Re: Schismas, meantone, well-temperaments (for Graham Breed)

Hello, there, Graham Breed and everyone, and please let me acknowledge
my real doubts about my ability to do full justice in replying to your
response at once rich with practical experience and offering a wealth
of new ideas with a relevance going far beyond the rather specialized
area of neo-Gothic music.

Your article (TD 1068:19) spurred me on to a post about 2-3-7
optimizations and neo-Gothic cadences in meantone, but that doesn't begin
to respond to all your valuable observations, many relating to themes such
as 11-limit harmony about which others can comment with a better basis in
theory and practice.

Your 41-note well-temperament looks to me like an idea whose time has
come, most appropriately at the opening of a new millennium. I hope
that others will join the dialogue, for example John deLaubenfels with
his enthusiasm for 11-limit and 13-limit harmony.

Here I'll try mainly to acknowledge some very valuable contributions
you're made, for example in clarifying the matter of "septimal
schisma" terminology, and to respond a bit further on your points
about optimization and neo-Gothic music in meantone.

However, this remains a very partial reply to ideas which I hope may
gain much attention and a receptive response from the alternative
tuning community at large, on this List and elsewhere.

> And that's another thing. When you take the 5s out, this is *the*
> septimal schisma. To reflect this, I suggest it be called the
> "Pythagorean-septimal schisma" or "2-3-7 schisma".

This is an excellent point, and you've convinced me that "2-3-7
schisma" is an ideal term because it communicates that we are talking
about the difference between the 2-3 or Pythagorean comma and the 3-7
or septimal comma. I really like it!

We could call the familiar schisma of ~1.95 cents the "2-3-5 schisma,"
and maybe the schisma at 5120:5103 the "3-5-7 schisma."

> But I find it conceptually easier to explain a temperament without
> using the complicated "bridge" intervals. I think of schismic
> temperament as being about commas. In the 5-limit case, the
> Pythagorean and syntonic commas are equivalent. The septimal comma
> 64:63 then joins the party. If 81:80 decides to leave, that's no
> problem!

With the 2-3-7 JI scheme, we could define it without using schismas at
all: it's simply two 12-note keyboards in Pythagorean tuning (Eb-G#) a
septimal comma apart.

> I don't even know the ratio for a Pythagorean comma offhand. I
> could work it out, of course. But all I need to know it that it's
> the amount by which 6 whole tones of 9:8 are sharp of a 2:1 octave.

In fact, that's one of the ways that Jacobus of Liege presents the
idea of the Pythagorean comma around 1325: six whole-tones of 9:8 give
us an interval called the _hexatonus_, _not_ equal to a pure octave.
The ratio of the comma is 531441:524288 (~23.46 cents).

Using the more modern device of rounded cents, we can approximate this
by reasoning that if 9:8 is about 204 cents, then six of these
whole-tones will be around 1224 cents -- a comma of about 24 cents. If
we take the 9:8 more accurately at 203.91 cents, then we get six of
these tones at 1223.46 cents -- and the comma at 23.46 cents, as
accurate an approximation as we're likely to need.

Curiously, this tends to get rounded off as 24 cents rather than the
slightly closer 23 cents, maybe a psychological and practical
attraction of an approximation in cents divisible by various
integers.

> In the same way, 12-equal is the borderline between meantone and
> schismic temperaments. So "meantone" would usually refer to scales
> generated by a fifth flatter than 7/12 octaves (although the
> definition is contentious :) and "schismic" to those with fifths
> sharp of 5/12 octaves.

Here I tend to use the term "negative tunings" for those with fifths
smaller than 700 cents, including historical meantones; and "positive
tunings" for those with fifths larger than 700 cents, including what I
tend to consider a rather small subset of "schismic temperaments."

However, your idea of a "schismic" temperament or tuning may be much
broader than mine because you're dealing with 11-limit harmony where
there may be more schismas to consider.

To me, a "schismic temperament" has the rather narrow meaning of a
tuning where the fifth is very slightly tempered to disperse either
the 2-3-5 schisma (e.g. Helmholtz/Ellis 1/8-skhisma, as they spell it,
with pure 5:4 thirds; or Sa/bat 1/9-schisma with pure 6:5 thirds); or
the 2-3-7 schisma (1/14-schisma for pure 7:4; 1/15-schisma for 7:6;
1/16-schisma for 9:7).

Additionally, there are "schismic untemperaments" such as two 12-note
keyboards a 3-7 or septimal comma apart (my 2-3-7 scheme), or a 3-5 or
syntonic comma apart, 81:80 or ~23.46 cents (for pure 2-3-5 ratios).

