Metastable intervals

Dear Tuning List Friends,

Please let me say how delighted I am to read the discussions about the
paper which David Keenan and I wrote in the year 2000 about metastable
intervals, an endeavor in which it was a great honor to participate.
It is a pleasure to see the paper very ably updated.

As it happens, the past seven years have given me an opportunity to
explore some of the issues raised in that paper further, for example
in the Peppermint 24 system (an offshoot of Keenan Pepper's famous
"Noble Fifth" tuning which so agreeably played a role in our paper)
and also the newer Zest-24 modified meantone system derived from
Zarlino's 2/7-comma temperament.

I hope that Dave's enthusiastic contributions here and elsewhere will
spark much constructive discussion, and look forward to joining in an
exploration of some of the ramifications of our paper.

Most appreciatively,

Margo Schulter
mschulter@calweb.com

Hello, everyone.

Given the recent discussions both on the meantone era and metastable
intervals, I thought I might post some links to musical pieces and
papers relating to a tuning system bringing these things together:
Zest-24, the Zarlino Extraordinaire Spectrum Tuning.

Before giving the links, I should quickly explain that in this tuning,
there are two 12-note modified meantone circles, each of which can
serve as a circulating temperament in itself. Within each circle,
eight fifths (F-C#) are tempered in Zarlino's 2/7-comma (695.81
cents), while the other four are tempered equally wide (708.38
cents). The two circles are placed at a distance equal to the
enharmonic diesis of 2/7-comma (50.28 cents). Note that many of the
intervals are identical to those of a regular 24-note version of
Zarlino's meantone, but others are different because of the irregular
temperament to close each circle.

Here are the links, followed by a Scala file of this circulating
system:

<http://www.bestII.com/~mschulter/IntradaFLydian.mp3>
<http://www.bestII.com/~mschulter/MMMYear001.mp3>
<http://www.bestII.com/~mschulter/InHoraObservationis.mp3>
<http://www.bestII.com/~mschulter/OElsa.mp3>
<http://www.bestII.com/~mschulter/Baran-GiftOfRain.mp3>
<http://www.bestII.com/~mschulter/LaPacifica.mp3>

<http://www.bestII.com/~mschulter/zest24-lattice.txt>
<http://www.bestII.com/~mschulter/zest24-septendecene.txt>
<http://www.bestII.com/~mschulter/zest24-RastBayyati.txt>

! zest24.scl
!
Zarlino Extraordinaire Spectrum Temperament (two circles at ~50.28c apart)
24
!
50.27584
25/24
120.94826
191.62069
241.89653
287.43104
337.70688
383.24139
433.51722
504.18965
554.46549
574.86208
625.13792
695.81035
746.08619
779.05173
829.32757
887.43104
937.70688
995.81035
1046.08619
1079.05173
48/25
2/1

Most appreciatively,

Margo Schulter
mschulter@calweb.com

Hello, everyone.

Many thanks to Dave Keenan for his list of metastable intervals. I
thought that I might see how many these are available in the Zest-24
system. Everything here follows his list, except that I've added a
"narrow supermajor sixth" at 923 cents. Also, I might term the
interval at 448c as both a "wide supermajor 3rd" and "narrow sub-4th,
but will here follow Dave's table.

If the nearest equivalent to one of these metastable intervals in
Zest-24 is more than about four cents from the Phi-mediant value, then
I have used brackets to emphasize this degree of variation. As we
point out in our paper, a Phi-mediant is not a precise point where we
expect something "special" to happen, but rather a guide to the region
of a plateau of complexity. However, I tend to assume that a
difference of five cents or more could cause a more appreciable change
in shading or color, and therefore is worthy of an alert to readers.

------------------------------------------------------------------
Limit of Cents Description Zest-24 locations
------------------------------------------------------------------
5:6 6:7 11:13 ... 284c wide subminor 3rd 287c 4
------------------------------------------------------------------
4:5 5:6 9:11 ... 339c narrow neutral 3rd 338c 2
------------------------------------------------------------------
4:5 7:9 11:14 ... 422c narrow supermajor 3rd 421c 4
------------------------------------------------------------------
3:4 7:9 10:13 ... 448c narrow sub-4th 446c 2
------------------------------------------------------------------
3:4 5:7 8:11 ... 560c wide super-4th 562c 2
------------------------------------------------------------------
2:3 5:7 7:10 ... 607c narrow diminished 5th [613c] [4]
------------------------------------------------------------------
2:3 5:8 7:11 ... 792c narrow minor 6th 792c 4
------------------------------------------------------------------
3:5 5:8 8:13 ... 833c narrow neutral 6th 829c 2
------------------------------------------------------------------
3:5 7:12 10:17... 923c narrow supermajor 6th 925c 4
------------------------------------------------------------------
3:5 4:7 7:12 ... 943c narrow subminor 7th or 946c 2
wide supermajor 6th
------------------------------------------------------------------
4:7 5:9 9:16 ... 1002c minor 7th [1008c] [14]
------------------------------------------------------------------
3:7 4:9 7:16 ... 1424c narrow supermajor 9th 1425c 2
------------------------------------------------------------------
2:5 3:7 5:12 ... 1503c narrow minor 10th 1500c 4
------------------------------------------------------------------
2:5 3:8 5:13 ... 1666c sub-11th [1671c] [2]
------------------------------------------------------------------
1:3 3:8 4:11 ... 1735c narrow super-11th 1737c 2
------------------------------------------------------------------
1:3 3:10 4:13 ... 2055c neutral 13th 2054c 2
------------------------------------------------------------------
2:7 3:10 5:17 ... 2109c wide major 13th or 2113c 4
narrow supermajor 13th
------------------------------------------------------------------
2:7 3:11 5:18 ... 2226c narrow neutral 14th 2229c 4
---------------------------------------------------------------------

With many thanks,

Margo Schulter
mschulter@calweb.com

Hello, everyone.

Please let me warmly take note of recent discussions on meantone
and related topics, which have lent me impetus to consider some
questions of "modal color" in the Renaissance and Manneristic
eras. As is very aptly observed in that thread, remote modal
transpositions involving "unusual" intervals around 1600 or
a bit later are very relevant to the question of "key character"
traditions of a later era oriented to a major/minor system.

For anyone seeking an understanding of meantone aesthetics, both
historical and modern, I would warmly recommend the writings of Doug
Leedy, which combine much erudition, fine musical judgement, and
eloquent advocacy.

While my main focus is on the late 15th to early 17th century, it
is fascinating how modal and tonal concepts interact in what might
be called the early tonal epoch around 1670-1730. Joel Lester's
book _Between Modes and Keys_ is a fascinating survey of the
situation in German from around 1592 to the mid-18th century.
The French literature around 1700 concerning "transposed modes"
that I mention below is intriguing, as are the modal organ
compositions of Chaumont (1695) for a temperament ordinaire
with nine regular and three wide fifths. Tempering the major
thirds slightly wide, with the regular fifths at around 1/5-comma
or 2/9-comma, would make the system circulate (i.e. no Wolf fifth).

A concern is that possibly I have written less of a coherent essay or
presentation than a table of contents for a treatise <grin>. Maybe the
best remedy is to say, early and often: "Please feel free to jump in
and ask questions or make comments at any point."

* * *

To me, the "meantone era" means especially the period from around 1450
to 1670 when this style of tuning (regular or sometimes modified)
nicely fit a style of modal polyphony based on a tertian ideal of
sonority (established by the 1520's or so), and also with many
characteristic accidental inflections often borrowed from the very
sophisticated medieval technique of the 14th century where the
complete trine (2:3:4) is the complete stable sonority.

However, I realize that meantone retained an important role in the
early tonal era (say 1670-1750), and indeed beyond, even as
circulating well-temperaments gained in popularity.

For me, the modal meantone era is a panorama of practices, a time when
traditional medieval patterns and concepts like intensive and
remissive cadences, and routine inflections, get used in new ways, and
sometimes reshaped to fit the new aesthetic of pervasive tertian
sonority, with subtly controlled shades of tension, as opposed to the
older dynamic contrast of stability and bold instability.

Thus the question of "modal color" can be an intricate one, because we
must consider not only the process of transposing the regular notes of
a mode, and the fluid nature of B/Bb, but also the routine or
sometimes not-so-routine inflections required to form intensive or
remissive cadences on various steps of a mode. This modal situation is
somewhat analogous to that of Near Eastern maqam and dastgah
traditions, where likewise certain accustomed inflections are part of
the routine "procedure" of a mode, nicely expressed by the Arabic
_sayr_ or Turkish _seyir_, i.e. the "path" followed the music.

Here we have a kaleidoscopic picture, in part captured by known
compositions themselves, in part reported by theorists (often of
diverse tastes and opinions!), and in part very imperfectly inferred
from this incomplete evidence. Why don't I outline a few points for
discussions, and eagerly invite anyone reading to seek clarification
of not-so-transparent points or definitions of unfamiliar terms and
concepts:

(1) Interestingly, the "standard" accidentals of Eb, Bb,
F#, C#, and G# seem common to much 14th-century and
16th-century music, in part because both periods tend
to use the same manners of cadencing, intensive or
remissive. on the same steps. Thus intensive cadences
with ascending semitones are common on F, C, G, D, and A;
and remissive semitones with descending semitones on E,
A, and sometimes D. In 1350 or 1550, a 12-note keyboard
tuning of Eb-G# nicely serves this system of modes or
octave-species and inflections -- albeit Pythagorean
in the earlier period and meantone in the latter.

(2) In the trinic focality of the 14th century, where the
restful and conclusive goal is a complete 2:3:4 trine
on the final of a mode or octave-species (e.g. D-A-D),
inflections serve mainly to alter unstable intervals
so as to make them "more closely approach" their stable
goals: for example, minor third before unison, major
third before fifth, major sixth before octave. Thus we
have intensive (E-G#-C# before D-A-D), or remissive
(Bb-D-G before A-E-A) manners. Many of these progressions
play a role in guiding the tertian textures of the 16th
century also, as does the intensive/remissive contrast,
noted by Tomas de Santa Maria in his treatise of 1565
on keyboard counterpoint and harmony. Of course, this
intensive/remissive theme of late medieval focality
takes new shapes in the meantone era.

(3) Additionally, by the 1520's, 16th-century technique
demands the regular use of accidentals to obtain
major thirds or tenths above the lowest voice of
sonorities of repose, as at final and also internal
cadences. This is one very strong reason for
adapting Eb-G# rather than Ab-C# as a usual 12-note
meantone: we need the major third E-G# for a close
in E Phrygian including the third. A love of what
Zarlino will describe as the harmonic division of
the fifth with the major third below the minor,
in frequency ratios 4:5:6, will invite many colorful
effects and inflections, including some forms of
direct chromaticism.

(4) One school of 16th-century and early 17th-century
thought is that ideally, at least, one should strive
to maintain smooth and "correct" meantone intonation
either by avoiding remote transpositions which will
place modes and especially their characteristic
cadences where the regular steps and inflections
will not be available on a given fixed-pitch
instrument; or, using an extended instrument
(anything from 13-16 to 31 notes) that will provide
those remote steps and accidentals in their usual
proportions. This is the view of Zarlino (1558), who
cautions that people writing modal transpositions
should consider the intended instrument or ensemble
and its possible limitations (not applying to singers,
of course). Praetorius, 60 years later, urges that
a meantone keyboard (1/4-comma) with 19 notes will
make available correct thirds such as F-Ab which are
missing on 12-note instruments in a usual Eb-G#.

(5) Another approach is to modify a 12-note tuning so
as to accommodate at least some locations where one
may be especially tempted to extend the system,
for example G#/Ab and/or Eb/D#. This approach is
documented as early as 1511 by Arnold Schlick, indeed
in the first practical guide on how to tune a keyboard
instrument (more specifically the organ) in a meantone
style of temperament. In Schlick's system, Ab is
somewhat compromised so as to serve as a tolerable
G# in ornamented intensive cadences on A. Praetorius
describes a similar compromise in 1618, a bit more
than a century later; and at the end of the 17th
century, Werckmeister exhorts those still enamoured
with the "Praetorian temperament" (i.e. 1/4-comma)
to at least consider this kind of finessing for
Eb/D#, for example.

(6) While authors such as Ramos (1482) and Denis (1643,
1650) emphasize that one must not play "bad" meantone
intervals, for example augmented seconds or diminished
fourths in place of regular thirds, others such as
Lanfranco (1533) tell us that C# and F#, for example,
are sometimes used as Db and Gb. This might imply
either the frank acceptance of some of the intervals
that Ramos and Denis warn us against, or a subtle
"mollification" of these intervals through a bit of
finessing (see point 5 above).

(7) Around 1700, the French literature speaks of
"transposed modes" in the sense of those involving
unusual intervals in a regular or modified meantone
tuning, and reports that some musicians find them
all the rage; the keyboard music of Couperin, as
analyzed by Lindley as neatly fitting a temperament
ordinaire, accords well with such observations.
While by this point we are moving into the realm
of early tonality, such a taste for unusual
intervals might also fit the sensibilities of late
Manneristic modality around 1600, along with the use
of new dissonances and chromatic idioms.

(8) In the performance on keyboard of a written
composition, unusual meantone intervals may occur
either if they are expressly written (e.g. D-Bb-F#
or E-G#-C), as may occur in conventional as well as
"experimental" 16th-century styles; or if sonorities
with regular spellings are performed with accidental
substitutions (e.g. written B-B-D#-F for an intensive
cadence to E realized B-B-Eb-F; written Ab-C-F-C for
a remissive cadence to G realized G#-C-F-C). How
often might performers accept the latter situation,
or possibly even relish its colorful consequences?
Lindley suggests that composers might sometimes
anticipate such a situation by writing D# to form
an intensive cadence on E, for example, but with
ornamentation to soften the vertical impact -- much
as Schlick had done with G# in his organ composing
when making an intensive cadence on A.

(9) From a modern xenharmonic perspective, neomedieval
or otherwise, the septimal approximations produced
by unusual meantone spellings can be a special
attraction. However, we should be aware that while
some musicians of the period such as Fabio Colonna
loved a range of unusual intervals and intonations,
a specific taste for septimal ratios and intervals
as an asset of meantone may first be articulated
by Huygens (1691) in his famous essay on the new
31-note harmonic cycle. It is a signal for due
caution when Vicentino (1555) finds the augmented
second or "minimal third" likely near 7:6 (in a
circulating 31-note 1/4-comma or 31-EDO) to tend
toward dissonance, and the augmented sixth likely
close to 7:4 as also on the dissonant side -- in
contrast to the neutral or "proximate minor" third
which he states is close to 11:9, and finds rather
concordant. Taking into account both Vicentino
and Huygens leaves open the possibility that
tastes might vary at any point in time.

(10) Looking at this situation from a 21st-century
perspective, we may be influenced in our view
of the meantone era by a familiarity with either
the earlier medieval or later tonal practice,
not to mention elements of other world musics.
It is fascinating to try and sort out both the
historical evidence as to the sonorous worlds
that may have obtained during this engaging
era, and our own "alternative histories" that
may be inspired by this era at once luminous
and nebulous.

Again, I welcome questions, comments, or requests for clarification on
any of these maybe not-so-easily comprehensible points, a great help
for me in trying to state things more coherently.

Most appreciatively,

Margo Schulter
mschulter@calweb.com

Re: Metastable Intervals and Tenney Complexity
--- In [34]tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
> > I find these intervals (among others) to fulfill my
> > requirements for non-Just, non-equal intervals with
> > a characteristic sound and internal cohesiveness/integrity
> > of their own- and I find them strangely "consonant".
>
> Well, we might have to agree not to call them "consonant" as such, as
> this could cause a lot of confusion, but they seem to have
> _something_, as Erv Wilson agrees.

> I've also been calling these and some other intervals "distant"
> or "soft" consonances. Strangely enough, when I first played them
> for him, a friend of mine who is a math teacher as well as
> professional contrabassist described these intervals in exactly
> the same manner I have, even eerily using similar hand gestures
> (hadn't seen him for a year)- soft, weeping and far-away. He and
> the singer he works found the chords of tonic, max.HE minor
> third, Phi-6th, and tonic, middle-3d, Phi-6th to be the cat's
> pajamas. There's more to this than just correspondence to HE
> maxima- hopefully I'll soon get a chance to post some
> demonstrations of this, but gotta run, and thanks for the kind
> words, Dave! More on Margot's zeta tuning asap, too, take care.

> -Cameron Bobro

Hi, Cameron and Dave.

Thanks to both of you for a discussion which has inspired me to a
project of recording some of these metastable intervals and their
typical resolutions in Zest-24. That may take a day or so, but in the
meantone, or meantime, why don't I comment briefly on what I suspect
the issue may be.

Here I'd want to make a distinction between a general discussion on
musical aesthetics or style, where distant or soft consonance sounds
to me often very evocative; and the rather narrow although interesting
context of a technical discussion on harmonic entropy or complexity.

In the latter, "consonance" tends to have a customary meaning of
"harmonic simplicity" or "coinciding partials" or the like, so that
indeed it could "cause a lot of confusion" if the same term were used
for maxima of complexity also. At least, I'd agree that one should be
aware of the potential conflict of meanings in this specialized arena
of discourse.

One simple solution which should avoid any potential confusion is to
use qualifying quotes. Thus:

While a 287-cent minor third is quite near the
zone of maximum complexity as suggested by the
Nobly Intoned (NI) value of around 284 cents,
I find it quite "consonant" and sweet.

Personally what I might advocate to avert this kind of semantic
complication is to use some other term than the often-charged
"consonance" for harmonic simplicity, coincidence, or smoothness.
However, given the very widespread technical use of this term by
Sethares and others, I'd suggest that in a harmonic entropy or
complexity discussion, we let metastable intervals be euphonious,
pleasing, or even concordant -- but "consonant" with the quotes.

In other contexts, where consonance/dissonance as well as
concord/discord are concepts perennially up for grabs, those quotes
seem to me unnecessary.

By the way, I wonder if maybe, in the harmonic entropy kind of
context, we might want a term like "assonance" (that came up once, I
seem to recall, in a discussion where Heinz Bohlen was involved) or
even "isonance," the latter meaning a kind of uniform or pervasive
non-coincidence of partials described by Professor Temes.

With many thanks,

Margo Schulter
mschulter@calweb.com

Dear Tom and Johnny,

Thank you both for a discussion which led to my own long post as a
kind of spinoff, but also raises other issues worthy of much
discussion.

First, Tom, I agree with your observation that different shades of
meantone temperament will have different qualities. The point about
the contrast between pure major thirds and slightly impure minor ones
in 1/4-comma is something Mark Lindley suggests could add special
interest to passages in parallel thirds.

As for 2/7-comma, which I mostly seem to be using in a modified form
(regular in the range of F-C#), I would agree with Lindley that it may
be ideal for rather moderately paced music, and also for music putting
a good deal of emphasis on modes favoring the arithmetic division of
the fifth where the minor third is placed below the major.

The big point about meantone I might emphasize, as does Jean Denis in
his treatise in the editions of 1643 and 1650, is the inequality of
the diatonic and chromatic semitones: 117/76 in 1/4-comma, and 121/71
in 2/7 comma. Even when no "special" intervals or direct chromatic
steps are used, as in a beautiful piece of Viadana I much enjoy, the
routine juxtaposition of different inflections of a step (e.g. F#/F)
can make this "enharmonic difference," as I recall Dave Keenan calls
it, vital.

For me, and also Denis, the difference in semitone sizes is of central
importance. Reacting to people who wanted keyboards tuned equally
(i.e. in 12-EDO), like the lute Denis said that rather this people
might better use their ingenuity to improve the tuning of the lute by
giving it proper major and minor semitones like the organ or
harpsichord!

Easley Blackwood also makes this point: meantone chromaticism has the
special charm of those dramatically unequal semitones, which just doesn't
translate into a 12-EDO rendition.

Johnny, I certainly agree that we shouldn't neglect irregular
temperaments other than "circulating well-temperaments." Lindley, for
example, emphasizes that Schlick's temperament would have had at least
what I'd call a "semi-Wolf" at C#-G#/Ab, about 8-10 cents wide, and
which Schlick himself urges we avoid. Finessing only one or two notes
in 1/4-comma like G#/Ab or Eb/D#, as described by Praetorius and
advocated by Werckmeister for those not interested in a circulating
well-temperament, would be unlikely to repair the Wolf fifth, although
it might alleviate it a bit, say to something like 1/6-comma.

The Grammateus temperament is a bit different, because that is
primarily based on Pythagorean tuning with diatonic whole tones
divided into two equal semitones -- that is, each equal to 9:8^(1/2),
or about 101.955 cents (e.g. F-F#-G); E-F and B-C are the
usual 256:243 or 90.224 cents. Jorgensen considers this as circulating,
with two fifths at 1/2 Pythagorean comma narrow, or just on the verge
of being Wolves by this definition. Maybe, like Schlick's, it could be
called "semi-circulating" -- but not necessarily by intent, as far as I
know. Dowland's irregular lute tuning is also interesting, and that has
at least one Wolf fifth, as I recall.

A final point I'd like to clarify is that 31 notes per octave in
1/4-comma is a very nice circulating system, much like 31-EDO, with
the subtle difference that some intervals will differ in size by about
6.07 cents, the difference between 18 pure octave and 31 1/4-comma
fifths. This variation will affect only a few major and minor thirds,
and is, of course, milder than that typically found even in quite
subtle 12-note well-temperaments.

Most appreciatively,

Margo Schulter
mschulter@calweb.com

Dear Shaahin,

Please let me suggest that if you are looking for a tetrachord
covering a tritone of some kind, the European Lydian mode might be one
example. Actually, these four notes are part of a pentachord covering
a perfect fifth:

--------------------
F G A B C
------------------------

Also, at first I thought you might want a tritone formed with four
_interval steps_, and there a Buzurg arrangement of the fifth might be
interesting, as described by Persian theorists around 1300, maybe
specifically Qutb-al-Din al-Shirazi, again with the tritone actually
part of a "pentachord" with a perfect fifth (here actually five
intervals or six notes):

0 128 359 498 626 702
1/1 14/13 16/13 4/3 56/39 3/2
14:13 8:7 13:12 14:13 117:112
128 231 139 128 76

Note that the lower tetrachord could be a permutation of Ibn Sina's
1/1-14/13-7/6-4/3, with its steps of 14:13-13:12-8:7.

In 96-EDO, we might do something like this for Buzurg:

0 125 362.5 500 625 700
125 237.5 137.5 125 75

or maybe this, with a large whole-tone step slightly narrow rather
than wide of 8:7,

0 137.5 362.5 500 625 700
125 225 137.5 125 75

More generally, we might notate the basic pattern of Buzurg, however
tuned, as this: C Dp Ep F F# G.

Most appreciatively,

Margo

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> Hi, Cameron and Dave.
>
> Thanks to both of you for a discussion which has inspired me to a
> project of recording some of these metastable intervals and their
> typical resolutions in Zest-24. That may take a day or so, but in
>the
> meantone, or meantime, why don't I comment briefly on what I
>suspect
> the issue may be.
>
> Here I'd want to make a distinction between a general discussion on
> musical aesthetics or style, where distant or soft consonance
>sounds
> to me often very evocative; and the rather narrow although
>interesting
> context of a technical discussion on harmonic entropy or
>complexity.

>
> In the latter, "consonance" tends to have a customary meaning of
> "harmonic simplicity" or "coinciding partials" or the like, so that
> indeed it could "cause a lot of confusion" if the same term were
>used
> for maxima of complexity also. At least, I'd agree that one should
be
> aware of the potential conflict of meanings in this specialized
>arena
> of discourse.
>
> One simple solution which should avoid any potential confusion is
to
> use qualifying quotes. Thus:
>
> While a 287-cent minor third is quite near the
> zone of maximum complexity as suggested by the
> Nobly Intoned (NI) value of around 284 cents,
> I find it quite "consonant" and sweet.

287 cents is very close to 13/11, but I also find it a sweet spot
below 13/11- whether that's from the detuning of these two
particularly characterful partials, or the proximity to the
strange spot at 284 cents, or a mixture of both, who knows?

>
> Personally what I might advocate to avert this kind of semantic
> complication is to use some other term than the often-charged
> "consonance" for harmonic simplicity, coincidence, or smoothness.
> However, given the very widespread technical use of this term by
> Sethares and others, I'd suggest that in a harmonic entropy or
> complexity discussion, we let metastable intervals be euphonious,
> pleasing, or even concordant -- but "consonant" with the quotes.

Hi Margot! "Distant" would lead us "Telephones", which has a nice
ring to it. :-) Personally, I'm going for "Skiophones"- the paper
I'm slowly putting together is called "Ab umbris, lumen".

>
> In other contexts, where consonance/dissonance as well as
> concord/discord are concepts perennially up for grabs, those quotes
> seem to me unnecessary.
>
> By the way, I wonder if maybe, in the harmonic entropy kind of
> context, we might want a term like "assonance" (that came up once,
>I
> seem to recall, in a discussion where Heinz Bohlen was involved) or
> even "isonance," the latter meaning a kind of uniform or pervasive
> non-coincidence of partials described by Professor Temes.

"Isophones" sounds very suitable, though I was thinking of the same
term for those skiaharmonic series consisting of harmonic means.
To be most correct, maximally inharmonic intervals would be truly
"xenharmonic"- xenophones. I don't find that the maxima of harmonic
entropy to be "the" yardstick, or "justify" intervals because
they're found by HE, rather, the reverse, for the HE charts
I have seen have a number of intervals I found by far
simpler means.

> With many thanks,

rather, thank you, Margot! Haven't had time to really dig into the
tunes you posted, but had a chance to poke at the zest24 tuning,
which is quite remarkable, not to mention playable in real life
with two manuals. I don't find the dark third or the middle third
functionally different from the ones I use, but drive of your high
third is more 9/7-y than the 422.323 cent high third I use. Also
interesting is how closely it corresponds to 88-equal- perhaps it
is possible to use 88 as a framework and thereby get a very similar
tuning with 7/4?

take care, gotta run,

Cameron Bobro

> Thank you, Margo, for your comments.

Thank you most warmly for your very helpful remarks and questions,
which have set me to considering carefully some aspects of what
"circulation" can mean, and also the statistics about "keys" or
locations for acceptable fifths and thirds that me meet in
Jorgensen and elsewhere.

I must apologize, because I've been writing for a good part of the
last 24 hours or so, and the best I've been able to come up with is a
reply of over 400 lines. Why can't I make it more concise?

Why don't I get your feedback as to whether you'd really want to read
something this long. I want the dialogue to be comfortable on all
sides. While I hope you'll take it as the deepest compliment that your
remarks prompted me to write at this length, I don't want to post
something that might upset the balance of discussion or overwhelm
rather than communicate with you and others in this thread.

If a shorter reply would be friendlier for you, I'd be glad to have
another go. Experience shows me that the better I have my argument in
order, the more concisely I should be able to express it: so length
may reflect imperfect formulation as well as sheer intellectual
excitement. On the other hand, sometimes writing at length and getting
helpful critiques is the best way to refine a concept so that it can
be more concisely expressed -- if this meets the needs of the people
doing the critiquing also!

It's an honor with to share ideas with someone who has made as great a
contribution as you have to microtonality _and_ the appreciation of a
great theorist such as Werckmeister. I want to accept this hospitality
with the greatest respect, and seek your advice on whether another go
at editing might be in order.

With deepest appreciation,

Margo

> Hi Margo,
> I noticed that you were absent from this list for
> quite some time, and it's good to see you back again.
> -monz

Dear Monz,

Thank you for your warm welcome, and for your perennial presence and
contributions. I noticed that you were involved in a translation by
Leonardo Perretti of Zarlino's discussion on 2/7-comma meantone.
Thanks to both of you for making this available! Might we do some
annotations or the like, in which I'd be delighted to participate?

As an aside, I might mention that 2/7-comma is almost identical to
119-EDO, maybe something that could be added to the TonalSoft material
if it's not already there.

With many thanks,

Margo

>> While a 287-cent minor third is quite near the zone of maximum
>> complexity as suggested by the Nobly Intoned (NI) value of
>> around 284 cents, I find it quite "consonant" and sweet.

> 287 cents is very close to 13/11, but I also find it a sweet spot
> below 13/11- whether that's from the detuning of these two
> particularly characterful partials, or the proximity to the strange
> spot at 284 cents, or a mixture of both, who knows?

Hi, Cameron, and that's a valid point. The strange spot around 284
cents, by the way, coincides with 33:28 (284.447 cents), for whatever
reason. I tend to think of something around 287 cents as a kind of
tempered compromise between 33:28 and 13:11.

>Hi Margot! "Distant" would lead us "Telephones", which has a nice
>ring to it. :-) Personally, I'm going for "Skiophones"- the paper
>I'm slowly putting together is called "Ab umbris, lumen".

That's a great title -- maybe like the corona revealed by a total
solar eclipse.

>> quotes seem to me unnecessary. By the way, I wonder if maybe,
>> in the harmonic entropy kind of context, we might want a term
>> like "assonance" (that came up once, I seem to recall, in a
>> discussion where Heinz Bohlen was involved) or even "isonance,"
>> the latter meaning a kind of uniform or pervasive
>> non-coincidence of partials described by Professor Temes.

> "Isophones" sounds very suitable, though I was thinking of the
> same term for those skiaharmonic series consisting of harmonic
> means. To be most correct, maximally inharmonic intervals would
> be truly "xenharmonic"- xenophones. I don't find that the maxima
> of harmonic entropy to be "the" yardstick, or "justify" intervals
> because they're found by HE, rather, the reverse, for the HE
> charts I have seen have a number of intervals I found by far
> simpler means.

Considering this, I'd agree that "isophones" or "isonance" runs into a
possible confusion with isoharmonic sonorities or chords, which have a
ery clear and accepted meaning (e.g. 7:9:11:13). However, your
"xenophones," or maybe xenonances, looks perfect. These nobly intoned
(NI) intervals, as Dave has named them, are indeed "strange" ones at
ratios maximally distant by at least one possible mathematical measure
from the familiar realm of simple ratios with coinciding partials.
What I like about "xenonance" is that it tells us that this is
definitely something different, but doesn't impose any viewpoint as to
just how we should hear it -- as concord, discord, or something else.
It also has a pleasant association with the "xentonality" of Bill
Sethares.

>> With many thanks

> , rather, thank you, Margot! Haven't had time to really dig into the
> tunes you posted, but had a chance to poke at the zest24 tuning,
> which is quite remarkable, not to mention playable in real life with
> two manuals.

Please let me tell you how encouraging and energizing your interest
is: it helps keep me going on this. Your comment hits the nail right
on the head: above all, I wanted something new that would be playable
in real life as well as theory.

> I don't find the dark third or the middle third functionally
> different from the ones I use, but drive of your high third is more
> 9/7-y than the 422.323 cent high third I use.

Maybe I should just whether we're speaking of the 420.948-cent third,
very close to yours (F#/Gb-Bb and G#/Ab-C in each 12-note circle), or
the 433.517-cent third (C#/Db-F in each circle, with a few available
also by combining notes from the two circles)? Either way, what you're
saying is of great interest to me.

> Also interesting is how closely it corresponds to 88-equal- perhaps
> it is possible to use 88 as a framework and thereby get a very
> similar tuning with 7/4?

You're right, and I hadn't even considered an 88-EDO version. Looking
at this in Scala, I saw how indeed, while Zest-24 has closest
approximations to 7/4 at around 963 and 975 cents, with 88-EDO we get
968.182 cents, virtually just!

With such closely related versions of a tuning, people could be drawn
to subtly different shadings for a range of reasons -- as with the
quest for the "ideal" degree of meantone. One reason for sometimes
preferring a version right around 2/7-comma for the regular meantone
fifths is that Mark Lindley has suggested that this might be about the
high end of the range with maximum euphony, and gentler on the fifths
and major thirds than something further toward 1/3-comma. However,
Wilson's Metameantone is quite comparable to your 88-EDO version, and
that 7/4 makes 88-EDO a logical spot to choose in this neck or niche
of the woods.

Best,

Margo

More 88 edo info. here:

http://en.wikipedia.org/wiki/List_of_meantone_intervals

Charles Lucy lucy@lucytune.com

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On 15 Oct 2007, at 23:33, Margo Schulter wrote:

> >> While a 287-cent minor third is quite near the zone of maximum
> >> complexity as suggested by the Nobly Intoned (NI) value of
> >> around 284 cents, I find it quite "consonant" and sweet.
>
> > 287 cents is very close to 13/11, but I also find it a sweet spot
> > below 13/11- whether that's from the detuning of these two
> > particularly characterful partials, or the proximity to the strange
> > spot at 284 cents, or a mixture of both, who knows?
>
> Hi, Cameron, and that's a valid point. The strange spot around 284
> cents, by the way, coincides with 33:28 (284.447 cents), for whatever
> reason. I tend to think of something around 287 cents as a kind of
> tempered compromise between 33:28 and 13:11.
>
> >Hi Margot! "Distant" would lead us "Telephones", which has a nice
> >ring to it. :-) Personally, I'm going for "Skiophones"- the paper
> >I'm slowly putting together is called "Ab umbris, lumen".
>
> That's a great title -- maybe like the corona revealed by a total
> solar eclipse.
>
> >> quotes seem to me unnecessary. By the way, I wonder if maybe,
> >> in the harmonic entropy kind of context, we might want a term
> >> like "assonance" (that came up once, I seem to recall, in a
> >> discussion where Heinz Bohlen was involved) or even "isonance,"
> >> the latter meaning a kind of uniform or pervasive
> >> non-coincidence of partials described by Professor Temes.
>
> > "Isophones" sounds very suitable, though I was thinking of the
> > same term for those skiaharmonic series consisting of harmonic
> > means. To be most correct, maximally inharmonic intervals would
> > be truly "xenharmonic"- xenophones. I don't find that the maxima
> > of harmonic entropy to be "the" yardstick, or "justify" intervals
> > because they're found by HE, rather, the reverse, for the HE
> > charts I have seen have a number of intervals I found by far
> > simpler means.
>
> Considering this, I'd agree that "isophones" or "isonance" runs into a
> possible confusion with isoharmonic sonorities or chords, which have a
> ery clear and accepted meaning (e.g. 7:9:11:13). However, your
> "xenophones," or maybe xenonances, looks perfect. These nobly intoned
> (NI) intervals, as Dave has named them, are indeed "strange" ones at
> ratios maximally distant by at least one possible mathematical measure
> from the familiar realm of simple ratios with coinciding partials.
> What I like about "xenonance" is that it tells us that this is
> definitely something different, but doesn't impose any viewpoint as to
> just how we should hear it -- as concord, discord, or something else.
> It also has a pleasant association with the "xentonality" of Bill
> Sethares.
>
> >> With many thanks
>
> > , rather, thank you, Margot! Haven't had time to really dig into the
> > tunes you posted, but had a chance to poke at the zest24 tuning,
> > which is quite remarkable, not to mention playable in real life with
> > two manuals.
>
> Please let me tell you how encouraging and energizing your interest
> is: it helps keep me going on this. Your comment hits the nail right
> on the head: above all, I wanted something new that would be playable
> in real life as well as theory.
>
> > I don't find the dark third or the middle third functionally
> > different from the ones I use, but drive of your high third is more
> > 9/7-y than the 422.323 cent high third I use.
>
> Maybe I should just whether we're speaking of the 420.948-cent third,
> very close to yours (F#/Gb-Bb and G#/Ab-C in each 12-note circle), or
> the 433.517-cent third (C#/Db-F in each circle, with a few available
> also by combining notes from the two circles)? Either way, what you're
> saying is of great interest to me.
>
> > Also interesting is how closely it corresponds to 88-equal- perhaps
> > it is possible to use 88 as a framework and thereby get a very
> > similar tuning with 7/4?
>
> You're right, and I hadn't even considered an 88-EDO version. Looking
> at this in Scala, I saw how indeed, while Zest-24 has closest
> approximations to 7/4 at around 963 and 975 cents, with 88-EDO we get
> 968.182 cents, virtually just!
>
> With such closely related versions of a tuning, people could be drawn
> to subtly different shadings for a range of reasons -- as with the
> quest for the "ideal" degree of meantone. One reason for sometimes
> preferring a version right around 2/7-comma for the regular meantone
> fifths is that Mark Lindley has suggested that this might be about the
> high end of the range with maximum euphony, and gentler on the fifths
> and major thirds than something further toward 1/3-comma. However,
> Wilson's Metameantone is quite comparable to your 88-EDO version, and
> that 7/4 makes 88-EDO a logical spot to choose in this neck or niche
> of the woods.
>
> Best,
>
> Margo
>
>
>

> Thank you, Margo, for your comments.

Thank you for some very rich and energizing remarks. I realize that,
if you're so inclined, we could be in for a long conversation, and
will try to follow your points step by step.

> Margo: Lindley, forexample, emphasizes that Schlick's temperament
> would have had at least what I'd call a "semi-Wolf" at C#-G#/Ab,
> about 8-10 cents wide, and which Schlick himself urges we avoid.

> Johnny: It seems that there is a standard irregular tuning family,
> much like the meantone family. They commonly have 20 major and
> minor keys (such as with Fischer). This indicates there are 2
> "semi-wolves" as you call them.

May I assume that "keys" here means simply the number of locations in
a tuning with a fifth plus a major or minor third of an acceptable
size for a given style? What I'd like to do is to look at Schlick from
a few stylistic angles, including Schlick's as far as we can tell, in
a way that might address some of the issues of the counting process.
I must admit I don't know what Fischer's temperament from the Bach era
is, but I'm very interested to learn. Why don't I give a Scala file
for one reading of Schlick (1511), based largely on Lindley's analysis:

! schlick_1511.scl
!
One reading of Arnold Schlick's modified meantone (1511)
12
!
88.27000
196.09000
303.91000
392.18000
501.95500
589.24750
698.04500
799.02250
894.13500
1002.93250
1090.22500
2/1

First, let's consider the fifths: 11 locations have fifths within
about 5 cents of pure, with C#/Ab as the semi-Wolf or "dog" at about
8-10 cents wide according to Lindley. In a medieval style, where major
and minor thirds are unstable and flexible in their tuning, and thus
not an essential element in defining a stable polyphonic center, we
have 11 locations for transposing any desired mode -- as long as that
pesky "dog fifth" doesn't "bark" on a degree where we need a stable
fifth, like the confinal or reciting tone. Actually that canine might
be very exciting and charming if it occurs between two upper voices in
the unstable sonority E-G#/Ab-C# leading to a cadence on D-A-D!
Indeed, Lindley says (and I agree) that in this kind of medieval
cadence, the upper fourth of that unstable sixth sonority might be
impure by a full Pythagorean comma!

