider
than 81/64 or so.

What I quickly learned is that Werckmeister's general approach --
and whether or not it precisely expresses his view, it seems to
fit many German well-temperaments quite nicely -- addressed the
context of music in the era of 1680-1700 or so, with thirds as
stable concords, It did not address medieval or neomedieval
music. They are two different languages, and often invite
different intonational styles.

A major third around 33/26 or 14/11 in a setting around 1700
would be pushing the envelope, as evidently some French composers
did with their irregular temperaments with remote thirds around
this size (Mark Lindley has written of this). It is testing how
far one can go and still have something close to a "concordant
triad," with 4:5:6 as the ideal.

In a medieval or neomedieval setting, however, it is more of an
accentuated 81/64. The listener may either accept it as a matter
of course, or perhaps find the whole style uncongenial. But it is
a different universe of polyphony, and of melody.

Incidentally, I consider it an open question whether Ab and G#
are functionally different in an early 15th-century European
setting. I might guess, for example, that traditionalists around
1410 (including Prosdocimus de Beldemandis) would have played (or
more precisely tuned) E-G#-C# in a cadence to D-A-D with
Pythagorean sharps, as the notation suggests. However, many
"moderns" likely tuned E-Ab-Db, a comma lower, and very close to
a just 12:15:20. However, the cadence would have been
recognizable with either tuning: the "directedness" of major
third to fifth or major sixth to octave would have held with
either tuning. And it would even hold if some enthusiast for the
old style around 1460, say, played Machaut on a German instrument
in the new meantone style of tuning likely by now in use in some
places. This situation may reflect, in part, the fluidity of
early to middle 15th-century tuning; but the choice of G# or Ab
may have varied from keyboard to keyboard in Dufay's youth, I
might guess.

* * *

A major self-directed caution is that my own impressions of Near
Eastern maqam or dastgah music may be influenced by my medieval
European roots, and lead me to different conclusions than a Near
Eastern musician would reach. When I play Huseyni, am I really
playing an idiomatic maqam, or more like the Dorian mode on the
second step of a Rast tuning? Of course, I can listen to some
pieces in Huseyni, and study what in Turkey is called the typical
_seyir_ or melodic "road" or "path" for this maqam, as well as
usual modulations to other maqamat. But "language interference"
from my European background is still a factor, whether
undesirable (as when it hinders my recognizing a Near Eastern
pattern) or sometimes creative (as when it leads me into a line
of melody not unapt from a Near Eastern point of view).

Finally, what I can add is that the "Are neutral intervals really
distinct from major or minor ones?" discussion comes up from time
to time, and people do perceive this differently.

Thank you again for your links to your examples, which helped me
better to understand your approach.

Most appreciatively,

Margo Schulter
mschulter@...

🔗hstraub64 <straub@...>

🔗Marcel de Velde <marcel@...>

Dear Margo,

I'm glad you heard them now.

I'm quite convinced however that my example did represent Rast correctly.
I agree it did not sound much like Turkish or Arabic music, but this is due to simply walking the steps of the scale with major and minor triads. You have to use your imagination a little with that example :)
Also it is due to other stylistic elements like instrumentation and way of playing etc.
Many makamlar are simple diatonic scales exactly the same as we use in the west, and they are even harmonized the same way in some Turkish music and yet the results sound very Turkish nonetheless.

> However, I have not yet, of course, heard your Turkish Rast
> example that you mention, and so what I say might not apply
> to that -- I am only describing what I heard in the two
> examples to which you linked.
>

Many many more examples on the way :)
Have all day tomorrow to work on them.

> An interesting point is that what you are addressing was an
> aspiration of some 20th-century Near Eastern musicians and
> theorists such as Vaziri in Iran, who wished to arrive at a
> tuning system which would permit combining traditional maqam or
> dastgah with European harmony and orchestration. That was one
> argument in favor of tuning systems such as 24-EDO, although
> interestingly, in Turkey, it became an argument for Pythagorean
> (which can be mathematically simplified into the 53-EDO model
> very common there today).
>

This has arrived! I'm fully sure of this.
It is indeed to be found in Pythagorean.
And while I had my doubts for a long time whether sometimes far out enharmonics were indicated in certain makamlar, I'm now thinking this is likely not the case.
17 tone Pythagorean is probably all that's needed to express all makamlar.
The stronger neutral intervals still express augmented primes and diminished thirds to me. I will make demos to demonstrate this.