Of course, since you're dealing with more factors like 11, you have a
wider range of schismas and schismic temperaments to consider.

Let's discuss this more, because it gets into some fascinating
questions of 21st-century terminology where either 11-limit harmony or
neo-Gothic polyphony may call for a real reappraisal of 19th-century
concepts such as "positive/negative" or "schismic" tunings.

> If I remember these, I could try them. Although the difference
> between 1/14 and 1/15 schisma is likely to be much smaller than
> between either and Pythagorean intonation. 12:18 and 14:21 are
> both fifths, right? And 12:14 is 6:7. So these are fifth pairs a
> subminor third apart.

That's correct, and in the 2-3-7 system that is how they are played: a
fifth on each keyboard, for example 12:14:18:21 as E^3-G3-B^3-D4.

Also, in the most typical resolution, those two fifths proceed by
parallel motion, while the unstable 7-based intervals resolve by
contrary motion. This may be simplest in a 22-tET spelling, where all
these are regular intervals, and I'll also give it in my 2-3-7 JI
spelling:

D4 C4 D4 C4
B4 C4 B^3 C4
G3 F4 G3 F3
E3 F3 E^3 F3

22-tET 2-3-7 JI

> Usually I go with a 4:5:6:7 for testing, or 4:5:6:7:8 if my fingers
> stretch far enough.

For 7-limit, that seems a natural choice.

> I think a lot of the neo-Gothic concepts would work in meantone.
> You still have good-enough approximations to the 2-3-7 intervals.

Certainly, and I might add that 2-3-7 is only one side of neo-Gothic.
For example, from a 2-3-7 viewpoint, the diminished fourth in
1/4-comma at around 427 cents isn't an especially accurate 9:7; but
it's a perfectly fine neo-Gothic major third, somewhere between 17-tET
and 39-tET.

In fact, a major third of almost identical size occurs in a neo-Gothic
JI tuning I came up a few months ago based on pure 3:2 fifths and
14:11 major thirds. It's a 14:11 plus two commas of 896:891, the
amount by which the 14:11 exceeds the Pythagorean major third at 81:64
(about 9.69 cents).

As I discuss in the optimization article, if we want to combine
Renaissance and neo-Gothic styles in a single piece based on a single
tuning, then meantone is an _ideal_ solution, because for the
Renaissance sonorities we want those thirds close to pure ratios of 5.

For this I would go with two manuals in 1/4-comma tuning a diesis
apart, a favorite tuning of mine for "Xeno-Renaissance" generally. In
effect, it's a subset of Vicentino's 31-note meantone cycle on his
archicembalo of 1555.

> It happens that 1/4-comma meantone is both the optimal minimax
> 5-limit temperament, and the same optimum for the 7-limit. So there
> isn't the 7-5 dichotomy you get in schismic temperament where the
> two are optimised either side of Pythagorean. However, in meantone,
> the fifths are always bad relative to schismic if you optimize to
> either of these limits.

Once we modify the 2-3-7 optimization question by bringing ratios of 5
into it, I would say that meantone has the very important practical
advantage of giving us 5-limit thirds as regular intervals filling the
keyboard at the usual places.

A significant distinction: in neo-Gothic, "7-flavor" for me includes
the 9:7; but for many people "7-limit" means 7-odd, not including the
9:7 major third. If we're optimizing 2-3-5-7, not including the 9:7,
then I agree that meantone is comparably accurate for 5 and 7.

For a near-Pythagorean 2-3-5-7 solution (with or without 9:7), one
strategy might be to tune a fifth around 702.04 cents (~0.08 cents
wide), at which point ratios of 5 and 7 alike should be within about
2.72 cents of pure (5:4, 6:5, 7:4, 7:6, 9:7). The idea is balance out
the 2-3-5 schisma (~1.95 cents) and the 2-3-7 schisma (~3.80 cents) so
that 5-based and 7-based intervals are impure by about the same
amount.

> I think the fifths in, say, 31-equal are good enough for trines to
> function as stable chords. I find the tempering to be noticeable
> but tolerable. 7-based chords should be able to resolve onto trines
> the same as they do in a schismic temperament.

Certainly I'd agree: neo-Gothic progressions are recognizable,
acceptable, and intriguing in 1/4-comma meantone with the fifths
around 5.38 cents narrow; in 22-tET with the fifths around 7.14 cents
wide, a yet greater compromise; or, with the right timbres, in 20-tET
with trines at 0-720-1200 cents.

