Yahoo Tuning Groups Ultimate Backup tuning 2D tunings in use?

Hi there.

A few days ago, I was wondering about the progress of the idea of 2D (or rank two) tunings over time. The oldest ever record of a 2D tuning in use, AFAIK, is the use of Pythagorean tuning. And then, many centuries later, there were meantone temperaments which are also 2D tunings. And then, some centuries later again, there's now the huge set of 2D temperaments, some of which we're still experimenting with. The thing is that I have a bit of chaos in my knowledge. Most importantly, I'm not sure how the idea of 2D tunings evolved over time and I would be quite surprised if it had "frozen" in the meantone form for such a long time and then turned into such an "outburst" of 2D temperaments at "one particular point in time", simply said. What's more, as early as in the 18th century, there were suggestions for 3D JI models (like the Euler's monochord scale) along with that.
What do you think?

Petr

🔗Chris Vaisvil <chrisvaisvil@...>

Please tell me what is a 2D or 3D tuning?

This is the first time I heard mention of this - or I forgot -

Thanks,

Chris

2009/3/28 Petr Pařízek <p.parizek@...>

> Hi there.
>
> A few days ago, I was wondering about the progress of the idea of 2D (or
> rank two) tunings over time. The oldest ever record of a 2D tuning in use,
> AFAIK, is the use of Pythagorean tuning. And then, many centuries later,
> there were meantone temperaments which are also 2D tunings. And then, some
> centuries later again, there's now the huge set of 2D temperaments, some of
>
> which we're still experimenting with. The thing is that I have a bit of
> chaos in my knowledge. Most importantly, I'm not sure how the idea of 2D
> tunings evolved over time and I would be quite surprised if it had "frozen"
>
> in the meantone form for such a long time and then turned into such an
> "outburst" of 2D temperaments at "one particular point in time", simply
> said. What's more, as early as in the 18th century, there were suggestions
> for 3D JI models (like the Euler's monochord scale) along with that.
> What do you think?
>
> Petr
>
>
>

🔗Carl Lumma <carl@...>

Hi Petr,

> A few days ago, I was wondering about the progress of the idea
> of 2D (or rank two) tunings over time. The oldest ever record
> of a 2D tuning in use, AFAIK, is the use of Pythagorean tuning.
> And then, many centuries later, there were meantone temperaments
> which are also 2D tunings. And then, some centuries later again,
> there's now the huge set of 2D temperaments, some of which we're
> still experimenting with. The thing is that I have a bit of
> chaos in my knowledge. Most importantly, I'm not sure how the
> idea of 2D tunings evolved over time and I would be quite
> surprised if it had "frozen" in the meantone form for such a
> long time and then turned into such an "outburst" of 2D
> temperaments at "one particular point in time", simply said.
> What's more, as early as in the 18th century, there were
> suggestions for 3D JI models (like the Euler's monochord scale)
> along with that. What do you think?

Interesting question. Actually I think there are two questions
here; one about the evolution of just intonation harmony, and
another about the evolution of temperaments used to approximate
it.

First, the evolution of harmony. We know most musical
traditions simly didn't employ it, at least not in an organic
fashion. Outside of Europe and prior to the 12th century, music
was primarily monophonic or heterophonic.

Organum was the birth of 3-limit harmony, which is a 1-D system.
http://en.wikipedia.org/wiki/Organum
About 250 years later, the 5-limit was introduced:
http://en.wikipedia.org/wiki/Contenance_Angloise

Though Euler and others theorized about the 7-limit, tetradic
music didn't emerge in practice until the late 19th century,
in the French impressionist and African-American forms.
(I'm unaware of direct cross-pollination between these schools
prior to Gershwin, but suspect it was taking place from the
outset).

However it is hard to justify these styles as 7-limit styles,
or at least, their principle tetrads are not 7-limit o- and
utonalities. Rather, they are the most consonant tetrads found
in 12-ET -- approximations of 8:10:12:15, 10:12:15:18, 4:5:6:7,
5:6:7:9, among others.

Harry Partch is apparently the first musician to systematically
employ the 11-limit.

Now, about the rank of temperaments supporting this music.
Temperaments by definition must be of lower rank than the
harmony they support. So the above gives upper bounds on which
temperaments were in use. Chains of pure (or not systematically
tempered) 5ths were probably the most common tuning systems
throughout the world. Certain African and Eastern traditions
may have been employing 5-, 7-, and 12-ET for several hundred
years. In Europe, the first extant description of a systematic
tempering of these 5ths dates from the late 1500s; about
150 years after 5-limit triads were introduced in song.

I expect there were a string of theorists noticing non-meantone
temperings of the chain of 5ths (Huygens, Newton ... Helmholtz),
but the first thorough inventory of linear temperaments with
5th-based generators I know of was due to Bosanquet (1876).
Fokker was apparently the first to understand tempering as
reducing the rank of just intonation. Erv Wilson was apparently
one of the first to explore linear temperaments with non-5th
generators. Then, the recent explosion you mention, which
took place on this mailing list, in which the full universe of
temperaments was delineated (no pun intended). Paul Erlich,
Paul Hahn, Graham Breed, Dave Keenan, and Gene Ward Smith were
especially instrumental in this between about 1997 and 2006.

Does this answer your questions?

-Carl

🔗Carl Lumma <carl@...>

🔗Claudio Di Veroli <dvc@...>

Carl Lumna wrote:
>Hi Petr,
> A few days ago, I was wondering about the progress of the idea
> of 2D (or rank two) tunings over time. The oldest ever record
> of a 2D tuning in use, AFAIK, is the use of Pythagorean

Your historical synthesis is a welcome introduction to "non-temperati",
Carl.
Unlike other writings on the matter (e.g. mine which specialise in the
Western classical tradition)
yours add interesting insights into non-Western and modern music.
I have two comments:

> In Europe, the first extant description of a systematic
> tempering of these 5ths dates from the late 1500s;

You mean early 1500s (Schlick 1511, Aron 1516 and 1523)

> I expect there were a string of theorists noticing non-meantone
> temperings of the chain of 5ths (Huygens, Newton ... Helmholtz),
> but the first thorough inventory of linear temperaments with
> 5th-based generators I know of was due to Bosanquet (1876).