However, for medieval or neomedieval purposes where "circulation"
means simply that all 12 fifths are within maybe 7 cents or so of pure
(as in 19-EDO, or 22-EDO with tempering in the opposite direction),
it's easy to achieve full circulation simply by lowering Ab/G# about
two or three cents, so that Ab-C and E-G# are about equally impure,
each somewhere 404 cents. This doesn't have to be precise, and here
I've simply picked one possibility that still favors Ab-C a bit:

! Schlick1511_variation.scl
!
Variation on Schlick (1511), all 5ths within 7c of pure
12
!
88.27000
196.09000
303.91000
392.18000
501.95500
589.24750
698.04500
797.00000
894.13500
1002.93250
1090.22500
2/1

Now let's go back to Schlick's temperament itself, viewing it from his
own stylistic perspective in an early 16th-century context. Now thirds
as well as fifths become factors in defining a good modal center. By
Schlick's own criteria, we have 8 locations with reasonably agreeable
major thirds, Ab-C less smooth than the others but still acceptable
for sustained use (maybe around 12-EDO size).

At around Pythagorean, E-G# is unstable but tolerable for use in
ornamented intensive cadences to A (with an ascending semitone G#-A);
B-Eb, F#-Bb, and C#-F are like typical meantone diminished fourths,
here maybe 413 cents or so, in the zone Secor has discussed between
Pythagorean and the region of 14:11 (418 cents) or a tad wider where
listeners might still hear some semblance of relatedness to 5:4 in
styles where it is the ideal. Writing in a 16th-century context, not
surprisingly, Schlick tells us that they are "little esteemed and
seldom used" -- but maybe also, one is tempted to guess, not
categorically excluded, at least for the more adventurous transposers
whom certain theorists either mention without reproach (Lanfranco
1533) or passionately warn us against (Denis 1643, 1650).

Thus we have 11 stable fifths, and 8 stable centers for modes
featuring major thirds above the final (Lydian, Ionian, Mixolydian) --
assuming that the C#-Ab/G# "dog" or those 4 major thirds at
Pythagorean or larger don't disrupt other steps where we need stable
tertian sonorities. How about modes with minor thirds above the final
(Dorian, Aeolian, Phrygian)?

Here things are more problematic, because we don't know Schlick's
criteria for minor thirds. If we embrace one modern view that the
tuning of these intervals is very flexible -- like major thirds also
in a medieval rather than Renaissance/Manneristic context -- with
anything from around 7:6 to 6:5 acceptable, then we have potentially
11 locations for these modes, everything except for C# with that dog
fifth. Again, once we consider other steps of the mode and cadences,
it will get more complicated.

If we instead say that a minor third shouldn't be "substantially"
smaller than Pythagorean, based on theorists who outright reject
meantone augmented seconds used as normal minor thirds (e.g. Denis),
or at least urge great caution (e.g. Vicentino, 1555), then we have 8
such minor thirds at locations with acceptable fifths, here including
Ab/G#-B-Eb with the minor third at around 291 cents, or three cents
narrow of Pythagorean. This leaves Eb-F# and Bb-C# at 285 cents or so,
outside the range of musical interchangeability under this criterion.

It's interesting to ask how a composer like Josquin might take to
playing something like his beautiful _Mille regretz_ in Eb Phrygian
with the third above the final Eb-F# at around 285 cents. Would it
sound too far from the new meantone ideal -- or possibly quite
congenial to the older Pythagorean tradition still in evidence around
1500, where it could add to the melancholy quality of the piece: a
touch of "modal color" analogous to the key color that French writers
about 200 years later describe for very small minor thirds in a
temperament ordinaire?

Unfortunately, Eb Phrygian becomes rather academic because of that
pesky dog fifth on C#, the step below the final, although that step is
less prominent than in lots of Phrygian settings. If we're going to
try it, I'd say let's lower Ab/G# a bit as in my variation, which
wouldn't affect the size of Eb-F# or Bb-C#, our smallest minor thirds.

My point is that counting acceptable modal or tonal centers can be
very sensitive to what people are assuming is "acceptable," and also
that having "8 locations with an acceptable major third and fifth; and
9 with an acceptable minor third and fifth" is only the first step in
deciding how many of those locations would also accommodate the
confinals, usual internal cadences, and so forth for a typical piece
in Dorian, say.

Let's illustrate this moving ahead a bit in the 16th century, to the
1520's and later. How about untransposed E Phrygian, or A Dorian for
that matter? With either, we encounter a very basic problem: a closing
sonority with a third above the final -- or likewise a sectional close
on some other step of the mode -- is by this point expected to be
major, by inflection if necessary. A Pythagorean E-G# thus makes an
up-to-date Phrygian untenable, and likewise modes like A Dorian which
often have internal cadences on E (here the confinal).

Anyway, as far as the fifths go, there's only one "semi-Wolf" in
Schlick's temperament, C#-Ab/G#. It's a situation where we have a
harmonic excess from the 11 notes or 10 fifths Eb-C#, with the six
diatonic fifths F-B maybe around 1/5-comma and the other four around
1/6-comma according to Lindley, of something like 12 cents. The
"finessing" involves the single note Ab/G#, with Schlick evidently
making Ab-Eb about 2-4 cents wide (thus comparable to some of the
narrow fifths), and leaving 8-10 cents for our C#-Ab/G# "dog." A
division of this excess more like 5/7 cents or 6/6 cents would make
all fifths fully circulating.

Again, I'm very curious about the Fischer temperament and how it might
compare to this one.

> I have seen the use of the term "dog-fifths" for these (for being
> more tame than wolves...instead of howling, they merely bark).

Yes, I have seen someone, Bradley Lehman maybe, compare the Wolf in
1/6-comma to a "border collie." By the way, in discussing
"circulation," maybe it would be helpful to separate the wolves and
dogs from the goats, so to speak.

My idea is to use the canicular terms for fifths somewhat more than
1/3-comma from just: maybe "dogs" in the Schlick range of 8-10 cents;
"semi-Wolves" around 1/2 comma or a bit more (Jorgensen goes as far as
15 cents or so in certain circumstances); and outright Wolves beyond
that.

In contrast, "goats" are thirds appreciably larger than Pythagorean
(if major) or smaller (if minor). My inspiration for this is a French
term, _ton de la chevre_ or "goat tone or mode," evidently used by
around 1700 to describe a transposition like C# Dorian or Aeolian
(moving by this point into minor) with "bleating" major thirds in a
regular or modified meantone. While the term might actually have
originated to describe some kind of lute effect involving timbre
rather than supra-Pythagorean major thirds, it came to describe a
keyboard transposition with those extra-wide thirds.

Thus in a conventional well-temperament, the idea is to avoid both
wolves or dogs and goats; in a "fifth-circulating" temperament, as I
might call it, we might specifically seek to have meantone color over
a bit more than half of the 12-note circle while making the remote
region a peaceful pasture where goats may safely graze. The variation
on Schlick that I describe would thus be a fifth-circulating
temperament of the goat-friendly variety, but certainly _not_ a
"well-temperament"!

> Margo: Finessing only one or two notes in 1/4-comma like G#/Ab or
> Eb/D#, as described by Praetorius and advocated by Werckmeister for
> those not interested in a circulating well-temperament, would be
> unlikely to repair the Wolf fifth, although it might alleviate it a
> bit, say to something like 1/6-comma.

> Johnny: I could not find this in Werckmeister's Musical
> Temperament, which I have just typed out to put with a new book I
> have written on Bach's tuning. Do you recall where you saw this?

I'm pretty sure that it was in this article, only available via the
Internet through a subscription but very likely still at my library:

Bach's keyboard temperament
Internal evidence from the Well-Tempered Clavier
BARNES Early Music.1979; 7: 236-249

This is a very well-written article that considers the evidence in a
calm and balanced way, and shows an appreciation for the fine points
of unequal temperament. As I recall, the relevant passage (possibly
from the Orgelprobe?, if not from the work you mention) occurs in a
discussion where he mentions Zarlino's 2/7-comma, and shows that even
1/7-comma will have a dog fifth, as we're calling it, so that the
direction to circulation lies elsewhere. Then he describes how people
who are wedded to the Praetorian temperament could still benefit from
the concepts he is presenting by adjusting Eb (if I recall correctly)
so that it is also usable as D# -- in other words, a very rudimentary
temperament ordinaire. This is my recollection, at any rate. A trip to
the library could clear this up, and I'll try to do it in the next day
or two.

> Margo: The Grammateus temperament is a bit different, because that
> is primarily based on Pythagorean tuning with diatonic whole tones
> divided into two equal semitones -- that is, each equal to
> 9:8^(1/2), or about 101.955 cents (e.g. F-F#-G); E-F and B-C are
> the usual 256:243 or 90.224 cents.

> Johnny: Doesn't it have 20 keys only like other irregular tuning?

The count of "keys" or locations, for a modal or tonal style, could
depend on two main things.

First, we have 10 pure fifths plus two (Bb-F and B-F) impure by 1/2
Pythagorean comma. For Jorgensen, a "Wolf fifth" is anything _more_
impure than precisely this amount (11.73 cents) -- so these two make
it just under the wire, and he considers it a circulating
temperament.

Second, since all fifths are pure or narrow, we don't have any goat
thirds to consider. For Jorgensen, a major third is serviceable if not
significantly larger than Pythagorean, and a minor third at just about
any size from around 7:6 to 6:5 -- so we have all 24 locations
covered, 12 major and 12 minor thirds.

If we consider those semi-Wolves or dog-fifths too assertively barking
to be stable, then we have 10 rather than 12 locations available as
stable centers in a medieval type of context; as in lots of early
15th-century music, the different shadings of unstable thirds will add
lots of modal color, but are more or less freely interchangeable.

In part. Grammateus is offering a mathematical demonstration or
"amusing reckoning" for the sheer intellectual joy of showing how a
superparticular ratio like 9:8 can be equally divided by geometrical
means. Musically, however, the tuning seems to me to fit with an early
15th-century kind of outlook where the idea is to have a basically
Pythagorean system, but with some schismatic or allied thirds deployed
at certain strategic locations. In this view, fifth-circulation is not
the point, but splitting the Wolf fifth into two parts to combine the
remaining ten pure fifths with a larger number of "mollified" thirds
including one of these two semi-Wolves in their chains. An English
organ tuning of 1373 with an 18:17:16 division of diatonic whole tones
(e.g. C-C#-D) in a generally Pythagorean scheme might address similar
concerns, and would well comport with the penchant for 5-limit or
nearby tunings of thirds shown in some strains of medieval English
theory and practice, of which you have also spoken.

Putting the two semi-Wolves on steps a semitone apart (Bb-F, B-F#)
means that no fewer than 8 of the 12 major thirds will be so tempered
at about 396 cents, but creates a semi-Wolf at Bb-F which can create
problems with lots of late medieval music. Relocating these fifths to
locations like Eb-Bb and F#-C# (7 major thirds smaller than 400 cents)
or F#-C# and G#-Eb (6 major thirds smaller than 400 cents) gives many
of the same advantages with less disruption.

It is worth emphasizing that these variations, like the original
Grammateus tuning, really belong more to the musical style current
around 1400 rather than 1500, and do not address the needs of a style
calling for some variety of meantone. The diatonic major thirds F-A,
C-E, and G-B, for example, remain Pythagorean -- so that the system is
not in the same musical category as Schlick's.

This said, we may note that the problem for someone wishing to make
these systems fully fifth-circulating are quite different, also. With
the Grammateus modified Pythagorean system, we face the problem that
the Wolf simply cannot be shared between only two fifths while making
both fifths less impure than 1/2 Pythagorean comma -- one or both must
howl, or at least bark quite assertively! An equal division among
three fifths could be marginally acceptable, with four fifths much
better. The result would be akin to a well-temperament, but with pure
fifths and Pythagorean thirds in the nearer part of the circle, and
tempered fifths and smoother thirds in the more remote portion. This
pattern fits with 15th-century Pythagorean variations rather than with
either 16th-century meantone or an 18th-century well-temperament.

With Schlick, the harmonic excess is something on the order of
about 12 cents, so it's easy to have both fifths within 7 cents or so
of pure if that's the goal -- which it wasn't for Schlick, who
evidently wanted to compromise Ab-C as little as possible while very
partially taming the goat third at E-G#.

> Margo: Jorgensen considers this as circulating, with two fifths at
> 1/2 Pythagorean comma narrow, or just on the verge of being Wolves
> by this definition.

> Johnny: While modern ears might enjoy a bark or even a howl, the
> earliers (as opposed to the moderns) strictly avoided moving in
> their territory. And we are speaking of improvisation, as
> composition strictly avoided the canine keys. I must disagree
> with Jorgensen; there is no circulating without a circle, and that
> begins on paper with Werckmeister.

Here I'd urge that we sort out the wolves and goats. On the canine
side, I'd agree that it's very unlikely that composers are going to
cadence on a fifth which is assumed to be half a comma or more
impure. If we see a fifth Ab-Eb, I'd take that as a hint that the
piece is intended for an instrument which actually has both notes!

There's an interesting partial exception that seems likely in the
earlier 15th century, when we find keyboard pieces in places like the
Buxheimer Organ Book with cadences like this:

B C
F# G
D C

Totally routine -- until we note, as Lindley does, other stylistic
indications that the piece is intended for a regular Pythagorean
tuning with written sharps tuned as flats, Gb-B, so that the upper
fourth F#-B, or actually the augmented third Gb-B, is wide by a full
comma! However, given the consonances formed by both upper voices with
the lowest one, and also the cadential context where we _expect_
instability and excitement when major third and sixth expand to fifth
and octave, he and I agree that it can be quite pleasing in practice.

Maybe that's the exception that proves the rule, as far as these
canine fifths go. It might fit with the 14th-century observation of
Johannes Boen (1357) that either an augmented or diminished fourth can
become a "consonance by circumstance" (_consonantia per accidens_)
when an appropriate third is placed below it (D-F-B or E-G#-C) -- and
so, apparently, an early 15th-century Wolf fourth.

However, depending on the concept of "circulation" at issue, we could
be talking about only wolves and dogs, or also goats. Here are two
possible definitions I'd consider relevant to this discussion:

-----------------
Fifth-circulation
-----------------

A 12-note tuning may be considered fifth-circulating if all 12 fifths
are within about 1/3-comma or 7 cents of pure: major and minor thirds
may be of any size as long as this condition is met. In a very
colorful modified meantone, for example, they may often range from
roughly 5-limit or pental to septimal: 5:4-9:7 and 7:6-6:5 for major
and minor thirds respectively.

-----------------------------------------
Tertian circulation or "well-temperament"
-----------------------------------------

A 12-note tuning may be considered tertian-circulating or
well-tempered when these three conditions are all met:

(1) The tuning meets the requirements for fifth-circulation,
with all 12 fifths no more than about 7 cents from just;

(2) The diatonic major thirds F-A, C-E, and G-B are not
substantially further from a just 5:4 than the 400 cents of
12-EDO.

(3) There are no "goat" major thirds substantially larger than
Pythagorean, with around 409-410 cents (say 19:15, used in a
temperament by Aaron Johnson) as a possible bounding region.

Getting fifth-circulation in a modified meantone is easy, and for a
more moderate degree of temperament like Schlick's irregular scheme
can easily be done by finessing only one note -- he could have done it
if he'd wanted to, and the same goes for a regular 1/6-comma meantone
where we adjust one note to split the Wolf about equally between two
fifths. Around 1/5-comma or 2/9-comma, we need to adjust two notes and
have three fifths share the excess; while for 1/4-comma or 2/7-comma,
this involves three adjusted notes or four comparably wide fifths.

However, I'd agree that for tertian circulation or well-temperament,
Werckmeister is a great innovator -- the only recorded earlier schemes
that I can think of that would qualify would be proposals or purported
proposals (like the one attributed by Doni to Frescobaldi) to tune
keyboards in 12-EDO. Very basically, even if Grammateus had split a
Pythagorean wolf into three or yet better four equal parts, achieving
full fifth-circulation, it would address a medieval kind of style
rather than the imperatives of meantone so central to Werckmeister.

Here the crucial imperative is requirement (2): the nearest major
thirds should be 12-EDO or smoother, not the Pythagorean size so
effective for medieval or neomedieval music but exactly what Schlick
and others tuning regular or modified meantone around 1500 are seeking
to get away from! To people like Zarlino and Lippius a Pythagorean
third is not only unstable but just about "dissonant," although
Praetorius gives us evidence that some people were ready to tolerate
it in order to have some approximation of both Eb and D# on a 12-note
instrument, for example. It's one thing to hit this now and then for
some remote sonority or cadence, another to have it at some of your
most frequently used locations for a 16th-century modal or
18th-century tonal style.

As for colorful fifth-circulating modified meantones, emphatically
including mine, as well as Schlick's rather more restrained approach,
we encounter requirement (3) of well-temperament: no goat thirds.
Note that this rule does _not_ exclude all fifths wider than pure, as
long as they are discreetly deployed, as in some of Neidhardt's
temperaments or Aaron Johnson's, so that thirds don't go substantially
outside Pythagorean limits.

On the subject of goats, as opposed to wolves and dogs, these
indubitably _are_ used in conventional as well as experimental
16th-century writing where they are explicitly spelled as diminished
fourths, for example. Here there is a fine and calculated balance,
whether in a Spanish Psalm setting or in an avant garde madrigal by
Monteverdi. The interval is "cushioned" by occurring between two upper
voices; occurs in passing; or otherwise is treated with due
discretion, rather like a more conventional suspension dissonance.

The open question is how often people made keyboard transpositions
where these goat thirds were substituted for regularly spelled ones in
a composition, and also whether composers, for example of the English
virginals school, either desired or expected this to happen for some
pieces going beyond the spellings available on a 12-note
instrument. Denis goes to great length to warn about the perils of
such transposed modes, which suggests that the practice wasn't unknown
in early 17th-century Paris, at any rate. However, he associates it
with organists too eager to accommodate singers, a motivation not so
relevant to solo keyboard performances then or now.

> Margo: Maybe, like Schlick's, it could be called "semi-circulating"
> -- but not necessarily by intent, as far as I know. Dowland's
> irregular lute tuning is also interesting, and that has at least
> one Wolf fifth, as I recall.

> Johnny: Dowland's tuning has different pitch distinctions for the
> same pitch, making it an irregular irregular tuning. As a lute
> tuning that does not need to make a full circle, it may best be
> categorized as a different can of beans.

Certainly I'd agree that this is in a different category from a
12-note modified meantone or the like, and maybe related to the
_tastini_ or alternative frets that Vincenzo Galilei tells us that
some lutenists used to improve the thirds.

> What do you think?

Mainly I think that we're discussing topics of great mutual interest,
from different perspectives, of course. Yours, as a performer and
advocate, may be the most valuable of all. I'm not sure if any of this
is intelligible and accessible, but if it is, I would be delighted.

> best, Johnny

With many thanks,

Margo

Hi Margo,

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> > Hi Margo,
> > I noticed that you were absent from this list for
> > quite some time, and it's good to see you back again.
> > -monz
>
> Dear Monz,
>
> Thank you for your warm welcome, and for your perennial
> presence and contributions. I noticed that you were
> involved in a translation by Leonardo Perretti of
> Zarlino's discussion on 2/7-comma meantone. Thanks to
> both of you for making this available! Might we do some
> annotations or the like, in which I'd be delighted to
> participate?

Absolutely. Please feel free to write any annotations
you'd like, and i'll be happy to include them into the
webpage.

> As an aside, I might mention that 2/7-comma is almost
> identical to 119-EDO, maybe something that could be
> added to the TonalSoft material if it's not already there.

I do have a quite extensive page about 2/7-comma meantone
which i wrote myself as a "regular" entry in the
Encyclopedia:

http://tonalsoft.com/enc/number/2-7cmt.aspx

That information should go on that page. If you want
to add anything else about it, please post it here
and i'll include it on the page. Thanks.

-monz
http://tonalsoft.com
Tonescape microtonal music software

> Hi Margo,
> welcome back

Dearest Leonardo,

Please let me send my warmest greetings from California in the USA.

>> Thank you for your warm welcome, and for your perennial presence
>> and contributions. I noticed that you were involved in a
>> translation by Leonardo Perretti of Zarlino's discussion on
>> 2/7-comma meantone. Thanks to both of you for making this
>> available! Might we do some annotations or the like, in which
>> I'd be delighted to participate?

> I will be happy to collaborate, of course.
> The translation needs some small corrections, so this would be a good
> chance to do them too.

That would be wonderful, and it is a great pleasure to be in touch
with you as the author of this fine contribution. What I'll do is to
look over the translation in the next few days and make notes to share
with you. I very much enjoy your style, and am fascinated by how the
idioms of our two languages might interact in translating from either
to the other.

> Greetings from Italy
> Leonardo

With many thanks,

Margo

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
> By the way, I wonder if maybe, in the harmonic entropy kind of
> context, we might want a term like "assonance" (that came up once, I
> seem to recall, in a discussion where Heinz Bohlen was involved) or
> even "isonance," the latter meaning a kind of uniform or pervasive
> non-coincidence of partials described by Professor Temes.

Dear Tuners,

Soon after this thread began, a week or so ago, I attempted to google
an email address or other means of contacting Lorne Temes to let him
know about the current interest in his work of more than 30 years ago.

I was unsuccessful, but the parallel with George Secor, and our
rediscovery of his "miracle" temperament after 25 years, did not
escape me.

I couldn't help wondering if, were Lorne Temes to appear on the tuning
list, we would be faced with another madman who would later claim he
was really a Greek god merely _impersonating_ Lorne Temes. :-)

If you are unfamiliar with the George Secor case see 'Gift of the Gods' at
http://dkeenan.com/sagittal/gift/Episode1.htm

Regards,
-- Dave Keenan

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> --- In tuning@yahoogroups.com, Margo Schulter <mschulter@> wrote:
> > By the way, I wonder if maybe, in the harmonic entropy kind of
> > context, we might want a term like "assonance" (that came up
once, I
> > seem to recall, in a discussion where Heinz Bohlen was involved)
or
> > even "isonance," the latter meaning a kind of uniform or pervasive
> > non-coincidence of partials described by Professor Temes.
>
> Dear Tuners,
>
> Soon after this thread began, a week or so ago, I attempted to
google
> an email address or other means of contacting Lorne Temes to let him
> know about the current interest in his work of more than 30 years
ago.
>
> I was unsuccessful, but the parallel with George Secor, and our
> rediscovery of his "miracle" temperament after 25 years, did not
> escape me.
>
> I couldn't help wondering if, were Lorne Temes to appear on the
tuning
> list, we would be faced with another madman who would later claim he
> was really a Greek god merely _impersonating_ Lorne Temes. :-)
>
> If you are unfamiliar with the George Secor case see 'Gift of the
Gods' at
> http://dkeenan.com/sagittal/gift/Episode1.htm
>
> Regards,
> -- Dave Keenan

Madman indeed! Dave, how thoughtless of you! This is the last
straw! :-(

After all we've been through, and after the painstaking effort I made
to explain the many obstacles to my being taken seriously, I'm really
disappointed that you would stoop so low.

Furthermore, your dogged insistence on firming up the seemingly
endless details of minas, tinas, and innumerable other things
relating to the semantics of the Sagittal notation have kept me from
a much more important task: that of resuming work on the truly
remarkable story of how the notation was developed. I must take this
opportunity, therefore, to express my profuse apologies to those who
faithfully followed the story through the end of Episode Two, only to
be left hanging (for more than 3 years!) with the yet unfulfilled
promise that the account was "to be continued." My patience has
completely run out. I can't tolerate this situation any longer!

Dave, there's still a way to redeem yourself. I've worked around the
clock on a temporary short conclusion to the story that will help to
set the record straight. I'll be sending that to you very shortly
after I post this message, and I'll forgive you for your previous
impositions if you'll put it on the Sagittal website, where everyone
can read it.

We'll all be looking for it soon, so don't let us down!

--"George"

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> >> While a 287-cent minor third is quite near the zone of maximum
> >> complexity as suggested by the Nobly Intoned (NI) value of
> >> around 284 cents, I find it quite "consonant" and sweet.
>
> > 287 cents is very close to 13/11, but I also find it a sweet spot
> > below 13/11- whether that's from the detuning of these two
> > particularly characterful partials, or the proximity to the
strange
> > spot at 284 cents, or a mixture of both, who knows?
>
> Hi, Cameron, and that's a valid point. The strange spot around 284
> cents, by the way, coincides with 33:28 (284.447 cents), for
whatever
> reason.

Also 5:6 from the half-octave... and has some other "coincidental"
features. The pure minor third from the half-octave is a
real plus for me using a 34 framework- I have 12 pure noble dark
thirds and 12 pure 6:5s in the tuning I'm using at the moment.
The "tempered" versions of both (it's a "WT") are consistent in
sign, with all but two dark thirds coinciding with the third I
measured in a whale song, 2 cents lower than the noble dark
third. The tuning has two clunkers for these intervals, 4.6 cents
off of the ideal but with the right sign at least.

>I tend to think of something around 287 cents as a kind of
> tempered compromise between 33:28 and 13:11.

If you prefer it to either, I'll just call it Margot's Dark Third,
if you don't mind. If it's what you want, it's not a
compromise, is it. :-)

>
> >Hi Margot! "Distant" would lead us "Telephones", which has a
nice
> >ring to it. :-) Personally, I'm going for "Skiophones"- the
paper
> >I'm slowly putting together is called "Ab umbris, lumen".
>
> That's a great title -- maybe like the corona revealed by a total
> solar eclipse.

Hadn't thought of it that way, but I'll quote you on that
image, if I may, in discussing the strength of
resolving/dissolving from maximum to minimum "dissonance".

>
> >> quotes seem to me unnecessary. By the way, I wonder if
maybe,
> >> in the harmonic entropy kind of context, we might want a term
> >> like "assonance" (that came up once, I seem to recall, in a
> >> discussion where Heinz Bohlen was involved) or
even "isonance,"
> >> the latter meaning a kind of uniform or pervasive
> >> non-coincidence of partials described by Professor Temes.
>
> > "Isophones" sounds very suitable, though I was thinking of the
> > same term for those skiaharmonic series consisting of harmonic
> > means. To be most correct, maximally inharmonic intervals
would
> > be truly "xenharmonic"- xenophones. I don't find that the
maxima
> > of harmonic entropy to be "the" yardstick, or "justify"
intervals
> > because they're found by HE, rather, the reverse, for the HE
> > charts I have seen have a number of intervals I found by far
> > simpler means.
>
> Considering this, I'd agree that "isophones" or "isonance" runs
into a
> possible confusion with isoharmonic sonorities or chords, which
have a
> ery clear and accepted meaning (e.g. 7:9:11:13). However, your
> "xenophones," or maybe xenonances, looks perfect. These nobly
intoned
> (NI) intervals, as Dave has named them, are indeed "strange" ones
>at
> ratios maximally distant by at least one possible mathematical
>measure
> from the familiar realm of simple ratios with coinciding partials.
> What I like about "xenonance" is that it tells us that this is
> definitely something different, but doesn't impose any viewpoint
>as to
> just how we should hear it -- as concord, discord, or something
else.
> It also has a pleasant association with the "xentonality" of Bill
> Sethares.

Xenonance is a great word, LOL.
>
>
>
> >> With many thanks
>
> > , rather, thank you, Margot! Haven't had time to really dig into
the
> > tunes you posted, but had a chance to poke at the zest24 tuning,
> > which is quite remarkable, not to mention playable in real life
with
> > two manuals.
>
> Please let me tell you how encouraging and energizing your interest
> is: it helps keep me going on this. Your comment hits the nail
>right
> on the head: above all, I wanted something new that would be
playable
> in real life as well as theory.

This is a very serious matter for me as well. I will at some
point have to get a generalized keyboard, for I want to put
pure 7s into my tuning via a 68 framework.
>
> > I don't find the dark third or the middle third functionally
> > different from the ones I use, but drive of your high third is
more
> > 9/7-y than the 422.323 cent high third I use.
>
> Maybe I should just whether we're speaking of the 420.948-cent
third,
> very close to yours (F#/Gb-Bb and G#/Ab-C in each 12-note circle),
or
> the 433.517-cent third (C#/Db-F in each circle, with a few
available
> also by combining notes from the two circles)? Either way, what
you're
> saying is of great interest to me.

The 433 third has that wild drive, the 421 is functionally
the same of course and I probably couldn't even hear the difference
there except when composing with familiar timbres. I think I
mentioned some time ago what a bull in the china shop 9/7 can be.
>
> > Also interesting is how closely it corresponds to 88-equal-
perhaps
> > it is possible to use 88 as a framework and thereby get a very
> > similar tuning with 7/4?
>
> You're right, and I hadn't even considered an 88-EDO version.
>Looking
> at this in Scala, I saw how indeed, while Zest-24 has closest
> approximations to 7/4 at around 963 and 975 cents, with 88-EDO we
get
> 968.182 cents, virtually just!

In a WT or irregular scheme in can actually be pure of course.
Without pure 7/4s, I'm concentrating on the harmonic or noble
mean between 7/4 and 9/5 as an ideal (both sound great, IMO)

>
> With such closely related versions of a tuning, people could be
drawn
> to subtly different shadings for a range of reasons -- as with the
> quest for the "ideal" degree of meantone. One reason for sometimes
> preferring a version right around 2/7-comma for the regular
meantone
> fifths is that Mark Lindley has suggested that this might be about
the
> high end of the range with maximum euphony, and gentler on the
fifths
> and major thirds than something further toward 1/3-comma. However,
> Wilson's Metameantone is quite comparable to your 88-EDO version,
and
> that 7/4 makes 88-EDO a logical spot to choose in this neck or
niche
> of the woods.

Wilson's Metameantone sounds superb, I think I failed to mention
this when Kraig brought it up a little while ago but now I've
corrected my lapse.

take care,

Cameron Bobro

Yes dear tuners, mad as a hatter. But I've put his crazy story up anyway.

It's a bit like Star Wars since you're getting the last episode before
the third. And I must apologise for a couple of things.

1. Some of the beautiful images of ancient vases and coins showing the
relevant gods and their implements, are not available at the moment as
the Perseus Server at Tufts University seems to be having problems.

2. The music that streams on loading the page, while excellent, is
rather a large file (about 3 MB). I had a shorter piece planned but am
still waiting for permission.

I thought I'd better not let these things hold it up any longer, in
case George got even sillier.

By the way, George has actually finished something useful too -- the
long awaited extreme Sagittal notation (half cent resolution). We just
need to put it into a presentable form. Hopefully _some_ time this year.

Anyway, enjoy:
http://dkeenan.com/sagittal/gift/FinalEpisode.htm

-- Dave Keenan

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
> Madman indeed! Dave, how thoughtless of you! This is the last
> straw! :-(
>
> After all we've been through, and after the painstaking effort I made
> to explain the many obstacles to my being taken seriously, I'm really
> disappointed that you would stoop so low.
>
> Furthermore, your dogged insistence on firming up the seemingly
> endless details of minas, tinas, and innumerable other things
> relating to the semantics of the Sagittal notation have kept me from
> a much more important task: that of resuming work on the truly
> remarkable story of how the notation was developed. I must take this
> opportunity, therefore, to express my profuse apologies to those who
> faithfully followed the story through the end of Episode Two, only to
> be left hanging (for more than 3 years!) with the yet unfulfilled
> promise that the account was "to be continued." My patience has
> completely run out. I can't tolerate this situation any longer!
>
> Dave, there's still a way to redeem yourself. I've worked around the
> clock on a temporary short conclusion to the story that will help to
> set the record straight. I'll be sending that to you very shortly
> after I post this message, and I'll forgive you for your previous
> impositions if you'll put it on the Sagittal website, where everyone
> can read it.
>
> We'll all be looking for it soon, so don't let us down!
>
> --"George"
>

Problems solved.

I found a working mirror for the Tufts Perseus server, in Berlin
Germany. So you can now see the intended images. And I got permission
for the music we really wanted, from Maribor, Slovenia (thanks
Cameron). So you can now enjoy the whole experience as intended.

http://dkeenan.com/sagittal/gift/FinalEpisode.htm

-- Dave Keenan

That's a lousy ending.

Oz.

----- Original Message -----
From: "Dave Keenan" <d.keenan@bigpond.net.au>
To: <tuning@yahoogroups.com>
Sent: 19 Ekim 2007 Cuma 10:46
Subject: [tuning] Re: Final Episode of the Sagittal story

> Problems solved.
>
> I found a working mirror for the Tufts Perseus server, in Berlin
> Germany. So you can now see the intended images. And I got permission
> for the music we really wanted, from Maribor, Slovenia (thanks
> Cameron). So you can now enjoy the whole experience as intended.
>
> http://dkeenan.com/sagittal/gift/FinalEpisode.htm
>
> -- Dave Keenan
>

> Hello Margo, and everyone. After checking out the canines in
> Schlick, I found some variance in the calculations.

Hi, Johnny. Looking over your post, I'd say that two factors may
be in play here. The first is that Schlick's temperament can have
different shades of interpretation, so people's numbers may
differ a bit. The second is that we may be using the term "dogs"
or "canines" a bit differently -- and each use has its own logic.

> Firstly, there are indeed 2 dogs in Schlick�s temperament, not
> one. The larger dog is from C# to G# at 711 cents according
> to Lindley's numbers, and the smaller dog is three cents sharp
> from just between G# and D#, as exists on the Halberstadt
> keyboard.

OK, what's happening here is that we were defining our "dogs"
differently. You are counting _wide_ fifths, by however much.
I am counting fifths far enough from pure in either direction to
constitute a "navigational hazard" to the style of music in
question.

Your usage has a natural logic behind it. A wide Wolf in meantone
is a kind of wild or howling canine. Thus even if a fifth in a
modified system like Schlick's is only three cents wide, we can
still think of it as a "domesticated canine," although it's no
hazard at all to navigation.

How about "poodle fifth"?

Maybe we could agree that Schlick has "one semi-Wolf and one
poodle fifth."

> J. Murray Barbour found different values for Schlick, making
> the smaller dog a bit bigger at four cents sharp of just.

Yes, he interpreted the tuning as fully fifth-circulating, and a
logical step toward 12-tET.

> With all other fifths flatter than just, this smaller dog is
> even more noticeable. This is the reason I consider Schlick an
> irregular tuning musician, someone that relished a variety of
> keys, is that I read this about him. Sadly, I do not know of a
> full translation for Schlick. (It really is hard to trust
> second and third hand sources.)

Certainly we agree that this is an irregular tuning; and if we
use "key" in a kind of neutral way, as Morley did, to mean simply
a location or transposition of a piece (modal or tonal), a bit
like the 17th-century French use of _ton_, variegation of keys or
modal transpositions is indeed what happens in Schlick.

> And Fischer, too, had only 20 ueable keys that were easily
> distinguishable with 2 dogs.

A quick comment: in describing a tuning, I'd want to distinguish
my Wolves or semi-Wolves from my gentle poodles. A fifth 9 cents
off in either direction is going to stand out in a typical
meantone or WT context, and not pleasantly. A fifth 4 cents wide
is in my view no problem at all, and often about as far off from
pure as the usual narrow fifths in the other direction; if it
causes any problems, it might do so by breeding goat thirds
inappropriate to the style. (How's that for a mixed metaphor?)

> And Werckmeister IV/Trost as well. In contrast, meantone is
> indistinguishable in chord function interval quality, with
> only non-harmonic tones providing color/or meantone variant
> distinction as per Tom. My sense is that meantone provides
> the greatest real impetus to ET, while deriving as well from
> a homogenization of well temperaments, while taking the
> ultimate advantage of the properties of the number 12, and
> the development of the modern piano.

Here I'm looking at this from what I'd take to be one meantone
era perspective. Where a meantone around 1/4-comma or 2/7-comma
differs from 12-tET dramatically is in the unequal semitones,
which really come out in 16th-century chromaticism; and also in
the "special effects" intervals like diminished fourths or
augmented fifths, etc. People around 1550-1650 were familiar both
with meantone on keyboards and 12-tET on lutes, and argued about
the virtues or vices of each system (Galilei, Mersenne, Doni,
Denis, etc.).

Certainly I understand your point that both meantone and 12-tET
provide uniform interval sizes as to the "usual" intervals, such
as regular major and minor thirds. As someone to whom "modern"
often means "around 1600," I give weight to those "unusual"
intervals, and to chromatic progressions as well. From a
perspective of 1750 or 1850, I can see your point. Someone who
likes the uniform sweetness of a meantone organ might conclude
that if we need a 12-note circulating system, let's at least make
everything as uniform as possible -- thus 12-tET.

[...]

> I loved the use of dog and goat intervals in your writing. It
> makes for easier reading and understanding. Using this turn
> of speech, I guess the major third in C# major in sixth comma
> meantone at 416 cents is a goat third, because it bleets. How
> about the large major second in that same key, at 220 cents;
> perhaps it is a frog second because it jumps higher?

That's a good question. Getting back to our "dogs," maybe we
could say that a "canine" is simply any fifth that is tempered in
the opposite direction from the prevailing one; or is
significantly impure in either direction in a just tuning. A Wolf
is deemed unusable; a semi-Wolf generally avoided; and a "poodle"
about as well-behaved as the other fifths tempered in the
opposite direction, or maybe even more so.

Actually I do have a term for a major second around 220 cents:
that would be a largish major second or whole tone approaching
the ratio of 8:7 (231 cents). To me 10:9 (182 cents) is small;
9:8 (204 cents) is middle; and 8:7 is large. I'm not sure about
animal metaphors, however.

> Yes, I do appreciate Barnes's work. Jorgensen has done a lot
> of good, but I don't understanding how he could mislead
> listeners to Anthony Newman's harpsichord performance of The
> Well-tempered Clavier by describing the tuning as
> Aron-Neidhardt II, rather than as Kirnberger III. As to no
> one tuning accurate ET in the Baroque, how about Neidardt
> doing so quite publicly in Jena in 1703, using a monochord.
> (Where there's one there's many.)

Please let me say that I'm not a member of the "12-tET was
impossible in the 18th century on keyboards" school, and I have
great respect for Neidhardt. It's only fair to add that I'm more
familiar with earlier eras; but Neidhardt's craft, like Bach's
counterpoint, is something that almost anyone might admire.

> Back to meantone alternatives, it is the variegation of the
> harmonic notes of keys that distinguished the irregular
> tunings. Even though these irregular tunings (to include
> both Grammateus and Schlick) have different derivations, the
> ear only hears the variety of harmonic connections in
> irregulars. With every key a different sounding tuning, no
> one can tell them apart!

Do you mean that given that Grammateus and Schlick are both
variegated, no one can tell these two tunings apart?

If so, I'd say let's take a keyboard prelude in an F mode, for
example, around 1500, and compare the 408-cent thirds at F-A,
C-E- and G-B in Grammateus with the same thirds somewhere around
390-394 cents in Schlick (say 1/5-1/6 comma). The first is
appropriate to a late medieval style based on some variation of
Pythagorean tuning, the second to a Renaissance style of the kind
that assumes meantone temperament.

> It was funny to read today in a biography of Frederick the
> Great how horrified the King was of the irregulars on the
> other side in war. In his meantone court, the soldiers were
> clearly intended to be regular. As you may know, Quantz built
> meantone distinctive keys into his new flutes/ D#-Eb, for
> example has a different key for each chromatic identity.