While Vicentino didn't design his archicembalo to play neo-Gothic
progressions, a 24-note version of his 38-note design makes these
intervals quite conveniently available -- about as simply or
intricately as one of my usual 24-note neo-Gothic tunings for ratios
of 2-3-7. There are a couple of examples in my post on neo-Gothic
progressions in meantone.

Also, from an historical point of view, I'd very much want to confirm
your point that meantone fifths are still stable concords, and can be
more conclusive than thirds in much early Renaissance keyboard music.

During the first epoch of meantone, say 1450-1500, thirds and sixths
were still typically considered "not-quite-conclusive" intervals, with
lots of cadences on sonorities identical to medieval trines. If fifths
weren't stable concords in meantone, there would have been lots of
problems.

In playing some Spanish vocal music around 1500 on a meantone
keyboard, I like the way that concluding fifth-plus-octave sonorities
(the term "trine" sounds a bit medievalistic for this Renaissance era)
have a bit of "bounce." Of course, meantone optimizes the sonorities
with thirds and sixths at most other points.

Also, 16th-century keyboard composers like Cabezon use lots of
sonorities with fifths and no thirds, and the tempering is no problem
at all for me. Mark Lindley likes the effect of these meantone fifths
in the lute music of Milan from around the 1530's, and finds this
tuning more pleasing than 12-tET (which theorists report was becoming
standard on the lute by around the middle of the century).

For neo-Gothic music, I would say that 1/4-comma meantone has fifths
tempered rather more heavily than in what I would consider the most
characteristic tunings (roughly Pythagorean to 17-tET), but
considerably less than in 22-tET, let along 20-tET. It's one more
alternative, and a very creative one.

> The near-pure thirds are something of a red-herring. Yes, either
> 5:4 or 6:5 can be pure in meantone, but so can either 7:6 or 7:4.
> The trade-off between meantone and schismic is the same whether you
> take 2-3-5 or 2-3-7. Schismic temperament will give you much purer
> intervals, but with a more complex scale.

Here I might agree with what you say while stating it a bit
differently from my point of view: a quasi-Pythagorean tuning fills
the keyboard manuals with regular Pythagorean-like intervals, and
meantone with regular meantone intervals.

For neo-Gothic music, I find that the Pythagorean and 7-based
intervals form a very nice and happy family, so that in effect a
24-note tuning for 2-3-7 JI gives me a regular Pythagorean scale plus
the special "7-flavor" steps, plus the ideal optimization.

For Renaissance music, meantone gives regular 5-limit intervals of a
conventional 16th-century style plus the 7-based intervals and other
treats such as near-11:9 neutral thirds (regarded as consonances by
Vicentino, who, however, found the near-7:6 and near-7:4 as leaning
toward dissonance, along with the near-9:7).

With either system, and I'd tune either in 24 notes, our 7-based
intervals are something other than regular thirds, sixths, and
sevenths, etc.

For a 2-3-7 approximation with real simplicity, I'd go for 22-tET,
where a 9:7 is a regular major third of four fifths up, etc., although
this is overall less accurate than either more intricate system --
with the fifth notably tempered more heavily than in meantone!

> I'm interested in how neo-Gothic would work in meantone. I haven't
> followed the details of the theory as you outlined it. I tend to
> switch off when I see harmonic progressions discussed, because I'm
> not familiar with the traditional theory. Perhaps a summary of
> characteristic progressions would be useful, or I could go back
> through the archives.

In my longer post, I give a few examples, including some which can be
played on a usual 12-note meantone keyboard with a range of Eb-G#.

Please let me maybe reassure you on one point about neo-Gothic music:
it doesn't really fit traditional theory as many people define it,
based mostly on 18th-19th century European harmony. It's more like
13th-14th century European theory, something quite different, with a
very different set of progressions.

Why don't I give URL's for my "Gentle introduction to neo-Gothic
progressions," and invite your questions or feedback. One thing I
tried to do was to explain some of the style without assuming
familiarity with medieval theory, but only readers can say how closely
I approached this ideal:

/tuning/topicId_15038.html#15038 (1/Pt 1)
/tuning/topicId_15630.html#15630 (1/Pt 2A)
/tuning/topicId_15685.html#15685 (1/Pt 2B)
/tuning/topicId_16134.html#16134 (1/Pt 2C)

> If enough intervals with 7 and 11 are involved, it may be possible
> to remove the Pythagorean ones altogether. So an 11-limit aware
> meantone notation could be used for variable pitched instruments.

From a neo-Gothic perspective, I want very much to keep those
Pythagorean intervals, which nicely contrast with the 7-based ones to
make a happy family; but for 11-limit styles, meantone could be the
ideal choice.