> To me, a neutral interval is not just a major or minor interval
> that is recognizable but out of tune -- it is something quite
> else, and exactly what I want when playing Rast or Huseyni or
> whatever.
>
Yes exactly!
But this is what I demonstrate in my example as well.
I did the normal major progression with minor and major semitones and the Rast progression with augmented primes and diminished thirds.
These are distinctly not minor and major seconds so this does not counter your previous observations.

> These issues became familiar to me what I first began to
> experiment with neomedieval tunings involving wide fifths,
> essentially the kinds of systems we are now discussing. A friend,
> a piano technician admirably skilled in tuning meantone and
> various historical well-temperaments and in advocating for these,
> asked about Werckmeister's precept that a major third should be
> no more than a comma wide of 5/4 -- in other words, not wider
> than 81/64 or so.
>
I agree.
The temperaments we're discussing here, "superpythagorean" and "gentle" are not well suited to polyphonic music in my opinion.
Somehow when we make the major fifths flatter they sweeten up, but to make them wider makes them sound out of tune very very fast and noticeable.
I cannot stand normal classical music in 46edo, or 703.91 fifths, or 29edo etc.
While meantones are somewhat acceptable to me (and I can understand many find them harmonically sweeter than Pythagorean and 12tet).
This is a harmonic thing btw, melodically I really dislike 1/4 comma meantone. It's pretty badly out of tune to me. 46edo is much better here, and on top of that it makes the difference between minor seconds and augmented primes extremely clear and gives it a lot of color.

> Incidentally, I consider it an open question whether Ab and G#
> are functionally different in an early 15th-century European
> setting. I might guess, for example, that traditionalists around
> 1410 (including Prosdocimus de Beldemandis) would have played (or
> more precisely tuned) E-G#-C# in a cadence to D-A-D with
> Pythagorean sharps, as the notation suggests. However, many
> "moderns" likely tuned E-Ab-Db, a comma lower, and very close to
> a just 12:15:20. However, the cadence would have been
> recognizable with either tuning: the "directedness" of major
> third to fifth or major sixth to octave would have held with
> either tuning.
>
Well I can tell you that in the context of Pythagorean a sudden E-Ab-Db sounds very out of tune. So on an instrument where this is noticeable I don't think they did this a lot..
In any case, even if they did It's not going to be perceived as an E-Ab-Db in the music of that period and in that cadence but simply still as a E-G#-C# only tuned off a bit (or colored is perhaps a better name).

> Finally, what I can add is that the "Are neutral intervals really
> distinct from major or minor ones?" discussion comes up from time
> to time, and people do perceive this differently.
>

Well that depends on the context which may explain why people are perceiving it differently.
If the music indicates a minor second in a clear way and tunes it to a neutral second instead then it is not distinct from a minor second only tuned differently.
But an augmented prime is always distinct from a minor second functionally.

Hear the difference between:
C-E-A > B-E-G# and C-E-A > C-Eb-Ab or C-F-Ab (best is to play it in Pythagorean of course, but 12tet will work fine too)
That A to Ab is surely not a minor second, and doesn't sound like one either. (and needs only a slight exaggeration of the fifths to become a neutral second tuning wise)

Kindest regards,
Marcel de Velde

🔗Marcel de Velde <marcel@...>

Hello Hans,

> > I've uploaded a simple harmonization of the western major scale vs
> > makam Rast:
> > https://soundcloud.com/justintonation/demo-makam-rast-harmonization
> > Here you can hear that regardless of tuning they are 2 functionally
> > different scales.
> > The second time I've added a viola that plays the scale clearly
> > C-D-E-F-G-A-B-C vs C-D-Fb-F-G-A-Cb-C and one can hear that the E
> > and Fb and Cb and B are different tones.
> >
>
> Interesting to hear, indeed. But I got to say that in the harmonized > versions the difference between the C-E and the C-Fb was hardly > audible to me - just "major", both of them.
>
Hmm.. are you sure??
I mean tuning wise the difference is subtle, but functionally the Fb is different from the E.
I can't imagine somebody not hearing the distinction at least subconsciously. Perhaps if you focus your attention on it you can pick it up.