Around 1600, meantone might not necessarily be associated with
"regularity" if we look at all the keyboard pieces celebrating
_durezze_ ("little dissonances") and augmented or diminished
intervals. People with split keyboards were often taking the
opportunity to relish rather than avoid these intervals.
Similarly, Artusi regarded Monteverdi as undisciplined for
writing C#-F (thus spelled) in one of his madrigals, although he
considered it fine on a 12-tET lute, and suggested that this
composer didn't know the difference! Meantone may mean different
things to different people in different periods.

> Margo, you might want to revisit your idea that "If we see a
> fifth Ab-Eb, I'd take that as a hint that the piece is
> intended for an instrument which actually has both notes!�
> This is exactly true, unless it is a circular well
> temperament, and of course, 12-tET. Werckmeister was
> explicit, though, that chromatics like Eb and D# were to be
> intended as enharmonic identities, while recognizing it might
> not take. Obviously, I think is ideas took big time,
> contrary to some on this List.

Please let me quickly and concisely (for once!) clarify that I
was speaking specifically of a meantone instrument around
1/4-comma or 2/7-comma temperament. In a context where enharmonic
identity applies, as for many 16th-century lutes in 12-tET as
well as 12-note WT a la Werckmeister, my statement obviously
doesn't apply -- as you're very right to point out.

We much agree that enharmonic equivalence for keyboards is an
idea "which took big time" -- maybe a bit of an understatement.

> I guess I don't care for the very idea of modified meantone.
> It creates a hazy distinction between the 2 categories I am
> looking at.

Of course, we'll each have our own tastes, in tunings and in
categorizations of them.

> It's understandable that there were individuals that tuned
> every which way. However, I am against imaginative
> contemporary interpretations as akin to the way
> anthropologists whitewashed their studies of native American
> music.

Of course the example of the music of the First Nations has a
special force because here we're talking about a context of
ethnocide, ethnic cleansing, and often outright genocide.

More generally, as someone who strives both to teach history and
to create new "alternative history" (George Secor and Daniel Wolf
have written eloquently on this latter concept), I do feel a
responsibility in my writing not to confuse the second with the
first.

With many thanks,

Margo

Although this conclusion seems rather to "gallop to a conclusion,"
which may be appropriate for an arrow-armed horseman, I think it's
great that this entertaining and educational presentation of this
important microtonal notation has "come full circle" so to speak... :)

Joseph Pehrson

P.S. Caught the satire that the "battle of the gods" was all in good
fun...

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@...> wrote:
>
> Yes dear tuners, mad as a hatter. But I've put his crazy story up
anyway.
>
> It's a bit like Star Wars since you're getting the last episode
before
> the third. And I must apologise for a couple of things.
>
> 1. Some of the beautiful images of ancient vases and coins showing
the
> relevant gods and their implements, are not available at the moment
as
> the Perseus Server at Tufts University seems to be having problems.
>
> 2. The music that streams on loading the page, while excellent, is
> rather a large file (about 3 MB). I had a shorter piece planned but
am
> still waiting for permission.
>
> I thought I'd better not let these things hold it up any longer, in
> case George got even sillier.
>
> By the way, George has actually finished something useful too -- the
> long awaited extreme Sagittal notation (half cent resolution). We
just
> need to put it into a presentable form. Hopefully _some_ time this
year.
>
> Anyway, enjoy:
> http://dkeenan.com/sagittal/gift/FinalEpisode.htm
>
> -- Dave Keenan
>
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> > Madman indeed! Dave, how thoughtless of you! This is the last
> > straw! :-(
> >
> > After all we've been through, and after the painstaking effort I
made
> > to explain the many obstacles to my being taken seriously, I'm
really
> > disappointed that you would stoop so low.
> >
> > Furthermore, your dogged insistence on firming up the seemingly
> > endless details of minas, tinas, and innumerable other things
> > relating to the semantics of the Sagittal notation have kept me
from
> > a much more important task: that of resuming work on the truly
> > remarkable story of how the notation was developed. I must take
this
> > opportunity, therefore, to express my profuse apologies to those
who
> > faithfully followed the story through the end of Episode Two,
only to
> > be left hanging (for more than 3 years!) with the yet unfulfilled
> > promise that the account was "to be continued." My patience has
> > completely run out. I can't tolerate this situation any longer!
> >
> > Dave, there's still a way to redeem yourself. I've worked around
the
> > clock on a temporary short conclusion to the story that will help
to
> > set the record straight. I'll be sending that to you very
shortly
> > after I post this message, and I'll forgive you for your previous
> > impositions if you'll put it on the Sagittal website, where
everyone
> > can read it.
> >
> > We'll all be looking for it soon, so don't let us down!
> >
> > --"George"
> >
>

OH... I also forgot to mention that I am grateful that I was not
present in the cottage with the livestock, despite their utility as a
potent new source of energy...

JP

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@...> wrote:
>
> Although this conclusion seems rather to "gallop to a conclusion,"
> which may be appropriate for an arrow-armed horseman, I think it's
> great that this entertaining and educational presentation of this
> important microtonal notation has "come full circle" so to
speak... :)
>
> Joseph Pehrson
>
> P.S. Caught the satire that the "battle of the gods" was all in
good
> fun...
>
>
>
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@> wrote:
> >
> > Yes dear tuners, mad as a hatter. But I've put his crazy story up
> anyway.
> >
> > It's a bit like Star Wars since you're getting the last episode
> before
> > the third. And I must apologise for a couple of things.
> >
> > 1. Some of the beautiful images of ancient vases and coins
showing
> the
> > relevant gods and their implements, are not available at the
moment
> as
> > the Perseus Server at Tufts University seems to be having
problems.
> >
> > 2. The music that streams on loading the page, while excellent, is
> > rather a large file (about 3 MB). I had a shorter piece planned
but
> am
> > still waiting for permission.
> >
> > I thought I'd better not let these things hold it up any longer,
in
> > case George got even sillier.
> >
> > By the way, George has actually finished something useful too --
the
> > long awaited extreme Sagittal notation (half cent resolution). We
> just
> > need to put it into a presentable form. Hopefully _some_ time
this
> year.
> >
> > Anyway, enjoy:
> >
http://dkeenan.com/sagittal/gift/FinalEpisode.htm
> >
> > -- Dave Keenan
> >
> >
> > --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> > > Madman indeed! Dave, how thoughtless of you! This is the last
> > > straw! :-(
> > >
> > > After all we've been through, and after the painstaking effort
I
> made
> > > to explain the many obstacles to my being taken seriously, I'm
> really
> > > disappointed that you would stoop so low.
> > >
> > > Furthermore, your dogged insistence on firming up the seemingly
> > > endless details of minas, tinas, and innumerable other things
> > > relating to the semantics of the Sagittal notation have kept me
> from
> > > a much more important task: that of resuming work on the truly
> > > remarkable story of how the notation was developed. I must
take
> this
> > > opportunity, therefore, to express my profuse apologies to
those
> who
> > > faithfully followed the story through the end of Episode Two,
> only to
> > > be left hanging (for more than 3 years!) with the yet
unfulfilled
> > > promise that the account was "to be continued." My patience
has
> > > completely run out. I can't tolerate this situation any longer!
> > >
> > > Dave, there's still a way to redeem yourself. I've worked
around
> the
> > > clock on a temporary short conclusion to the story that will
help
> to
> > > set the record straight. I'll be sending that to you very
> shortly
> > > after I post this message, and I'll forgive you for your
previous
> > > impositions if you'll put it on the Sagittal website, where
> everyone
> > > can read it.
> > >
> > > We'll all be looking for it soon, so don't let us down!
> > >
> > > --"George"
> > >
> >
>

--- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@...> wrote:
>
> Although this conclusion seems rather to "gallop to a conclusion,"
> which may be appropriate for an arrow-armed horseman, I think it's
> great that this entertaining and educational presentation of this
> important microtonal notation has "come full circle" so to speak... :)

Thanks, Joseph, for being one of our most loyal fans. Zeus sends his
blessings.

--"George"

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> As an aside, I might mention that 2/7-comma is almost identical to
> 119-EDO, maybe something that could be added to the TonalSoft material
> if it's not already there.

Euclid gives us 12, 19, 50, 119, 764, 883, 1647, 2530, 6707, 250689...
Between 50 and 119 is 69, a semiconvergent. I'd recommend 50 as a
surrogate for 2/7 comma.

Hello, everyone.

Please let me share a link to a new paper on a style of 12-note
circulating temperament which balances the Pythagorean, syntonic,
and septimal comma to seek a variety of thirds from 5-limit to
septimal: the temperament extraordinaire (TE).

<http://www.bestII.com/~mschulter/TE1.txt>

While some historical versions of the temperament ordinaire
around 1700 might fit structurally into this category, I'd consider
it mainly a 21st-century concept (unless people know of other
precedents) because of the specific goal of some septimal
approximations in the far part of the circle.

As discussed in the paper, this kind of 12-note circle might be
seen as a kind of fusion between historical meantone and the
"17-tone revolution" which George Secor heralded with his 17-tone
well-temperament of 1978, and in which I have delighted to
participate with him. While George certainly shouldn't be held
responsible for any dubious concepts or implementations on my
part, his art and most admirable craft in scale design have been
a great inspiration to me, and a principal source for whatever
merits my concepts might have.

This paper describes some basic aspects of the temperament
extraordinaire and its strategy for obtaining a variegation
of thirds through "harmonic recycling." Eight regular meantone
fifths somewhere in the range from around 1/4-comma to 2/7-comma
or even 3/10-comma do an amount of narrow tempering considerably
overshooting the Pythagorean comma; and this "harmonic surplus"
is "recycled" by four wide fifths which disperse the septimal
comma for some approximations of 7:6, 9:7, 7:4, etc.

While much of the paper focuses on musical purposes and stylistic
contexts for this tuning approach, there's also an appendix
that looks at some mathematical aspects of harmonic recycling,
and how John Brombaugh's temperament units (TU) can be used in
tracking the interplay of the three commas.

Many thanks to all of you whose ideas and suggestions over the
years have helped shape this project.

By the way, I would be glad also to post the full paper here if
people would like this, which might make for easier quoting
(carefully selective, I would entreat!), but thought that given
the length over 800 lines, a URL might be the best option, at
least for now.

Most appreciatively,

Margo Schulter
mschulter@calweb.com

Hey George and Dave,

When are the gods going to give speech to the mortals? How 'bout some
phonemes for those squiggles? I want to sing in Sagittal.

Regards,

Robin

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@> wrote:
> >
> > Although this conclusion seems rather to "gallop to a conclusion,"
> > which may be appropriate for an arrow-armed horseman, I think it's
> > great that this entertaining and educational presentation of this
> > important microtonal notation has "come full circle" so to
speak... :)
>
> Thanks, Joseph, for being one of our most loyal fans. Zeus sends his
> blessings.
>
> --"George"
>

--- In tuning@yahoogroups.com, "Robin Perry" <jinto83@...> wrote:
>
> Hey George and Dave,
>
> When are the gods going to give speech to the mortals? How 'bout
some
> phonemes for those squiggles? I want to sing in Sagittal.
>
> Regards,
>
> Robin

Hi Robin,

Believe it or not, I was already working on a system of assigning
short names to Sagittal symbols and brought this up to Dave shortly
after your recent discussion with him about this. We discussed it
only briefly, since there were other important pending issues that we
needed to resolve.

Dave did forward to me a copy of your correspondence with him, from
which I made several observations and conclusions. My approach to
the problem was very similar to one of those you tried. Although
it's possible to come up with enough one-syllable names for all of
the single-shaft Sagittal symbols, solmisation (which is what I think
you're interested in) would require a one-syllable name for every
combination of natural nominal (A thru G) plus conventional
accidental (sharp and flat alteration) plus Sagittal accidental
(counting up and down as separate accidentals). I don't believe
there are enough one-syllable combinations of phonemes to achieve
that. Unless you're willing to restrict the one-syllable names to a
small subset of Sagittal accidentals, I don't think that what you're
asking for is feasible.

3) I came to the conclusion that multi-syllable names for the
Sagittal accidentals would be easier to remember than single-syllable
ones (and also less likely to misunderstand).

I recall that Fokker devised one-syllable Dutch names for all of the
pitches of 31-equal, so that could serve as a starting point. I have
Leigh Gerdine's translation of Fokker's _New Music With 31 Notes_,
where I saw it. I'll look this up and report back.

Best,

--George

That's good to hear! Dave had an idea of being able to string
together string-togetherable consonants.. I originally balked at
that idea, but have reconsidered it..

What if you just assign consonants to the elements and let people
insert the vowels of choice wherever they want? It would make it
much easier to rhyme in Sagittal that way. Or, maybe there aren't
enough consonants even for that..hmm.. just idle pondering..

Good luck..

Robin

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Robin Perry" <jinto83@> wrote:
> >
> > Hey George and Dave,
> >
> > When are the gods going to give speech to the mortals?
How 'bout
> some
> > phonemes for those squiggles? I want to sing in Sagittal.
> >
> > Regards,
> >
> > Robin
>
> Hi Robin,
>
> Believe it or not, I was already working on a system of assigning
> short names to Sagittal symbols and brought this up to Dave
shortly
> after your recent discussion with him about this. We discussed it
> only briefly, since there were other important pending issues that
we
> needed to resolve.
>
> Dave did forward to me a copy of your correspondence with him,
from
> which I made several observations and conclusions. My approach to
> the problem was very similar to one of those you tried. Although
> it's possible to come up with enough one-syllable names for all of
> the single-shaft Sagittal symbols, solmisation (which is what I
think
> you're interested in) would require a one-syllable name for every
> combination of natural nominal (A thru G) plus conventional
> accidental (sharp and flat alteration) plus Sagittal accidental
> (counting up and down as separate accidentals). I don't believe
> there are enough one-syllable combinations of phonemes to achieve
> that. Unless you're willing to restrict the one-syllable names to
a
> small subset of Sagittal accidentals, I don't think that what
you're
> asking for is feasible.
>
> 3) I came to the conclusion that multi-syllable names for the
> Sagittal accidentals would be easier to remember than single-
syllable
> ones (and also less likely to misunderstand).
>
> I recall that Fokker devised one-syllable Dutch names for all of
the
> pitches of 31-equal, so that could serve as a starting point. I
have
> Leigh Gerdine's translation of Fokker's _New Music With 31 Notes_,
> where I saw it. I'll look this up and report back.
>
> Best,
>
> --George
>

Hello, everyone.

Please let me comment very briefly on some issues where I would
consider it far more important for us each to be true to our own
musical traditions and practices, old and new, than to seek any
"standard" terminology.

Johnny, to me a "Wolf" or "semi-Wolf" means a fifth rather more than
1/3 syntonic comma impure -- with 1/3 Pythagorean comma (as in
Werckmeister IV) sort of pushing the envelope, but still just within a
"circulating" category. This is under the Scala reading where the most
heavily tempered fifths are around 694 and 710 cents; Paul Erlich was
willing to consider going this far under very different circumstances
to optimize septimal ratios for his "paultones" with fifths around
22-EDO, or possibly a bit larger.

If it's within 7 cents or so of just, I generally would simply call it
a "fifth" -- noting the direction of temperament where that is
relevant ("wide" or "narrow"). The new term "poodle" sounds very nice,
and I _would_ use it for a 707-cent fifth in a tuning where 697 cents
is the norm. As far as I'm concerned, the two are equally circulating.

Maybe this reflects my custom of quite routinely tempering fifths in
either or both directions -- all regular fifths wide in Peppermint or
a 17-note circulating temperament; all narrow in regular meantone; and
a mixture of eight narrow and four wide in my modified meantones.

Similarly, Charles, I regard a major third around 9:7 as routine and
desirable, and at the same time distinct from 5:4, rather as some fine
ornate typeface is distinct from a simple and smooth one. Juxtaposing
the two indiscriminately, in music or typesetting, is not necessarily
the formula for aural or visual concord. In a modified meantone like
the one I use, where all the fifths are comparably impure, I would
consider Db-F at close to 9:7 as the glory of Eb Dorian or C Phrygian,
where this third respectively contracts to a unison or expands to a
fifth; these modalities almost by definition call for a medieval or
neomedieval style. Of course, it also makes a striking and compelling
diminished fourth in a Renaissance meantone setting. The fun question
is which transitions or juxtapositions are pleasant and desirable.

Then, again, I'm ready to go rather further than 9:7 when it comes to
cadential major thirds expanding to fifths, with an article by Jay
Rahn about Marchettus of Padua as the basis for the following:

<http://www.bestII.com/~mschulter/PythEnharImprov01.mp3>

What I've decided is that making new music should be my first
priority, and offering some curious verbalizations about what I'm up
to, however imperfect and incomplete, a good next priority. That means
discussing lots of history, and making some alternative history, which
with a passionate enough immersion in both need be distinguished only
when convenient for certain pedagogic purposes. Those purposes often
_are_ relevant and convenient, by the way.

A quick final note on Werckmeister V: I'd call it a nice circulating
temperament, and note with curiosity that it achieves one goal also
important to my 2/7-comma temperament extraordinaire in a 16th-century
context: keeping major thirds on the six untransposed modal finals no
larger than about 396 cents. Of course, they are radically different
systems, and Werckmeister was presumably focusing mainly on tonality
around the end of the 17th century rather than polyphonic modality of
one kind or another.

Most appreciatively,

Margo Schulter
mschulter@calweb.com

Margo: Johnny, to me a "Wolf" or "semi-Wolf" means a fifth rather more than
1/3 syntonic comma impure -- with 1/3 Pythagorean comma (as in
Werckmeister IV) sort of pushing the envelope, but still just within a
"circulating" category. This is under the Scala reading where the most
heavily tempered fifths are around 694 and 710 cents; Paul Erlich was
willing to consider going this far under very different circumstances
to optimize septimal ratios for his "paultones" with fifths around
22-EDO, or possibly a bit larger.

Johnny: If that was true for Werckmeister, he would never have suffered the
grief that he did in promoting WIII ahead of WIV. I can certainly accept
that you feel this way, and I speak more regularly with Paul as I’m planning to
do my Odysseus in Boston on May 4 and Paul is playing.
While I await a possible correction from Paul on WIV in C major, I believe
the historical situation for Werckmeister, who inherited his positions using
what he now call WIV, but which he credited to Trost and his unnamed late
cousin. My suspicion it is because Andreas was primarily recognized as an
improviser rather than a composer. I have been performing every Werckmeister piece
I can find, more recently his Praeludium in G from the Halle dissertation.
Most famously, the Christmas cantata Wo ist die neugeborne Konig auf den
Juden (PITCH 200202 Early).

Margo: If it's within 7 cents or so of just, I generally would simply call
it
a "fifth" -- noting the direction of temperament where that is
relevant ("wide" or "narrow"). The new term "poodle" sounds very nice,
and I _would_ use it for a 707-cent fifth in a tuning where 697 cents
is the norm. As far as I'm concerned, the two are equally circulating.

Johnny: I suspect you have stretched ears. Werckmeister actually used the
word Circul to indicate the difference between WIII and WIV. (Sorry everyone
that the names of these prominent scales are so arcane.) What you like to
hear, or consider sounding nice, doesn’t change the facts. If one is allergic
to dogs, even the nicest, sweetest poodle would still cause an allergic
response. It may be one of the most amazing challenges, the ability to get into
someone’s head. Actors do it all the time. Musicians are often actors,
usually the best of them. Getting into the mind of a long dead person through
their writing may be toughest of all. As I said to Paul, I do not think the
literature portends the full story. At best it is a part of the story. The
literature is more interesting in its influence of later generations.

Margo: Maybe this reflects my custom of quite routinely tempering fifths in
either or both directions -- all regular fifths wide in Peppermint or
a 17-note circulating temperament; all narrow in regular meantone; and
a mixture of eight narrow and four wide in my modified meantones.

Johnny: Exactly, you have stretched ears. Me, too. That’s why I recognize
the condition. It’s no different than musicians being over sensitive to
sound in general. We have stretched our minds in to encompass dissonance as
consonance. The French take the just third of 386 cents and recoil in horror at
its dissonance (anecdotal).
Your later comments: “What I've decided is that making new music should be
my first
priority, and offering some curious verbalizations about what I'm up
to, however imperfect and incomplete, a good next priority” make total sense
as regards your intervallic reactions.

************************************** See what's new at http://www.aol.com

Hello, Leonardo and Monz and all.

Please let me bring you up to date: I'm now in the middle of the process
of going through the translation and making suggestions. I hope to have a
first draft of proposed revisions within a week or so, maybe sooner. It's
only fair to admit that I'm shaky on the geometric procedures, but find
the discussions of interval ratios friendly and familiar -- so that any
suggestions I have regarding passages discussing the former might be read
with special caution!

Thank you for the opportunity to share in a most worthy project.

With many thanks,

Margo

--- In tuning@yahoogroups.com, "Robin Perry" <jinto83@...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > ... I came to the conclusion that multi-syllable names for the
> > Sagittal accidentals would be easier to remember than single-
syllable
> > ones (and also less likely to misunderstand).
> >
> > I recall that Fokker devised one-syllable Dutch names for all of
the
> > pitches of 31-equal, so that could serve as a starting point. I
have
> > Leigh Gerdine's translation of Fokker's _New Music With 31
Notes_,
> > where I saw it. I'll look this up and report back.
> >
> > Best,
> >
> > --George
>
> That's good to hear!

It turns out the names are in German, but here they are (for what
it's worth):

c C
ci C-semisharp
cis C-sharp
des D-flat
dèh D-semiflat
d D
di D-semisharp
dis D-sharp
es E-flat
èh E-semiflat
e E
eï E-semisharp
fèh F-semiflat
f F
fi F-semisharp
fis F-sharp
ges G-flat
gèh G-semiflat
g G
gi G-semisharp
gis G-sharp
as A-flat
àh A-semiflat
a A
ai A-semisharp
ais A-sharp
B B-flat
hèh B-semiflat
h B
hi B-semisharp
cèh C-semiflat

In case the above special characters don't come out right:
à = a with grave accent
è = e with grave accent
ï = i with umlaut (double dotted i)

> Dave had an idea of being able to string
> together string-togetherable consonants.. I originally balked at
> that idea, but have reconsidered it..

Yes, that's possible.

> What if you just assign consonants to the elements and let people
> insert the vowels of choice wherever they want? It would make it
> much easier to rhyme in Sagittal that way. Or, maybe there aren't
> enough consonants even for that..hmm.. just idle pondering..

Why do they need to rhyme? I thought that perhaps a consonant could
be used to indicate the direction of alteration (and a leading
consonant could be used for a left-accent), but that was for names
for the Sagittal accidentals only. If we're attempting to map single
syllables to pitches (nominals plus accidentals), then it might be
better to use seven consonants for the seven nominals (A thru G) and
use consonants for the accidentals (since there are more consonants
than vowels).

--George

Dear Johnny,

Please let me happily say that today I found the passage
in Werckmeister that I had mentioned about the "Praetorian"
temperament and Zarlino; see Orgel-Probe (1698), ch. 32.

We can discuss it more; but that's the reference.

Most appreciatively,

Margo
mschulter@calweb.com

-------------------------------------------------
Nanotemperament and Brombaugh's temperament units
A quick essay for Brad Lehman
-------------------------------------------------

Whatever mixed blessings synthesized sound may have brought to the
world of early European music and its modern performances and
derivative styles, this technology has introduced a quirk into the
process of tuning with curious ramifications: _nanotemperament_.

Here I'll try to explain this quirk, and show how John Brombaugh's
temperament units (TU) can sometimes help in evaluating the
consequences. Since it was a commentary by Brad Lehman to his
temperament spreadsheet that made me aware of these convenient units,
I warmly dedicate this modest essay to him.

There may be a certain paradox, Brad, in my dedicating this essay
about synthesizer tuning to you, above all a paragon of the craft of
tuning historical instruments by ear. Please let me emphasize that
this dedication is a token of deepest appreciation and indebtedness
which, however, should carry no implications as to how you might view
my curious methods and musical goals.

---------------------------
1. What is nanotemperament?
---------------------------

For many microtunable hardware synthesizers, the units of tuning are
rather coarse, on the order of 1-2 cents. Theoretically "regular"
temperaments often become in practice necessarily irregular, in order
to balance out the synthesizer tuning units.

A very practical example is my temperament extraordinaire based on
Zarlino's 2/7-comma meantone, with eight fifths (F-C#) in this regular
tuning, and the other four tempered equally wide. For more information
on this general style of temperament, people might wish to visit:

<http://www.bestII.com/~mschulter/TE1.txt>

In theory, at least, the meantone fifths should be narrow by about
6.145 cents, and the wide ones by 6.424 cents -- not that any tuning
by ear could be anywhere near this precise!

As it happens, I could not be this precise on my Yahama TX-802
synthesizer either. Here there are 1024 tuning steps per octave, so
that each is 1.171875 cents. An interval of 599 steps is a virtually
just 3:2 fifth (actually about 0.002 cents narrow), so that for all
practical purposes one may regard this fifth as pure.

Unfortunately, the desired narrow temperament for the meantone fifths
of 6.145 cents (2/7-comma) is not an even multiple of 1.171875 cents.
Our nearest approximations are either 5 steps narrow (5.859 cents);
or 6 steps narrow (7.0325 cents).

A serviceable compromise, however, is available: _nanotemperament_, or
introducing very small variations in the sizes of the theoretically
regular fifths in order to have them "average out" at about 2/7-comma
narrow. Specifically, a chain of three fifths each 5 steps narrow,
plus one 6 steps narrow, means an average tempering of about 6.152
cents, almost identical to the desired 6.145 cents.

Another way to gauge the accuracy of this "averaging out" is to
consider that a regular major third in 2/7-comma is about 383.241
cents, or 1/7 comma narrow of 5:4. Our nanotemperament produces a
third of 383.203 cents. Since there are eight of these narrow fifths
to be tuned, we can arrange them in two groups of four, so that any
major third with both notes within this accidental range will have
this size virtually identical to that of Zarlino's meantone. Thus:

F C G D A E B F# C#
-5 -5 -6 -5 -5 -5 -6 -5

The theoretically uniform tempering of the wide fifths, 6.424 cents,
likewise is not an even multiple of the 1.171875-cent synthesizer
unit, or very close to such a multiple. Again, the nearest available
amounts of temperament are 5 steps (5.857 cents) or 6 steps (7.029
cents).

However, a chain with two fifths at 5 steps and two fifths at 6 steps
wide produces an average temperament of about 6.443 cents wide --
again, almost identical to the desired size. Together, these four
fifths produce a major third (or meantone diminished fourth) at around
433.594 cents, as compared with 433.517 cents in the theoretical
version.

Now we come to an interesting decision -- or rather accident -- that
occurred in implementing this concept. For whatever reason, rather
than arranging the pattern of wide 5-step and 6-step fifths as evenly
as possible, I found myself with this situation:

C# G# D#/Eb Bb F
+5 +6 +6 +5

Here the two fifths tempered by 6 steps are placed adjacent to one
another. What I seem to recall is that I was thinking along the lines
of a traditional meantone chain, and going by the impression that the
more "remote" fifths on each end of the chain (G#-D#/Eb and Eb-Bb)
should be the more heavily tempered.

Only later did I note this slight "unevenness" and consider the
implications for the tuning. Such a consideration invites a convenient
measure which happens almost exactly to mesh with the synthesizer
tuning system: Brombaugh's temperament units.

-----------------------------
2. A (nano)temperament recipe
-----------------------------

In Brombaugh's system, there are 720 temperament units (TU) in a
Pythagorean comma, 531441:524288 or 23.460 cents. A syntonic comma at
81:80 (21.506 cents) is virtually equal to 660 TU; and a septimal
comma at 64:63 (27.264 cents) to about 837 TU, and almost exactly
to 836.75 TU.

On a synthesizer like the TX-802 dividing the octave into 1024 equal
steps, 20 of these steps are almost precisely equal to a Pythagorean
comma, or 720 TU; and one synthesizer step to 1/20 of this comma, or
36 TU (actually about 35.965 TU).

A tempering of 5 synthesizer steps will thus be equivalent to 1/4 of a
Pythagorean comma or 180 TU; and 6 steps to 3/10 of the comma or 216 TU.

We can use negative values to show temperament in a narrow or meantone
direction, and positive values to show temperament in a wide direction.
The following temperament recipe results:

F C G D A E B F# C# G# D#/Eb Bb F
-180 -180 -216 -180 -180 -180 -216 -180 +180 +216 +216 +180

Here 6 fifths at -180 TU plus two at -216 TU do a total of 1512 TU of
negative temperament, exceeding the 720 TU of the Pythagorean comma by
792 TU. The 2 wide fifths at +180 cents, plus the 2 at +216 cents, do
792 TU of positive temperament -- thus balancing the circle.

The obvious compromise in this arrangement, of course, is that some
narrow and wide fifths alike will be tempered at just over 7 cents
impure, a degree of temperament comparable to 1/3-comma meantone or
the almost identical system of 19-tone equal temperament (19-tET),
also known as a 19-note equal division of the octave (19-EDO). Among
regular tunings with wide fifths, 22-EDO is also comparable.

Given that this compromise is unavoidable on such a synthesizer if we
wish to approximate the 2/7-comma temperament extraordinaire, we might
ask if any benefits are derived from this necessity. Before addressing
this question, we should quickly mention an alternative.

To avoid having any fifths more impure in either direction than 180 TU
(5 synthesizer units), we could simply temper the 8 meantone fifths at
180 TU narrow each, and the other four fifths at 180 TU wide -- so
that each is tempered by 1/4 Pythagorean comma. This procedure would
yield a very nice temperament while keeping all fifths within 6 cents
of pure. Further, it would not require any nanotemperament, since a
fifth tempered at 5 synthesizer steps narrow or wide is for all
practical purposes tempered by 1/4 Pythagorean comma or 180 TU in the
chosen direction.

However, I must admit that using Zarlino's 2/7-comma as the basis for
a circulating system does has a special historical appeal to me: this
is the first known meantone (1558) to be described in precise
mathematical terms. Thus the above temperament recipe is an invitation
to a meeting between the old and the new, with the "seasoning" of
nanotemperament having two especially interesting consequences.

---------------------------------------
3. Optimizing the smallest minor thirds
---------------------------------------

In a temperament extraordinaire, one of the primary goals is to obtain
some small minor and large major thirds at or close to the septimal
ratios of 7:6 and 9:7, or about 267 and 435 cents; these will be the
smallest minor and largest major thirds of the tuning, generated
respectively by a chain of three or four wide fifths.

To see how my inadvertant nanotemperament scheme for the remote
portion of the circle served to promote this quest, let us consider
the relevant region of the tuning recipe:

C#/Db G#/Ab D#/Eb A#/Bb F
+180 +216 +216 +180

Since a minor third is generated from a chain of three fourths up
(F-Bb-Eb-Ab) or three fifths down, we can read this diagram from right
to left in order to locate the two minor thirds generated from three
wide fifths (or narrow fourths): F-Ab and Bb-Db.

Having the two more heavily tempered fifths at 3/10 Pythagorean comma
or 216 TU placed together in the central portion of this wide-fifth
region, with the two fifths at 1/4 Pythagorean comma or 180 TU at the
borders of the region, ensures that any minor third formed from three
wide fifths will get both of the 216 TU fifths, together with one
"border" fifth at 180 TU. This adds up to 612 TU of temperament in the
direction of 7:6.

Thus here is the situation for F-Ab:

F-Ab +612 TU
|----------------------------|
C#/Db G#/Ab D#/Eb A#/Bb F
+180 +216 +216 +180

and likewise for Bb-Db:

Bb-Db +612 TU
|------------------------------|
C#/Db G#/Ab D#/Eb A#/Bb F
+180 +216 +216 +180

How close to 7:6 are these small minor thirds? A minor third from a
chain of four pure 3:2 fifths would have a Pythagorean size of 32:27
or 294.135 cents. To obtain a just 7:6 third (266.871 cents), we must
do a full septimal comma (64:63) of tempering in the wide-fifth
direction, 27.264 cents or about 837 TU (almost precisely 836.75 TU).

Here we have done 612 TU of tempering for each of our smallest minor
thirds, leaving them about 225 TU wide of a just 7:6 -- or 7.323
cents. These thirds have a size of about 274.194 cents.

In the theoretical model for a 2/7-comma temperament extraordinaire,
these same thirds would have a size of 274.862 cents; so the necessity
of nanotemperament, along with the quirk of my placing those two wide
fifths at 216 TU together, has resulted in "optimizing" these
approximations of 7:6.

------------------------------------------
4. Outlying meantone thirds and E Phrygian
------------------------------------------

A temperament extraordinaire is meant for playing in modal styles of
music: typically Renaissance and Manneristic styles premised on
meantone in the nearer portion of the circle; and medieval or
neomedieval styles in the remote part of the circle premised on
Pythagorean intonation and its accentuated modern variations and
offshoots.

For music in Renaissance and Manneristic styles, having restful major
thirds quite close to 5:4 is an important consideration. This is
obviously no problem in the F-C# region of the tuning, identical to
Zarlino's regular 2/7-comma temperament -- apart from small and mostly
inconsequential variations on synthesizer due to nanotemperament. The
most "consequential" variation is that the fifths G-D and B-F# are
narrow by a tad more than 7 cents, rather than the 6.145 cents of
Zarlino's 2/7-comma. In effect, they are approximate 1/3-comma fifths:
tolerable, but not necessarily ideal.

Of more concern, from the perspective of major third quality, is the
"outlying meantone" region occupied by the thirds Bb-D and E-G#,
formed from three narrow fifths plus one wide fifth. In a theoretical
model of a 2/7-comma temperament extraordinaire, these thirds would be
at 395.810 cents, or 9.497 cents wide of 5:4. This is comparable to
the regular major thirds of 1/7-comma meantone (395.531 cents), and
also to the smallest and mildest major thirds of Werckmeister V and
some Neidhardt well-temperaments, for example, at 1/2 Pythagorean
comma smaller than the Pythagorean 81:64 (407.820 cents), or 396.090
cents.

Granted that using these considerably compromised major thirds in
place of their regular meantone counterparts is less than ideal, we
might wish, at least, to keep these "outlying meantone" thirds Bb-D
and E-G# as close to 5:4 as reasonably possible. This can be
especially desirable for E-G#, routinely used in closing sonorities in
untransposed E Phrygian, a common and very beautiful modality.

Here nanotemperament can contribute a small but helpful nudge in the
right direction. Let us focus on the nearer of meantone portion of the
tuning recipe, plus the two wide fifths Bb-F and C#-G# that border it.

Bb F C G D A E B F# C# G#
+180 -180 -180 -216 -180 -180 -180 -216 -180 +180

The two outlying meantone major thirds Bb-D and E-G# each have a
generating chain of four fifths including one fifth narrow by 216 TU,
two narrow by 180 TU, and one wide by 180 TU. The wide fifth, in
effect, "cancels out" the narrow tempering done by one of the 180 TU
fifths. This leaves a narrow tempering of 216 TU plus 180 TU, or a
total of 396 TU.

A chain of four pure 3:2 fifths would produce a Pythagorean major
third at 81:64; to obtain a pure 5:4 (386.314 cents), we must do a
full syntonic comma (81:80, 21.506 cents) or 660 TU of narrow
tempering. Here we have done 396 TU, or 3/5 of that comma, leaving
Bb-D and E-G# wide of 5:4 by 2/5 comma, or about 8.603 cents -- a size
of about 394.916 cents.

This is almost a cent closer to 5:4 than the theoretically expected
value of 395.810 cents, and curiously comparable to the smallest
minor thirds in the Prelleur temperament at about 394.669 cents. It is
still hardly an ideal situation, but a viable compromise slightly
improved.

Interestingly, this modestly more pleasant result with Bb-D and E-G#
owes itself in part to the same cause as the closer approach to 7:6 of
the small minor thirds F-Ab and Bb-Db: the arrangement of the four
wide fifths.

Arranging them with the two fifths at +180 TU, Bb-F and C#-G#, both
placed on the borders of the meantone region, as shown in the last
diagram, means that Bb-D and E-G# will each have in their generating
chains a wide fifth cancelling out only 180 TU rather than 216 TU of
the desired narrow temperament provided by the other three fifths.

Instead, the two more heavily tempered wide fifths G#-D#/Eb and Eb-Bb
are concentrated at the center of their region, as discussed in
Section 3, where rather than undesirably moving outlying meantone
major thirds further from 5:4, they can agreeably draw our smallest
minor thirds nearer to 7:6.

-----------------------------------
5. Conclusion, or postlude for Brad
-----------------------------------

Each tuning form, and musical style, may have its own constraints and
preferences. Writing this essay has been an opportunity to reflect on
some of the historical contrasts and playful paradoxes that can arise
when, as Vicentino proposed in his famous title, "ancient music" is
"reduced" or "adapted" to "modern practice."

I have devoted considerable space to the quest for 7:6 and 9:7 thirds
in a modern circulating temperament, because it is for me a high
musical priority -- but by no means necessarily a priority for those
of the meantone era, with interesting exceptions (notably Huygens).

Indeed, as Paul Poletti recently noted, Michael Praetorius reported a
name that early 17th-century musicians had given to the meantone
augmented second F-G#: "the wolf," a term now especially associated
with the diminished sixth or augmented third (e.g. G#-Eb or Eb-G#) in
a regular meantone tempered by about 1/6-comma or more.

Since Praetorius specifically favored 1/4-comma meantone with pure 5:4
major thirds -- the "Praetorian" temperament to later writers such as
Werckmeister -- we know that F-G# was very close to a just 7:6. Yet he
very clearly describes it as a "false minor third" -- an obstacle to
free harmonic navigation encountered in a usual 12-note temperament,
rather than an adornment, as Huygens would declare it in 1691.

It is thus a somewhat humorous modern predicament to be debating how
much to compromise both the narrow and wide fifths of a temperament
extraordinaire in order to "optimize for 7:6" -- that is, in terms
like those of Praetorius, to keep the "wolves" at F-G# and Bb-C# as
"wolfish" as possible while getting the fifths to circulate.

Moving ahead to the era of Bach and well-temperaments which you have
studied and explored so intimately, Brad, I also reflect on the
possible distinctions between modal and tonal approaches to the
question of the major third E-G#.

In a 16th-century modal style, where it is customary often to have
closing sonorities including a major third above the modal final,
obtained by inflection if necessary, E-G# is basic to E Phrygian, and
is one of my first criteria for evaluating any 12-note circulating
system. More generally, I would like each of the untransposed modal
finals (C, D, E, F, G, A) to have a major third within 9-10 cents of a
pure 5:4; and likewise Bb-D, which may occur often in various modes.

A temperament extraordinaire can meet this goal with fewer compromises
than tonal 12-note well-temperaments, because an equally important
goal is to obtain major thirds in the more remote portion of the
circle at sizes such as 421 cents or 434 cents. These are "excellent
neomedieval thirds" -- a rather different description than might be
applied in a tonal context around 1706 or 1722!

Thus each art form, contrapuntal or intonational, may have its own
peculiar rules. Brad, my warmest thanks to you as an artist and
scholar who has brought to my attention the noble ingenuity of the
temperament unit, and inspired me to seek creative by-ways for
applying it.