> I'm also wondering how much Common Practice harmony could work with
> 9-limit intervals. The occasional 6:7:9 minor triad, or 21:20
> leading tone, could do it a power of good.

Someone more informed about common practice -- "key-based era," I
might say, basically Corelli-Wagner or the like (say 1680-1900) --
might have better answers, and I suspect that John deLaubenfels might
be very supportive of this approach, since he loves the 6:7:9 for
settings of music from this era. It's interesting how these 7-based or
higher ratios can apply to different kinds of styles.

> Ah, right. So this is like a well-temperament for schismic scales.
> As such, it would favor some keys over others. Although mostly,
> yes, likely a conceptual difference.

Technically, yes, we could call RAST (Rational Adaptive Schisma
Tuning) with two Pythagorean keyboards a septimal comma apart a
"well-temperament" in the sense that G#-D# is a 3.80-cent septimal
schisma wider than pure, with the other 22 fifths all pure.

Within the base range of Eb-G#, it's a JI system for pure ratios of
2-3-7 (ratios of 7 formed from 14, 15, or 16 fifths up or down).

However, for Pythagorean-like intervals in the more remote part of the
gamut including the wide, 17-tET-like fifth in their chains, it is a
subtle kind of well-temperament, with such intervals altered by the
3.80-cent schisma. For example, the major third F#-A# (or F#-Bb^ in my
keyboard notation) is enlarged from a usual Pythagorean 81:64 (~407.82
cents) to around 411.62 cents, or a bit more than halfway to 29-tET.

Using my 2-3-7 schisma notation, I'd write this F#-A#', with the
apostrophe (') showing a note raised by a 2-3-7 schisma from its usual
Pythagorean position.

Here I might do well to add that for many people a "well-temperament"
implies circulation, which for this scheme would call for 106 notes;
but certainly the 24-note version involves a slight shift of "color"
for some more remote intervals because of the one 17-tET-like fifth.

>> Of course, as the phrase goes, this is a "device-specific"
>> issue. If we had a synthesizer in 1350-tET, then a regular
>> 1/14-schisma temperament with all fifths at 790 units would be
>> the way to go.

> You only need 135-tET for that.

Of course you're right, and I may have indulged in a bit of overly
obscure humor by picturing a new synthesizer standard dividing each
135-tET step into ten equal parts, or 1350-tET, for a resolution a bit
better than one cent.

By the way, on your point regarding precision vs. accuracy, I wonder
if there have been any accuracy tests on sythesizers by microtonal
advocates.

> 12 note well temperaments work as they do because some parts of the
> keyboard favour thirds, and some fifths. This compromise comes from
> the ideal tunings for 3:2 and 5:4 being either side of 12-equal. In
> the same way, a 41 note RAST might balance the 3 and 7 on one side,
> and the 11 on the other. Perhaps you could look at this in the
> light of your theories. (Or perhaps you already have.)

Actually, my first impressions is that this looks like a great idea
which is a bit outside of usual neo-Gothic theory, but potentially a
most creative solution for 11-limit musics.

For example, in neo-Gothic, a 24-note e-based tuning at ~704.61 cents
optimizes the ratios of 2-3-7, 11:7, and 11:8 _as they're typically
used in this musical style_, but not necessarily as they'd be used in
11-limit. The 11:8, for example, is a kind of special "Wolf fourth"
occurring between a diminished fourth and major sixth of a sonority
expanding to a trine like a usual major sixth sonority:

G#4 A4
Eb4 E4
B3 A3

In an 11-limit system of harmony, I suspect that these intervals might
be used in quite different ways, and that the e-based temperament
might be much less relevant than your neat 41-note well-temperament
idea, which looks really worthwhile to me in theory and practice.

Please let me conclude by saying that such a well-temperament could be
relevant to various musics, and has very much a "21st-century" ring to
it.

Most appreciatively,

Margo Schulter
mschulter@value.net

Margo Schulter" wrote:

> > I don't even know the ratio for a Pythagorean comma offhand. I
> > could work it out, of course. But all I need to know it that it's
> > the amount by which 6 whole tones of 9:8 are sharp of a 2:1
octave.
>
> In fact, that's one of the ways that Jacobus of Liege presents the
> idea of the Pythagorean comma around 1325: six whole-tones of 9:8
give
> us an interval called the _hexatonus_, _not_ equal to a pure octave.

It's the way I've seen it in books as well. Certainly the easiest to
remember.