If you play the example I gave Margo in my previous email.
C-E-A > B-E-G# vs C-E-A > C-Eb-Ab or C-F-Ab
Do you hear the A to G# as the same interval functionally as the A to Ab?
Preferably played in Pythagorean, but in 12tet you should still hear the functional difference even thought there is no tuning difference.
You will find the same kind of things in my Rast progression only it is perhaps less easy to pick up when playing a scale like this. (in music you can make it quite dramatic and very apparent)

> Why did you use a minor chord for the first note? It appears to me > that a major-style chord (with he C-Fb interval) would be better.
>
Well I've been saying all along that one cannot simply play a diminished fourth like C-Fb and interpret it as such, it gets interpreted as a major third unless the music clearly indicates a diminished fourth (like for instance in an augmented chord in its proper place in a harmonic minor scale which then gets resolved).
So one can play a C-E on the first step but it makes the harmonization only more complex when 2 steps later there is a Fb instead of an E.
Btw my harmonization is a simple one with only major and minor chords, and one of several possible harmonizations of Rast using oly major and minor chords.

> Here is an arabic Rast scale with its lower third, equipped with a > counterpuntal line according to my personal counterpoint rules, in > 17edo tuning.
>
> https://dl.dropboxusercontent.com/u/102870769/RastCounterpoint17edotest1.mp3
>

Well that counterpoint line sounds fine to me :)
A first on these lists haha.
You follow my rules which say no C-Fb btw ;) You do the C-E thing.

Kindest regards,
Marcel de Velde

🔗Marcel de Velde <marcel@...>

Hi Hans,

I forgot to say something.
Since your makam is in 17tet, it may be more likely to interpret it not as C-D-Fb-F-G-A-Cb-C Rast but as C-D-D#-F-G-A-A# C instead (no idea what this makam would be called, it has Bayati tetrachord on D).
The 17tet neutral second is closer to an augmented prime. And you also use a major chord on the first step making C-D-D#-E-F more likely it seems.
Btw I've seen this used in makam practice as well. Even maqamworld.com has examples of makam Rast posted online (under specific examples for Rast, go figure) of which one is clearly not Rast but C-D-D#-F instead.

Kind regards,
Marcel de Velde

> Hello Hans,
>
>> > I've uploaded a simple harmonization of the western major scale vs
>> > makam Rast:
>> > https://soundcloud.com/justintonation/demo-makam-rast-harmonization
>> > Here you can hear that regardless of tuning they are 2 functionally
>> > different scales.
>> > The second time I've added a viola that plays the scale clearly
>> > C-D-E-F-G-A-B-C vs C-D-Fb-F-G-A-Cb-C and one can hear that the E
>> > and Fb and Cb and B are different tones.
>> >
>>
>> Interesting to hear, indeed. But I got to say that in the harmonized >> versions the difference between the C-E and the C-Fb was hardly >> audible to me - just "major", both of them.
>>
> Hmm.. are you sure??
> I mean tuning wise the difference is subtle, but functionally the Fb > is different from the E.
> I can't imagine somebody not hearing the distinction at least > subconsciously. Perhaps if you focus your attention on it you can pick > it up.
>
> If you play the example I gave Margo in my previous email.
> C-E-A > B-E-G# vs C-E-A > C-Eb-Ab or C-F-Ab
> Do you hear the A to G# as the same interval functionally as the A to Ab?
> Preferably played in Pythagorean, but in 12tet you should still hear > the functional difference even thought there is no tuning difference.
> You will find the same kind of things in my Rast progression only it > is perhaps less easy to pick up when playing a scale like this. (in > music you can make it quite dramatic and very apparent)
>
>> Why did you use a minor chord for the first note? It appears to me >> that a major-style chord (with he C-Fb interval) would be better.
>>
> Well I've been saying all along that one cannot simply play a > diminished fourth like C-Fb and interpret it as such, it gets > interpreted as a major third unless the music clearly indicates a > diminished fourth (like for instance in an augmented chord in its > proper place in a harmonic minor scale which then gets resolved).
> So one can play a C-E on the first step but it makes the harmonization > only more complex when 2 steps later there is a Fb instead of an E.
> Btw my harmonization is a simple one with only major and minor chords, > and one of several possible harmonizations of Rast using oly major and > minor chords.
>
>> Here is an arabic Rast scale with its lower third, equipped with a >> counterpuntal line according to my personal counterpoint rules, in >> 17edo tuning.
>>
>> https://dl.dropboxusercontent.com/u/102870769/RastCounterpoint17edotest1.mp3
>>
>
> Well that counterpoint line sounds fine to me :)
> A first on these lists haha.
> You follow my rules which say no C-Fb btw ;) You do the C-E thing.
>
> Kindest regards,
> Marcel de Velde
>