Most appreciatively,

Margo Schulter
mschulter@calweb.com

[On my view that Werckmeister IV might be marginally circulating]

> Johnny: If that was true for Werckmeister, he would never have
> suffered the grief that he did in promoting WIII ahead of WIV. I
> can certainly accept that you feel this way, and I speak more
> regularly with Paul as I'm planning to do my Odysseus in Boston on
> May 4 and Paul is playing.

Hi, Johnny, and thanks for a very friendly and pleasant dialogue.
Maybe I should clarify that from my 21st-century perspective I'd
regard WIV as _marginally_ circulating: 1/3-Pythagorean comma
temperament of the fifths is a lot further than I'd like to go in a
usual as opposed to "xentonal" style (a la Sethares).

However, there's a much more basic reason I shouldn't overlook why WIV
is _not_ circulating in a tonal context, nor in a 16th-century modal
one: those thirds at 416 cents, and very possibly also those at 278 or
286 cents from a period perspective. My gut reaction is to run up and
embrace those thirds as old friends! -- but _not_ evidently the
general period view. Here WIII has tonal circulation; WIV doesn't.

By the way, so that people can compare notes and versions, here's
Scala's version of WIV in an octave of C-C:

! werck4.scl
!
Andreas Werckmeister's temperament IV
12
!
82.40600
196.09000
32/27
392.18000
4/3
1024/729
694.13500
784.36100
890.22500
1003.91000
4096/2187
2/1

> While I await a possible correction from Paul on WIV in C major, I
> believe the historical situation for Werckmeister, who inherited
> his positions using what he now call WIV, but which he credited to
> Trost and his unnamed late cousin. My suspicion it is because
> Andreas was primarily recognized as an improviser rather than a
> composer. I have been performing every Werckmeister piece I can
> find, more recently his Praeludium in G from the Halle
> dissertation. Most famously, the Christmas cantata Wo ist die
> neugeborne Konig auf den Juden (PITCH 200202 Early).

This is the kind of detail and intimacy in studying any period that I
deeply treasure and respect. I must admit that I'm quite unfamiliar
with this era, but your writing makes it intriguing. Of course,
"historically informed" views can often differ -- as can views of
contemporary writers in a given period also!

> Johnny: I suspect you have stretched ears. Werckmeister actually
> used the word Circul to indicate the difference between WIII and
> WIV. (Sorry everyone that the names of these prominent scales are
> so arcane.) What you like to hear, or consider sounding nice,
> doesn�t change the facts. If one is allergic to dogs, even the
> nicest, sweetest poodle would still cause an allergic response.

Here you make a _critical_ point: what someone like Vicentino,
Werckmeister, or whoever writes _does_ tell us a lot about their
viewpoint, and should be duly noted!

Thus Schlick clearly wanted his poodle (or Lehmanian "prairie dog"?)
at Ab-Eb to be used, because that's the whole point of the compromise
with Ab/G#. The idea is to have Ab-C-Eb usable as a sustained
sonority, as he does in his organ composition, with Ab-C maybe similar
to 12-EDO, while still being able to use E-G# at around Pythagorean
for ornamented cadences to A. An allergy to Ab-Eb would defeat that
purpose. So it appears that Schlick is a "poodle-friendly" tuning.

However, I seem to remember reading about someone around 1600
suggesting that 3:2 is "as wide as bearable," or the like, for a fifth
-- which _would_ mean the kind of poodle allergy we're discussing.

> It may be one of the most amazing challenges, the ability to get
> into someone�s head. Actors do it all the time. Musicians are
> often actors, usually the best of them. Getting into the mind of a
> long dead person through their writing may be toughest of all. As
> I said to Paul, I do not think the literature portends the full
> story. At best it is a part of the story. The literature is more
> interesting in its influence of later generations.

Please let me agree that while period literature, like any attempt to
verbalize about music and musical taste, is necessarily incomplete, it
is not unuseful. Thus reading Praetorius and Denis will at least alert
me to the fact that they considered meantone F-G# a "wolf" or "false
minor third" rather than "a beautiful and near-pure 7:6." Thanks to
Paul Poletti for reminding me that Praetorius discussed the use of the
term "wolf" _specifically_ to mean this third.

In the early 14th-century Italy of Marchettus, 7:6 or so _may have_
been a favored tuning of a cadential minor third contracting to a
unison (e.g. C#-E resolving to D), but we can't be sure. Christopher
Page concludes that it should often be considerably smaller than
Pythagorean, but leaves the details to the singers.

[On my taste for wide fifths]

> Johnny: Exactly, you have stretched ears. Me, too. That's why I
> recognize the condition. It's no different than musicians being
> over sensitive to sound in general. We have stretched our minds in
> to encompass dissonance as consonance. The French take the just
> third of 386 cents and recoil in horror at its dissonance
> (anecdotal).

This is very true, and can affect my reactions both to the intervals
themselves, and to seeing their sizes in cents: numbers like 274, 287,
416, or 427 cents make me feel a certain affection for a tuning. Of
course, I still know that these sizes should not be taken as freely
interchangeable with meantone thirds <grin> -- although sometimes I
try curious things anyway.

It's a bit like the question of parallel fifths, routine in a
13th-century style, but generally excluded in a 16th-century style
(apart from relaxed styles like certain dance settings and the
villanella).

By the way, I recall some years ago playing in a usual 14th-century
Pythagorean (Eb-G#) and hearing a diminished fourth (384 cents) as
indeed something of a "wolf." Acoustically, to borrow from Gertrude
Stein, it may be true that "a 5:4 is a 5:4 is a 5:4," and likewise
"a 7:6 is a 7:6 is a 7:6." In varying musical contexts, however, they
can indeed sound different, fitting or otherwise.

> Your later comments: "What I've decided is that making new music
> should be my first priority, and offering some curious
> verbalizations about what I'm up to, however imperfect and
> incomplete, a good next priority" make total sense as regards your
> intervallic reactions.

This is a direction in which George Secor encouraged me during our
very fruitful dialogue of 2001-2002 concerning his 17-tone
well-temperament: seeking out to compose and explore an "alternative
history" where the music would, he predicted, likely sound less and
less like anything historically familiar. Of course, there's a rich
spectrum of degrees from something "musicologically correct" for a
given period to something strikingly xentonal or the like.

Most appreciatively,

Margo

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

>
> By the way, so that people can compare notes and versions, here's
> Scala's version of WIV in an octave of C-C:
>
> ! werck4.scl
> !
> Andreas Werckmeister's temperament IV
> 12
> !
> 82.40600
> 196.09000
> 32/27
> 392.18000
> 4/3
> 1024/729
> 694.13500
> 784.36100
> 890.22500
> 1003.91000
> 4096/2187
> 2/1
>

What an awkward manner of expressing a temperament! Mixing fractions
with cents! Man, just stick to the one or the other. Or do it all in
ratios to 13 decimals, which is how I do my calculations. It gives you
the advantage of fairly simple numbers for consonances (like 1,5 or
1,25) AND the consistency of units, which is generally regarded as a
desirable thing. Plus it assures a maximum of accuracy and freedom
from rounding errors. I see no advantage to using 1024/729 for f#,
especially as it returns a value which is INCORRECT by almost 1 cent
(0,977 cents to be more exact). Even reducing the ratio to only 6
decimal places (the same number of digits), 1,405457, the answer is
far closer to the real theoretical value than that returned by this
fraction. It's precisely the avoidance of this sort of error which
causes real scientists to avoid mixing units.

It's also the desire to avoid this kind of error that caused me long
ago to reject all precalculated values and sit down and do the numbers
myself, using the original sources. It's not so difficult, now that we
have XL to do the number crunching for us. I use but one ratio for any
given tempered fifth, and I test all of my temperaments for cumulative
error and except nothing less than a very long train of zeros after
the 2 after going around the circle of fifths. Not that I think it is
important audibly, it just keeps you from starting down that slippery
slope which leads to really wanky values like Johnny gave us.

Ciao,

P

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> I see no advantage to using 1024/729 for f#,
> especially as it returns a value which is INCORRECT by almost 1 cent
> (0,977 cents to be more exact). Even reducing the ratio to only 6
> decimal places (the same number of digits), 1,405457, the answer is
> far closer to the real theoretical value than that returned by this
> fraction.

Oops. Turns out I grabbed the wrong ratio from the wrong column in my
master spreadsheet that allows me to manipulate values between an
original and a slightly altered version of a temperament. I accidently
grabbed a manipulated value instead of the theoretically correct
value. As it turns out, 1024/729 does indeed return the correct ratio
for f#.

That said, I would posit that my own blunder is but furhter support
for my statement:

> It's precisely the avoidance of this sort of error which
> causes real scientists to avoid mixing units.

;-)

Seriously, remember the mars probe which crashed into the surface of
the planet because some software engineer grabbed the wrong conversion
for feet to meters?

Ciao,

P

My, you are edgy nowadays, aren't you?

I'll get back to you on your response.

Oz.

----- Original Message -----
From: "Paul Poletti" <paul@polettipiano.com>
To: <tuning@yahoogroups.com>
Sent: 28 Ekim 2007 Pazar 11:53
Subject: [tuning] Re: Wolves, poodles, goats, and styles

> --- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
> >
> > I see no advantage to using 1024/729 for f#,
> > especially as it returns a value which is INCORRECT by almost 1 cent
> > (0,977 cents to be more exact). Even reducing the ratio to only 6
> > decimal places (the same number of digits), 1,405457, the answer is
> > far closer to the real theoretical value than that returned by this
> > fraction.
>
> Oops. Turns out I grabbed the wrong ratio from the wrong column in my
> master spreadsheet that allows me to manipulate values between an
> original and a slightly altered version of a temperament. I accidently
> grabbed a manipulated value instead of the theoretically correct
> value. As it turns out, 1024/729 does indeed return the correct ratio
> for f#.
>
> That said, I would posit that my own blunder is but furhter support
> for my statement:
>
>
> > It's precisely the avoidance of this sort of error which
> > causes real scientists to avoid mixing units.
>
> ;-)
>
> Seriously, remember the mars probe which crashed into the surface of
> the planet because some software engineer grabbed the wrong conversion
> for feet to meters?
>
> Ciao,
>
> P
>
>

Thank you, Margo, for reevaluating WIV from a different angle. And thank
you for posting the mix of ratio and cents that you retrieved from Scala. The
more information given, in all its guises, the better for a full evaluation,
not only of the truth, but of different errors that are circulating. This
List is for exploration, not necessarily for circulation.

The more I think of the literature, the less I think is its percentage of
full revelation of a historical musical practice. Besides the translation
problems, and Werckmeister's translations are more difficult in the regularly
recurring Latin than in the German, few tend to read what gets published.

Please correct me if warranted, but few read Grammateus in his life time.
Werckmeister, on the other hand, was ingenious in getting his work published
and distributed throughout the German speaking world. It would be interesting
to list important writings on tuning according to the category of
dissemination. More important, for most, is the value place on the writings due to
influence on later generations, and here we might place Kirnberger.

Still, more important, much of musical action is not written about in books,
especially by musicians. Note how few microtonal musicians are on this
List. If you are unfamiliar with them, I can make a list for you later.
Musicians, as should be commonly understood, regularly deal with the ineffable.
Simply, they are not verbal. In this I am a notable exception. (At Columbia I
was called by my advisor "a talking chimpanzee."

>Thus Schlick clearly wanted his poodle (or Lehmanian "prairie dog"?)
at Ab-Eb to be used, because that's the whole point of the compromise
with Ab/G#. The idea is to have Ab-C-Eb usable as a sustained
sonority, as he does in his organ composition, with Ab-C maybe similar
to 12-EDO, while still being able to use E-G# at around Pythagorean
for ornamented cadences to A. An allergy to Ab-Eb would defeat that
purpose. So it appears that Schlick is a "poodle-friendly" tuning.

Margo, because I have been so busy teaching this year, I had to pause on my
Bach's Tuning work, sometimes finding myself confused as to where I got
something. I had a source on Schlick where he wrote of what to avoid and yearn to
find it again.

Most dramatic in my college experiences was taking a Renaissance music
graduate course with Prof. Lippman at Columbia University. Lippman was known for
his book on ancient Greek tuning. When I appeared in his class, as an
ethnomusicology grad student on fellowship, I was asked to leave the class. It had
been determined that as an ethnomusicologist, there would be no reason for
me to be there. Intervals were to written on a blackboard in ratios. When I
brought in a Korg tuner to play the intervals, they shuddered and shrank.

Now that I have the WIV numbers correctly crunched thanks in part to my good
friend at NYU, I see that there are 2 dogs at 710. Dogs are clearly, for me
and maybe for others in earlier times, simply "over the top." :)

Margo: However, I seem to remember reading about someone around 1600
suggesting that 3:2 is "as wide as bearable," or the like, for a fifth
-- which _would_ mean the kind of poodle allergy we're discussing.

Johnny: Exactly my view expressed.

> It may be one of the most amazing challenges, the ability to get
> into someoneâs head. Actors do it all the time. Musicians are
> often actors, usually the best of them. Getting into the mind of a
> long dead person through their writing may be toughest of all. As
> I said to Paul, I do not think the literature portends the full
> story. At best it is a part of the story. The literature is more
> interesting in its influence of later generations.

Please let me agree that while period literature, like any attempt to
verbalize about music and musical taste, is necessarily incomplete, it
is not unuseful. Thus reading Praetorius and Denis will at least alert
me to the fact that they considered meantone F-G# a "wolf" or "false
minor third" rather than "a beautiful and near-pure 7:6." Thanks to
Paul Poletti for reminding me that Praetorius discussed the use of the
term "wolf" _specifically_ to mean this third.

Johnny: On a lighter point, we might substitute "indicate" for "mean" in the
previous sentence. :)

Margo: By the way, I recall some years ago playing in a usual 14th-century
Pythagorean (Eb-G#) and hearing a diminished fourth (384 cents) as
indeed something of a "wolf." Acoustically, to borrow from Gertrude
Stein, it may be true that "a 5:4 is a 5:4 is a 5:4," and likewise
"a 7:6 is a 7:6 is a 7:6." In varying musical contexts, however, they
can indeed sound different, fitting or otherwise.

Johnny: Again relating to that Columbia class, I wanted to set the
terminology that a ditone was not a major third (to no avail). Certainly they come
from different perspectives. They were used at different times, with ditone
being the earlier term. About the 7, there is Tartini (by repute used in his
playing...though maybe you know of some specific usage of 7-limit intervals),
and Kirnberger (who notated with the letter i in his Flute Sonata for a 7/4
(969) cent interval).

Margo: This is a direction in which George Secor encouraged me during our
very fruitful dialogue of 2001-2002 concerning his 17-tone
well-temperament: seeking out to compose and explore an "alternative
history" where the music would, he predicted, likely sound less and
less like anything historically familiar. Of course, there's a rich
spectrum of degrees from something "musicologically correct" for a
given period to something strikingly xentonal or the like.

Most appreciatively,

Margo

Johnny: La Monte Young had a similar idea, that one must compose music that
sounded like it had a great history behind it, ala Bach. Pioneers don't
necessary last forever in contradistinction. You certainly have it under
control. Happy composing!

joyfully, Johnny

************************************** See what's new at http://www.aol.com

Hello, everyone, and thanks to Tom, Johnny, Paul, and others for
prompting me to an idea concerning one possible variant on
Werckmeister IV.

First, based on Werckmeister V, one might guess that Werckmeister
regards wide fifths about as playable in themselves as narrow fifths
sharing the same degree of impurity.

Secondly, in Werckmeister IV, the fifths tempered by 1/3 comma are not
restricted to remote portions of the circle: rather they occur in such
prominent positions as C-G, D-A, and E-B.

Looking at that beautiful fascimile of Werckmeister's 1681 organ
tempering diagram, Paul (many thanks!), it occurred to me that if the
"comma" may be either Pythagorean or syntonic, with the schisma
regarded as not so important, then why not 1/3-syntonic comma
tempering for the narrow and wide fifths alike?

Here are two Scala files for this variant reading, the first in usual
Scala style with just intervals indicated as interval ratios; and the
second, as you've suggested, Paul, with all ratios given in cents.
As well said by Thomas Morley (1597) on such points of preference:
There be more ways to the wood than one.

! werckmeisterIV_variant.scl
!
Werckmeister IV with 1/3 syntonic comma temperings
12
!
85.00995
196.74124
32/27
393.48248
4/3
45/32
694.78624
785.01123
891.52748
1003.25876
15/8
2/1

! WerckmeisterIV_variant_c.scl
!
Werckmeister IV variation, 1/3-SC, all intervals in cents
12
!
85.00995
196.74124
294.13500
393.48248
498.04500
590.22372
694.78624
785.01123
891.52748
1003.25876
1088.26871
2/1

As I recall, Werckmeister IV is said to be best for a diatonic style
sticking mainly to familiar locations. The 1/3-comma variation seems
to fit this model -- at least if, like Costeley (1570) and Salinas
(1577), we are ready to accept fifths tempered by this great a
quantity, as in a regular 1/3-comma meantone or 19-EDO. Zarlino (1571)
found 1/3-comma temperament "languid," in contrast to either 1/4-comma
or his own 2/7-comma -- but not, as I recall, outright unacceptable.

If one can bear with these fifths, then one advantage for usual
diatonic transpositions (modal, or by this epoch tonal also) is that
each of the seven major thirds formed from two _musica recta_ notes
(the diatonic steps plus Bb) will be uniformly at 1/6-comma meantone
quality -- except for E-G#, which will actually be a schisma closer to
5:4 because of the "schisma fifth" in its chain at C#-G#.

At Eb-G we have a fifth at about 401 cents, considerably less than
ideal but still tenable, in modes like G Dorian or F Lydian or Ionian,
a compromise possibly rather like Schlick's (1511) with Ab-C.

The schisma adjustment of C#-G# in this 1/3-syntonic comma version may
offer a small but notable advantage in pieces calling for B-D# as a
usual major third: this third is a schisma smaller than Pythagorean,
or around 406 cents, the size which Easley Blackwood has suggested may
mark the limit for a stable major triad in a tonal style.

In contrast, C#-F, F#-Bb, and G#-C remain appreciably wider than
Pythagorean, at about 413-415 cents, also the evident upper limit for
major thirds or diminished fourths in Schlick's temperament and some
versions of the temperament ordinaire in favor at the time of
Werckmeister's treatise of 1681. Brad might note that the augmented
fifth C-G# gives a nice approximation to a just 11:7, by the way --
this is something that I look for, also.

These three diminished fourths would nicely fit their role as "special
effects" intervals in meantone, and might sometimes be used as
"marginally interchangeable" with more usual major thirds, as Mark
Lindley describes in French keyboard music of Couperin. George Secor's
modern temperament ordinaire also explores this territory.

However, if we take the main focus of Werckmeister IV to be a
conventional diatonic style, then the 1/3-syntonic-comma variant does
seem an attractive solution if one can accept this much temperament of
the fifths, a la Costeley or Salinas.

Please let me conclude, Tom, by thanking you for your at once diligent
and creative insight that Werckmeister's Septenarius temperament
(Werckmeister VI) might better fit his own declared musical goals and
standards if the monochord value printed as "176" were read as "175."
I looked at your proposal, and quickly saw how much sense it made!
That gave me the idea of considering whether, if either a Pythagorean
or a syntonic comma is a possible reading of Werckmeister's diagrams,
an interpretation of the "comma" in Werckmeister IV as syntonic might
not likewise promote his musical purposes.

With many thanks,

Margo Schulter
mschulter@calweb.com

Margo wrote (inter alia):

> Brad, my warmest thanks to you as an artist and
> scholar who has brought to my attention the noble ingenuity of the
> temperament unit, and inspired me to seek creative by-ways for
> applying it.

You're welcome! And it seems the thanks should really go back to Brombaugh himself for coming up with that convenient unit, and the way he explained it to me years ago. I'm just an outspoken fan of its usefulness. May I forward your essay to Brombaugh?

Brad Lehman

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> Here I'll try to explain this quirk, and show how John Brombaugh's
> temperament units (TU) can sometimes help in evaluating the
> consequences.

I'll repeat my recommendation that TUs not be used, since they lack a
precise definition. Anything you can use TUs for you can also use flus,
for; with exactly 46032 flus to an octave, 900 flues to a Pythagorean
comma, and 825 to a Didymus comma.

Of course, you can define TUs so that the Pythagorean comma is
*exactly* 720 TU, but now larger intervals don't come out evenly. With
flus you know precisely what to call any 5-limit interval in flu terms.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> Of course, you can define TUs so that the Pythagorean comma is
> *exactly* 720 TU, but now larger intervals don't come out evenly.
With
> flus you know precisely what to call any 5-limit interval in flu
terms.

If you absolutely must use TUs, I suggest defining them so that the
octave is *exactly* 36825.6 TU, with the fifth taken as 21541.6 TU and
the major third taken as 11855.2 TU. It's bizarre, but it works for
general intervals, which TUs as ordinarily "defined" do not.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:

> If you absolutely must use TUs, I suggest defining them so that the
> octave is *exactly* 36825.6 TU, with the fifth taken as 21541.6 TU
and
> the major third taken as 11855.2 TU. It's bizarre, but it works for
> general intervals, which TUs as ordinarily "defined" do not.
>

The following intervals come out evenly in terms of TUs:

atom: |161 -84 -12> 0
whoosh comma |37 25 -33> 16
schisma 32805/32768 60
diaschisma 2048/2025 600
Didymus comma 81/80 660
Pythagorean comma |-19 12> 720
diesis 128/125 1260
major diesis 648/625 1920

The problem is, most intervals do not come out evenly. Since
the "definition" of TU gives integer values for the two commas,
anything which is a product of these comes out evenly, but that
excludes most intervals.

> > Of course, you can define TUs so that the Pythagorean comma is
> > *exactly* 720 TU, but now larger intervals don't come out evenly.
> With
> > flus you know precisely what to call any 5-limit interval in flu
> terms.
>
> If you absolutely must use TUs, I suggest defining them so that the
> octave is *exactly* 36825.6 TU, with the fifth taken as 21541.6 TU
> and the major third taken as 11855.2 TU. It's bizarre, but it works
> for general intervals, which TUs as ordinarily "defined" do not.

I suspect you're missing the main reason of existence for TUs. It
has nothing to do with octaves. And they work perfectly for the
purpose they were designed for.

Brombaugh chose 720 *because* 720, 660, and the difference 60 are all
so easily divisible by 2, 3, 4, 5, and 6...which are the main types
of comma splits in historical temperaments (having nothing to do with
any computers or synthesizers, or even for that matter with
theoretical speculations either). He wanted a simple system with
which he could explain on paper -- with no calculators or computers --
the various distributions of those two main commas, and the
schisma. It's the easy stuff one can draw onto a napkin at dinner,
or mentally in the shower or on a bike ride, dealing with simple
arithmetic.

When everything works out so neatly as integers, it's plenty clear
down to (and way beyond) the level that any human being can hear any
of these intervals. It's a system for measuring comma distributions,
and it's logarithmic, so the everyday operations get reduced to
addition/substraction instead of multiplication/division. It's not
designed for use beyond the width of (say) 2000 or 3000 TU. It
scarcely has application beyond 720, frankly, unless the topic of
interest is a diesis.

If anybody wants to do more precise yet less practical things, they
can go make up their own unit that is additionally divisible by 7 and
11 and 13 and 17 if they care to. Or define the whole octave as "a
bazillion" logarithmic units and then start chopping it up, with or
without any round-off errors down below the threshold of pitch
discrimination of any living being. The main point is: Brombaugh
wanted something finer-toothed and more comma-centered than cents
are, for basic explanations on paper. And he succeeded.

Brad Lehman

> Brombaugh chose 720 *because* 720, 660, and the difference 60 are all
> so easily divisible by 2, 3, 4, 5, and 6...which are the main types
> of comma splits in historical temperaments (having nothing to do with
> any computers or synthesizers, or even for that matter with
> theoretical speculations either). He wanted a simple system with
> which he could explain on paper -- with no calculators or computers --
> the various distributions of those two main commas, and the
> schisma. It's the easy stuff one can draw onto a napkin at dinner,
> or mentally in the shower or on a bike ride, dealing with simple
> arithmetic.

Oh, wait, I'll take part of that back. Mental processing of tuning
concepts in the shower *is* theoretical speculation. My bad. Carry
on. The rest of my point remains, for what it's worth.

If we're trying to do 1/7 syntonic comma on a non-electric instrument,
94 TU is close enough, and then some. How big is the major 3rd in
regular 1/7 syntonic comma? Tally up four 94s to 376. Divide by 660.
We're about 57% of a syntonic comma sharp, above a pure 5:4 major
3rd. See, that's all we *really* need to know in practice. Pencil
and paper with arithmetic any 12-year-old should know.

Brad Lehman

> If we're trying to do 1/7 syntonic comma on a non-electric instrument,
> 94 TU is close enough, and then some. How big is the major 3rd in
> regular 1/7 syntonic comma? Tally up four 94s to 376. Divide by 660.
> We're about 57% of a syntonic comma sharp, above a pure 5:4 major
> 3rd. See, that's all we *really* need to know in practice. Pencil
> and paper with arithmetic any 12-year-old should know.

What, nobody has kicked me in the head yet for the error in that
illustration, from my haste at typing it? OK, I'll fill in the
missing step myself. Tally up four 94s to 376. That's the amount
flat from a Pythagorean ditone. To get the amount sharp from a pure
5:4, we have to subtract from 660...getting 284. Then divide *that*
by 660 to see what percent of comma we're sharp, and it's 43%.

Brad Lehman

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:

> > If you absolutely must use TUs, I suggest defining them so that
the
> > octave is *exactly* 36825.6 TU, with the fifth taken as 21541.6
TU
> > and the major third taken as 11855.2 TU. It's bizarre, but it
works
> > for general intervals, which TUs as ordinarily "defined" do not.
>
> I suspect you're missing the main reason of existence for TUs. It
> has nothing to do with octaves. And they work perfectly for the
> purpose they were designed for.

Wrong. I do know what they are used for, but I think a system to
analyze 5-limit intervals which can only deal with a small subset is
intelletual crippleware.

> Brombaugh chose 720 *because* 720, 660, and the difference 60 are
all
> so easily divisible by 2, 3, 4, 5, and 6...which are the main types
> of comma splits in historical temperaments (having nothing to do
with
> any computers or synthesizers, or even for that matter with
> theoretical speculations either).

Then he could at least have made a real definition, by for instance
setting the apotome to 3488.8 TU. As it is, YUs are not defined
except on a small subset of 5-limit intervals. Hell of a way to run a
railroad.

> When everything works out so neatly as integers, it's plenty clear
> down to (and way beyond) the level that any human being can hear
any
> of these intervals.

And using flus, or defining the apotome as 3488.8 TUs, doesn't do
this because...?

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:

> If we're trying to do 1/7 syntonic comma on a non-electric instrument,
> 94 TU is close enough, and then some.

But it's too small if you want more exact answers.

How big is the major 3rd in
> regular 1/7 syntonic comma? Tally up four 94s to 376. Divide by 660.
> We're about 57% of a syntonic comma sharp, above a pure 5:4 major
> 3rd. See, that's all we *really* need to know in practice. Pencil
> and paper with arithmetic any 12-year-old should know.

This is mathematical voodoo. Depending on how you set up the
calculation, *even without using approximations*, you come up with
different answers. The size of the major third in 1/7-comma meantone
using the intellectual crippleware you advocate is undefined. Do it my
way, and it is 15172 4/7 flus, which is 12138 2/35 TUs. The major third
is 14819 flus, which is 11855.2 TUs. The difference is 355 2/7, which
is *exactly* 3/7 of a syntonic comma of 825 flus. You get the same
answer if you use my suggestion for defining TYs for all 5-limit
intervals. No vagaries about "about 57%" need apply; which anyway
whould be about 43%. And while there is a tiny imprecision from
ignoring the atom, the conclusion that it is exactly 3/7 of a syntonic
comma sharp is correct.

>> Brad, my warmest thanks to you as an artist and
>> scholar who has brought to my attention the noble ingenuity of the
>> temperament unit, and inspired me to seek creative by-ways for
>> applying it.

> You're welcome! And it seems the thanks should really go back to
> Brombaugh himself for coming up with that convenient unit, and the
> way he explained it to me years ago. I'm just an outspoken fan of
> its usefulness. May I forward your essay to Brombaugh?

Hi, Brad. Please feel free indeed to forward my essay to him with
warmest thanks from another outspoken fan!

With many thanks,

Margo

> This is mathematical voodoo. (...) The size of the major third in
> 1/7-comma meantone using the intellectual crippleware you advocate
> is undefined. Do it my
> way, and it is 15172 4/7 flus, which is 12138 2/35 TUs. The major
> third is 14819 flus, which is 11855.2 TUs. The difference is 355
2/7, > which is *exactly* 3/7 of a syntonic comma of 825 flus.

"Intellectual crippleware"? "Mathematical voodoo"? Anybody should
get the *exactly* 3/7 of a syntonic comma answer, without having to
calculate down to any of these units, or pick any amount of decimal
round-off. We've taken 1/7 comma off each of four 5ths, such as
C-G-D-A-E to build a C-E of some size. There are therefore 3/7 of
that comma remaining. The major 3rd is therefore 3/7 of a syntonic
comma sharp from 5:4. Exactly. Coming down from a Pythagorean ditone
toward a pure 5:4 major 3rd, we've succeeded in knocking off 4/7 of
the error, the syntonic comma.

The "intellectual crippleware" you're complaining about (the TU that
you're rubbishing) is not designed to measure the entire size of the
interval, but only the distance of the error from pure 5:4. There is
nothing "crippling" about this. It only answers a different type of
question than you would pose to it. Listening to a real organ or a
real harpsichord, the normal listener is not going to give a flying
fig how big the major 3rd is in absolute size, measured in any
numerical units from bottom to top, either in some logarithmic units
or as numerical frequencies per unit of time. Rather, the perception
(well, at least for me!) is: "about how much" is the interval
wavering, being not quite on a pure integer ratio spot? It's some
amount sharp. How much? A little less than half a comma, or 3/7, or
about 43%. Or a smidge under size 5 on the 11-unit metric from Sorge
et al, where 11 is a Pythagorean ditone, and a 12et major 3rd works
out to exactly 7. How big is the entire major 3rd on that 11-unit
metric? It's not designed to answer that question. It's designed,
rather, to tell us how many 1/11ths of a syntonic comma we're off from
pure in a given major 3rd. All the TU system does is multiply this by
60, and it answers similar questions.

There exists a scale of Scoville Units to measure the pungency of
chili peppers.
http://www.chemsoc.org/exemplarchem/entries/mbellringer/scoville.htm
But when you go to a restaurant and peruse the menu, does it give you
any numerical measurements in Scoville units? No, you get something
like Mild, Medium, Spicy, and maybe a fourth option, and you pick one,
and then the chef tries to give you something accordingly to the taste
you said you wanted. The chef probably doesn't even know or care what
absolute number of Scoville units his dish would register, nor does
the customer; only that it tastes enjoyable and as ordered. Is the
Mild-Medium-Spicy scale "intellectual crippleware", or does it really
help the customer get what he wants to eat?

Let's ask the flu system some question it's not designed to answer.
How much is 1/3 of a major 3rd, in flus? Well, let's see. The entire
major 3rd, presumably a pure 5:4, is (according to you) 14819 flus.
And it's not evenly divisible by 3. It comes out to a repeating
decimal and can never be rounded off accurately! Nor is 14819
divisible by 2, 3, or 5. It *is* divisible by 7 and 2117, so maybe
the flu is designed to hack major 3rds into seven equal pieces? What
kind of system assigns a big unit that is not divisible by 2, 3, 4, 5,
6, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, or 20 but *only* by
7? Well, I guess it's just not designed to answer such questions
expecting the answer of an integer. Does that make it "intellectual
crippleware", or is it merely a tool being misapplied to a wrong task?
How big is a major 3rd in Fahrenheit?

Brad Lehman

Spurred by Margo's recent postings, I've tried some experimentation
with 2/7 syntonic comma on harpsichord. Here are some preliminary
comments from that. These are all from working it by ear at the
instrument, not calculating anything.

I set up regular 2/7 all the way around from Eb up to G#. The major
3rds are a little bit narrow, the minor 3rds (6:5) scarcely wide. The
F# to Bb, B to Eb, C# to F, and G# to C work out to hit almost on a
9:7, and the F to G#, Bb to C#, Eb to F# are almost on 7:6. With this
installed I played through a bunch of random pages of medieval and
Renaissance music, mostly vocal, from the _Historical Anthology of
Music_ (HAM). Margo's point, as I recall, is to set up something
suitable for accompanying this type of repertoire.

Then I tried her recommendation of converting this to _temperament
extraordinaire_ (her nomenclature, not mine; I'd already co-opted the
phrase "extraordinary temperament" to mean something different, a
couple of years ago...). That is: stretch F-Bb-Eb wide each, and
F#-C#-G# wide each, so it eradicates the wolf G#-Eb and reduces it to
four other lesser canines, all the same as one another. In practice
at harpsichord this is simply a process of cranking the Bb and Eb
downward in turn, and the C# and G# upward in turn, until we have
similar 5ths F#-C#-G#-D#-A#-E#. They're all a little bit rougher than
the 2/7 comma 5ths, and of course they're rough in the opposite
direction, but given that all these 5ths/4th are growly anyway they
don't sound too bad in context...playing them as open 5ths/4ths,
without any 3rds (whether major or minor) in them.

But, does it work? I played some more from HAM, some of the same
compositions and some others randomly. To the extent that the notes
Bb, Eb, G#, and C# came up, which was rarely, they always stood out to
me as not fitting their context when played in major or minor 3rds.
Neither fish nor fowl. Especially in the case of Bb, the most
frequent of these four notes, it seemed to me there's too much
difference in character between the Bb major triad and the F major
triad, and between the A-Bb semitone vs the E-F semitone. The Bb-D is
scarcely wide as a major 3rd, and pleasant enough, but it doesn't have
that tight center of gravity like the major 3rds on the naturals.
Similarly, A-C# is only a little bit wide but it's lost its gravity.
And Eb-G and E-G# are watered down even more, each. They're still
smaller than Pythagorean ditones and they're pleasant enough, in
isolation, but they don't seem to fit their context.

Worse yet: we haven't really gained any usable B-D#, F#-A#, Ab-C, or
Db-F. They were kind of OK and attractive in their own way (while
exotic) when they were hitting nearly 9:7, tuned regularly. But now,
having these modified 5ths, these misspelled major 3rds are in a
no-man's-land. They just sound like "wrong notes" giving nothing for
my ear to grab onto, or to take me past the wincing. I kept itching
to put them back down (sharps) and up (flats) where they belong.

And in doing so, I found that I was presented with several viable
places to put each of those stinkers Bb, Eb, C#, and G#.

Well, first, the Bb almost had to go back to its regular 2/7 position
so it's not sticking out as mentioned above.

Having done that, there are four or five arguably decent places to put
the others, all pretty close to one another when turning the tuning
pin and sliding through all possibilities.

C#: back to its regular 2/7 comma place from F#, or tune it as a pure
5:4 from A-C#, or tune it as pure 7:6 with Bb, or tune it as pure 9:7
from F. Or, simply as a pure 3:2 from F#. All five of these spots
are arguably good for C#. Following Margo's lead I favored the two
septimal spots. Why hit only near a 7:6 or a 9:7 when you can hit
directly on one of them? It makes the minor triads, diminished
triads, and tritones lock in that way, too.

G# similarly has four or five spots: its 2/7 comma tempering from C#
(before we moved C#), or as 5:4, 7:6, 9:7, or 3:2 from its neighbors.

And Eb similarly has those options with respect to Bb, B, F#, and G.

All of this writes off the G#-Eb wolf as a loss...but it's not going
to come up in this medieval and Renaissance music anyway. And by
going back to regular 2/7, or nearly so (choosing septimal placements
on C#, G#, and Eb), we've got a fantastic set of eleven minor triads,
eight tightly attractive major triads, three septimal major triads,
and all the other related emoluments. The 5ths F#-C#-G# and Eb-Bb
work out decently (for context) no matter what we do, given that the
2/7 comma 5ths on the naturals are already rough.

If the point is to accompany medieval or Renaissance [vocal] music,
regular 2/7 works better (at least for me) than the 5th-stretching
_extraordinaire_ step does. If the point was to get those septimal
resonances going, why move off them by widening the 5ths?

I could change my mind tomorrow, but that's how I hear it today. And
all of this might work better (or worse?) on organ as opposed to
harpsichord. On harpsichord, at least, the septimal intervals are
clear and pungent.

And if the notes D#, A#, Ab, and Db aren't going to come up in the
music anyway...why not just stick with regular 1/4 instead of 2/7?
It's sort of a trade-off here between the major 3rds and minor 3rds.
But if we wanted pure minor 3rds, why not go all the way to 1/3 comma
then? 2/7 goes down between these, giving the strengths of both...and
because those 3rds aren't pure, they're (arguably) more attractive
just by virtue of being livelier, not sitting there dead-on. Every
listener might of course have a different opinion about that.

Brad Lehman

Hi Brad,

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:
>
> Spurred by Margo's recent postings, I've tried some
> experimentation with 2/7 syntonic comma on harpsichord.
> Here are some preliminary comments from that. These
> are all from working it by ear at the instrument,
> not calculating anything.

Since you are describing empirical observations, i
thought i'd supplement your post with some observations
about the calculations.

;-)

> And if the notes D#, A#, Ab, and Db aren't going to
> come up in the music anyway...why not just stick with
> regular 1/4 instead of 2/7? It's sort of a trade-off
> here between the major 3rds and minor 3rds. But if we
> wanted pure minor 3rds, why not go all the way to 1/3 comma
> then? 2/7 goes down between these, giving the strengths
> of both...and because those 3rds aren't pure, they're
> (arguably) more attractive just by virtue of being
> livelier, not sitting there dead-on. Every listener
> might of course have a different opinion about that.

2/7-comma meantone is in fact the variety of meantone
which both minimizes and equalizes the error for all
of the main ratio-of-5 intervals simultaneously:
ratios 5:4 & 8:5, and 6:5 & 5:3. You can see this
in the last graphic at the bottom of my webpage:

http://tonalsoft.com/enc/number/2-7cmt.aspx

In that graphic, the blue plots represent the temperament
of these four 5-limit ratios. These graphic comes from
a javascript applet i have on another page:

http://tonalsoft.com/enc/m/meantone-error.aspx

If you mouse over the meantone names on the left side
of that graphic, you can see the plots of each meantone
move as your mouse points over the different names.

I could have made these graphs much better by including
labels showing the ratios ... so i'll give them here:

* the red plot below the thick zero-error line is the
amount of error of the 3:2 ratio, and the red plot above
that line is the error for 4:3,

* the dark blue plots, from left to right, represent
the error for ratios 8:5, 6:5, 5:3, and 5:4.

* the yellow plots represent selected ratios-of-7 and
the light blue plots represent selected ratios-of-11,
as tabulated directly under the graphic.

Note also that the fraction-of-a-comma error is exact
for the ratios-of-3 and ratios-of-5, but only approximate
for the ratios-of-7 and ratios-of-11. In case the reason
for that is not obvious, it's because the syntonic-comma
*is* a ratio-of-5.