> > In the same way, 12-equal is the borderline between meantone and
> > schismic temperaments. So "meantone" would usually refer to
scales
> > generated by a fifth flatter than 7/12 octaves (although the
> > definition is contentious :) and "schismic" to those with fifths
> > sharp of 5/12 octaves.
>
> Here I tend to use the term "negative tunings" for those with fifths
> smaller than 700 cents, including historical meantones; and
"positive
> tunings" for those with fifths larger than 700 cents, including
what I
> tend to consider a rather small subset of "schismic temperaments."

22-equal is a positive tuning, but not schismic.

> However, your idea of a "schismic" temperament or tuning may be much
> broader than mine because you're dealing with 11-limit harmony where
> there may be more schismas to consider.

11-limit harmony isn't part of the definition of schismic
temperaments, because there isn't one approximation that covers the
whole range of schismic tunings. The 5-limit approximation provides
the real definition, like with meantone. The 7-limit approximation is
fairly consistent in both, unlike the 22, 34, 46, ... family. In this
context, the 7-limit takes center stage. So 118-equal is schismic,
but not "septimal schismic".

> To me, a "schismic temperament" has the rather narrow meaning of a
> tuning where the fifth is very slightly tempered to disperse either
> the 2-3-5 schisma (e.g. Helmholtz/Ellis 1/8-skhisma, as they spell
it,
> with pure 5:4 thirds; or Sa/bat 1/9-schisma with pure 6:5 thirds);
or
> the 2-3-7 schisma (1/14-schisma for pure 7:4; 1/15-schisma for 7:6;
> 1/16-schisma for 9:7).

That's the one. It's implicit in Arabic music as well. I worked it
out from one of the files in Manuel's archive that is written in
Pythagorean, but turns out to be suspiciously close to a JI diatonic
scale.

> Additionally, there are "schismic untemperaments" such as two
12-note
> keyboards a 3-7 or septimal comma apart (my 2-3-7 scheme), or a 3-5
or
> syntonic comma apart, 81:80 or ~23.46 cents (for pure 2-3-5 ratios).

The schismic nature of these tunings is compelling, so they do belong
under that umbrella even though they aren't temperements. Getting
intervals so close to just leads to the temptation to make them just.
So if the logic of the keyboard is

> Of course, since you're dealing with more factors like 11, you have
a
> wider range of schismas and schismic temperaments to consider.
> Let's discuss this more, because it gets into some fascinating
> questions of 21st-century terminology where either 11-limit harmony
or
> neo-Gothic polyphony may call for a real reappraisal of 19th-century
> concepts such as "positive/negative" or "schismic" tunings.

In the 11-limit, it makes less sense to think of positive scales as a
coherent system. The mechanics of keyboard mappings can be
generalized, but the 11-limit intervals have to be thrown in ad-hoc.
I don't think there's any "grand unified theory" that covers 41, 29,
22, 46 and 17-equal, all of which work with some 11-limit intervals in
their own special way.

The full 11-limit does tend towards equal temperaments. There are so
many compromises, and the intervals get so complex, that you may as
well simplify it all by collapsing the tuning into one dimension. The
exception is that gift of a 2-3-6-11 approximation, which is almost
just but does need a lot of fifths to work.

However, you can invent new temperaments based on different subsets of
the 11-limit. The only ones I've found so far are the "neutral third
scales". They also have counterparts in Arabic music, and are
mentioned in that Carey/Clampitt paper that came out around when I was
looking at them. I explain them at
<http://x31eq.com/7plus3.htm>. The idea is that a perfect
fifth divides into two equal thirds, which can be identified with
11:9. You get a 7-note scale coming out, that fits the ETs 10, 17,
24, 31, 38, etc and 41. It also happens that, whereas the 2-3-11
approximation is common to the system, there are different mappings to
7 and 5.

> > I think a lot of the neo-Gothic concepts would work in meantone.
> > You still have good-enough approximations to the 2-3-7 intervals.
>
> Certainly, and I might add that 2-3-7 is only one side of
neo-Gothic.
> For example, from a 2-3-7 viewpoint, the diminished fourth in
> 1/4-comma at around 427 cents isn't an especially accurate 9:7; but
> it's a perfectly fine neo-Gothic major third, somewhere between
17-tET
> and 39-tET.

It's good enough for a 6:7:9 chord to work. The inverse, "major"
chord sound distinctly automitive, but perhaps it does in JI as well.