🔗hstraub64 <straub@...>

🔗Margo Schulter <mschulter@...>

Dear Marcel,,

Please let me quickly explain that D# and Fb in 17-EDO are
equivalent, and both 353 cents above C. Whether the third step of
Rast gets interpreted as one or the other seems rather academic
to me, since the spelling depends mainly on the temperament.

From around 29-EDO to 17-EDO, a usual Rast tetrachord would be
C-D-Fb-F or the like, with Fb higher than D#. However, beyond
17-EDO, for example at 39-EDO, D# would be higher than Fb, and
the likely choice for Rast (which tends to have the larger
neutral second precede the smaller).

And G# and Ab will definitely be distinct, because the difference
may range from 41 cents (29-EDO) up to a full 70 cents (a regular
semitone) in 17-EDO. Of course, these notes can be conceptually
and musically distinct even when they are realized by the same
pitch, as in 12-EDO, but here the difference is acoustical as
well.

Also, any version of Rast will have a Bayyati tetrachord on the
second step: that is part of a usual definition of Rast, with a
lower tetrachord plus a tone at T J J T (with T showing a tone or
"tanini," and J a "mujannab" or middle interval smaller than a
tone but larger than a limma). So in the second degree of Rast we
have J J T, which is a general definition of Bayyati.

However, at a closer intonational level, J J T at the second step
of Rast is not quite an ideal Bayyati, which is often preferred
to have the smaller neutral step first, as in Persian Shur also.
This sometimes leads either to instruments with two versions
of this third step of Rast, called sikah in Arabic and segah in
Persian and Turkish: a higher one for Rast, and a lower one for
Maqam Bayyati -- say at around 16/13 and 39/32 above Rast, or
around 128/117 or 12/11 and 13/12 above the second step, dukah in
Arabic and dugah (or sometimes dogah) in Persian and Turkish.

Some fixed-pitch instruments tend to use one note for both
flavors of sikah, although the more fastidious Arab musicians may
retune between pieces or improvisations to get a higher sikah for
Rast and a lower one for Bayyati.

In Turkish practice, it's common to play segah at around 26/21,
or even close to 5/4, as the usual step, but when approaching a
final cadence to lower this step by around half a comma, say; I
do it by a full comma, e.g. 26/21 vs. 11/9. We could think of
these interpretations as B-Eb and B-D# or the like, if we're
assuming a temperament in the 704-cent neighborhood; in 39-EDO,
it would be the opposite (B-D# higher than B-Eb).

But the comparison is between larger and smaller neutral thirds,
with the difference often equal to the 17-note comma, which is
tempered out in 17-EDO, so that B-Eb or B-D# is 353 cents, or
precisely half a fifth.

By the way, Hans, 17-EDO is an interesting choice because, like
24-EDO, it does neutralize the distinction between larger and
smaller neutral intervals, while also providing a very nice set
of neutral intervals in a compact circulating package.

At the same time, a circulating 17-note system like George
Secor's 17-WT is also quite compact, but offers more contrast
between larger and smaller neutral intervals.

Best,

Margo
mschulter@...