Keeping those labels in mind, you can see by
simply following the movement of the various colors
of the plots, that:

* 1/3-comma meantone gives 6:5 & 5:3 with no error,
and the same amount of error, namely 1/3-comma, for
that meantone's approximation of the two pairs of
ratios 5:4 & 8:5 and 3:2 & 4:3;

* 1/4-comma meantone does something similar, but gives
JI ratios for 5:4 & 8:5 (the other pair of ratios-of-5)
with no error, and the same amount of error, 1/4-comma,
for 6:5 & 5:3 and 3:2 & 4:3;

* 2/7-comma meantone equalizes the amount of error
for the two pairs of ratios-of-5: all four of them
have 1/7-comma error, while the ratios-of-3 (3:2 & 4:3)
have 2/7-comma error.

Paul Erlich noted that it was perhaps the the new
preference for composers to feature ratios-of-5 in
their work during the Renaissance, along with this
aspect of 2/7-comma meantone (that it provides the
lowest aggregate error for the ratios-of-5), which
may have led to its advocacy during that time.
While vague descriptions of meantone began to appear
about 50 years earlier, 2/7-comma was the earliest
mathematically-described meantone to be advocated.

It can also be seen from viewing the various different
meantones on this graphic that, according to this
particular error measurement:

* 1/4-comma gives the lowest overall error for all
5-limit intervals (that is, including also the
3-limit 3:2 & 4:3);

* 3/13-comma (and its close relative 136-edo) gives
the lowest overall error for all of the 11-limit
ratios i used in my evaluation.

As it became more and more important for composers to
once again emphasize the harmonic paradigm of ratios-of-3
in their music, the natural tendency would be a desire
to reduce the error for those ratios as well (at the
expense of the accuracy of the ratios-of-5), which would
lead to a preference for 1/4-comma meantone, and eventually,
to meantones with even less amount of tempering, such
as 1/5-comma and 1/6-comma and, finally, 1/11-comma
(which is essentially indistinguishable from 12-edo).

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:

> "Intellectual crippleware"? "Mathematical voodoo"? Anybody should
> get the *exactly* 3/7 of a syntonic comma answer, without having to
> calculate down to any of these units, or pick any amount of decimal
> round-off.

Indeed they can, but that's irrelevant to the question of making use
of tuning units.

> The "intellectual crippleware" you're complaining about (the TU that
> you're rubbishing) is not designed to measure the entire size of the
> interval, but only the distance of the error from pure 5:4.

It cannot do this without further definitions. The problem is this:
TUs are defined by giving the value 660 to a syntonic comma and 720
to a diatonic comma. From this it follows that the atom, calculated
from 12*660-11*720, is set to zero. However, nothing follows for the
value of a pure 5:4, since both the syntonic and diatonic comma are
set equal to zero by 12-et. Consequently, all products are set equal
to zero also. It can only represent intervals which are in
the "kernel" of the 12-et map, and 5/4 is not in the kernel, being
four steps of 12-equal.

If we assume a regular mapping ("val") <a b c| gives the value of 5-
limit intervals in terms of TUs, then solving for what gives the
correct values for the syntonic and diatonic commas tells us that
<12t 19t+60 28t-420| will work for any value of t. However, we want a
decent map. The best we can do using integer values for t is t=36828,
which is indeed a possible definition for the TU. However, while it is
5-limit consistent, it's rather cheesy compared to 46032, which is
my "flu" recommendation.

Note, however, that until it is defined in some way, ***the TU is
undefined except for a special class of intervals***. That's the
problem with it--it doesn't mean anything very specific.

> Let's ask the flu system some question it's not designed to answer.
> How much is 1/3 of a major 3rd, in flus?

You need to ask this in a tempering context because...? Flus multiply
TUs by 5/4, and so exchange a factor of 4 for a factor of 5. Since
divisibility by 2 is so easy, and by 5 not much harder, this isn't
actually a big difference. In any case, obviously TUs aren't doing
the job at all, whereas anything TUs can do, flus can do and with
about the same degree of ease.

Hi Gene and Brad,

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:
>
> The "intellectual crippleware" you're complaining about
> (the TU that you're rubbishing) is not designed to measure
> the entire size of the interval, but only the distance
> of the error from pure 5:4. There is nothing "crippling"
> about this. It only answers a different type of
> question than you would pose to it.

(For proper formatting of the following tables, if
viewing on the stupid Yahoo web interface, use the
"Option | Use Fixed Width Font" links on the upper-right
of the page, under the date.)

For the definition of TU which makes it a logarithmic
1/720 of the pythagorean-comma, here are the TUs for
various divisions of both the syntonic and pythagorean
commas, to an accuracy of 3 decimal places:

TUs:

division . syntonic . pythagorean
... 1 .... 660.039 .... 720.000
... 2 .... 330.020 .... 360.000
... 3 .... 220.013 .... 240.000
... 4 .... 165.010 .... 180.000
... 5 .... 132.008 .... 144.000
... 6 .... 110.007 .... 120.000
... 7 ..... 94.291 .... 102.857
... 8 ..... 82.505 ..... 90.000
... 9 ..... 73.338 ..... 80.000
.. 10 ..... 66.004 ..... 72.000
.. 11 ..... 60.004 ..... 65.455
.. 12 ..... 55.003 ..... 60.000

Here are the same divisions, measured in flus:

flus:

division . syntonic . pythagorean
... 1 .... 824.981 .... 899.926
... 2 .... 412.491 .... 449.963
... 3 .... 274.994 .... 299.975
... 4 .... 206.245 .... 224.981
... 5 .... 164.996 .... 179.985
... 6 .... 137.497 .... 149.988
... 7 .... 117.854 .... 128.561
... 8 .... 103.123 .... 112.491
... 9 ..... 91.665 ..... 99.992
.. 10 ..... 82.498 ..... 89.993
.. 11 ..... 74.998 ..... 81.811
.. 12 ..... 68.748 ..... 74.994

Gene, i'm curious as to why you claim that TU
is "undefined" ... i've always understood it to
mean a logarithmic 1/720 of a pythagorean-comma.

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> Gene, i'm curious as to why you claim that TU
> is "undefined" ... i've always understood it to
> mean a logarithmic 1/720 of a pythagorean-comma.

That would indeed be a precise definition. But then it would be false
that the syntonic comma was 660 TUs; it would acually be
660.0392862471599369... TUs, as you indicate above. And then, you would
not be ignoring the atom; the atom is now .471434965919245629... TUs,
which if the TU is our unit is not really negligable.

I take the TU system to be one which assigns 720 to the P7ythagorean
comma and 660 to the Didymus. It is therefore defined for all intervals
which are products of these, which is precisely the kernel of 5-limit
12-equal.

> Spurred by Margo's recent postings, I've tried some experimentation
> with 2/7 syntonic comma on harpsichord. Here are some preliminary
> comments from that. These are all from working it by ear at the
> instrument, not calculating anything.

Dear Brad,

Please let me start with a short reply to say how excited I am that
you have done this. It was a special delight to read how your
experiment led you to your own variation on 2/7-comma tempered to a
very experienced taste!

> All of this writes off the G#-Eb wolf as a loss...but it's not going
> to come up in this medieval and Renaissance music anyway. And by
> going back to regular 2/7, or nearly so (choosing septimal
> placements on C#, G#, and Eb), we've got a fantastic set of eleven
> minor triads, eight tightly attractive major triads, three septimal
> major triads, and all the other related emoluments. The 5ths
> F#-C#-G# and Eb-Bb work out decently (for context) no matter what we
> do, given that the 2/7 comma 5ths on the naturals are already rough.

Here I consider this kind of empirically fine-tuned solution as a
creative side of things that must have often shaped the historical
temperaments that are now all too easy to approach mainly as neat
mathematical constructs.

I'm honored to have played some role in this process, and would like
to say so in a short post where your new variation, admirably suited
to historical Renaissance music, can enjoy center stage.

In a longer post, I've responded to your comments about the
_temperament extraordinaire_ at some length, and mainly to clarify
that this system is meant mainly for _new music_ which mixes or
alternates Renaissance styles using meantone colors in the nearer
portion of the circle with medieval or especially neomedieval
styles using modes transposed to the remote portion of the circle
with its large major and small minor thirds. This is a kind of
21st-century manneristic tuning, combining two very different
styles of intonation in one 12-note circle, with many inevitable,
striking, and hopefully artful distortions.

However, your septimal refinement of a usual 2/7-comma has the great
advantage for period performances of leaving the usual meantone
structure intact while gently optimizing 7:6 and 9:7. It might be
called "neo-classical" in the best sense: keeping the advantages of
the original scheme while subtly nuancing it for the ears of the
discerning.

Please let me thank you for taking the time and effort for such a
thoughtful musical exploration, and congratulate you on your choice
variation!

With many thanks,

Margo

> Here's a relevant calculation that TU are particularly messy at
> dealing with, if they can even be used at all:
> What is the temperament of the fifth in a regular meantone such that
> F-G# is a pure 7:6?
> Answers in any comprehensible form.

Hello, Tom.

Happily I find that the TU is a good tool to solve this problem, and
will try to explain my process for solving it in a reasonably
comprehensible way, I hope.

To start with, let's consider that in a regular 12-note meantone, F-G#
will be formed from a chain with one Wolf fifth plus two regular
meantone fifths. For example:

F - Bb - Eb - G#
regular regular Wolf

Now we know that the net tempering of those three fifths has to be
equal to the septimal comma or Comma of Archytas (thanks to George
Secor for the latter name!) at 64:63, 27.26 cents, or 836.75 TU.
More specifically, this net temperament of the fifths must be in
the wide direction.

Let's take that as an even 837 TU for most practical purposes. If we
use "t" for the temperament of the regular fifths (the unknown to be
found), and "w" for the impurity of the Wolf fifth in the wide
direction ("w" standing for either Wolf or wide), we have:

w - 2t = 837

Next, let us consider the relationship between t and w. If we take t
as the amount by which regular fifths are tempered narrow, and a
positive value for w to mean the amount by which the Wolf fifth is
wide of a pure 3:2, then we have:

w = 11t - 720

This means that we take the total amount of negative temperament done
by the 11 regular meantone fifths, and subtract 720 (the Pythagorean
comma) to find out the amount by which the Wolf exceeds 3:2, and so
can contribute toward a septimal size for an augmented second like
F-G#.

Now, substituting our second equation for the size of the Wolf fifth
into the first concerning the net temperament of F-G#, we have:

11t - 720 - 2t = 837

This simplifies to:

9t - 720 = 837

and further to:

9t = 1557

and thus:

t = 173

This seems intuitively a reasonable result, since we know that
1/4-comma at 165 TU comes very close, but with F-G# a bit larger than
a just 7:6. This is a regular fifth at about 696.32 cents, or 5.64
cents wide, as an approximation based on 837 TU for the septimal
comma.

It would be more accurate to do the same calculation with a value of
836.75 TU for this comma. Instead of dividing 1557 by 9, we divide
1556.75 by 9 -- otherwise, the process is the same. We get about
172.972 TU, at least according to my Orpie calculator program.

The result is a meantone fifth of about 696.319 cents, or 5.636 cents
narrow of 3:2. Scala shows the 7:6 minor third or augmented second
accurate to 0.001 cents.

! pure7-6mnt.scl
!
Meantone with pure 7:6, based on 836.75 TU for 64:63
12
!
74.23296
192.63799
311.04302
385.27598
503.68101
577.91396
696.31899
770.55195
888.95698
1007.36201
1081.59497
2/1

Of course I would be glad to explain any unclear points if I can, or
generally tidy up my presentation a bit.

In the meantime, as the saying goes, Q.E.D.

Most appreciatively,

Margo Schulter
mschulter@calweb.com

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> --- In tuning@yahoogroups.com, Margo Schulter <mschulter@> wrote:
>
>
> >
> > By the way, so that people can compare notes and versions, here's
> > Scala's version of WIV in an octave of C-C:
> >
> > ! werck4.scl
> > !
> > Andreas Werckmeister's temperament IV
> > 12
> > !
> > 82.40600
> > 196.09000
> > 32/27
> > 392.18000
> > 4/3
> > 1024/729
> > 694.13500
> > 784.36100
> > 890.22500
> > 1003.91000
> > 4096/2187
> > 2/1
> >
>
> What an awkward manner of expressing a temperament! Mixing fractions
> with cents! Man, just stick to the one or the other.

Why, when some of the values are tuned as rationals, and others are
tempered from them?

-A.

> Spurred by Margo's recent postings, I've tried some experimentation
> with 2/7 syntonic comma on harpsichord. Here are some preliminary
> comments from that. These are all from working it by ear at the
> instrument, not calculating anything.

Dear Brad,

Thank you again for devoting this kind of creative energy to a tuning
system in a very hands-on way, exploring its nuances and indeed
customizing it to your own very experienced tastes.

In my response, I'll try to keep to this spirit by avoiding too much
mathematical complexity and focusing on the sound of the intervals and
their intended musical effect. Your impressions are fascinating to
read, and come from someone who can illuminate all kinds of practical
angles that my experience simply doesn't cover.

While it appears that our recipes for tuning the _extraordinaire_ may
have varied a bit, as explained below, most of your comments are
equally applicable to my recipe. They neatly highlight some aspects of
the tuning, and questions about its purpose, which I'll try to address
here, and ask for your feedback on how well I'm doing.

I've decided to rewrite my original long response, because I realize
that the biggest issue needing clarification here may be the _purpose_
of the tuning. As will appear below, it is a hybrid circulating system
meant mainly for _new_ music often alternating between or juxtaposing
Renaissance/Manneristic meantone styles in the nearer portion of the
cycle with medieval or especially neomedieval styles using transposed
modes focusing on the remote portion of the circle.

Here's a piece I did in a mostly 16th-century style emphasizing the
nearer portion of the circle, with a few septimal touches:

<http://www.bestII.com/~mschulter/IntradaFLydian.mp3>

Here are some neomedieval pieces and improvisations focusing on the
remote portion of the circle:

<http://www.bestII.com/~mschulter/MMMYear001.mp3>
<http://www.bestII.com/~mschulter/InHoraObservationis.mp3>
<http://www.bestII.com/~mschulter/OElsa.mp3>

Following your admirable lead of giving first place to actual sound
and musical experience, I'm offering this sample of how the
_temperament extraordinaire_ sounds with music designed for or adapted
to it. In some of what follows, I'll refer back to these pieces to
illustrate some of your very well-taken points and explain why I find
certain compromises acceptable or even attractive for my purposes --
but not necessarily for others!

> I set up regular 2/7 all the way around from Eb up to G#. The major
> 3rds are a little bit narrow, the minor 3rds (6:5) scarcely wide.
> The F# to Bb, B to Eb, C# to F, and G# to C work out to hit almost
> on a 9:7, and the F to G#, Bb to C#, Eb to F# are almost on 7:6.

Brad, you've hit on one of the attractions of 2/7-comma: if the idea
is to optimize both 9:7 and 7:6, this is a "sweet spot." A small
clarification: in 2/7-comma, the minor thirds are also "a little bit
narrow" of 6:5, as you have noted the major ones are of 5:4.

> With this installed I played through a bunch of random pages of
> medieval and Renaissance music, mostly vocal, from the _Historical
> Anthology of Music_ (HAM). Margo's point, as I recall, is to set up
> something suitable for accompanying this type of repertoire.

For Renaissance music, I would indeed say that a regular 2/7-comma (or
1/4-comma) is hard to outdo, in comparison to _any_ 12-note
circulating scheme! For medieval music, I'd likewise recommend the
usual period tuning of a regular Pythagorean -- or possibly, for a
discreetly "modern" variation, something like Peppermint where all
regular fifths are a tad more than 2 cents (66 TU) wide of 3:2.

My circle, in comparison, can hardly be ideal if we are aiming
consistently for either type of period sound. As compared with a
regular meantone, some very important pure or near-pure thirds are
compromised; as compared with Pythagorean or even a modern variant
such as the gentle Peppermint, the fifths are impure by 6-7 cents!
This will come out more in your remarks below and my responses.

The purpose the _temperament extraordinaire_, rather, is primarily to
accommodate new music which is free to transpose any mode to any step,
or cadence on any of the 12 fifths, suiting the style to a given
region of the circle -- now Renaissance meantone, now neomedieval,
sometimes with striking transitional colors or "fringe effects."

Actually I _do_ use this tuning for playing historical Renaissance
pieces in the nearer portion of the circle (Bb-G#, with Eb calling for
discretion as discussed below), and like the result -- but a regular
meantone would, as you note, be more idiomatic and consistent both
harmonically and melodically with a period ethos.

> Then I tried her recommendation of converting this to _temperament
> extraordinaire_ (her nomenclature, not mine; I'd already co-opted
> the phrase "extraordinary temperament" to mean something different,
> a couple of years ago...).

Please let me join you in noting the difference between the French
term _temperament extraordinaire_, which I've been using since 2003
</tuning/topicId_43016.html#43016> to
describe a modified 1/4-comma or 2/7-comma meantone with eight narrow
and four equally wide fifths; and "Bach's Extraordinary Temperament,"
the title of your articles of 2005.

I've never thought of translating my French term literally into
English, and your personal trademark now clearly stamped on the phrase
"extraordinary temperament" is another reason not to do so.

> That is: stretch F-Bb-Eb wide each, and F#-C#-G# wide each, so it
> eradicates the wolf G#-Eb and reduces it to four other lesser
> canines, all the same as one another. In practice at harpsichord
> this is simply a process of cranking the Bb and Eb downward in turn,
> and the C# and G# upward in turn, until we have similar 5ths
> F#-C#-G#-D#-A#-E#.

Please let me clarify that my recipe for the _extraordinaire_ is a bit
different in that only three notes are adjusted: Bb, Eb, and G#. The
following diagram using our beloved TU may make this clearer:

F C G D A E B F# C# G# D#/Eb A# F
-189 -189 -189 -189 -189 -189 -189 -189 +198 +198 +198 +198

One aural recipe for use at the harpsichord might be to tune the eight
fifths F-C# in a regular 2/7-comma, thus setting nine of the 12 notes.
Then, _without changing any of these notes_, temper G#, D#/Eb, and Bb
so that the near-9:7 major third or diminished fourth C#/Db-F already
set is derived from a chain of four equally wide fifths.

Specifically, in my recipe, C# is not adjusted, so that A-C# stays a
regular meantone third and C#/Db-F stays at its near-9:7 size. This
changes the details of the compromises involved, but not their basic
nature, which you well describe.

You make a very important point about the wide fifths:

> They're all a little bit rougher than the 2/7 comma 5ths, and of
> course they're rough in the opposite direction, but given that all
> these 5ths/4th are growly anyway they don't sound too bad in
> context...playing them as open 5ths/4ths, without any 3rds (whether
> major or minor) in them.

Thank you for this invaluable aural feedback, which for me is really a
critical point: in a medieval or neomedieval styles with their remote
modal transpositions, those wide fifths are indeed going continually
to be used as stable consonances without any thirds, in this context
definitely unstable intervals.

> But, does it work? I played some more from HAM, some of the same
> compositions and some others randomly. To the extent that the notes
> Bb, Eb, G#, and C# came up, which was rarely, they always stood out
> to me as not fitting their context when played in major or minor
> 3rds. Neither fish nor fowl. Especially in the case of Bb, the
> most frequent of these four notes, it seemed to me there's too much
> difference in character between the Bb major triad and the F major
> triad, and between the A-Bb semitone vs the E-F semitone.

Here I might be curious as to whether these were mainly medieval or
Renaissance pieces from HAM, or some mixture. I'll focus here on a
Renaissance style, since typically with medieval pieces I would make
remote transpositions (e.g. D Dorian to Eb or Bb) that would bring
other intervals and issues into play, some addressed below.

So that people can hear some of the things you discuss in relation
both to altered third sizes and melodic semitones in a Renaissance
meantone kind of style, here's a link repeated for convenience to my
_Intrada in F Lydian_, which prominently uses lots of the accidentals
and intervals you mention. Apart from the matter of C#, which as I
noted is not modified in my recipe, all of your comments apply:

<http://www.bestII.com/~mschulter/IntradaFLydian.mp3>

Thus while usual meantone thirds are at 383 cents, a tad narrow of 5:4
as you have said, Bb-D is around 396 cents, and Bb-D-F often occurs
close to a near-pure sonority such as F-A-C or D-F#-A. While I might
call this "a stimulating touch of modal color," it's definitely a
compromise for a period sound where the uniformly pure or near-pure
thirds of regular meantone are the norm.

Also, the major third Eb-G, used here in the sixth sonority Eb-G-C, is
actually at 408 cents a tad larger than Pythagorean; while I much like
the color of this sonority and also C-Eb-G in a Renaissance as well as
neomedieval context, it is in the former setting a dramatic departure
from the smooth 5:4 sound again offered by a regular meantone.

Your point about the different melodic sizes of A-Bb and E-F is also
right on. While E-F is a regular 2/7-comma semitone at 121 cents, the
altered A-Bb is about 108 cents (comparable to 1/5-1/6 comma), while
Eb-D, featured in some cadences, is 96 cents. One can relish this
variegation, or have more mixed feelings, but it is again a departure
from a regular meantone where all these steps would be 121 cents.

> The Bb-D is scarcely wide as a major 3rd, and pleasant enough, but
> it doesn't have that tight center of gravity like the major 3rds on
> the naturals.

Certainly the different is there -- 384 vs. 396 cents. From a
21st-century perspective, I might argue that in a Renaissance kind of
style with restful thirds, Bb is a less likely modal final than the
naturals (other than B), and so, in a circulating system, can better
afford some compromise. However, that argument is double-edged, as
will appear shortly! In any event, for a period sound, regular
meantone would, of course, leave Bb-D near-pure like the naturals.

> Similarly, A-C# is only a little bit wide but it's lost
> its gravity. And Eb-G and E-G# are watered down even more, each.
> They're still smaller than Pythagorean ditones and they're pleasant
> enough, in isolation, but they don't seem to fit their context.

While A-C# is in my scheme unaltered, what you say about it definitely
could apply to E-G# at 396 cents, the "double edge" of my possible
argument regarding Bb-D, since E _is_ a natural, and the untransposed
final for one of my favorite modes, E Phrygian, which comes up
continually. While some other things are going in the piece which
follows, I'd like people to hear the compromise of closing in Phrygian
on that 396-cent third, which I accept, but wouldn't happen in a
regular 2/7-comma:

<http://www.bestII.com/~mschulter/Invocation-ToneIV.mp3>

> Worse yet: we haven't really gained any usable B-D#, F#-A#, Ab-C, or
> Db-F. They were kind of OK and attractive in their own way (while
> exotic) when they were hitting nearly 9:7, tuned regularly. But
> now, having these modified 5ths, these misspelled major 3rds are in
> a no-man's-land. They just sound like "wrong notes" giving nothing
> for my ear to grab onto, or to take me past the wincing.

Certainly I'd agree, as a septimal fan myself, that having only one
near-9:7 major third at Db-F, rather than four, is a compromise --
although one not so unpleasant for me, since in a neomedieval setting
I also relish thirds around 408 or 421 cents, or anywhere along the
spectrum from Pythagorean to septimal. This can be very much a matter
of taste, with one person's "pleasantly variegated neomedieval thirds"
being another person's "wrong notes."

This more remote terrain in the circle is rather like a distinctively
modern circle of another kind: a 17-tone well-temperament such as
George Secor's of 1978, with major thirds at 418-429 cents. While
Marchettus of Padua (1318) and the modern scholar and performer of
medieval music Christopher Page provide a basis for concluding that
thirds of around this size may have occurred in 14th-century
performances of vocal music, this is above all a 20th-21st century
keyboard concept rather than an historical one!

Why don't I repeat for convenience the links to neomedieval pieces so
that people can hear these thirds at 408 and 421 as well as 434 cents.

<http://www.bestII.com/~mschulter/MMMYear001.mp3>
<http://www.bestII.com/~mschulter/InHoraObservationis.mp3>
<http://www.bestII.com/~mschulter/OElsa.mp3>

It might not hurt for me to emphasize that these thirds are generally
intended for use in a neomedieval context where they contrast with and
often expand to stable fifths; or in a Renaissance context as meantone
diminished fourths (B-Eb, C#-F, F#-Bb, G#-C). Apart from Eb-G at about
408 cents or Pythagorean, which I might use in a meantone setting in
Eb-G-C or C-Eb-G, they are _not_ meant for use in a Renaissance style
as usual thirds.

Also, your remarks point to a basic consequence of circulation. The
extreme values for major thirds, 383 and 434 cents, or very close to
5:4 and 9:7, are identical to those in a regular 2/7-comma. However,
averaging out that Wolf fifth into four wide fifths results also in
the intermediate values of 396, 408, and 421 cents. While I like these
intermediate shades, they inevitably result in fewer major thirds at
either extreme, at a near-just 5:4 or 9:7.

[On your own preferred modification of 2/7-comma]

> C#: back to its regular 2/7 comma place from F#, or tune it as a
> pure 5:4 from A-C#, or tune it as pure 7:6 with Bb, or tune it as
> pure 9:7 from F. Or, simply as a pure 3:2 from F#. All five of
> these spots are arguably good for C#. Following Margo's lead I
> favored the two septimal spots. Why hit only near a 7:6 or a 9:7
> when you can hit directly on one of them? It makes the minor
> triads, diminished triads, and tritones lock in that way, too.

This is very creative: to take my septimal theme, but realize it in a
different way that leaves the tuning quite close to its regular
historical form! I wonder if your having altered C# in your version of
the _extraordinaire_ recipe for circulation might have facilitated
this happy result.

> All of this writes off the G#-Eb wolf as a loss...but it's not going
> to come up in this medieval and Renaissance music anyway.

Here the different, of course, is that for my remote neomedieval
transpositions Ab-Eb is going to come up all the time in Ab Mixolydian
or Eb Dorian or Gb Lydian, etc. Where this kind of circulation isn't
relevant, however, your solution very subtly optimizes the septimal
intervals while leaving the smooth Renaissance meantone color
undisturbed.

> And by going back to regular 2/7, or nearly so (choosing septimal
> placements on C#, G#, and Eb), we've got a fantastic set of eleven
> minor triads, eight tightly attractive major triads, three septimal
> major triads, and all the other related emoluments. The 5ths
> F#-C#-G# and Eb-Bb work out decently (for context) no matter what we
> do, given that the 2/7 comma 5ths on the naturals are already rough.

It's wonderful that your exploration led you to this happy conclusion
bearing your own distinctive stamp!

> If the point is to accompany medieval or Renaissance [vocal] music,
> regular 2/7 works better (at least for me) than the 5th-stretching
> _extraordinaire_ step does. If the point was to get those septimal
> resonances going, why move off them by widening the 5ths?

Another way of phrasing this question: "What price circulation?" If I
wanted to play in a Renaissance style around the circle, it wouldn't
make sense. If something like Ab-C-Eb did occur, say in Schlick or an
extended meantone piece assuming more than 12 notes per octave, having
Ab-C at 421 cents would be just as much a barrier as a Wolf at G#-Eb.

The answer is neomedieval styles in remote transpositions -- at the
cost of fewer septimal resonances: Db-F at a near-9:7 as in the
regular meantone, plus F-Ab and Bb-Db at around 275 cents, or not too
far from 7:6. Now we have a 12-note circle where Bb-Db-F, for example,
can cadence beautifully to Ab-Eb in Ab Dorian or Mixolydian, say. I'm
quite happy with Eb-Gb and C-Eb at 287 cents also, another favorite
size, so it's an amenable compromise to me -- but still a compromise!

In short, from the viewpoint of septimal optimizations, I'd consider
this one of the best 12-note circulating systems -- but not so good as
a regular meantone!

> I could change my mind tomorrow, but that's how I hear it today.
> And all of this might work better (or worse?) on organ as opposed to
> harpsichord. On harpsichord, at least, the septimal intervals are
> clear and pungent.

Again, thank you for your honesty and directness, as well as your
experience and musical sensitivity. You also bring up an interesting
question regarding timbres: do I tend to select timbres on
synthesizer, for example (which I often consider a kind of organ)
where major thirds at 421 cents, say, might have a different effect
than on harpsichord? Anyway, I'm delighted that your septimal
variation fits your instrument well.

> And if the notes D#, A#, Ab, and Db aren't going to come up in the
> music anyway...why not just stick with regular 1/4 instead of 2/7?
> It's sort of a trade-off here between the major 3rds and minor 3rds.
> But if we wanted pure minor 3rds, why not go all the way to 1/3
> comma then? 2/7 goes down between these, giving the strengths of
> both...and because those 3rds aren't pure, they're (arguably) more
> attractive just by virtue of being livelier, not sitting there
> dead-on. Every listener might of course have a different opinion
> about that.

Your appreciation of 2/7-comma may be rather like mine: a certain
balance for major and minor thirds with both gently beating. Also, I
love those just 25:24 minor semitones (kept at C-C# and F-F# in my
_extraordinaire_). They're beautiful in 16th-century chromatic
progressions, and excellent as regular neomedieval semitones, very
close to George Secor's ideal size of around 67 cents.

Please forgive the length of my response, but take it as a measure of
how thoughtful and significant I find your comments.

Most appreciatively,

Margo Schulter
mschulter@calweb.com

> Happily I find that the TU is a good tool to solve this problem, and
> will try to explain my process for solving it in a reasonably
> comprehensible way, I hope.
> (...)
> t = 173

Margo rocks. I've confirmed with my spreadsheet: 173/720 PC meantone
should certainly be close enough to the mark for anybody with human
hearing. Running 173/720 around to all twelve notes, the Eb-F# below
middle C works out to beat less than 0.0052 per second (as a
misspelled 7:6 minor 3rd) based on an A anywhere near 440. The eight
major 3rds are each about -0.53 on Sorge's 11-point metric (where a
Pythagorean ditone is 11). The four diminished 4ths are about 22.07
each, which is a smidge wider than a full diesis of 21.

As for the phrase "extraordinary temperament" in the way I use it,
here's my explanation from more than two years ago:
http://www-personal.umich.edu/~bpl/larips/ordinary.html
I use it to refer to circulating temperaments that happen to have E-G#
wider than Ab-C.

My page about TU and the Neidhardt/Sorge model of 21 schismas in a diesis:
http://www-personal.umich.edu/~bpl/larips/tunits.html

Each stack of three major 3rds has to add up to 21 of those. 1/4
syntonic comma meantone gives major 3rds of 0, 0, 21. 1/4 PC gives
-1, -1, 23. 1/6 PC gives major 3rds of 3, 3, 15. The above-mentioned
septimal thingy of 173/720 PC is about "halfway" between 1/4 SC and
1/4 PC, with its major 3rd sizes of -0.5, -0.5, 22.

In practice on harpsichord that's such a small smidge in the major
3rds that it's hard to hear the impurity. And at A=440, the F under
that A (middle of the treble clef) is narrow to it by only about 1
beat per second. Easier to hear by playing major 10ths instead of
major 3rds.

Brad Lehman

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:
> ...
> As for the phrase "extraordinary temperament" in the way I use it,
> here's my explanation from more than two years ago:
> http://www-personal.umich.edu/~bpl/larips/ordinary.html
> I use it to refer to circulating temperaments that happen to have E-
G#
> wider than Ab-C.
>
> Brad Lehman

Hi Brad,

I haven't been following this thread, but I just happened to read
this because "Margo rocks" caught my eye in the message listing
summary.

We may have a terminology conflict brewing. I first used the term
_temperament extraordinaire_ a couple of years ago to refer to a 12-
tone circulating temperament containing 2 or 3 fifths tempered wide
by a small amount (<4 cents):
/tuning/topicId_59689.html#59999
That was posted at the same time as the Lehman-Bach temperament was
being discussed on this list, and I don't know whether you were using
the term "extraordinary" at the time. (You needn't read the entire
message, because I've summarized its essential characteristics below.)

My original temperament was subsequently refined into one of a group
of rationally-defined proportional-beating temperaments (both well
and _extraordinaire_), which are listed here:
/tuning/topicId_59689.html#66222
The updated original (and still best of the lot, IMO) is my 5/23-
comma _temperament extraordinaire_ (which is probably not audibly
different from the original version).

A key feature of the _extraordinaire_ (as I use the term) is the <1/4-
comma tempering of the narrowest fifths. In the 5/23-comma version,
the fifths in the root-position C and G major triads beat at the same
rate as the major 3rds, which hides the beating of the major 3rds --
so it sounds as if the major 3rds are just (as in 1/4-comma
meantone), while the narrowest 5ths (tempered <4.7 cents) beat more
slowly than in 1/4-comma meantone. (The minor 3rds, which beat
exactly 4 times as fast as the 5ths, are also masked to some degree
by the beating 5ths.) Since the 5ths are tempered less than 1/4
comma on the near side of the circle, there will be less error on the
far side of the circle than with a comparable temperament having
narrow 1/4-comma 5ths. In a nutshell, that's what makes it
_extraordinaire_.

However, in searching through back messages for "extraordinaire", I
find that Margo may have used the term _extraordinaire_ for a
temperament earlier than both of us (March 2003):
/tuning/topicId_42903.html#42903
She indicates that she likes major and minor 3rds that approximate
11:14 and 11:13, respectively. The narrowest minor 3rds in my 5/23-
comma temperament are all close to 11:13 (without getting
significantly narrower), while the widest major 3rds are a few cents
less than 11:14, so I may have met at least some of her objectives.
In the last paragraph she mentioned a temperament of mine (of the
_ordinaire_ class) that I previously suggested (in a personal
communication a couple of years earlier) would be suitable for
medieval music, if the far side of the circle of 5ths were used.
That particular temperament has since been replaced by my 5/23-comma
_extraordinaire_, so I believe we were thinking along the same lines
when we (independently!) appropriated the term .

--George

Dear George and Brad,

> Hi Brad,
> I haven't been following this thread, but I just happened to read
> this because "Margo rocks" caught my eye in the message listing
> summary.

Please let me say what a pleasure it is for us to be in touch,
George. Today, before I saw your message, I was reading over your
_Xenharmonikon_ 18 article, which brought me great pleasure and a bit
of nostalgia for old times.

> We may have a terminology conflict brewing. I first used the term
> _temperament extraordinaire_ a couple of years ago to refer to a 12-
> tone circulating temperament containing 2 or 3 fifths tempered wide
> by a small amount (<4 cents):
> </tuning/topicId_59689.html#59999>

Yes, one lesson of this may be that the phrase _temperament ordinaire_
naturally prompts the idea _temperament extraordinaire_. Maybe the
ways we've each used it might give curious clues about us <grin>.

Brad and George, I might say that for each of you, the aspect of the
tuning that makes it _extraordinaire_ might be rather subtle.

Brad, I very much enjoy your explanation of how the essence of the
"extraordinary" in your usage is having E-G#, traditionally a regular
meantone major third, smaller than Ab-C, traditionally a diminished
fourth -- a heritage reflected in many well-temperaments, and also in
the French _ordinaire_ where the fifths tempered wide tend to be on
the flat side (e.g. Ab-Eb-Bb-F in a very simple modification of
regular 1/5-comma, say). As developed in your reconstruction of Bach's
temperament, the idea that E-G# should be larger than Ab-C has some
fascinating ramifications.

George, while any tuning system you design is likely to be
"extraordinary," your _extraordinaire_ seems especially so in the
matter of coordinated beat rates. Again, this is a very subtle but
musically significant detail, a mark of craftspersonship possibly more
notable for its depth than its flamboyance.

In contrast, my _extraordinaire_ seems rather less refined. Take a
regular 1/4-comma or 2/7-comma meantone in 12 notes (Eb-G#), and
simply distribute the Wolf equally over four wide fifths, leaving F-C#
as is. The _extraordinaire_ aspect is not some fine nuance in the
gradation of sharps and flats, or some pleasant synchrony of beat
rates, but the sheer "colorfulness" of the thing: major thirds at
386-427 cents and minor thirds at 279-310 cents in 1/4-comma; and
respective ranges of 383-434 cents and 275-313 cents in 2/7-comma.

Of course, in March 2003 or now, to get excited about something like
this you have to a certain curious sensibility: for example,
applauding interval sizes like 421 cents as "delicious." But the idea
of my "extraordinaire" is mostly having a full spectrum of thirds
running most if not all of the way from 5-limit through Pythagorean
through 14:11 or 13:11 country and beyond to septimal.

> </tuning/topicId_42903.html#42903

> She indicates that she likes major and minor 3rds that approximate
> 11:14 and 11:13, respectively. The narrowest minor 3rds in my 5/23-
> comma temperament are all close to 11:13 (without getting
> significantly narrower), while the widest major 3rds are a few
> cents less than 11:14, so I may have met at least some of her
> objectives.

Yes, I'd agree that our systems share this connecting theme. A
distinction might be that yours does this as part of a very subtle
balance within the circle. In contrast, my approach is to seek a
spectrum reaching out into at least the "suburbs" of the septimal
realm, as reflected in my remarks in that same post about
approximations of 4:6:7:9 -- but without the kind of nuances and
careful planning that might be the mark of a more discriminating
designer.

> In the last paragraph she mentioned a temperament of mine (of the
> _ordinaire_ class) that I previously suggested (in a personal
> communication a couple of years earlier) would be suitable for
> medieval music, if the far side of the circle of 5ths were used.
> That particular temperament has since been replaced by my 5/23-comma
> _extraordinaire_, so I believe we were thinking along the same lines
> when we (independently!) appropriated the term .

Yes, your comments in our correspondence of 2001-2002 about that
earlier temperament were germinal to my 1/4-comma and 2/7-comma
variations. Even a regular meantone, of course, can demonstrate the
basic idea, as when in 1998 or 1999 I not surprisingly found myself at
a 1/4-comma keyboard playing

Eb E Eb F
Bb B Bb C
F# E or F# F

Your communication as part of our discussion as a "Committee of
Correspondence for the 17-tone revolution," however, was what
presented me with the realization that this kind of thing could be the
basis for a 12-note circulating system.

Anyway, I can offer my part of a solution to "disambiguate" things a
bit on "extraordinary" or _extraordinaire_ tunings. Since the
distinctive thing to me about my modified 1/4-comma and 2/7-comma
tunings is the "encompassing spectrum" of thirds from 5-limit to
near-septimal in a 12-note circle, I'm drawn to the names:

Quest: QUarter-comma Encompassing Spectrum Tuning (based on 1/4-comma)
Zest: Zarlino Encompassing Spectrum Tuning (based on 2/7-comma)

How does that sound, George and Brad?

Or, maybe I could still let the "e" in Quest or Zest stand for
"Extraordinaire," but with an understanding that I would use Quest
or Zest or whatever as the specific name for mine, George, so that
"temperament extraordinaire" or TX could remain a more distinctive
name for yours while the common roots would still be explicit. That
might also be influenced by how the two of you approach your own
usages of "extraordinare" and "extraordinary."

By the way, Brad, I must say that your dialogue with me about your
2/7-comma harpsichord experiments has been most educational for me,
and makes me resolve to strive for accounts of these spectrum tunings
as lucid as yours about "extraordinary" ones. In fact, reading your
explanation on "ordinary" and "extraordinary" has given me some ideas
I might share here.