> > It happens that 1/4-comma meantone is both the optimal minimax
> > 5-limit temperament, and the same optimum for the 7-limit. So
there
> > isn't the 7-5 dichotomy you get in schismic temperament where the
> > two are optimised either side of Pythagorean. However, in
meantone,
> > the fifths are always bad relative to schismic if you optimize to
> > either of these limits.
>
> Once we modify the 2-3-7 optimization question by bringing ratios
of 5
> into it, I would say that meantone has the very important practical
> advantage of giving us 5-limit thirds as regular intervals filling
the
> keyboard at the usual places.
>
> A significant distinction: in neo-Gothic, "7-flavor" for me includes
> the 9:7; but for many people "7-limit" means 7-odd, not including
the
> 9:7 major third. If we're optimizing 2-3-5-7, not including the 9:7,
> then I agree that meantone is comparably accurate for 5 and 7.

Yes, I used a sleight of hand to bring "7-limit" tuning into
contention. Meantones aren't as good in the full 9-limit, but the
optimum is around the same point, as discussed in the other thread.

> For a near-Pythagorean 2-3-5-7 solution (with or without 9:7), one
> strategy might be to tune a fifth around 702.04 cents (~0.08 cents
> wide), at which point ratios of 5 and 7 alike should be within about
> 2.72 cents of pure (5:4, 6:5, 7:4, 7:6, 9:7). The idea is balance
out
> the 2-3-5 schisma (~1.95 cents) and the 2-3-7 schisma (~3.80 cents)
so
> that 5-based and 7-based intervals are impure by about the same
> amount.

Oh no! I've been into this before. For whatever reason, balancing 5
and 7 doesn't work. So you optimize for 7, and let 5 take care of
itself.

> During the first epoch of meantone, say 1450-1500, thirds and sixths
> were still typically considered "not-quite-conclusive" intervals,
with
> lots of cadences on sonorities identical to medieval trines. If
fifths
> weren't stable concords in meantone, there would have been lots of
> problems.

So this was a crossover period, which in retrospect looks like a
mixture of Medieval and more modern harmony?

> Also, 16th-century keyboard composers like Cabezon use lots of
> sonorities with fifths and no thirds, and the tempering is no
problem
> at all for me. Mark Lindley likes the effect of these meantone
fifths
> in the lute music of Milan from around the 1530's, and finds this
> tuning more pleasing than 12-tET (which theorists report was
becoming
> standard on the lute by around the middle of the century).

Isn't there a lute tutorial with a section on tuning that says simply
"tune the strings as tight as they can get without breaking"? Even if
the technology had moved on from there, it's possible that the lutes
wouldn't have kept perfect equal temperament for long, and so the
effect of the fifths might have been more like those of a precisely
tuned meantone harpsichord.

> For neo-Gothic music, I find that the Pythagorean and 7-based
> intervals form a very nice and happy family, so that in effect a
> 24-note tuning for 2-3-7 JI gives me a regular Pythagorean scale
plus
> the special "7-flavor" steps, plus the ideal optimization.
>
> For Renaissance music, meantone gives regular 5-limit intervals of a
> conventional 16th-century style plus the 7-based intervals and other
> treats such as near-11:9 neutral thirds (regarded as consonances by
> Vicentino, who, however, found the near-7:6 and near-7:4 as leaning
> toward dissonance, along with the near-9:7).

Yes, it does seem that the 7- and 11-limit pull in the same direction.
So the "7" and "11" flavour would become equivalent in 31-equal. And
the 3-flavour would be replaced by the 5-flavour. So there's no way
of getting away from the 5-limit intervals in meantone. But there's
still a contrast between "neat" and "flavoured".

> > I'm interested in how neo-Gothic would work in meantone. I
haven't
> > followed the details of the theory as you outlined it. I tend to
> > switch off when I see harmonic progressions discussed, because I'm
> > not familiar with the traditional theory. Perhaps a summary of
> > characteristic progressions would be useful, or I could go back
> > through the archives.
>
> In my longer post, I give a few examples, including some which can
be
> played on a usual 12-note meantone keyboard with a range of Eb-G#.
>
> Please let me maybe reassure you on one point about neo-Gothic
music:
> it doesn't really fit traditional theory as many people define it,
> based mostly on 18th-19th century European harmony. It's more like
> 13th-14th century European theory, something quite different, with a
> very different set of progressions.

There doesn't seem to be as strong a contrast as some people suggest.
The progressions may be different, but they're still progressions.
And there's still the idea of particular intervals resolving in
particular directions. Such ideas, from what I've seen, are unique to
the European tradition. Not that I don't belong to that tradition,
but I don't have much education in this area. I am trying to learn,
at my own speed.