Thanks to both of you as teachers and people who make things more fun.

Most appreciatively,

Margo

Thanks to Margo for her persistence in arithmetic. However, one should
note two things:

In order to start off the whole calculation, one had to look up the
cent value of 64:63, and then convert it to TU via some rather
unmemorable string of decimals.

At the end of the calculation the value of the fifth, and the
resulting meantone scale, are again presented in cents, using the same
ugly conversion factor in reverse.

So yes one can use TU, but they seem here to be completely equivalent
to doing the calculations in cents with two extra steps to convert
both ways. Like measuring things in centimetres, then calculating in
hundredths of an inch, then converting back to centimetres to test the
calculation.

The point is, can one determine 64:63 *directly* in TU without any
help from cents?

There must be a formula, but I expect it's pretty complicated. To deal
with it at all conveniently one would have to memorize the fraction
that defines the Pythagorean comma.

Cents is easy enough: 1200*log(64/63)/log(2).

~~~T~~~

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> we know that the net tempering of those three fifths has to be
> equal to the septimal comma or Comma of Archytas (thanks to George
> Secor for the latter name!) at 64:63, 27.26 cents, or 836.75 TU.

(...)

> t = 173
>
> This seems intuitively a reasonable result, since we know that
> 1/4-comma at 165 TU comes very close, but with F-G# a bit larger than
> a just 7:6. This is a regular fifth at about 696.32 cents, or 5.64
> cents wide, as an approximation based on 837 TU for the septimal
> comma.
>
> It would be more accurate to do the same calculation with a value of
> 836.75 TU for this comma. Instead of dividing 1557 by 9, we divide
> 1556.75 by 9 -- otherwise, the process is the same. We get about
> 172.972 TU, at least according to my Orpie calculator program.
>
> The result is a meantone fifth of about 696.319 cents, or 5.636 cents
> narrow of 3:2. Scala shows the 7:6 minor third or augmented second
> accurate to 0.001 cents.
>

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

>
> My original temperament was subsequently refined into one of a group
> of rationally-defined proportional-beating temperaments (both well
> and _extraordinaire_), which are listed here:
> /tuning/topicId_59689.html#66222
> The updated original (and still best of the lot, IMO) is my 5/23-
> comma _temperament extraordinaire_

Mmm, interesting, but I have to say I wonder about one thing; if you
are going to "rationalize," why not go whole-hog a truly rationalize,
instead of using a bunch of fifths ostensibly similarly tempered but
all slightly different in sizes?

! Secor5_23TX.scl
!
George Secor's rational 5/23-comma temperament extraordinaire
12
!
390/371
3321/2968
4397/3710
929/742
2476/1855
8325/5936
555/371
9365/5936
621/371
9904/5565
5559/2968
2/1

Which works out to be:

C-G 1,4959568733154 or 697,2823 cents
G-D 1,4959459459460 or 697,2697
D-A 1,4959349593496 or 697,2569
A-E 1,4959742351047 or 697,3024
E-B 1,4959634015070 or 697,2899

When one size fifth of 1,4959535062432 or 697,2784 cents would have
done the job quite well. And if part of your definition of
"rationalization" includes the use rational numbers instead of a more
precise method of representation, then it would have been 7209/4819 or
27542/18411, or even 868220/580379... though I don't see the point in
this archaic methodology. Rather like doing dental surgery with a shot
of whiskey, a pair of pliers and a soldering iron.

> A key feature of the _extraordinaire_ (as I use the term) is the <1/4-
> comma tempering of the narrowest fifths. In the 5/23-comma version,
> the fifths in the root-position C and G major triads beat at the same
> rate as the major 3rds, which hides the beating of the major 3rds --
> so it sounds as if the major 3rds are just (as in 1/4-comma
> meantone), while the narrowest 5ths (tempered <4.7 cents) beat more
> slowly than in 1/4-comma meantone.

Umm, I tried this temperament this morning in Absynth using both
Goerge's values and re-rationalized values with constant fifths, with
both a regal sound and a sampled Silbermann open diapason, and I don't
find that there is any "masking" to speak of. I still hear both the
thirds and the fifths beating, and the thirds certainly do NOT sound
like the pure thirds of straight-up meantone. Granted, the fifths
don't beat as fast, but I think the claims about the thirds are inflated.

> In a nutshell, that's what makes it
> _extraordinaire_.

What?! All this mucking about with numbers just for TWO major chords
that display this supposedly extra-ordinary characteristic, and only
when played in root position? Oh well, I guess it IS the era of
reduced expectations...

:-(

Ciao,

P

Hi Gene, Margo, and Brad,

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

/tuning/topicId_73833.html#74175?var=0&l=1

> Please let me clarify that my recipe for the
> _extraordinaire_ is a bit different in that only
> three notes are adjusted: Bb, Eb, and G#. The
> following diagram using our beloved TU may make
> this clearer:
>
F C G D A E B F# C# G# D#/Eb A# F
-189 -189 -189 -189 -189 -189 -189 -189 +198 +198 +198 +198
>
> One aural recipe for use at the harpsichord might be
> to tune the eight fifths F-C# in a regular 2/7-comma,
> thus setting nine of the 12 notes. Then, _without changing
> any of these notes_, temper G#, D#/Eb, and Bb so
> that the near-9:7 major third or diminished fourth
> C#/Db-F already set is derived from a chain of four
> equally wide fifths.
>
> Specifically, in my recipe, C# is not adjusted, so that
> A-C# stays a regular meantone third and C#/Db-F stays
> at its near-9:7 size. This changes the details of the
> compromises involved, but not their basic nature, which
> you well describe.

/tuning/topicId_73833.html#74180

> Anyway, I can offer my part of a solution to "disambiguate"
> things a bit on "extraordinary" or _extraordinaire_ tunings.
> Since the distinctive thing to me about my modified 1/4-comma
> and 2/7-comma tunings is the "encompassing spectrum" of
> thirds from 5-limit to near-septimal in a 12-note circle,
> I'm drawn to the names:
>
> Quest: QUarter-comma Encompassing Spectrum Tuning
> (based on 1/4-comma)
> Zest: Zarlino Encompassing Spectrum Tuning
> (based on 2/7-comma)

Margo, i think those names are great. I'd like to make
entries for these tunings for the Encyclopedia. If
you'd be willing to write up the text for me, that
would be great ... otherwise i can put the pages
together from your tuning list posts.

Gene, i'm trying to figure out how to make a Tonescape
.tuning file for the Zest tuning, but could use some
help. The chain-of-5ths which are -189 TUs are clearly
tempered from 5-limit JI by 1/7, 2/7, or 3/7 of a
syntonic-comma.

But i'm not sure how to set a generator for the 5ths
which are +198 TUs. Margo stipulates that the interval
C#/Db:F is close to 9/7, but since they are already both
embedded in 3,5-space via the 2/7-comma tempering, i
can't figure out how to set up the 7-space component.

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com
Tonescape microtonal music software

Hi Gene,

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Gene, Margo, and Brad,
>
>
> --- In tuning@yahoogroups.com, Margo Schulter <mschulter@> wrote:
>
> /tuning/topicId_73833.html#74175?var=0&l=1
>
> > Please let me clarify that my recipe for the
> > _extraordinaire_ is a bit different in that only
> > three notes are adjusted: Bb, Eb, and G#. The
> > following diagram using our beloved TU may make
> > this clearer:
> >
F C G D A E B F# C# G# D#/Eb A# F
-189 -189 -189 -189 -189 -189 -189 -189 +198 +198 +198 +198
> >
> > One aural recipe for use at the harpsichord might be
> > to tune the eight fifths F-C# in a regular 2/7-comma,
> > thus setting nine of the 12 notes. Then, _without changing
> > any of these notes_, temper G#, D#/Eb, and Bb so
> > that the near-9:7 major third or diminished fourth
> > C#/Db-F already set is derived from a chain of four
> > equally wide fifths.
>
> <snip>
>
> Gene, i'm trying to figure out how to make a Tonescape
> .tuning file for the Zest tuning, but could use some
> help. The chain-of-5ths which are -189 TUs are clearly
> tempered from 5-limit JI by 1/7, 2/7, or 3/7 of a
> syntonic-comma.
>
> But i'm not sure how to set a generator for the 5ths
> which are +198 TUs. Margo stipulates that the interval
> C#/Db:F is close to 9/7, but since they are already both
> embedded in 3,5-space via the 2/7-comma tempering, i
> can't figure out how to set up the 7-space component.

Specifically, i'm approaching this two different ways
in Tonescape:

1) a .tonespace file which sets up 7-limit JI and then
tempers it via 2/7-syntonic-comma and whatever the
other unison-vector should be (i'm guessing that it's
some fraction of 64:63);

2) a .tuning file which uses two tempered generators:
one for the -189 TU chain-of-5ths, and another one
for the +198 TU chain. This one is easy enough for
me to do myself -- it's #1 that i need help with.

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com
Tonescape microtonal music software

Hi Gene,

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> Specifically, i'm approaching this two different ways
> in Tonescape:
>
> 1) a .tonespace file which sets up 7-limit JI and then
> tempers it via 2/7-syntonic-comma and whatever the
> other unison-vector should be (i'm guessing that it's
> some fraction of 64:63);
>
> 2) a .tuning file which uses two tempered generators:
> one for the -189 TU chain-of-5ths, and another one
> for the +198 TU chain. This one is easy enough for
> me to do myself -- it's #1 that i need help with.

Actually, i'd appreciate whatever you have to say about
#2 as well. When using tempered generators, there's
always some ambiguity about how chains of different
generators intersect.

In this case, i've used "A" as the origin, and the
2/7-comma chain extends from "F" at -5 to "C#" at +4,
but the other chain can extend from either of those
endpoints, and in fact is supposed to connect both
of them ... but i can only do that with #1.

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com
Tonescape microtonal music software

Hi Gene,

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> Actually, i'd appreciate whatever you have to say about
> #2 as well. When using tempered generators, there's
> always some ambiguity about how chains of different
> generators intersect.
>
> In this case, i've used "A" as the origin, and the
> 2/7-comma chain extends from "F" at -5 to "C#" at +4,
> but the other chain can extend from either of those
> endpoints, and in fact is supposed to connect both
> of them ... but i can only do that with #1.

Duh ... i guess i had a brain fart. It wasn't that
difficult to figure this out after all: since it's
a 12-tone tuning intended to have the same structure
as many historical 12-tone tunings, it's obvious that
there should be two small +198 TU chains, one extending
in the negative direction from F to give Bb and Eb,
and the other extending in the positive direction from
C# to give G#. D#/Eb is the unison-vector.

But i do still need help figuring out how to incorporate
7-space into #1 (the .tonespace file).

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
>
> >
> > My original temperament was subsequently refined into one of a
group
> > of rationally-defined proportional-beating temperaments (both
well
> > and _extraordinaire_), which are listed here:
> > /tuning/topicId_59689.html#66222
> > The updated original (and still best of the lot, IMO) is my 5/23-
> > comma _temperament extraordinaire_
>
> Mmm, interesting, but I have to say I wonder about one thing; if you
> are going to "rationalize," why not go whole-hog a truly
rationalize,
> instead of using a bunch of fifths ostensibly similarly tempered but
> all slightly different in sizes?

If you make all of the narrowest fifths exactly the same size (as you
suggest below), you've *irrationalized* them. That's the way I
started out, in fact, but I soon discovered that the beating in the
*minor* triads was not exactly proportional.

> ! Secor5_23TX.scl
> !
> George Secor's rational 5/23-comma temperament extraordinaire
> 12
> !
> 390/371
> 3321/2968
> 4397/3710
> 929/742
> 2476/1855
> 8325/5936
> 555/371
> 9365/5936
> 621/371
> 9904/5565
> 5559/2968
> 2/1
>
> Which works out to be:
>
> C-G 1,4959568733154 or 697,2823 cents
> G-D 1,4959459459460 or 697,2697
> D-A 1,4959349593496 or 697,2569
> A-E 1,4959742351047 or 697,3024
> E-B 1,4959634015070 or 697,2899
>
> When one size fifth of 1,4959535062432 or 697,2784 cents would have
> done the job quite well.

In my brief explanation I didn't point out that other major triads
(from Eb to E) and many of the minor triads also have *exact*
proportional beat rates. This is not the case with a single size
fifth, in which they're approximate. Exact proportional beat rates
tend to make the beating sound more unified (or periodic), like a
vibrato, as if the instrument is "singing". With non-proportional
beating, the interactions between the differing beat rates sounds
more disturbing.

And if part of your definition of
> "rationalization" includes the use rational numbers instead of a
more
> precise method of representation, then it would have been 7209/4819
or
> 27542/18411, or even 868220/580379... though I don't see the point
in
> this archaic methodology. Rather like doing dental surgery with a
shot
> of whiskey, a pair of pliers and a soldering iron.

The rational numbers are precise, i.e., they result in *exact*
proportional beat rates within the various triads. The rational
approximations to the irrational number you suggested are inexact.

> > A key feature of the _extraordinaire_ (as I use the term) is the
<1/4-
> > comma tempering of the narrowest fifths. In the 5/23-comma
version,
> > the fifths in the root-position C and G major triads beat at the
same
> > rate as the major 3rds, which hides the beating of the major
3rds --
> > so it sounds as if the major 3rds are just (as in 1/4-comma
> > meantone), while the narrowest 5ths (tempered <4.7 cents) beat
more
> > slowly than in 1/4-comma meantone.
>
> Umm, I tried this temperament this morning in Absynth using both
> Goerge's values and re-rationalized values with constant fifths,
with
> both a regal sound and a sampled Silbermann open diapason, and I
don't
> find that there is any "masking" to speak of. I still hear both the
> thirds and the fifths beating, and the thirds certainly do NOT sound
> like the pure thirds of straight-up meantone. Granted, the fifths
> don't beat as fast, but I think the claims about the thirds are
inflated.

The thirds beat if they are sounded separately, but within a triad
they're masked by the beating fifths, because the beat rate of both
is the same. I think it's preferable to have both intervals beat at
the same rate, as opposed to one beatless and the other beating,
because the contrast in beating vs. beatless intervals tends to make
the beating more conspicuous.

You need to compare the C and G triads in the ~5/23-comma rational
temperament (not an oxymoron!) with 1/4-comma meantone temperament
(which also has proportional-beating triads, BTW). The question to
ask is whether one sounds better than the other. If the difference
is judged to be very small or insignificant, then the rational
temperament will be a better choice as a starting point for a
circulating temperament, because there will be less total error on
the far side of the circle of fifths.

> > In a nutshell, that's what makes it
> > _extraordinaire_.
>
> What?! All this mucking about with numbers just for TWO major chords
> that display this supposedly extra-ordinary characteristic, and only
> when played in root position?
>

No, and no. The major-3rd-to-fifth beating for those two triads is
1:1 in all (close) triad positions, except that you need to
substitute the fourth for the fifth and (for the 1st inversion) the
minor 6th for the major 3rd.

In the other major triads the proportions are different, yet still
fairly simple; e.g., the ratio of the minor-3rd-to-major-3rd rates is
2:1 in the E and F major triads and 3:1 in D major. In the minor
triads the proportions are not quite as simple, but the effect is
still better than if the ratios between the rates are unplanned
(i.e., haphazard).

> Oh well, I guess it IS the era of
> reduced expectations...
>
> :-(

Or an era of increasing explanations. :-)

--George

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:
>

>
> If you make all of the narrowest fifths exactly the same size (as you
> suggest below), you've *irrationalized* them. That's the way I
> started out, in fact, but I soon discovered that the beating in the
> *minor* triads was not exactly proportional.

>
> In my brief explanation I didn't point out that other major triads
> (from Eb to E) and many of the minor triads also have *exact*
> proportional beat rates. This is not the case with a single size
> fifth, in which they're approximate. Exact proportional beat rates
> tend to make the beating sound more unified (or periodic), like a
> vibrato, as if the instrument is "singing". With non-proportional
> beating, the interactions between the differing beat rates sounds
> more disturbing.

Well, I have to admit that after a half hour of staring at my large
monitor (19" wide) and my big chart which graphically shows the beat
rates of every major and minor triad over two octaves (thirds and
fifths), I haven't got the foggiest notion what you are on about. When
I switch back and forth between the regular-sized fifth version and
your irregular sized fifth version, the movement of the traces is so
small that it can hardly be seen (full scale is 25 b/s, so we're
talking about 6mm or so for every 1 b/s - pretty darn easy to see
changes that would be nigh to impossible to hear). If I set the trace
value indicators to 2 decimal places, it also tells me that the change
only affects a very few triads, and the largest change is in the area
of than 0,01 b/s. That means you are going to have to hold a triad
for about 100 seconds while you do nothing else musically to distract
the ear from the extremely subtle gradual drift of phase between third
and fifth or third and third beat rates. Do you REALLY think this is
an advantage?

Just wonderin'...

Ciao,

p

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> So yes one can use TU, but they seem here to be completely
equivalent
> to doing the calculations in cents with two extra steps to convert
> both ways.

If the TU is simply a logarithmic measure, as Monz tells me it is and
as Margo's calculation assumes, yes. I was under the impression the
definition was a completely different one, in which case, however,
the calculation could not be done at all.

> The point is, can one determine 64:63 *directly* in TU without any
> help from cents?
>
> There must be a formula, but I expect it's pretty complicated.

If something is a logarithm, it's a logarithm and that's all here is
to it. If you want a system which gives integer values for 7-limit
intervals and which tempers out the atom, those can be found.

Hi Tom and Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
>
> > So yes one can use TU, but they seem here to be
> > completely equivalent to doing the calculations
> > in cents with two extra steps to convert
> > both ways.
>
> If the TU is simply a logarithmic measure, as Monz
> tells me it is and as Margo's calculation assumes, yes.
> I was under the impression the definition was a
> completely different one, in which case, however,
> the calculation could not be done at all.

I always understood the TU to be defined as a logarithmic
1/720 of a pythagorean-comma. Perhaps there is a more
ambiguous definition, but i don't know about it.

> > The point is, can one determine 64:63 *directly*
> > in TU without any help from cents?
> >
> > There must be a formula, but I expect it's
> > pretty complicated.
>
> If something is a logarithm, it's a logarithm and
> that's all here is to it. If you want a system which
> gives integer values for 7-limit intervals and which
> tempers out the atom, those can be found.

It shouldn't be any more complicated than calculating
cents, because it's the same operation. The only
complicating aspect is that you have to take the log
to base-pythagorean-comma instead of base-2.

However, when i actually did the calculation, i got
the wrong answer. So maybe Gene can help.

Here's how i did it (wrong) in Open Office Calc:

=log(64/63;pythagorean_comma)*720

where "pythagorean_comma" is a reference to a cell
which contains =(2^-19)*(3^12).

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> Here's how i did it (wrong) in Open Office Calc:
>
> =log(64/63;pythagorean_comma)*720
>
> where "pythagorean_comma" is a reference to a cell
> which contains =(2^-19)*(3^12).

Looks like it ought to work. What was your numberical result? It should
be 720 log(64/63)/log(P), where P is the Pythagorean comma and log is
log base anything. As Margo obtained, the result is ~ 836.75

By the way, if you want to temper out the atom and get 7-limit
intervals accurately, 11664, 21480 and 33144 are all recommendable. If
all three commas are supposed to have nice divisibility properties,
that adds another twist, but for instance you can assign 1728 to the
Pythagorean, 1584 to the Didymus, and 2008 to the septimal commas by
using 88380-et. 21480 has 420, 385, 488, and etcetera.

OK, my mathematical training tells me that 64/63 in TU should be

720 * log(64/63) / log(531441/524288)

compare cents :

1200 * log(64/63) / log(2).

The TU formula requires eleven extra characters, which is why it is
unlikely to catch on. Can't help with Open Office, I'm afraid.

My main objection to TU is that they are absurdly and spuriously tiny
for any realistic acoustic situation. What is the point of an
organbuilder using a unit which is probably at least ten times smaller
than any interval he could reliably or usefully adjust his temperament
by? Thinking in TU carries a great risk of attributing a
pseudo-scientific precision to what is, in reality, the rather messy,
unpredictable and artistic process of actually tuning organ pipes.

Didn't Brad tell us not too long ago about tuning a small organ for
continuo accompaniment ... one of the remarks there was that the
reading on an electronic pitch meter was chronically unstable, and
thus useless for calibration or setting temperament, and this was a
normal situation for certain types of register. Also, I know that if
one tunes the pipe with the lid of the portative organ off, then
replaces the lid, some relevant intervals may change noticeably -
probably by dozens of TU. So what use are TU there? Can one ever hear
or control *one* TU?

Cents have probably stayed in use partly because they do correspond
quite closely to the minimum audible difference between intervals in
some relevant situations - for example tuning fifths on a keyboard
instrument.

By the way, my solution was as follows. Nine fifths F-G# have to have
a total (flatwards) tempering of one comma plus 64/63. In my
third-(syntonic)-comma / septimal scheme that is seven steps, since
64/63 is represented by 4/3 comma. Hence the tempering of each fifth
is one-ninth of 7/3 comma, ie 7/27 (syntonic) comma. So far, not a
single logarithm or decimal in sight... This then comes to 5.58 cents,
which is about 0.05 cents less than the exact result. My error on 7:6
is about half a cent. As a musician: forget it.

The next exercise: Take the intervals 5:4, 6:5, 7:6 and 9:7, along
with their meantone approximations via a chain of 4, 3, 9 and 8 of the
regular fifths respectively. What meantone minimises the maximum error
of these intervals?

My nearly-exact method gives, relatively simply, 5/18-comma, which is
interesting given that Smith didn't care about septimal intervals. The
four thirds are tempered by about 1/9, 1/6, 1/6 and 1/9 comma
respectively.

Has there been some systematic study of how to find the meantone that
minimizes the maximum error on some given collection of intervals?

~~~T~~~

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
>
> Hi Tom and Gene,
>
>
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@>
> wrote:
>
> I always understood the TU to be defined as a logarithmic
> 1/720 of a pythagorean-comma. Perhaps there is a more
> ambiguous definition, but i don't know about it.
>
>
> > > The point is, can one determine 64:63 *directly*
> > > in TU without any help from cents?
> > >
> > > There must be a formula, but I expect it's
> > > pretty complicated.
> >
> > If something is a logarithm, it's a logarithm and
> > that's all here is to it. If you want a system which
> > gives integer values for 7-limit intervals and which
> > tempers out the atom, those can be found.
>
>
> It shouldn't be any more complicated than calculating
> cents, because it's the same operation. The only
> complicating aspect is that you have to take the log
> to base-pythagorean-comma instead of base-2.
>
> However, when i actually did the calculation, i got
> the wrong answer. So maybe Gene can help.
>
> Here's how i did it (wrong) in Open Office Calc:
>
> =log(64/63;pythagorean_comma)*720
>
> where "pythagorean_comma" is a reference to a cell
> which contains =(2^-19)*(3^12).
>

>> Quest: QUarter-comma Encompassing Spectrum Tuning
>> (based on 1/4-comma)
>> Zest: Zarlino Encompassing Spectrum Tuning
>> (based on 2/7-comma)

> Margo, i think those names are great. I'd like to make entries for
> these tunings for the Encyclopedia. If you'd be willing to write up
> the text for me, that would be great ... otherwise i can put the
> pages together from your tuning list posts.

Thank you for this feedback, and I will be delighted to write entries
for these tunings in their 12-note and 24-note versions (the latter
with two circles at the distance of the regular meantone's enharmonic
diesis). At the conclusion of this message, I'll provide some Scala
files, although I'm not sure about the best representations for your
Tonescape software.

Please let me emphasize that your editorial ideas would be invaluable,
because my desire is above all to communicate some real-world musical
qualities of these tunings. Brad's post about his impressions of
2/7-comma variants is very helpful, and of course he and anyone else
interested would be welcome to join in the dialogue.

While I'll write some drafts for these texts, I feel that it would
best be an active collaboration, with you as an especially logical
participant because you know the style of the Encyclopedia. I want
something that people coming from different backgrounds can
understand. They may or may not like the tunings, but a good
presentation should help in understanding some possible reasons why or
why not.

In the meantime, why don't I give a few links, one discussing what I
then called the "Temperament extraordinaire" but should now be called
the "Encompassing Spectrum Temperament" in 12 notes:

<http://www.bestII.com/~mschulter/TE1.txt>

The other links deal with some specific characteristics of Zest-24,
including a lattice diagram that you might enjoy:

<http://www.bestII.com/~mschulter/zest24-lattice.txt>
<http://www.bestII.com/~mschulter/zest24-septendecene.txt>
<http://www.bestII.com/~mschulter/zest24-RastBayyati.txt>

Also, in the next few days I'll be putting up some musical examples,
especially of intervals approximating David Keenan's Noble Intonation
(NI) as determined by phi-weighted mediants. This might be an
interesting link to go with the page.

A related project I'm concering is a FAQ on some of these
temperaments, with Brad's pages and posts a helpful guide to some
cardinal points.

However, writing -- with some editorial advice here, I hope -- some
pages for the Encyclopedia seems one fine place to start,

With many thanks,

Margo

! quest12.scl
!
QUarter-comma Encompassing Spectrum Tuning (F-C# 1/4-c, other 5ths ~4.888c wide)
12
!
76.04900
193.15686
289.73529
5/4
503.42157
579.47057
696.57843
782.89214
889.73529
996.57843
1082.89214
2/1

! quest24.scl
!
Two circles of Quest-12 at 128:125 diesis (~41.059c) apart
24
!
128/125
76.04900
117.10786
193.15686
234.21572
289.73529
330.79415
5/4
32/25
503.42157
544.48043
579.47057
620.52943
696.57843
737.63729
782.89214
823.95100
889.73529
930.79415
996.57843
1037.63729
1082.89214
1123.95100
2/1

! zest12.scl
!
Zarlino Encompassing Spectrum Tuning, F-C# 2/7-comma, other 5ths ~6.424c wide
12
!
25/24
191.62069
287.43104
383.24139
504.18965
574.86208
695.81035
779.05173
887.43104
995.81035
1079.05173
2/1

! zest24.scl
!
Zarlino Encompassing Spectrum Tuning (two circles at ~50.276c apart)
24
!
50.27584
25/24
120.94826
191.62069
241.89653
287.43104
337.70688
383.24139
433.51722
504.18965
554.46549
574.86208
625.13792
695.81035
746.08619
779.05173
829.32757
887.43104
937.70688
995.81035
1046.08619
1079.05173
48/25
2/1

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

>
> My main objection to TU is that they are absurdly and spuriously tiny
> for any realistic acoustic situation.

Amen to that! That's why I gave it a skip long ago when somebody,
probably Brad, brought it up on the harpsichord list. I just didn't
see the point. As far as a Pythag/Sintonic comma eq, the old 11/12
rule is good enuf for jazz, and then some.

> What is the point of an
> organbuilder using a unit which is probably at least ten times smaller
> than any interval he could reliably or usefully adjust his temperament
> by?

None whatsoever.

> Thinking in TU carries a great risk of attributing a
> pseudo-scientific precision to what is, in reality, the rather messy,
> unpredictable and artistic process of actually tuning organ pipes.

You can say that again, in spades!
>
> Didn't Brad tell us not too long ago about tuning a small organ for
> continuo accompaniment ... one of the remarks there was that the
> reading on an electronic pitch meter was chronically unstable, and
> thus useless for calibration or setting temperament, and this was a
> normal situation for certain types of register.

Depends on a many different factors, far too many to list here, but
yeah, organ pipes are notoriously unstable in any sort of super
precise terms. You don't need any electronic device to see it, either.
Just compare the sound of any pipe to an electronically generated tone
(with a reasonable amount of harmonics) by listening to the pipe
through a pair of non-isolating headphones. The slight wavers produced
by random phase shifting is readily audible.

Furthermore, don't forget that the frequency of a pipe changes by
about 3 cents for every degree C of temperature! So you'd better move
fast when tuning an organ. See:

http://hyperphysics.phy-astr.gsu.edu/hbase/waves/opecol.html

BTW, I don't know what sort of "electronic pitch meter" Brad uses, but
if it is a Korg, forget it! Those things are utterly worthless,
inherently unstable, and so poorly programmed that even the octaves
are often out of tune.

> Also, I know that if
> one tunes the pipe with the lid of the portative organ off, then
> replaces the lid, some relevant intervals may change noticeably -
> probably by dozens of TU.

A common problem, due to the lid shading the pipe. Worse yet, in
portatives, the pipes are packed so close to one another that they
interact acoustically, so tuning one pipe can mess up the tuning of
another already tuned, merely because the process changes the
impedances of the ambient field.

> So what use are TU there? Can one ever hear
> or control *one* TU?

How many angels can dance on a pinpoint?
>
> Cents have probably stayed in use partly because they do correspond
> quite closely to the minimum audible difference between intervals in
> some relevant situations - for example tuning fifths on a keyboard
> instrument.

Yeah, there is a pretty good theory/practice correlation with cents,
though the main problem is that we hear beats and not interval size,
and beats are linear while interval size is log. That's why we can
tolerate a relatively large detuning (in cents). Often, when the
tuning of an organ goes all wanky during a rehearsal and I've only got
45 minutes to clean it up AND retune the harpsi before the concert, I
don't even touch the bass pipes. Nobody knows.

TU, like so much of what seems to go on here, is nothing more than
having fun with numbers. Great way to pass the time if you're so
inclined, but in the real world is has no application.

Ciao,

P

> BTW, I don't know what sort of "electronic pitch meter" Brad uses, but
> if it is a Korg, forget it! Those things are utterly worthless,
> inherently unstable, and so poorly programmed that even the octaves
> are often out of tune.

Let's nip this in the bud right now. I have never had
*any* "electronic pitch meter", Korg-esque or otherwise. All my tuning
is by ear, from any of several individual tuning forks.

A couple of months ago I listened patiently to a professional full-time
piano tuner showing me and enthusing about his Accu-Tuner, an
approximately $2000 device that is supposedly programmable in some
zillion different ways, but I still don't want one.

Brad Lehman

I see that Tom complains because TUs are too precise for practical use,
and Gene complains because TUs are not precise enough.

While I decide which position is Scylla and which is Charybdis, I need
to go take two ibuprofen tablets for a headache. Supposedly that's a
total of 400mg as the label prescribes two tablets of 200mg each. Now,
if the measurement of a milligram is too small to be useful (who can
control a single milligram of the substance, accurately enough?), or
too large for some other purpose, all we can really do is to hope that
*on average* the medicine manufacturer has succeeded in getting *about*
200mg into each tablet within some decent tolerance.

Brad Lehman

Hi Margo,

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> >> Quest: QUarter-comma Encompassing Spectrum Tuning
> >> (based on 1/4-comma)
> >> Zest: Zarlino Encompassing Spectrum Tuning
> >> (based on 2/7-comma)
>
> <snip>
>
> Thank you for this feedback, and I will be delighted
> to write entries for these tunings in their 12-note
> and 24-note versions (the latter with two circles at
> the distance of the regular meantone's enharmonic
> diesis).

Wow, i didn't even know they came in 24-note versions!
I'm glad you answered my post! ;-)

> At the conclusion of this message, I'll provide
> some Scala files, although I'm not sure about the best
> representations for your Tonescape software.

That's no problem ... i can make the Tonescape .space
file when Gene (or whomever else) helps me with the
question i asked about how Zest tempers the 7-space.

And i already have a .tuning file for it. It's easy
enough to create both of these types of files for all
the other versions of these tunings, once i'm sure
about how they work.

If you have access to a Windows XP computer, you can
now download Tonescape free and hopefully get it
installed properly. (Lots of people have had problems
installing it because of incompatibilities between
different versions of DirectX.) Then you could compose
your own pieces in it using these tunings. Anyway,
i'll certainly make a few, or if you have any scores
of period pieces which you think would illustrate them
that would be even easier for me.

I'll make all of the these Tonescape files available
via links on the Zest and Quest webpages.

Thanks for the links and the willingness to help out
with the Encyclopedia.

-monz

email: joemonz@yahoo.com
http://tonalsoft.com
Tonescape microtonal music software

Hi Paul,

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> --- In tuning@yahoogroups.com, "Tom Dent" <stringph@> wrote:
>
> >
> > My main objection to TU is that they are absurdly and spuriously tiny
> > for any realistic acoustic situation.
>
> Amen to that! That's why I gave it a skip long ago when somebody,
> probably Brad, brought it up on the harpsichord list. I just didn't
> see the point. As far as a Pythag/Sintonic comma eq, the old 11/12
> rule is good enuf for jazz, and then some.

I'm inclined to agree with that ... except for the fact
that you're most likely talking about 12-edo when you
invoke it, and that is precisely *not* the topic of
this list!

;-P

(Just joking ... i write stuff about 12-edo here all
the time.)

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com
Tonescape microtonal music software

--- In tuning@yahoogroups.com, "Brad Lehman" <bpl@...> wrote:
>
> I see that Tom complains because TUs are too precise for practical
use,
> and Gene complains because TUs are not precise enough.

No, my complaint was that the definition as I understood it was
unsatisfactory. If it's just another logarithmic measure like cents,
that's precise enough. Whether it helps is another issue.

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:
> > Amen to that! That's why I gave it a skip long ago when somebody,
> > probably Brad, brought it up on the harpsichord list. I just didn't
> > see the point. As far as a Pythag/Sintonic comma eq, the old 11/12
> > rule is good enuf for jazz, and then some.
>
>
> I'm inclined to agree with that ... except for the fact
> that you're most likely talking about 12-edo when you
> invoke it, and that is precisely *not* the topic of
> this list!

Looks suspiciously like 612-et to me.

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> That's no problem ... i can make the Tonescape .space
> file when Gene (or whomever else) helps me with the
> question i asked about how Zest tempers the 7-space.

It's not clear to me what you want me to do.

Hi Gene and Paul,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> [Paul Poletti]
> > > Amen to that! That's why I gave it a skip long ago
> > > when somebody, probably Brad, brought it up on the
> > > harpsichord list. I just didn't see the point. As
> > > far as a Pythag/Sintonic comma eq, the old 11/12
> > > rule is good enuf for jazz, and then some.
> >
> >
> > I'm inclined to agree with that ... except for the fact
> > that you're most likely talking about 12-edo when you
> > invoke it, and that is precisely *not* the topic of
> > this list!
>
> Looks suspiciously like 612-et to me.

Yup, exactly! That's the one i was thinking of.

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com
Tonescape microtonal music software

Hi Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
>
> > That's no problem ... i can make the Tonescape .space
> > file when Gene (or whomever else) helps me with the
> > question i asked about how Zest tempers the 7-space.
>
> It's not clear to me what you want me to do.

Please refer back to these posts:

/tuning/topicId_73833.html#74184

>> Gene, i'm trying to figure out how to make a Tonescape
>> .tuning file for the Zest tuning, but could use some
>> help. The chain-of-5ths which are -189 TUs are clearly
>> tempered from 5-limit JI by 1/7, 2/7, or 3/7 of a
>> syntonic-comma.
>>
>> But i'm not sure how to set a generator for the 5ths
>> which are +198 TUs. Margo stipulates that the interval
>> C#/Db:F is close to 9/7, but since they are already both
>> embedded in 3,5-space via the 2/7-comma tempering, i
>> can't figure out how to set up the 7-space component.

/tuning/topicId_73833.html#74185

>> Specifically, i'm approaching this two different ways
>> in Tonescape:
>>
>> 1) a .tonespace file which sets up 7-limit JI and then
>> tempers it via 2/7-syntonic-comma and whatever the
>> other unison-vector should be (i'm guessing that it's
>> some fraction of 64:63);
>>
>> 2) a .tuning file which uses two tempered generators:
>> one for the -189 TU chain-of-5ths, and another one
>> for the +198 TU chain. This one is easy enough for
>> me to do myself -- it's #1 that i need help with.

As you'll see if you follow my subsequent posts on this,
it was easy to do #2, the .tuning file, since i know that
Margo intends to spell the 12-tone chain with nominals
from Eb to G#.

What i want to do for #1 is set up a 2,3,5,7-space
and then use Tonescape to temper each dimension so
that it will warp the final lattice into the 3-D
"crumpled napkin" geometry.

Assuming octave-equivalence, 2 is already out of the
picture. Tempering the 3,5 component is easy because
we already know that Zest is based on 2/7-comma meantone,
so i simply tell Tonescape to temper each generator
along those axes by 2/7 comma.

What i can't figure out is how to temper the other chain
(the +198 TU generators) which covers the wide 5ths between
the notes C# - G#/Ab - Eb - Bb - F.

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com
Tonescape microtonal music software

Hi Gene,

--- In tuning@yahoogroups.com, "monz" <monz@...> wrote:

> Assuming octave-equivalence, 2 is already out of the
> picture. Tempering the 3,5 component is easy because
> we already know that Zest is based on 2/7-comma meantone,
> so i simply tell Tonescape to temper each generator
> along those axes by 2/7 comma.
>
> What i can't figure out is how to temper the other chain
> (the +198 TU generators) which covers the wide 5ths between
> the notes C# - G#/Ab - Eb - Bb - F.

And that chain of wide 5ths will also involve the 7 component.

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com
Tonescape microtonal music software

Dear Monz,

Please let me say that having seen some of your posts asking for
assistance as to how Zest-12 might be mapped to a JI-style lattice, I
would like at least to share my views as the designer on some of the
ratios that might be represented. How you would best map this in your
application sounds like a more complicated question, because you know
the technical practicalities and I don't.

Why I don't a give quick summary of the categories of major and minor
thirds? Certainly we can agree that the nearest range F-C# represents
5-limit meantone, while the thirds C#/Db-F, Bb-C#/Db, and F-G#/Ab
represent ratios of 2-3-7, the first 9:7 and the other two 7:6. It
gets more complicated when we consider how a combination of small and
large fifths in a chain can, in effect, disperse other kinds of
commas.

I'm not sure if I'm doing you a favor here by mentioning yet more factors
to consider diagramming. However, my purpose is only to explain how I tend
to interpret the different third sizes, leaving to you the consideration
what is possible, realistic, or elegant to include in a lattice diagram.

------------
MAJOR THIRDS
------------

Obviously the regular meantone thirds F-A, C-E, G-B, D-F#, and A-C# at
383 cents, built exclusively from small fifths, represent 5:4.

The thirds Bb-D and E-G# at 396 cents, built from three small and one
large fifths, may be considered rather compromised 5-limit thirds
also. Indeed I often quote the usual "meantone-like" range as Bb-G#,
which implies that these thirds are meant to be interchangeable with
the near-pure ones.

The thirds Eb-G and B-D# at 408 cents, built from two small and two
large fifths so that the two directions of temperament almost cancel
out, are in effect representations of the Pythagorean 81:64, and
generally are meant for a neomedieval rather than meantone style.
Sonorities like Eb-G-C or C-Eb-G, however, do come up as touches of
"modal color" in a meantone setting.