> Why don't I give URL's for my "Gentle introduction to neo-Gothic
> progressions," and invite your questions or feedback. One thing I
> tried to do was to explain some of the style without assuming
> familiarity with medieval theory, but only readers can say how
closely
> I approached this ideal:
>
> /tuning/topicId_15038.html#15038 (1/Pt 1)
> /tuning/topicId_15630.html#15630 (1/Pt 2A)
> /tuning/topicId_15685.html#15685 (1/Pt 2B)
> /tuning/topicId_16134.html#16134 (1/Pt 2C)

Yes, they can always be listed again. I did find them and go through
them after sending that original message.

It seems the general case is for a 7th chord on the VIIth degree to
resolve onto a 3-limit chord on the tonic. And all parts move by
either a tone or semitone. It looks like each chord could be given a
meantone tuning, and the flavoured versions tend to have those smaller
semitones and larger tones.

I've written the examples onto a few cards, and I'll try them with
different tunings sometime.

> > If enough intervals with 7 and 11 are involved, it may be possible
> > to remove the Pythagorean ones altogether. So an 11-limit aware
> > meantone notation could be used for variable pitched instruments.
>
> From a neo-Gothic perspective, I want very much to keep those
> Pythagorean intervals, which nicely contrast with the 7-based ones
to
> make a happy family; but for 11-limit styles, meantone could be the
> ideal choice.

Yes, there's no way you can fake Pythagorean intervals in meantone
because they don't exist.

> For example, in neo-Gothic, a 24-note e-based tuning at ~704.61
cents
> optimizes the ratios of 2-3-7, 11:7, and 11:8 _as they're typically
> used in this musical style_, but not necessarily as they'd be used
in
> 11-limit. The 11:8, for example, is a kind of special "Wolf fourth"
> occurring between a diminished fourth and major sixth of a sonority
> expanding to a trine like a usual major sixth sonority:
>
> G#4 A4
> Eb4 E4
> B3 A3
>
> In an 11-limit system of harmony, I suspect that these intervals
might
> be used in quite different ways, and that the e-based temperament
> might be much less relevant than your neat 41-note well-temperament
> idea, which looks really worthwhile to me in theory and practice.

There isn't any other theory of 11-limit music I know of that says
anything about how such intervals should behave. So, if you're
finding progressions that work in one context, they might be adaptable
to another. The B-Eb is a like B-D# and a 5:4 major third. Then
Eb-G# is an 11:8, and the B-G# is the major sixth. Unfortunately, as
that's a Pythagorean interval, we don't end up with a genuine 11-limit
chord. I see.

However, if we move the major sixth from the 3- to 5-flavour, the
major third becomes neutral. That could be:

D/ E
A B
F/ E

This progression can be played on the white notes of my neutral third
mapping. It looks like it could be promising as a cadence. The
"11-limit" chord still can't be made just, but it does contain only
approximations to 11-limit intervals. And it also contains the two
intervals considered dissonant in this context -- the 11:8 fifth and
the 6:5 third (or 5:3 here). So, it's worth a look.

Graham

Rummaging around the archives, I found this:

--- In tuning@y..., "M. Schulter" <MSCHULTER@V...> wrote:

/tuning/topicId_18146.html#18146

> Date: Tue Jan 30, 2001 3:31pm
>
>> [Graham Breed:]
>> I don't even know the ratio for a Pythagorean comma offhand.
>> I could work it out, of course. But all I need to know it
>> that it's the amount by which 6 whole tones of 9:8 are sharp
>> of a 2:1 octave.
>
>
> In fact, that's one of the ways that Jacobus of Liege presents
> the idea of the Pythagorean comma around 1325: six whole-tones
> of 9:8 give us an interval called the _hexatonus_, _not_ equal
> to a pure octave. The ratio of the comma is 531441:524288
> (~23.46 cents).
>
> Using the more modern device of rounded cents, we can
> approximate this by reasoning that if 9:8 is about 204 cents,
> then six of these whole-tones will be around 1224 cents --
> a comma of about 24 cents. If we take the 9:8 more accurately
> at 203.91 cents, then we get six of these tones at 1223.46
> cents -- and the comma at 23.46 cents, as accurate an
> approximation as we're likely to need.

Hello Margo,

I suppose, since you seem to be quite familiar with Boethius,
that you are probably aware that this demonstration of the
"Pythagorean Comma" (in quotes because I believe its recognition
is likely far older than Pythagoras's) dates from long before
Jacobus, c. 1325. But if not, and for those who don't know,
here's an investigation of it.