The thirds F#-A# and Ab-C at 421 cents, from three large and one small
fifth, might be taken as not-so-accurate representations of 14:11,
although 23:18 is also a possibility. These are very characteristically
neomedieval, and not too far from 17-EDO -- but 14:11 or 23:18 seems
to me the likely JI interpretation.

The third Db-F at 434 cents from four large fifths is a near-9:7, and
here the lattice clearly enters septimal territory.

------------
MINOR THIRDS
------------

The thirds D-F, A-C, E-G, B-D, F#-A, and C#-E at 313 cents are again
regular meantone intervals representing 6:5, from three small fifths
(or large fourths).

The thirds G-Bb and G#-B at 300 cents built from two small and one
large fifth have an ambiguous interpretation, with an inexact 6:5 in
meantone or 32:27 in a neomedieval as likely historical associations.
Sometimes 19:16 is also a possibility, as for example with G-Bb-D used
in a style around 1500 as a closing sonority for G Dorian, say; this
third might to some modern listeners, at least, actually be more
conclusive than a near-just 6:5, although Bb-D is 8 cents narrow of
the 24:19 in a just 16:19:24. Also, something like D-F-Ab/G#-B might
be interpreted as 15:18:21:25, with G#-B at 300 cents as 25:21.

The thirds C-Eb and Eb-Gb at 287 cents, built from two large and one
small fifth, closely represent 13:11, or also 33:28, and might also
represent a variation on the traditional medieval 32:27.

The thirds F-Ab and Bb-Db at 275 cents, from three large fifths (or
small fourths), represent 7:6, and are almost precisely at 75:64.
Again, we are clearly in septimal territory, albeit not without some
compromise.

To sum up, here are the ratios I might suggest are likeliest
candidates, trying to focus on a few main themes:

MAJOR THIRDS MINOR THIRDS

383 cents 313 cents
5:4 6:5

396 cents 300 cents
5:4 6:5, 32:27
(more impure)

408 cents 287 cents
81:64 13:11, 32:27

421 cents 275 cents
14:11, 23:18 7:6

434 cents
9:7

With many thanks,

Margo

--- In tuning@yahoogroups.com, "Paul Poletti" <paul@...> wrote:
>
> --- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@> wrote:
> >
> > If you make all of the narrowest fifths exactly the same size (as
you
> > suggest below), you've *irrationalized* them. That's the way I
> > started out, in fact, but I soon discovered that the beating in
the
> > *minor* triads was not exactly proportional.
> >
> > In my brief explanation I didn't point out that other major
triads
> > (from Eb to E) and many of the minor triads also have *exact*
> > proportional beat rates. This is not the case with a single size
> > fifth, in which they're approximate. Exact proportional beat
rates
> > tend to make the beating sound more unified (or periodic), like a
> > vibrato, as if the instrument is "singing". With non-
proportional
> > beating, the interactions between the differing beat rates sounds
> > more disturbing.
>
> Well, I have to admit that after a half hour of staring at my large
> monitor (19" wide) and my big chart which graphically shows the beat
> rates of every major and minor triad over two octaves (thirds and
> fifths), I haven't got the foggiest notion what you are on about.
When
> I switch back and forth between the regular-sized fifth version and
> your irregular sized fifth version, the movement of the traces is so
> small that it can hardly be seen (full scale is 25 b/s, so we're
> talking about 6mm or so for every 1 b/s - pretty darn easy to see
> changes that would be nigh to impossible to hear). If I set the
trace
> value indicators to 2 decimal places, it also tells me that the
change
> only affects a very few triads, and the largest change is in the
area
> of than 0,01 b/s. That means you are going to have to hold a triad
> for about 100 seconds while you do nothing else musically to
distract
> the ear from the extremely subtle gradual drift of phase between
third
> and fifth or third and third beat rates. Do you REALLY think this is
> an advantage?

If you're saying, for all practical purposes, there's no audible
difference between slightly varying fifths with rational ratios and
like fifths with irrational ratios, then yes, you're right. OTOH, I
could turn that around and say that there's no advantage of the like
fifths with irrational ratios over slightly varying fifths with
rational ratios.

Using rational fifths enables me *to define the tuning* in such a way
that the proportions between the beat rates are *exact (usually
simple) rational values* in both major and minor triads, so there is
a *theoretical* advantage for doing this. (Call it "eye candy", if
you like.)

--George

Hello, everyone.

Many thanks to Tom Dent, who in a recent discussion of possible
tunings for certain Baroque pieces, noted that in some circumstances
meantone diminished fourths might be used in conventional tonal
progressions. Applying this insight to my own modal style in its
"Manneristic" aspects, I came upon a cadence that might be of
interest.

<http://www.bestII.com/~mschulter/zest12-cadence001.mp3>
<http://www.bestII.com/~mschulter/zest12-cadence001.pdf>

Tom's idea was a "dominant" sonority spelled in his message as
C#-B-F-A, suggesting to me also the possibility of C#-F-G#-B in the
rather different kind of musical syntax I often follow.

In Zarlino's 2/7-comma meantone, and also in my encompassing spectrum
tuning of 2003 (then called a "temperament extraordinaire") based on
this regular tuning, C#-F at about 433.52 cents is very close to
9:7. Thus C#-F-G#-B gives a rather good approximation of 14:18:21:25,
a sonority which George Secor called to my attention in October 2001
as a pleasing use of 9:7. More specifically, he favored 14:18:21:25:28
including the octave, with its many mixed isoharmonic differences of
3, 4, and 7; but my progression in four voices uses the simpler
14:18:21:25.[1]

Here is the progression, notated again in a usual meantone spelling,
with C5 indicating middle C; as it happens, all notes and intervals
used are identical to those of Zarlino's regular tuning. See also the
link above to a PDF score:

1 2 + | 1 + 2 | 1 2 + | 1 2 ||
B5 B5 B5 C6 C6 C6 C6
G#5 G#5 G#5 G5 A5 A5 A5
F5 F5 F5 E5 D5 E5 F5 F5 F5
C#5 C#5 C#5 C5 F4 F4 F4

A curious feature of this cadence is that the outer minor seventh of
the opening sonority, which we might often expect to contract to a
fifth by stepwise contrary motion, insteads expands in this fashion to
an octave! Likewise the upper minor third, instead of contracting to a
unison, expands to a fourth.

From this point of view, the seventh B5 might actually be viewed as a
kind of surrogate for Bb5, which would yield a major sixth sonority at
approximately 14:18:21:24. With this more conventional note, and the
progression otherwise identical, we would have a major sixth C#5-Bb5
expanding to an octave, and an upper major second G#5-Bb5 expanding to
a fourth.

While the "substitution" of the minor seventh for the major sixth
might be considered a kind of rhetorical figure -- see below -- in
other ways this is typical "Xeno-Mannerism." A sonority with a tritone
or augmented fourth -- here F5-B5 -- characteristically serves as an
antepenultimate leading up to a cadential 4-3 suspension. This might
contrast with tonal harmony, where tritonic sonorities often play a
penultimate or "dominant" role.

Also, the motion of the bass, descending first by a step and then by a
fifth, is very characteristic both of historical Renaissance and
Manneristic cadences, and of 21st-century offshoots such as this.
Typically the stepwise descent is part of a resolution from major
sixth to octave with an upper voice ascending by a whole tone, leading
into a 4-3 suspension between the bass and another voice, which then
ascends by a semitone while the bass descends a fifth.

This cadence varies only in that instead of an initial expansion from
major sixth to octave, as already noted, we move from a minor seventh
to an octave, which both outer voices moving by a semitone -- the
minor semitone of 25:24 or about 71 cents in the bass, and the regular
diatonic semitone of Zarlino's meantone at 121 cents in the soprano!

A fine point of contrapuntal style might involve the parallel fifths
between bass and alto moving from the first to the second measure
(C#-G# to C-G). Possibly one might argue that since the minor or
chromatic semitone could be considered only a matter of "coloring,"
this might be excused as analogous to repetition of a fifth without
melodic motion! However, a better approach might be frankly to
recognize that some of the normal liberties of historical medieval and
neomedieval writing, where parallel fourths and fifths are freely
used, are sometimes "imported" into Xeno-Manneristic idioms.
Conventional meantone progressions would be more likely to observe
16th-century proprieties.

Now for the concept of a rhetorical figure. In defending Monteverdi's
bold uses of the minor seventh around 1600, treated rather more
cautiously in a conventional 16th-century style (where it appears
prominently mainly as a suspension), an academic writing under the
name of "l'Ottuso" argues as follows:

"[Y]ou allow an excellent poet the metaphor
purposely used; similarly the seventh is taken
in place of the octave."[2]

Here a minor seventh (C#-B) is taken in place of an expected
near-septimal major sixth (C#-A#, or C#-Bb in a usual meantone
spelling). Possibly this experiment in modern mannerism could serve as
a tribute to the Monteverdi brothers (Claudio and his literary partner
Giulio Cesare) for the 400th anniversary of their manifesto of 1607
defending the modern practice of the previous six decades or so
against those who could see Zarlino's classic rules of counterpoint as
the only ones conceivable.

While the free uses of the seventh by Monteverdi and his contemporaries
became one ingredient of major/minor tonality in the later part of the
17th century, he and his brother in 1607 notably championed a style
drawing on a variety of modes as might be required to express the
ideas and emotions of a given text. It is interesting that precisely
this concept had been championed in 1555 by Nicola Vicentino, who with
his archicembalo and 31-note meantone cycle might be described as the
greatest xenharmonicist of the 16th century.

It is thus as a meeting place of old and new that I share this
cadence.

-----------
Notes
-----------

1. Interestingly, George's colleague in so much of the development of
the Sagittal notation system, David Keenan, briefly mentions
14:18:21:25 as a possible tuning for the tonal dominant seventh in
a Tuning List post of 1999.

2. Claude V. Palisca, "The Artusi-Monteverdi Controversy," in Denis
Arnold and Nigel Fortune, eds. _The Monteverdi Companion_
(W. W. Norton, New York, 1972), pp. 133-166, at p. 155.

Most appreciatively,

Margo Schulter
mschulter@calweb.com

Hello, everyone.

Here are some new PDF scores for two pieces in the Zest-12 tuning,
along with links to mp3 files which I've previously posted. Of
course, these pieces could be played on _either_ 12-note keyboard
of a full Zest-24.

I hope that Monz, Brad, and others may find the scores helpful in
following the different shadings of thirds from regular 2/7-comma
meantone to septimal, for example.

<http://www.bestII.com/~mschulter/IntradaFLydian.pdf>
<http://www.bestII.com/~mschulter/IntradaFLydian_regbars.pdf>
<http://www.bestII.com/~mschulter/IntradaFLydian.mp3>

<http://www.bestII.com/~mschulter/OElsa.pdf>
<http://www.bestII.com/~mschulter/OElsa_BbDorian.pdf>
<http://www.bestII.com/~mschulter/OElsa.mp3>

Each PDF score has two versions. For _Intrada in F Lydian_, I have
provided a version with more flexible barring, and one with relatively
more regular barring (including however some alternating 2/2 and 6/4),
in an effort at least partially to accommodate different tastes.

For _O Elsa_, I have provided a transposed version in Bb Dorian which
is played in the mp3 recording, thus bringing the piece into the
remote portion of the Zest-12 circle; and also the original version in
D Dorian, fine for routine singing or playing in medieval Pythagorean
intonation, for example, or a regular neomedieval variant like the
Peppermint temperament.

In the score for _Intrada in F Lydian_, I sought to follow the
expected augmented or diminished spellings for near-septimal intervals
like Bb-C# at 274 cents or F-C# at 766 cents, so that people playing
in a regular meantone around 1/4-comma or 2/7-comma would likewise get
septimal colors. Indeed, when I play the piece in regular 2/7-comma it
sounds fine, and likewise in regular 1/4-comma or Quest-12 (the latter
corresponding to Zest-12 for 2/7-comma).

The most dramatic differences concern sonorities involving Eb, which
in the score retains its usual spelling although it is actually
located precisely halfway between Eb and D# in a regular 2/7-comma.
Thus F-A-Eb (opening of the second page in either version of the
score) is about 0-383-983 cents, a 4:5:7 approximation comparable to
that of 22-EDO, but a usual 0-383-1008 cents in a regular Eb-G# tuning.
Also, Eb-G-C is 0-408-913 cents, having a Pythagorean flavor rather
than the usual 5-limit smoothness of 0-383-887 cents in the regular
tuning. These distinctions of color "go with the territory" in moving
from one system to the other, and tastes may vary.

Please enjoy.

With many thanks,

Margo

Hey Margo,

Thanks for these...very nice..I like the Intrada in particular.

Are these new pieces...I'm not certain I've heard them before.

Can you post the scala file for Zest-12?

Best,
Aaron.

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Hello, everyone.
>
> Here are some new PDF scores for two pieces in the Zest-12 tuning,
> along with links to mp3 files which I've previously posted. Of
> course, these pieces could be played on _either_ 12-note keyboard
> of a full Zest-24.
>
> I hope that Monz, Brad, and others may find the scores helpful in
> following the different shadings of thirds from regular 2/7-comma
> meantone to septimal, for example.
>
> <http://www.bestII.com/~mschulter/IntradaFLydian.pdf>
> <http://www.bestII.com/~mschulter/IntradaFLydian_regbars.pdf>
> <http://www.bestII.com/~mschulter/IntradaFLydian.mp3>
>
> <http://www.bestII.com/~mschulter/OElsa.pdf>
> <http://www.bestII.com/~mschulter/OElsa_BbDorian.pdf>
> <http://www.bestII.com/~mschulter/OElsa.mp3>
>
> Each PDF score has two versions. For _Intrada in F Lydian_, I have
> provided a version with more flexible barring, and one with relatively
> more regular barring (including however some alternating 2/2 and 6/4),
> in an effort at least partially to accommodate different tastes.
>
> For _O Elsa_, I have provided a transposed version in Bb Dorian which
> is played in the mp3 recording, thus bringing the piece into the
> remote portion of the Zest-12 circle; and also the original version in
> D Dorian, fine for routine singing or playing in medieval Pythagorean
> intonation, for example, or a regular neomedieval variant like the
> Peppermint temperament.
>
> In the score for _Intrada in F Lydian_, I sought to follow the
> expected augmented or diminished spellings for near-septimal intervals
> like Bb-C# at 274 cents or F-C# at 766 cents, so that people playing
> in a regular meantone around 1/4-comma or 2/7-comma would likewise get
> septimal colors. Indeed, when I play the piece in regular 2/7-comma it
> sounds fine, and likewise in regular 1/4-comma or Quest-12 (the latter
> corresponding to Zest-12 for 2/7-comma).
>
> The most dramatic differences concern sonorities involving Eb, which
> in the score retains its usual spelling although it is actually
> located precisely halfway between Eb and D# in a regular 2/7-comma.
> Thus F-A-Eb (opening of the second page in either version of the
> score) is about 0-383-983 cents, a 4:5:7 approximation comparable to
> that of 22-EDO, but a usual 0-383-1008 cents in a regular Eb-G# tuning.
> Also, Eb-G-C is 0-408-913 cents, having a Pythagorean flavor rather
> than the usual 5-limit smoothness of 0-383-887 cents in the regular
> tuning. These distinctions of color "go with the territory" in moving
> from one system to the other, and tastes may vary.
>
> Please enjoy.
>
> With many thanks,
>
> Margo
>

> Hey Margo,

> Thanks for these...very nice..I like the Intrada in particular.

Hi, Aaron. I'm delighted that it's brought you enjoyment, and hope
that things are going well with your own musicmaking after the recent
festival. It seems that you had quite a cast of characters, to say the
least! Congratulations.

> Are these new pieces...I'm not certain I've heard them before.

Actually I posted the Intrada to MMM around mid-2005, judging from the
file date, and _O Elsa_ in early 2006 as I recall. However, I'm
delighted if these pieces, whether novel or familiar, bring pleasure
to those who hear them.

> Can you post the scala file for Zest-12?

Most certainly, and please note that Zest-12 is simply the new name
for what I have called my "Zarlino-based 2/7-comma temperament
extraordinaire" -- given that the name _extraordinaire_ seems to
attract use for other systems.

! zest12.scl
!
Zarlino Encompassing Spectrum Tuning, F-C# 2/7-comma, other 5ths ~6.424c wide
12
!
25/24
191.62069
287.43104
383.24139
504.18965
574.86208
695.81035
779.05173
887.43104
995.81035
1079.05173
2/1

While a temperament might be _extraordinaire_ in various ways, the
"encompassing spectrum tuning" concept might convey the idea of a
"rainbow" of major and minor thirds ranging from approximately 5-limit
to septimal -- 383-434 cents for major thirds, and 275-313 cents for
minor thirds.

> Best,
> Aaron.

With many thanks,

Margo

Margo-

I get it now--yes, Zest is a perhaps better name, given that people
might get confused with 'temperament ordinaire'.

Have you tried a variant of this with 5th size ~695.63? (see the
recent G. Secor post on MMM re:metameantone-19)

Best,
Aaron.

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> > Hey Margo,
>
> > Thanks for these...very nice..I like the Intrada in particular.
>
> Hi, Aaron. I'm delighted that it's brought you enjoyment, and hope
> that things are going well with your own musicmaking after the recent
> festival. It seems that you had quite a cast of characters, to say the
> least! Congratulations.
>
> > Are these new pieces...I'm not certain I've heard them before.
>
> Actually I posted the Intrada to MMM around mid-2005, judging from the
> file date, and _O Elsa_ in early 2006 as I recall. However, I'm
> delighted if these pieces, whether novel or familiar, bring pleasure
> to those who hear them.
>
> > Can you post the scala file for Zest-12?
>
> Most certainly, and please note that Zest-12 is simply the new name
> for what I have called my "Zarlino-based 2/7-comma temperament
> extraordinaire" -- given that the name _extraordinaire_ seems to
> attract use for other systems.
>
> ! zest12.scl
> !
> Zarlino Encompassing Spectrum Tuning, F-C# 2/7-comma, other 5ths
~6.424c wide
> 12
> !
> 25/24
> 191.62069
> 287.43104
> 383.24139
> 504.18965
> 574.86208
> 695.81035
> 779.05173
> 887.43104
> 995.81035
> 1079.05173
> 2/1
>
> While a temperament might be _extraordinaire_ in various ways, the
> "encompassing spectrum tuning" concept might convey the idea of a
> "rainbow" of major and minor thirds ranging from approximately 5-limit
> to septimal -- 383-434 cents for major thirds, and 275-313 cents for
> minor thirds.
>
> > Best,
> > Aaron.
>
> With many thanks,
>
> Margo
>

Hi Margo,

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> ! zest12.scl
> !
> Zarlino Encompassing Spectrum Tuning, F-C# 2/7-comma, other 5ths
~6.424c wide
> 12
> !
> 25/24
> 191.62069
> 287.43104
> 383.24139
> 504.18965
> 574.86208
> 695.81035
> 779.05173
> 887.43104
> 995.81035
> 1079.05173
> 2/1

The Tonescape .tuning file i made uses "A" as the
origin, with the 2/7-comma chain-of-5ths running
from -4 = F to +4 = C#. The chain of wide fifths
is actually split into two parts: F:Bb:Eb on the
negative meantone side, and C#:G# on the positive side.

Here's an ASCII lattice with integer cents values:

(use the "Option | Use Fixed Width Font" link to view
properly on the stupid Yahoo web interface.)

G#
1092
|
F --- C --- G --- D --- A --- E --- B --- F# --- C#
817 313 1008 504 0 696 192 887 383
|
Bb
108
|
Eb
600

If i use exactly these same pitches but call "C"
the 1/1 instead, i get the values in your Scala file.

So now that i have all that straight, i can use
the .pdf scores you posted to create some .tonescape
Musical Piece files of your pieces.

But i'd still like to have the other version of
the Tonescape .tuning files, which i've asked Gene
to help me with, and which use tempered 3, 5, and 7
as the generators.

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com
Tonescape microtonal music software

> Margo-

> I get it now--yes, Zest is a perhaps better name, given that people
> might get confused with 'temperament ordinaire'.

Dear Aaron,

Please let me explain that as I originally saw this, describing this
as a special form of temperament ordinaire would be quite correct, and
thus "temperament extraordinaire." Indeed we have most of the fifths
in a regular shading of meantone (2/7-comma), with a few tuned wide of
pure.

However, I've concluded that indeed the technical resemblance might be
confusing, as you say, as to the outlook and purpose of this
21st-century approach.

Specifically, a temperament ordinaire, like a more moderate 12-note
well-temperament, will typically strive to make more major thirds
"usable" _in a 5-limit style_ than would be so in a regular 12-note
meantone, where there are eight available.

In contrast, the Zest-12 tuning actually _reduces_ the number from
eight to seven, with the traditional four diminished fourths staying
Pythagorean or larger and Eb-G also becoming a tad larger than
Pythagorean. To put it another way: we get five great neomedieval
major thirds! That, however, evidently wasn't the goal of Couperin and
his colleagues and successors in the art of the temperament ordinaire,
although it might be fun to imagine them launching a revival of
Machaut around 1700 and relishing the remote transpositions for this
purpose!

Thus Zest may better fit something new, and keep alternative history a
bit more distinct from actual history where confusion could indeed
result.

> Have you tried a variant of this with 5th size ~695.63? (see the
> recent G. Secor post on MMM re:metameantone-19)

Actually I discussed some of the resources of this variant in a paper
on what I was then still calling the temperament extraordinaire, where
Wilson's Metameantone is treated as a very significant point on the
spectrum from 1/4-comma to 88-EDO or so:

<http://www.bestII.com/~mschulter/TE1.txt>

The main complication on a 1024-EDO synthesizer like the TX-802 is
that this division is much kinder to Zarlino's 2/7-comma than to
Wilson's Metameantone, so that the former actually becomes one not
unreasonable approximation of the latter. In brief, the problem is
that the right combination of fifth sizes will produce a major third
almost identical to Zarlino's; Wilson's is located much closer to
halfway between the nearest two available values.

A MIDI-based realization or the like using 49152-EDO, for example,
obviously wouldn't have this problem. Here's a Scala file for West-12,
the Wilson Encompassing Spectrum Tuning.

! west12.scl
!
Wilson Encompassing Spectrum Tuning (Metameantone-based)
12
!
69.41306
191.26087
286.89131
382.52175
504.36956
573.78262
695.63044
778.15219
886.89131
995.63044
1078.15218
2/1

> Best, Aaron.

Most appreciatively,

Margo

Hi - thanks to Monz for pointing out that the wheel has been
reinvented several times ... it was Neidhardt who did it first, you
know, though whether he would have approved of recasting it as 1/600
octave is unclear. He was only concerned with temperings, not interval
sizes per se.

Now, Gene (or anyone), can you explain why 612-edo is so good with
5-limit? It is equivalent to approximating the Pythagorean comma with
1/51 octave, but is this relevant?

~~~T~~~

--- In tuning@yahoogroups.com, "Gene Ward Smith" <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, "monz" <monz@> wrote:
> > > Amen to that! That's why I gave it a skip long ago when somebody,
> > > probably Brad, brought it up on the harpsichord list. I just didn't
> > > see the point. As far as a Pythag/Sintonic comma eq, the old 11/12
> > > rule is good enuf for jazz, and then some.
> >
> >
> > I'm inclined to agree with that ... except for the fact
> > that you're most likely talking about 12-edo when you
> > invoke it, and that is precisely *not* the topic of
> > this list!
>
> Looks suspiciously like 612-et to me.
>

Hi Tom,

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> Now, Gene (or anyone), can you explain why 612-edo is
> so good with 5-limit? It is equivalent to approximating
> the Pythagorean comma with 1/51 octave, but is this relevant?

Here's my answer:

53-edo is a decent approximation to 5-limit JI, and
a superb approximation to 3-limit pythagorean. Its
chief drawback in approximating 5-limit JI is that it
does not distinguish the skhisma (~2 cents). 612-edo
retains all of the closeness of approximation of
53-edo but also distinguishes the skhisma.

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com/support/tonescape/help/tonescape-overview.aspx
Tonescape microtonal music software

(not on my computer so a bit difficult)
one feature that i thought might be useful is that while metameantone
converges on a particular size, it is also a recurrent sequence which
could be seeded with other meantones. my own observation has been that
i actually prefer many of the recurrent sequence before they actually
reach too near a convergent point. Wilson provides an example of
metameantone being seeded with a just major scale in the archives

--- In tuning@yahoogroups.com, "Tom Dent" <stringph@...> wrote:

> Now, Gene (or anyone), can you explain why 612-edo is so good with
> 5-limit? It is equivalent to approximating the Pythagorean comma with
> 1/51 octave, but is this relevant?

What's relevant is that it is extremely accurate. If we call one step
of 612 "sk", then the fifth is 357.997 sk, the majr third 197.020 sk,
and the minor third 160.977 sk. Hence these can be rounded off to
integer values and iterated multiple times, and still give the best
values. Also, the Pythagorean comma is 11.965 sk whereas the Didymus
comma is 10.968, so the approximate 12:11 relationship is captured. The
schisa of 32805/32768 is 0.9964 sk, so the unit is an apprimiate
schisma and quite close to the correct value.

Dear Justin,

Please let me warmly thank you for this opportunity to learn more
about Japanese music and its intonation. A possible just tuning has
occurred to me which is not too far from the melodic steps you
describe, and on which I would warmly invite your comments.

First, however, I would like to express my clear and strong opinion
that some genres of Japanese music are indeed polyphonic. In gagaku,
the classical court orchestral music, various kinds of complex
sonorities with several voices sounding at once are a basic feature of
the style. It is said that these sonorities are built from
superimposed fifths or fourths, as also happens in some 20th-21st
century European music, both classical and more popular.

This is also true for music played on the sheng or "mouth organ," also
common to the music of China, Laos (where it is known as the khene and
is a treasured national instrument), and in certain Cambodian
traditions.

Further, Japanese koto music often is polyphonic, in that it uses two
or more notes sounding at the same time. A very beautiful sonority,
for example, may involve the fifth at 3:2 or 702 cents in JI, and a
major ninth at 9:4 or 1404 cents; interestingly, this happens both in
medieval European music for voices and in modern Japanese koto music.

Here is the seven-note tuning which I mentioned above:

! ForJustin001.scl
!
Scale for Justin, possibly applicable to Japanese modes
7
!
22/21
33/28
4/3
3/2
11/7
39/22
2/1

If I understand correctly, one five-note or pentatonic mode often used
is as follows, and may be found within this seven-note tuning:

! ForJustin-pentatonic001.scl
!
Pentatonic mode for Justin, possibly applicable to Japanese styles
5
!
22/21
4/3
3/2
11/7
2/1

As I hope these scales will show, just intonation often presents
simple and beautiful musical solutions. The question remains, however,
as to whether a solution fits a particular kind of melodic or harmonic
style.

Here I picked a semitone at 22:21, around 81 cents and thus slightly
larger than the 75 cents you mentioned, but maybe not too much
larger. One possible advantage of this semitone is that it produces a
minor sixth at the just ratio of 11:7 or 782 cents, which might, for
example, be heard if the use of a drone were made.

In the five-note or pentatonic scale, all fifths and fourths will be
at their pure ratios of 3:2 and 4:3. If I understand correctly, these
are the primary consonances in styles of Japanese music such as
gagaku, sheng, or koto playing where two or more simultaneous voices
are common.

In the seven-note scale, there are some compromises, but not, I hope,
too severe. In the six locations for fifths, we find that four of them
remain at a pure 3:2, but two of them have intervals at about five
cents wider at the almost identical ratios of 182:121 or 176:117.
These two wide fifths differ by the very small amount of 10648:10647,
or 0.163 cents, less than a sixth of a cent!

In practice, except on a computer or the like, the tuning of course
would be not nearly this precise. I am not sure if fifths about five
cents wide would fit this kind of Japanese style.

In the pentatonic or five-note scale, however, all fifths and fourths
are pure; I would be curious how you like either scale.

With many thanks,

Margo
mschulter@calweb.com

Hi Margo
Thank you for your comments.
Is it possible that you could write it a bit like how Herman did?
Because I can't understand the fractions, but I do understand cents.
Herman wrote it with the nearest letter-names fro mWestern music, and
with the cent value, something like this:
D
0: 1/1 0.000 unison, perfect prime

Eb
1: 25/24 70.672 classic chromatic semitone,

F
2: 32/27 294.135 Pythagorean minor third

etc.

Would that format be possible? Sorry to trouble you!

Also, did you notice the chart I made of the 4 sets of transpositoins
of the scale? It is here, but I'm afraid you need to click on "reply"
in order to see the chart properly, due to this website changing the
displayed format:
/tuning/post?act=reply&messageNum=74302

And, yes Gagaku is quite different from Edo period koto music,
musically and culturally. I don't know much about it I'm afraid. The
sho chords are quite strange! Nice effect. Takemitsu Toru wrote a
modern gagaku composition which is very nice. There is a whole CD of
it. Somehow seems to me much more interesting and alive than the
traditional gagaku!

Justin

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:
>
> Dear Justin,
>
> Please let me warmly thank you for this opportunity to learn more
> about Japanese music and its intonation. A possible just tuning has
> occurred to me which is not too far from the melodic steps you
> describe, and on which I would warmly invite your comments.
>
> First, however, I would like to express my clear and strong opinion
> that some genres of Japanese music are indeed polyphonic. In gagaku,
> the classical court orchestral music, various kinds of complex
> sonorities with several voices sounding at once are a basic feature of
> the style. It is said that these sonorities are built from
> superimposed fifths or fourths, as also happens in some 20th-21st
> century European music, both classical and more popular.
>
> This is also true for music played on the sheng or "mouth organ," also
> common to the music of China, Laos (where it is known as the khene and
> is a treasured national instrument), and in certain Cambodian
> traditions.
>
> Further, Japanese koto music often is polyphonic, in that it uses two
> or more notes sounding at the same time. A very beautiful sonority,
> for example, may involve the fifth at 3:2 or 702 cents in JI, and a
> major ninth at 9:4 or 1404 cents; interestingly, this happens both in
> medieval European music for voices and in modern Japanese koto music.
>
> Here is the seven-note tuning which I mentioned above:
>
>
> ! ForJustin001.scl
> !
> Scale for Justin, possibly applicable to Japanese modes
> 7
> !
> 22/21
> 33/28
> 4/3
> 3/2
> 11/7
> 39/22
> 2/1
>
>
> If I understand correctly, one five-note or pentatonic mode often used
> is as follows, and may be found within this seven-note tuning:
>
>
> ! ForJustin-pentatonic001.scl
> !
> Pentatonic mode for Justin, possibly applicable to Japanese styles
> 5
> !
> 22/21
> 4/3
> 3/2
> 11/7
> 2/1
>
>
> As I hope these scales will show, just intonation often presents
> simple and beautiful musical solutions. The question remains, however,
> as to whether a solution fits a particular kind of melodic or harmonic
> style.
>
> Here I picked a semitone at 22:21, around 81 cents and thus slightly
> larger than the 75 cents you mentioned, but maybe not too much
> larger. One possible advantage of this semitone is that it produces a
> minor sixth at the just ratio of 11:7 or 782 cents, which might, for
> example, be heard if the use of a drone were made.
>
> In the five-note or pentatonic scale, all fifths and fourths will be
> at their pure ratios of 3:2 and 4:3. If I understand correctly, these
> are the primary consonances in styles of Japanese music such as
> gagaku, sheng, or koto playing where two or more simultaneous voices
> are common.
>
> In the seven-note scale, there are some compromises, but not, I hope,
> too severe. In the six locations for fifths, we find that four of them
> remain at a pure 3:2, but two of them have intervals at about five
> cents wider at the almost identical ratios of 182:121 or 176:117.
> These two wide fifths differ by the very small amount of 10648:10647,
> or 0.163 cents, less than a sixth of a cent!
>
> In practice, except on a computer or the like, the tuning of course
> would be not nearly this precise. I am not sure if fifths about five
> cents wide would fit this kind of Japanese style.
>
> In the pentatonic or five-note scale, however, all fifths and fourths
> are pure; I would be curious how you like either scale.
>
> With many thanks,
>
> Margo
> mschulter@...
>

Does anybody know if these recordings by Ezra Sims are microtonal? 72edo?

Ezra Sims. Features "String Quartet No.2," "Elegie nach Rilke," and "Third Quartet." (CD)

James Dashow/Bruce Saylor/Ezra Sims. Featuring "Come Away." (CD)

All Done From Memory (LP)

Thanks in advance

Tony Salinas

I have just read this from the Sagittal notation doc.:

We have also notated even higher divisions such as 270, 282, 306, 311, 342, 388, 494, 612, and beyond, as well as many below 224

I would love to see if you guys have done any progress in these higher divisions like 612

It sounds a very exciting work.

I would also love to hear Johnny's opinion about these symbols if they get as accurate as his 1200edo notation.
Do you think Johnny Reinhard, that this could improve the sight-reading of your musicians if you had a chance
to teach them from scratch???

Thanks

Tony Salinas

-------------------------------------------
Noble Intonation Approximations in Zest-24
An mp3 sampler of metastable interval zones
-------------------------------------------

Hello, everyone.

Recently David Keenan and others have taken part in discussions about
the "metastable" or "Nobly Intoned" (NI) intervals which the two of us
described in a paper of 2000:

<http://dkeenan.com/Music/NobleMediant.txt>

Here I would like to share some mp3 files of cadential progressions
involving reasonably close approximations of many of these NI sizes as
found in the 24-note Zarlino Encompassing Spectrum Tuning, or Zest-24.
In short, an interval of around NI size may represent the zone of
maximum harmonic complexity between two simpler ratios. For example,
the 5:4 major third at 386 cents and 9:7 major third at 435 cents may
have an intervening region of "gravitational equipoise," or maximum
complexity, at around 422 cents.

The paper explains David Keenan's ingenious process for estimating
this NI value by using a Phi-weighted mediant of the two simpler
ratios. Here, however, the purpose is not so much to analyze this
process as to hear the audible results.

Zest-24 is called an "Encompassing Spectrum Tuning" because it surveys
many different regions and subregions of interval space. In my notes
for the mp3 examples which follow, I describe these regions in terms
which hopefully will be reasonably self-explanatory, but are more
fully developed in my newly completed paper exploring one possible
approach to the interval spectrum.

<http://www.bestII.com/~mschulter/IntervalSpectrumRegions.txt>

As mentioned at the beginning of this paper, I am much indebted to the
very thoughtful and musically sensitive writings of David Keenan on
interval names and regions and much else. Our counterpoint of ideas,
like a musical counterpoint, seems to proceed now by similar and now
by contrary motion.

<http://dkeenan.com/Music/>

The Zest-24 system uses two 12-note circles of a modified meantone
tuning based on Gioseffo Zarlino's 2/7-comma meantone. Each circle has
eight regular meantone fifths (F-C#), with the other four equally
wide; the two circles are at the distance of Zarlino's meantone
diesis, 50.276 cents. A Scala file is included at the end of this
article.

The name for each mp3 file starts with "NIA" for "Noble Intonation
Approximation," and includes the sizes in rounded cents for the
relevant NI approximations used in a given example. Here I have
specified the sizes which theoretically should occur in my specific
synthesizer realization of Zest-24 in 1024-EDO, which may sometimes
vary by something on the order of a cent from the values appearing in
the Scala file below.

----------
EXAMPLE 1.
----------

<http://www.bestII.com/~mschulter/NIA001-422.926.mp3>

This example includes two simple and very characteristic neomedieval
cadences using a 422-cent major third and 926-cent major sixth, the
first virtually coinciding with the estimated NI zone at 422 cents or
so, and the second slightly wide of this zone at around 923 cents.
The sonority Ab-C-F (or G#-C-F in a typical meantone spelling) found
in either 12-note circle includes these intervals. The resolution has
the major third expand to a fifth and the major sixth to an octave.

----------
EXAMPLE 2.
---------

<http://www.bestII.com/~mschulter/NIA002-337.829.mp3>

This example features the small neutral third and sixth at 337 and 829
cents, not too far from NI values of 339 and 833 cents -- these
intervals, in either shading, may also be called supraminor thirds and
sixths. Here the third expands to the fifth, and the sixth to the
octave. A helpful spelling might be C-Eb*-Ab*, with the asterisk (*)
showing a note on the upper keyboard raised by a 50-cent diesis.

----------
EXAMPLE 3.
----------

<http://www.bestII.com/~mschulter/NIA003-337.mp3>

This example shows the use of the 337-cent small neutral third in a
progression where it contracts to a unison; the unstable sonority is
C-Eb*-G.

----------
EXAMPLE 4.
----------

<http://www.bestII.com/~mschulter/NIA004-791.1500.mp3>

An interesting feature of the theory of metastable or Nobly Intoned
intervals is that sometimes these zones can involve quite familiar
sizes, here a 791-cent minor sixth almost identical with the
Pythagorean 128:81 or rounded NI value (both 792 cents), and a minor
tenth at an even 1500 cents which approximates the NI value of 1503
cents. These intervals (G-Eb-Bb in either circle) expand to the octave
and 12th.

----------
EXAMPLE 5.
----------

<http://www.bestII.com/~mschulter/NIA005-287.422.mp3>

This standard neomedieval cadence uses a minor third at 287 cents,
slightly larger than the NI value of around 284 cents, with the NI
major third at around 422 cents between the middle and upper voices
(0-422-708 cents), a tuning found at Eb-Gb-Bb in either circle.
Here minor third contracts to unison and major third expands to
fifth.

----------
EXAMPLE 6.
----------

<http://www.bestII.com/~mschulter/NIA006a-287.422.longer.mp3>

This example has the same final cadence as the last, but approached
from the diminished seventh sonority D-F-Ab-Cb (0-313-587-888 cents)
available in either circle (compare 15:18:21:25, 0-316-583-884 cents).
The diminished seventh contracts to the outer fifth of a sonority with
a 4-3 suspension, the fourth resolving to a _minor_ third that then
resolves to a unison, as in the previous example.

----------
EXAMPLE 7.
----------

<http://www.bestII.com/~mschulter/NIA007-791.mp3>

This cadence uses the sonority G-Eb-B with its two almost identical
intervals of 791 cents (G-Eb) and 792 cents (Eb-B) in a kind of
Manneristic meantone idiom treating Eb-B as an augmented fifth.
The very slight inequality of these two intervals is a quirk of the
synthesizer temperament.

----------
EXAMPLE 8.
----------

<http://www.bestII.com/~mschulter/NIA008-946.mp3>

This neomedieval cadence features what I term in my new paper an
_interseptimal_ interval of around 946 cents, located in the
fascinated region between the septimal ratios of 12:7 (933 cents) for
a large major sixth and 7:4 (969 cents) for a small minor seventh.
Here it is coupled with another dynamic interseptimal interval at
around 454 cents, virtually identical to a just 13:10, in the region
between the septimal 9:7 major third at 435 cents and the narrow 21:16
fourth at 471 cents. In this context, these intervals (C*-F-Bb) serve
as an "ultramajor" third and sixth expanding to fifth and octave with
the upper voices rising by 50-cent steps.

The 454-cent interval showcased in this example, by the way, is almost
identical to Gene Ward Smith's "ratwolf" equal to precisely to 13:10,
also his name for a regular meantone temperament differing minutely
from 2/7-comma which produces a "RAT(ional) WOLF" at this just ratio
from a chain of 11 fifths up. In Zest-24, the 2/7-comma "near-ratwolf"
is tempered out or "domesticated" within each 12-note circle, but
happily "reintroduced into the wild" by the 50-cent diesis between the
two circles.

----------
EXAMPLE 9.
----------

<http://www.bestII.com/~mschulter/NIA009-926.mp3>

This Manneristic cadence in a meantone style uses a 926-cent major
sixth (or here diminished seventh) in the sonority C#-E-G-Bb at around
0-313-626-926 cents, followed by a 4-3 suspension and resolution.

-----------
EXAMPLE 10.
-----------

<http://www.bestII.com/~mschulter/NIA010-422.2113.mp3>

An interesting feature of metastable or NI intervals underscored in a
recent list contributed by David Keenan is that octave extensions can
make a difference. Here the metastable or NI major third at 422 cents
is coupled with a near-NI major 13th at 2113 cents, close to the
estimated NI zone at around 2109 cents. These intervals (Gb3-Bb3-Eb5
with C4 as middle C), available within either circle, resolve to fifth
and 15th (or double octave) respectively.

-----------
EXAMPLE 11.
-----------

<http://www.bestII.com/~mschulter/NIA011-2055.mp3>

Another example of an NI interval larger than octave, this cadence has
a 2055-cent neutral 13th, here used together with a 2752-cent neutral
third plus two octaves or neutral 17th (Ab2-E*4-B*4). These intervals
expand to the 15th or double octave and 19th or fifth plus double
octave. This wide a spacing of voices is rather rare in a neomedieval
style; but this example and the last show how certain NI intervals
well beyond the octave may provide an incentive.

-----------
EXAMPLE 12.
-----------

<http://www.bestII.com/~mschulter/NIA012-422.mp3>

This concluding example shows how the 422-cent NI major third may be
used as an idiomatic meantone diminished fourth in a Manneristic
style. Here we start at D-G-Bb-D, with the fourth and minor sixth
above the bass suggesting a cadential approach. The fourth G then
proceeds to the major third F# -- forming the diminished fourth F#-Bb
at 422 cents, which leads to a 4-3 suspension and resolution.

----------

As promised, here is the Zest-24 tuning:

! zest24.scl
!
Zarlino Encompassing Spectrum Temperament (two circles at ~50.28c apart)
24
!
50.27584
25/24
120.94826
191.62069
241.89653
287.43104
337.70688
383.24139
433.51722
504.18965
554.46549
574.86208
625.13792
695.81035
746.08619
779.05173
829.32757
887.43104
937.70688
995.81035
1046.08619
1079.05173
48/25
2/1

Most appreciatively,

Margo Schulter
mschulter@calweb.com

> Hi Margo
> Thank you for your comments.
> Is it possible that you could write it a bit like how Herman did?

Hi, Justin.

Certainly I can write my suggested tunings in this way.

I should quickly explain that what I posted are Scala files; when used
with Scala, they would show the values in cents, and much other useful
information. Why don't I give a link to Scala if you might be curious,
and then give the scales as you have asked.

<http://www.xs4all.nl/~huygensf/scala>.

> Because I can't understand the fractions, but I do understand cents.
> Herman wrote it with the nearest letter-names fro mWestern music, and
> with the cent value, something like this:
> D
> 0: 1/1 0.000 unison, perfect prime
> Eb
> 1: 25/24 70.672 classic chromatic semitone,
> F
> 2: 32/27 294.135 Pythagorean minor third
> etc.

> Would that format be possible? Sorry to trouble you!

Certainly, and in fact I'm editing some output from Scala for this
purpose, changing or adding a couple of interval names for the sixth
and seventh steps of this 7-note set in the process, giving note names
both in a line before the note (as in your example above) and also at
the end of a line with the fraction and size in cents:

E
0: 1/1 0.000 unison, perfect prime (E)
F
1: 22/21 80.537 undecimal minor semitone (F)
G
2: 33/28 284.447 undecimal minor third (G)
A
3: 4/3 498.045 perfect fourth (A)
B
4: 3/2 701.955 perfect fifth (B)
C
5: 11/7 782.492 undecimal minor sixth (C)
D
6: 39/22 991.165 tridecimal minor seventh (D)
E
7: 2/1 1200.000 octave (E)

The steps are around 81-204-214-204-81-209-209 cents.

Here is the pentatonic or 5-note set I suggested from among these:

E
0: 1/1 0.000 unison, perfect prime (E)
F
1: 22/21 80.537 undecimal minor semitone (F)
A
2: 4/3 498.045 perfect fourth (A)
B
3: 3/2 701.955 perfect fifth (B)
C
4: 11/7 782.492 undecimal minor sixth (C)
E
5: 2/1 1200.000 octave (E)

The steps are around 81-418-204-81-418 cents.

> Also, did you notice the chart I made of the 4 sets of
> transpositoins of the scale? It is here, but I'm afraid you need to
> click on "reply" in order to see the chart properly, due to this
> website changing the displayed format:
> [35]/tuning/post?act=reply&messageNum=74302

For some reason, on this site there seem to be formatting problems
with this version of the chart and another no matter what I do,
including the "reply" option. I am very interested, and wonder if you
could please send me an e-mail of this. Or, could the sets be
described in a text format that wouldn't be sensitive to these site
problems?

For example, maybe:

Tuning: E0 F81 G284 A498 B702 C782 D991 E1200

Set 1: Start from 0:
E0 F81 A498 B702 C782 E1200

Set 2: Start from 284:
G284 A498 C782 D991 E1200 G284

Set 3: Start from 782:
C782 D991 F81 G284 A498 C702

Set 4: Start from 991
D991 E1200 G284 A498 C782 D991

This might not be anything like the actual sets, but possibly could be
a way of writing them for this site -- let's see how this message gets
formatted.

> And, yes Gagaku is quite different from Edo period koto music,
> musically and culturally. I don't know much about it I'm
> afraid. The sho chords are quite strange! Nice effect. Takemitsu
> Toru wrote a modern gagaku composition which is very nice. There is
> a whole CD of it. Somehow seems to me much more interesting and
> alive than the traditional gagaku! Justin

Personally I have found the sho chords -- thanks for reminding me of
the Japanese name, which I'm guessing might be equivalent to the
Chinese sheng -- very beautiful, and a great attraction of Gagaku. and
would agree that they are quite different from most other musics!

Anyway, I'm sure we'll find a way for me to read your transposition
chart.

With many thanks,

Margo Schulter
mschulter@calweb.com

Hi Margo
I haven't digested your mail yet, but just first to reply to a couple
of things:

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> Here is the pentatonic or 5-note set I suggested from among these:
>
> E
> 0: 1/1 0.000 unison, perfect prime (E)
> F
> 1: 22/21 80.537 undecimal minor semitone (F)
> A
> 2: 4/3 498.045 perfect fourth (A)
> B
> 3: 3/2 701.955 perfect fifth (B)
> C
> 4: 11/7 782.492 undecimal minor sixth (C)
> E
> 5: 2/1 1200.000 octave (E)

I think this is the same scale we are using.
If you look at the chart I made (did you manage to see it? You can see
it directly here:
/tuning/post?act=reply&messageNum=74302
) then I think it is the same pentatonic scale. I mentioned in the
neginning, that although there are 7 notes, it is actually 5 main
notes, with 2 secondary notes. Notes 1, 2, 4, 5, 6 are the main notes,
and notes 3 and 7 are the secondary notes.
On my chart the sets are in columns. If you look at the second column
(second set), D is actually the fifth note. But if you start the scale
there, and just use the 5 primary notes, you get:
D, D#-25, A, Ab-25, C.
Isn't that the same as your one? (Sorry I am not so good at
understanding the letter names! Takes me ages to figure out what they
are in our music!)

> Anyway, I'm sure we'll find a way for me to read your transposition
> chart.

Did you manage to? This link takes you straight there (it's the
"reply" page but you can see the chart properly there. 4 vertical
columns. Hope it makes sense?
/tuning/post?act=reply&messageNum=74302

Justin

>
> With many thanks,
>
> Margo Schulter
> mschulter@...
>

Hi Margo
A couple of things:

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

>
> E
> 0: 1/1 0.000 unison, perfect prime (E)
> F
> 1: 22/21 80.537 undecimal minor semitone (F)
> G
> 2: 33/28 284.447 undecimal minor third (G)
> A
> 3: 4/3 498.045 perfect fourth (A)
> B
> 4: 3/2 701.955 perfect fifth (B)
> C
> 5: 11/7 782.492 undecimal minor sixth (C)
> D
> 6: 39/22 991.165 tridecimal minor seventh (D)
> E
> 7: 2/1 1200.000 octave (E)
>
> The steps are around 81-204-214-204-81-209-209 cents.

Could you tell me why you prefer "33/28 284.447 undecimal minor third"
to Herman's suggestion "32/27 294.135 Pythagorean minor third"?
And "39/22 991.165 tridecimal minor seventh" to Herman's suggestion
"16/9 996.090 Pythagorean minor seventh"?

> > click on "reply" in order to see the chart properly, due to this
> > website changing the displayed format:
> >
[35]/tuning/post?act=reply&messageNum=74302
>
> For some reason, on this site there seem to be formatting problems
> with this version of the chart and another no matter what I do,
> including the "reply" option. I am very interested, and wonder if you
> could please send me an e-mail of this. Or, could the sets be
> described in a text format that wouldn't be sensitive to these site
> problems?

Sorry I didn't see that part of your mail.
Just to check, perhaps it is the way I wrote it, rather than what you
are seeing being the problem. What you should see is 5 columns, with
(sorry if untidy) dots filling up all the spaces (I did that the try
to keep the spacing proper, as this site automatically deletes blank
spaces!) So the left column is the letter-names of the pitches. Then
you have 4 columns on the right, with the 7 notes of the scale,
numbered. Really it is better to display it as a circle! For example
First column 1=D, 2=D#-25c. Second column 1=G, 2=Ab-25, etc.
If you can still not make sense of it, please let me know.

Thanks!
Justin

Hi Tony,

We have discussed this before on the List. My players are able to perform
best in cents notation, if only because it encompasses all manner of different
tunings. There really is little difference between polymicrotonal tuning
and 1200-tone equal temperament as I interpret it. For real time, seat of the
pants playing, cents notation, with only two new symbols (quarterflat and
quartersharp), deviations of 1-49 above or below any note are easily negotiated.
New memorization of different symbols is perhaps better suite for
electronic/keyboard based instruments, instruments with preset tuning as opposed to
flexible tuning.

Johnny
_tony@tonysalinas.com _ (mailto:tony@tonysalinas.com ) _jamsalinas _
(http://profiles.yahoo.com/jamsalinas)
Mon Nov 12, 2007 8:03 pm (PST)
I have just read this from the Sagittal notation doc.:

We have also notated even higher divisions such as 270, 282, 306,
311, 342, 388, 494, 612, and beyond, as well as many below 224

I would love to see if you guys have done any progress in these
higher divisions like 612

It sounds a very exciting work.

I would also love to hear Johnny's opinion about these symbols if
they get as accurate as his 1200edo notation.
Do you think Johnny Reinhard, that this could improve the sight-
reading of your musicians if you had a chance
to teach them from scratch???

Thanks

Tony Salinas

************************************** See what's new at http://www.aol.com

--- In tuning@yahoogroups.com, "J.A.Martin Salinas" <tony@...> wrote:
>
> I have just read this from the Sagittal notation doc.:
>
> We have also notated even higher divisions such as 270, 282, 306,
> 311, 342, 388, 494, 612, and beyond, as well as many below 224
>
> I would love to see if you guys have done any progress in these
> higher divisions like 612
>
> It sounds a very exciting work.

Yep, as it says there, we are indeed able to notate 612-ET with
Sagittal (as well as many beyond that, up to 2460-ET). In fact,
there's more than one way to do 612, but we haven't yet decided which
should be the standard notation, which is why we haven't released
anything specific. (Much the same could be said for the other
divisions on that list.)

Your question is very timely, because Dave Keenan and I have just
finished defining all of the symbols that will be used for just
intonation (which, in turn, will help us to determine how the above
ET's should be notated). We expect to be releasing documentation
very shortly regarding this, including a spreadsheet that will show
how to notate (with alternate spellings) any ratio in Sagittal
through prime limit 61 (using prime factor exponents, or "monzos").

> I would also love to hear Johnny's opinion about these symbols if
> they get as accurate as his 1200edo notation.

Since it's able to notate 2460-EDO, Sagittal is capable of roughly
twice the precision, though it's debatable whether that's really
necessary. For those don't want to cope with the many accidentals
this requires, we offer lower-precision JI options: e.g., extreme-
precision (olympian-level) JI corresponds to ~2460-EDO, whereas high-
precision (herculean-level and promethean-level) JI is comparable to
612-EDO and 494-EDO (respectively), and medium-precision (athenian-
level) JI to 224-EDO. If that's still too many accentals for you,
then we would recommend mapping your tones to either 130-ET (using
the spartan symbol set) or 72-ET (which uses less than half of the
spartan set).

> Do you think Johnny Reinhard, that this could improve the sight-
> reading of your musicians if you had a chance
> to teach them from scratch???

Johnny's notation is well suited to enabling players to arrive at the
desired pitches on conventional flexible-pitch instruments. Sagittal
is best suited to providing a meaningful notation for retuned fixed-
pitch instruments, for specially built microtonal wind instruments,
for the electronic medium, and for theoretical applications. The two
approaches are quite different and are, in fact, complementary: each
tends to excel in applications where the other is weakest.

--George

Hi Tony, Johnny, and George,

--- In tuning@yahoogroups.com, "George D. Secor" <gdsecor@...> wrote:

> > I would also love to hear Johnny's opinion about
> > these symbols if they get as accurate as his 1200edo
> > notation.
>
> Since it's able to notate 2460-EDO, Sagittal is capable
> of roughly twice the precision, though it's debatable
> whether that's really necessary.

Hmm ... first reading Johnny's answer to Tony, in which
he describes his 1200-edo notation as employing all the
regular accidentals, with additional quarterflat and
quartersharp symbols, and 0-49 cent markings to indicate
deviations from those ... and then George's description
here of 2460-edo (minas), got me to thinking ...

It's too bad that 2460 doesn't divide evenly into 24.
The regular-plus-quartertone accidental set and deviation
numbers 0-100 would make a nice reference tuning for
the notation.

Actually, Johnny, you don't even need the quatertone
symbols for 1200-edo because you can just use the
regular sharps and flats and +/-50 cents and still
get precision to 1200-edo, if you can do without the
enharmonic spellings of notes which would need the
quartertone symbols.

2460-edo could theoretically be acheived in a similar way
by using the regular sharps and flats plus numerical
indications +/-102.5 degrees, so you can still get
close enough for practical use by using +/-100. If
the lowernote-plus-100 is not equal to uppernote-minus-100,
so that there is one degree between those two values,
then you get 201 degrees per 12-edo semitone, which
is 2412-edo.

You could also employ quartertone symbols to fit between
these two values, without using any numerical deviations
when using the quartertone symbols, in which case you
get 202 degrees per semitone = 2424-edo.

Anyway, just some ideas i had. The system Johnny already
has in use feels very natural to a trained microtonal
musician, because the quarter-tones are "anchors" at
the 50-cent points, and so using +/-50 cents (without
the quartertone accidentals) is an easy way for a performer
to develop ear training to acheive the desired results.

> For those don't want to cope with the many accidentals
> this requires, we offer lower-precision JI options:
> e.g., extreme-precision (olympian-level) JI corresponds
> to ~2460-EDO, whereas high-precision (herculean-level
> and promethean-level) JI is comparable to 612-EDO and
> 494-EDO (respectively), and medium-precision (athenian-
> level) JI to 224-EDO. If that's still too many accentals
> for you, then we would recommend mapping your tones
> to either 130-ET (using the spartan symbol set) or 72-ET
> (which uses less than half of the spartan set).

Myself, i still like 72-edo HEWM. ;-P

I'm still looking forward to having HEWM and Sagittal
notations in Tonescape someday, so i can compare the
same piece notated both ways.

-monz

email: joemonz(AT)yahoo.com
http://tonalsoft.com/support/tonescape/help/tonescape-overview.aspx
Tonescape microtonal music software

Hi Joe, Tony, George,

Joe: Actually, Johnny, you don't even need the quatertone
symbols for 1200-edo because you can just use the
regular sharps and flats and +/-50 cents and still
get precision to 1200-edo, if you can do without the
enharmonic spellings of notes which would need the
quartertone symbols.

Johnny: Not ready to dump the tonal implications, some only psychological,
of sharps to flats, that a more pure number approach would seem to be an
improvement. The quartertone sharp and the quartertone flat are visual landmarks
(not audible landmarks). They allow the conservatory trained music reader to
follow a logic they come to expect as normal to playing. In addition,
numbers 1-49 are clearly more likely to be numbers less than 25. 50 breaks the
bank. Numbers larger than 50 are hard to negotiate on the hoof.

I agree with George.

Joe: Anyway, just some ideas i had. The system Johnny already
has in use feels very natural to a trained microtonal
musician, because the quarter-tones are "anchors" at
the 50-cent points,

Johnny: I think they are much more visual anchors for sight reading skill
expectations more than they are soundposts for hearing an interval.

Joe: and so using +/-50 cents (without
the quartertone accidentals) is an easy way for a performer
to develop ear training to acheive the desired results.

Johnny: Sorry to pick (or poke), but where is the ear training from the
following of visual symbology? They are apples and oranges here. The hearing
of, and the initiating of, discrete microtonal intervals is independent of
actually playing it. In fast movement the eye goes to a quartertone specific
sector (quadrant?) and the deviation of a norm is quickly calculated. Even
with this, I use a notation shorthand for quick fingerings, but I still need the
number of cents to set the intellect of my mind. Both are necessary.

But there is also a difference between genuine sight reading, and
preparation preceding group reading. At a certain point, a player employs other
symbology; I use a pair of glasses to make me alert at a certain point in a piece
of music. I circle alternative fingerings to distinguish them from the much
more usual fingerings(generally yielding monophonic pivots, if not simply
alternative timbres). There are some fingerings for ease of motion, for
trilling, for dynamic, for timbre, for pitch sensitivity (one with open tone holes
and one with the gradual depression of a key). Of course, I know you know much
of this.

Joe: Myself, i still like 72-edo HEWM. ;-P
I'm still looking forward to having HEWM and Sagittal
notations in Tonescape someday, so i can compare the
same piece notated both ways.
-monz

Johnny: Great art is made out of the material found about it. There are
many ways to segregate. 72 does not yield just intervals that I miss. But I
still think it is the composer that makes the difference, not the tuning.
(But it does help some individuals over the hump of being derivative, the method
that is so abundant taught in contemporary music education.)

************************************** See what's new at http://www.aol.com

Margo!

!!!

I am completely blown away by your post and your wonderful article
surveying the interval spectrum. I hope to be able to address it in
some detail at a later date. But I must tell you that some of your
NI/JI cadence "examples" brought tears to my eyes, so powerful did I
find them.

And I must point out that any interval naming for which you mentioned
indebtedness to me, rightfully belongs to Adriaan Fokker.
http://www.xs4all.nl/~huygensf/english/fokker.html

And again: Wow!

-- Dave Keenan

> Here are the 4 sets (transpositions of the scale), layed out
> differently, in case my chart was confusing people! (Margo I hope you
> can read it better now).

> 1st set
> D, D#-25, F, G, A, Bb-25, C
> 2nd set
> G, Ab-25, Bb, C, D, D#-25, F
> 3rd set
> A, Bb-25, C, D, E, F-25, G
> 4th set
> C, C#-25, D#, F, G, Ab-25, Bb

Dear Justin,

Thank you so much! Arigatogozaimas! Now I can read your chart without
any problem

Also, please let me thank you very warmly for the _Haru-no-Kyoku_ mp3,
which sounds to me very often polyphonic, and sometimes seems to use
what is called in Europe "imitation" -- that is, having the same
melodic theme or pattern sounded first in one voice, and then in
another.

Your new listing of the four sets is very helpful in explaining a
possible reason for the tuning I have proposed. However, please let me
emphasize that _it is not necessarily better than your tuning above,
or Herman's_. Either of those might be better, and also closer to the
prevailing practice as you have described it!

Herman, please feel warmly welcome to step in at any point. I suspect
that we are illustrating two approaches to tuning, and I hope that I
am correctly interpretating your intentions. As I emphasize below,
your tunings of minor thirds and sevenths are just as good as mine,
and my idea is to experiment with the idea of a single chain of nine
fifths to express the four transposition sets -- six pure fifths and
three tempered by a tad less than five cents.

A possible advantage of my tuning is that it could be done on certain
instruments with fewer than 13 notes available per octave, since it
would allow us to use a single string, flute hole, or key on a
keyboard to represent what have traditionally been two separate
pitches. Thus we have in the usual style C# and C#-25, D# and D#-25,
and F and F-25.

Here I would prefer to spell C# as Db, since C to Db-25 in the 4th set
is a usual semitone, and Db-25 to Ab-25 is a regular fifth. However, I
understand that Western note spellings may sometimes be confusing, and
might actually myself be quite happy simply to use cents.

Why don't I first show my ten-note tuning in which we would need only
one version rather than two for Db, F, and Ab; and then the four sets
as they would be played in this tuning. Here I would add the Western
note names at the end of each line, not being sure if they are
helpful, and also column listing of where the note appears in each
set, with an "n" meaning it is not used in a given set. For example,
the notation "D:154n" tells us that D is the first step of the 1st
set, the fifth step of the 2nd, the fourth step of the 3rd, and is not
used in the 4th. I've tried to get the four columns to line up.

0: 1/1 0.000 unison, perfect prime (D:;154n)
1: 22/21 80.537 undecimal minor semitone (Eb:26n3)
2: 9/8 203.910 major whole tone (E:;nn5n)
3: 33/28 284.447 undecimal minor third (F:;3764)
4: 4/3 498.045 perfect fourth (G::4175)
5: 88/63 578.582 (Ab:n2n6)
6: 3/2 701.955 perfect fifth (A::5n1n)
7: 11/7 782.492 undecimal minor sixth (Bb:6327)
8: 39/22 991.165 tridecimal minor seventh (C:;7431)
9: 13/7 1071.702 16/3-tone (Db:nnn2)
10: 2/1 1200.000 octave (D:;154n)

We may need to discuss some concepts a bit, if you are curious,
because I realize that the theory can get rather complicated. However,
the basic idea is that if we'd like a note to be in two different
places in different sets, maybe we can put it somewhere between those
two places, and then leave it where it is for both sets. This is a
compromise, and if you find that it's better just to move that note as
convenient, then you would not be wrong!

Maybe the best way to start explaining is to see how each of the four
sets will be tuned with this scale of only ten notes in the octave.
Here I will keep the original numbers of the ten notes for each set.

1st set

0: 1/1 0.000 unison, perfect prime (D:;154n)
1: 22/21 80.537 undecimal minor semitone (Eb:26n3)
3: 33/28 284.447 undecimal minor third (F:;3764)
4: 4/3 498.045 perfect fourth (G::4175)
6: 3/2 701.955 perfect fifth (A::5n1n)
7: 11/7 782.492 undecimal minor sixth (Bb:6327)
8: 39/22 991.165 tridecimal minor seventh (C:;7431)
10: 2/1 1200.000 octave (D:;154n)

2nd set

4: 1/1 0.000 unison, perfect prime (G)
5: 22/21 80.537 undecimal minor semitone (Ab)
7: 33/28 284.447 undecimal minor third (Bb)
8: 117/88 493.120 fourth narrow by 4.925 cents (C)
10: 3/2 701.955 perfect fifth (D)
1: 11/7 782.492 undecimal minor sixth (Eb)
3: 99/56 986.402 undecimal minor seventh (F)
4: 2/1 1200.000 octave (G)

3rd set

6: 1/1 0.000 unison, perfect prime (A)
7: 22/21 80.537 undecimal minor semitone (Bb)
8: 13/11 289.210 tridecimal minor third (C)
0: 4/3 498.045 perfect fourth (D)
2: 3/2 701.955 perfect fifth (E)
3: 11/7 782.492 undecimal minor sixth (F)
4: 16/9 996.090 Pythagorean minor seventh (G)
6: 2/1 1200.000 octave (A)

4th set

8: 1/1 0.000 unison, perfect prime (C)
9: 22/21 80.537 undecimal minor semitone (Db)
1: 968/819 289.372 (Eb)
3: 121/91 493.282 (F)
4: 176/117 706.880 (G)
5: 3872/2457 787.417 (Ab)
7: 484/273 991.327 (Bb)
8: 2/1 1200.000 octave (C)

The secret is that we are using only a single chain of fifths for the
whole tuning, nine fifths in all to give our ten notes per octave. Six
of these fifths are pure, as in Herman's tuning also, where three of
them are made wide by about five cents each.

Note that the tunings vary a little from set to set, but that they
should all be reasonably tolerable, if my guess is correct -- this is
for you to judge!

Indeed many of the intervals will be identical to those of Herman's
fine tuning. Thus note that the 1st and 3rd sets have a pure fifth and
a pure fourth above the first note (D or A), and that in the 3rd set,
from A to G is a Pythagorean minor seventh at 16:9 or 996 cents.

At the same time, melodically, you will see that the semitone steps in
each of these four sets, found between the first and second steps of a
given set and also the fifth and sixth steps, are about 81 cents,
somewhat wider than the preferred 75 cents, but yet smaller than the
90 cents we would get tuning with all fifths and fourths pure.

This brings me to the idea of a "chain of fifths," which you also
mentioned in a message with questions for Herman. I will try and show
the chain of fifths in this tuning, starting with Db and going to E.

Db--+4.9--Ab--0--Eb--0--Bb--0--F--+4.8--C--+4.9--G--0--D--0--A--0--E

Here a "0" between two numbers means a pure fifth, and "+4.9" or
"+4.8" shows that a fifth is tempered wide by about 4.9 or 4.8 cents.

> Could you tell me why you prefer "33/28 284.447 undecimal minor
> third" to Herman's suggestion "32/27 294.135 Pythagorean minor
> third"?

Actually I think that these are both fine minor thirds, and very much
like both flavors or shadings, as they might be called. The possible
advantage of 33:28 is that, as one pleasing kind of minor third, it
might be closer to a pure fourth in relation to your preferred size of
minor sixth around 775 cents -- or, here, often 782 or 787 cents.
Actually, 284 is closer to a pure fourth below 782 cents, while either
284 or 294 cents (32:27) is comparably close to a pure fourth below
787 cents.

> And "39/22 991.165 tridecimal minor seventh" to Herman's suggestion
> "16/9 996.090 Pythagorean minor seventh"?

Either of these minor sevenths is fine! I'd be happy to use either. In
fact, if you are playing polyphony and play a note, its fourth, and
minor seventh all together -- about 0-498-996 cents -- then you might
prefer the pure 16:9 minor seventh of Herman's tuning, because both
fourths will be pure. With my version, if you play D-G-C in the first
set at the same time, D-G will be a pure fourth, but G-C will be
almost five cents narrow. I use this kind of sound often, but you
would not be wrong to prefer pure intervals.

What I may need to explain in a more careful and step-by-step way is
how a chain of fifths, or more than one chain, can be used to built a
tuning. The idea of my tuning is to have as many reasonably useful
fifths and possible with the smallest number of notes per octave; have
most of these fifths and fourths pure; and have scale steps like the
second and sixth in a given set fairly close to their usual sizes as
you have described them.

Most appreciatively,

Margo
mschulter@calweb.com

Dear Dave,

Thank you for your warm response, which beautifully affirms from
someone not the least when it comes to the numerical aspects of our
art that indeed we are talking about real music and about moving the
ears and emotions.

> And I must point out that any interval naming for which you mentioned
> indebtedness to me, rightfully belongs to Adriaan Fokker.
> <http://www.xs4all.nl/~huygensf/english/fokker.html>

Of course this is a point well taken, and you make it very strongly in
the paper on interval naming to which we are referring:

<http://dkeenan.com/Music/IntervalNaming.html>

My quick comment would be that just as Adriaan Fokker, a person of
peace, managed during a brutal military occupation of his country
to lay many of the foundations for the modern 31-tone system, so you,
also eminently a person of peace, have honored those foundations by
building upon them and contributing some most thoughtful further
perspectives.

With many thanks,

Margo
mschulter@calweb.com

On Tue, 13 Nov 2007, Margo Schulter wrote:

> Of course this is a point well taken, and you make it very strongly in
> the paper on interval naming to which we are referring:

> <http://dkeenan.com/Music/IntervalNaming.html>

Please let me correct this link, which shows that I should _always_
follow the procedure of validating a link before posting it by
temporarily postponing the message in my mail program, and then
reading the postponed message and testing the link. If I had done
that, this correction would not be needed, but I hope that people
will get the accurate message that this paper is indeed worth
reading:

<http://dkeenan.com/Music/IntervalNaming.htm>

With many thanks,

Margo
mschulter@calweb.com

Hi Margo and everyone
Is there anywhere on the internet where one might listen to samples of
Adriaan Fokker's music?
Thank you
Justin

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> My quick comment would be that just as Adriaan Fokker, a person of
> peace, managed during a brutal military occupation of his country
> to lay many of the foundations for the modern 31-tone system, so you,
> also eminently a person of peace, have honored those foundations by
> building upon them and contributing some most thoughtful further
> perspectives.
>
> With many thanks,
>
> Margo
> mschulter@...
>

Dear Justin,

Please let me share this link to some MIDI and score files of Adriaan
Fokker's compositions, as well as those of some other people:

<http://www.xs4all.nl/~huygensf/english/music.html>

Most appreciatively,

Margo Schulter
mschulter@calweb.com

Margo Schulter wrote:

> 0: 1/1 0.000 unison, perfect prime (D:;154n)
> 1: 22/21 80.537 undecimal minor semitone (Eb:26n3)
> 2: 9/8 203.910 major whole tone (E:;nn5n)
> 3: 33/28 284.447 undecimal minor third (F:;3764)
> 4: 4/3 498.045 perfect fourth (G::4175)
> 5: 88/63 578.582 (Ab:n2n6)
> 6: 3/2 701.955 perfect fifth (A::5n1n)
> 7: 11/7 782.492 undecimal minor sixth (Bb:6327)
> 8: 39/22 991.165 tridecimal minor seventh (C:;7431)
> 9: 13/7 1071.702 16/3-tone (Db:nnn2)
> 10: 2/1 1200.000 octave (D:;154n)

This is pretty close to a chain of fourths or fifths if 896;891 is tempered out. A couple of notes are different (since your scale has factors of 13 in it).

0: 1/1 0.000 unison, perfect prime
1: 82.356 cents 82.356
2: 206.959 cents 206.959
3: 289.315 cents 289.315
4: 496.274 cents 496.274
5: 578.630 cents 578.630
6: 703.232 cents 703.232
7: 785.589 cents 785.589
8: 992.547 cents 992.547
9: 1074.904 cents 1074.904
10: 1199.506 cents 1199.506

This is getting farther from the desired 75.0 cent step size, and it does include tempered intervals that beat slightly, but someone somewhere might find this useful. :-)

Hi Johnny,

Thanks for clarifying about your notation, and sorry I am still catching
up for many years away from the list.

Next question is:

How do you train your players?

Do they practice with an attached digital tuning meter giving the note name
and +/- cents deviation? That seems to me the most obvious way!

> Inspired by Margo, I came up with a splendid maqam tuning with at least one
> nobly intoned ratio.

> First, calculate the noble mediant between 27:25 and 14:13.
> (27+14)*phi / (25+13)*phi=131.55 cents

> In SCALA, type:
> equal 31
> copy 0 1
> move
> 131.55
> merge
> 1
> Voila!
> Oz.

Dear Oz and Dave,

Please let me diplomatically attempt here to mediate in a situation
which I suspect must occur many times in the realm of musical practice
and theory.

Oz, you have used a mathematical formula which Dave devised and he and
I documented for rather different purposes -- but to excellent effect,
deriving a fine size of small neutral second for maqam music. The step
of 131.55 cents should be especially welcome to Persian or Kurdish
ears, for example, attuned to maqam or dastgah styles where a step on
the order of 14:13 is very idiomatic. Here the mathematical noble
mediant between 14:13 and 27:25 -- or also 1:1 and 13:12, as Dave has
suggested -- happens to yield a fine melodic effect. I see no reason
not to use this mathematical formula, or any mathematical formula,
when it produces a result that fits the musical situation.

Dave, you are equally correct that the concept of Noble Intonation
was, if I am correct, conceived as especially applying to
_simultaneous_ intervals which represent regions of maximum complexity
between two simpler ratios. A classic illustration, of course, is the
pair of major thirds at a just 5:4 and 9:7, with the NI region located
somewhere around 422 cents.

Thus I warmly agree that it would be a misunderstanding of the
original NI concept to think that _any_ mathematical noble mediant
between a pair of integer ratios will produce an audible "NI" effect.
In fact, Oz's purpose seems to me quite different: to strike a kind of
artful compromise between two sizes for _melodic_ steps, 14:13 and
27:25. The mathematical noble mediant operation happens to produce a
most agreeable result -- although that wasn't its original purpose.

Dave, I would say that one of the hallmarks of really great concepts
and discoveries is that they may be used in various ways. Sometimes
applications not anticipated by the discoverer may arise -- for better
or worse -- and even be (mis)attributed to the discoverer.

Here I would say, "Oz -- congratulations on a fine maqam tuning!"; and
to Dave, "Indeed, while the formula was used here to pleasant effect,
people should understand that the NI concept which gave rise to the
formula is somewhat different."

With warmest wishes to you both, and all,

Margo Schulter
mschulter@calweb.com

[Brad]

> And to complete the picture of these several concerts, the rest of
> our first half (the 17th century half) was an Uccellini sonata in A
> minor, Venice 1649, which happened to have some B major triads in it
> (cadencing into E minor, repeatedly), and an F# major triad. And
> the other accompanied piece was a Schmelzer sonata in G minor from
> 1664, which happens to need the note A-flat. [And then he played
> the Biber unaccompanied Passacaglia, and then we went on to the
> Corelli as I mentioned above.]

Hi, Brad. Indeed this is a very interesting period, for me more or
less _ultra_-modern, and in one style of periodization we would say
that the 1640's might mark the end of the Manneristic era (1540-1640
or so) and the beginning of the Early Baroque. In another traditional
approach, the Baroque starts around 1600 with the rise of continuo
notation, opera, and oratorio, so the whole 17th century would be
included.

One small point, where I know that usages differ, although maybe not
so relevantly for the intonational questions we're considering: While
it's common to apply major/minor key labels to early 17th-century
music, I might prefer to follow the composer's designation, whatever
it is, for example a given tone or mode. Of course, if composers are
already using major/minor terminology, then that would be both
appropriate and very noteworthy -- for Uccellini, I'm not sure.

> Again: what would you have us do? 1/4 comma meantone doesn't really
> work for *any* of these pieces, unless we're supposed to put up with
> ridiculous-sounding B major and F# major in normal cadencing
> situations.

Actually my guess is that 1/4-comma meantone should be just fine, if
you use a tuning set which has accidentals with proper spellings for
sonorities like B-D#-F# and F#-A#-C#. This would often involve the
very authentic period choice of an instrument with more than 12 notes
per octave.

While an instrument with only a couple of split keys (G#/Ab, Eb/D#)
would address a lot of common situations, here we also need A#, which
might suggest an instrument with at least 17 notes per octave. A usual
_cembalo cromatico_ of the kind popular around Naples in the epoch of
Gesualdo, with 19 notes per octave likely tuned in 1/4-comma, would
neatly address the situation.

This is the kind of 19-note keyboard recommended by Praetorius, and
also taken by Fabio Colonna (1618) as the basis for tuning the first
19 notes of his 31 note _Sambuca Lincea_ evidently based on a cycle of
1/4-comma meantone closely approximating 31-EDO. Of course, Colonna is
championing a 31-note meantone cycle which goes back to Nicola
Vicentino (1555, 1561).

In 1666, Lemme Rossi discusses both extended 1/4-comma meantone and a
precise 31-EDO, with Christiaan Huygens in 1691 advocating "The New
Harmonic Cycle" based on 31-EDO, which he also shows differs only
minutely from the familiar 1/4-comma.

Of course, one could construct examples where extended 1/4-comma (or
31-EDO) would run into problems: specifically, if a composer relies on
enharmonic equivalence, as Vincenzo Galilei does in the late 16th
century specifically to show the "perfection" of 12-EDO and the lute
as an instrument customarily using it.

However, unless this kind of equivalence is demanded, my guess is that
the problem you are describing is not inherent to 1/4-comma, but
rather to instruments which don't cover a large enough range of it to
have available the needed accidentals for a given piece.

I realize, of course, that this is a rather obvious point; but since
period practices are such an important consideration in this thread,
maybe mentioning split-key accidentals isn't out of place.

This leaves the question, "What should we do if only a 12-note
keyboard is available?" That's an intriguing question. Of course, one
period solution (however controversial) is a regular tuning at or not
too far from 12-EDO, which Mersenne discusses in the 1630's, and Denis
heartily disapproves in his treatise on keyboard tuning and music
(1643, 1650). Doni tells the story that Frescobaldi recommended 12-EDO
for a new church organ which Doni persuaded those responsible to tune
in the usual meantone -- however accurate or otherwise this tale.

Another is some kind of temperament ordinaire based on something
around 1/4-comma. An interesting question is what context F#-A#-C#
appears in. If it's used as a cadential sonority leading to B, then a
bit of ornamentation might help (a la Schlick 1511). Mark Lindley
suggests that even an unaltered 1/4-comma diminished fourth at 32:25
(427 cents) might be used in ornamented contexts, although an
irregular scheme that ameliorates F#-A# to about Pythagorean could
further ease this dilemma.

Again, 1650 is really late for me, so I'd welcome comments from Tom or
others more conversant with this epoch.

Most appreciatively,

Margo Schulter
mschulter@calweb.com

> Another is some kind of temperament ordinaire based on something
> around 1/4-comma. An interesting question is what context F#-A#-C#
> appears in. If it's used as a cadential sonority leading to B, then a
> bit of ornamentation might help (a la Schlick 1511). Mark Lindley
> suggests that even an unaltered 1/4-comma diminished fourth at 32:25
> (427 cents) might be used in ornamented contexts, although an
> irregular scheme that ameliorates F#-A# to about Pythagorean could
> further ease this dilemma.

In the Uccellini piece we played, it had me sitting on B major for a
whole bar at a time while the violin played melodic/ornamental stuff,
and then E minor for the whole next bar. This happened several times.
Sure, I could have made up more ornamental stuff of my own during
some of that, and I probably did, but there's also the danger of
competing too much with the melodic part. And if I would have had to
do it every time there's a whole bar that is sounding cruddy, just
because a bad temperament has been set up...well, that's just not a
very good musical reason for that much ornamentation (in my opinion).
The ornamentation itself would have had so many wrong enharmonics in
it (if forced to use a 1/4 comma system), it would just be a mess.