The earliest written documentation I know of which
presents this proposition is the pseudo-Euclidian
_Sectio Canonis_, Proposition 9; in modern edition:
Barker 1989, p. 199.

This treatise cannot be dated with absolute accuracy,
but seems most likely to have been written c. 100 AD
during the neo-Pythagorean revival. It may possibly
date back as far as c. 300 BC, near the time of Euclid.

The proposition was elaborated by Boethius c. 505 AD.
Books 1 to 4 of his treatise are apparently a Latin
translation of a lost treatise by Nicomachus, and the
fragmentary Book 5 begins a translation of Ptolemy.

After the "Carolingian Renaissance" of c. 800 AD which
"rediscovered" Boethius's treatise and entrenched it
as the standard music-theory reference, it became
an integral part of European music-theoretic knowledge
until Boethius's treatise was supplanted by the actual
Greek sources in the late 1400s. So Boethius is the
most likely source for Jacobus de Liege's reference
to it in c. 1325.

The following table is that given in Boethius 2.31
(in modern edition: Bower, p. 86).

1 8 64 512 4096 32768 262144
> 32768
294912
> 36864
331776
> 41472
373248
> 46656
419904
> 52488
472392
> 59049
531441

Boethius explains (using different terminology) that the
numbers along the horizontal row are powers of 8, and that
the numbers on the right are the "eighth part" of those
in the column immediately preceding, and are to be successively
added to them. Thus,

262144 / 8 = 32768, and 262144 + 32768 = 294912, etc.

Boethius then explains that 262144 * 2 = 524288,
and 531441 is close to, but obviously *not* equal to,
524288. Although the ratio of these two numbers is not
given a name, neither here nor in _Sectio Canonis_, it
is none other than the Pythagorean Comma.

(Boethius recapitulates this with less explanation
in 4.2 [Bower, p. 121-122], at the end of his near-quoting
from _Sectio Canonis_ Propositions 1 - 9.)

Taking advantage of modern mathematics this can be simplified.

Using 8 and 9 as factors gives a clear presentation:

8^0 8^1 8^2 8^3 8^4 8^5 8^6[* 9^0]
> 8^5 * 9^0
8^5 * 9^1
> 8^4 * 9^1
8^4 * 9^2
> 8^3 * 9^2
8^3 * 9^3
> 8^2 * 9^3
8^2 * 9^4
> 8^1 * 9^4
8^1 * 9^5
> 8^0 * 9^5
[8^0 *]9^6

It's easy to see here that if 8 and 9 are our factors,
the exponents progress according to the vector addition
formula |-1 1|, giving a series of 6 successive 9:8 ratios.

In prime-factor notation:

2^0 2^3 2^6 2^9 2^12 2^15 2^18[* 3^0]
> 2^15 * 3^0
2^15 * 3^2
> 2^12 * 3^2
2^12 * 3^4
> 2^9 * 3^4
2^9 * 3^6
> 2^6 * 3^6
2^6 * 3^8
> 2^3 * 3^8
2^3 * 3^10
> 2^0 * 3^10
[2^0 *]3^12

Here the vector addition formula is |-3 2|.
(Can someone please explain how the actual addition works
in this case?)

Now, for the benefit of further historical research...

In Base-60 numbering system:

1 8 1,4 8,32 1,8,16 9,6,8 1,12,49,4
> 9,6,8
1,21,55,12
> 10,14,24
1,32,9,36
> 11,31,12
1,43,40,48
> 12,57,36
1,56,38,24
> 14,34,48
2,11,13,12
> 16,24,9
2,27,37,21

(1,12,49,4) * 2 = 2,25,38,8

So in base-60 the Pythagorean comma would thus be:

(2,27,37,21) / (2,25,38,8)

= 1,0,49,6,56,42,41,6,32,...

= ~ 1,0,49,7 ...a very good approximation

I presented the base-60 notation because I have a strong
hunch that the "Pythagorean comma" was known to the Sumerians.

Are there any Assyriologists or Sumerologists out there
who recognize this progression of numbers? Can someone
please forward this to Siemen Terpstra? Or to a Sumerology
list? (I'm sure one exists somewhere on the internet.)

REFERENCES
----------

Boethius. c. 505.
_De institutione musica_.

Barker, Andrew. 1989.
Greek Musical Writings, vol. 2: Harmonic and Acoustic Theory.
Cambridge University Press, Cambridge.

Bower, Calvin. 1989.
Boethius's _Fundamentals of Music_.
English translation of Boethius c. 505.
Yale University Press, New Haven.

-monz
http://www.monz.org
"All roads lead to n^0"