Rightly so! Temperaments of most fifths in non-meantone ways (i.e.
"irregular" temperaments) were introduced in the late 1600's, first as
meantone modifications, then as Pythagorean modifications, but only in the
1800's were their given the correct theoretical context.

Kind regards

Claudio

http://temper.braybaroque.ie/

🔗Graham Breed <gbreed@...>

Petr Pařízek wrote:
> Hi there.
> > A few days ago, I was wondering about the progress of the idea of 2D (or > rank two) tunings over time. The oldest ever record of a 2D tuning in use, > AFAIK, is the use of Pythagorean tuning. And then, many centuries later, > there were meantone temperaments which are also 2D tunings. And then, some > centuries later again, there's now the huge set of 2D temperaments, some of > which we're still experimenting with. The thing is that I have a bit of > chaos in my knowledge. Most importantly, I'm not sure how the idea of 2D > tunings evolved over time and I would be quite surprised if it had "frozen" > in the meantone form for such a long time and then turned into such an > "outburst" of 2D temperaments at "one particular point in time", simply > said. What's more, as early as in the 18th century, there were suggestions > for 3D JI models (like the Euler's monochord scale) along with that.
> What do you think?

There has been a kind of explosion over the past decade. It hasn't rocked the world yet but probably more of us are using different rank 2 temperaments now than ever before. That's probably a result of the internet. It lets the minority who might be interested in this kind of thing get in touch with each other. Retunable digital synthesizers help as well.

Yes, there are old records of Pythagorean tuning, from Mesopotamia, China, and the actual Pythagoreans. The idea persisted around Eurasia. Some of the references could be taken as implying schismatic temperament but there was nothing explicit until Riemann, or somebody, in the 19th Century.

It certainly looks like a string of great thinkers -- including Euler and Newton -- studied mathematical music theory without discovering any rank 2 tunings beyond meantone and maybe schismatic. I've compared the temperament explosion to a paradigm shift. During the meantone era, European theorists treated music as following meantone or 5-limit JI rules. Any conflicting system would have been dismissed as useless or unmusical. So, really, the idea was frozen -- meantone in Europe and Pythagorean outside.

You could maybe treat Vicentino as an exception. Though he didn't describe it this way, his enharmonic system is really rank 2 with a neutral third generator.

If anybody has other counter examples, let's see them!

Back to the 19th Century. A group of German theorists, including Riemann and most notably Helmholtz, understood schismatic temperament. They were also in touch with Bosanquet, who made schismatic organs, and developed a system of classifying equal temperaments according to fifth generators. Still, though, at the end of the 19th century rank 2 temperaments, and tunings in use, were exclusively variations of Pythagorean or meantone fifths.

Next is Shohe Tanaka. He obviously wasn't German but you can still put him in the German school. He showed that a kleisma "unison vector" (without that name) leads to a minor third generator. But this was only a step towards 53 note equal temperament.

The kleisma was rediscovered by Larry Hanson, who designed a keyboard around the implied temperament. Around the same time we have Erv Wilson adding more prime approximations to meantone and schismatic and designing keyboards around them and George Secor publishing an article on miracle temperament (without the name) as well as considering and rejecting other systems such as magic. I think these people constitute a school of thought. They knew enough about rank 2 temperaments to have started the explosion but interest seems to have dwindled.

There are other (obscure) examples before the internet school. I don't know much about them though. There's Negri (19&29) published in 1986. O'Connell's phi tuning, currently under discussion, is essentially rank 2 although he equalized it. Balzano and Zweifel I think used rank 2 constructions to get scales from equal temperaments. Rothenberg (earlier) did something similar as well but didn't publish.

Then there's Paul Erlich's paper on what we now call pajara. This is important because it describes a system where the period divides the octave (equivalence interval). A full theory of rank 2 temperaments has to consider such cases but they'd been ignored before. A group of us connected by the internet took it from there.

Graham

🔗Chris Vaisvil <chrisvaisvil@...>

I'll read it - thanks.

At this moment I don't know that the "D" stands for

On Sat, Mar 28, 2009 at 6:47 PM, Carl Lumma <carl@...> wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "Carl Lumma"
> <carl@...> wrote:
> >
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@> wrote:
> > >
> > > Please tell me what is a 2D or 3D tuning?
> > >
> > > This is the first time I heard mention of this - or I forgot -
> > >
> > > Thanks,
> > >
> > > Chris
> >
> > Chris, the answer is slightly involved, but has been given to
> > you several times in the past, by myself and others, both onlist
> > and offlist. What are we doing wrong?
>
> Have you read the introductory materials I referred you to?
> Such as,
>
> http://lumma.org/tuning/erlich/erlich-tFoT.pdf
>
> Maybe we can focus questions around what is presented therein.
>
> -Carl
>
>
>

🔗Graham Breed <gbreed@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Carl Lumma <carl@...>

Hi Claudio,

> Your historical synthesis is a welcome introduction to
> "non-temperati", Carl.
> Unlike other writings on the matter (e.g. mine which specialise
> in the Western classical tradition)
> yours add interesting insights into non-Western and modern music.
> I have two comments:

Success! :)

> > In Europe, the first extant description of a systematic
> > tempering of these 5ths dates from the late 1500s;
>
> You mean early 1500s (Schlick 1511, Aron 1516 and 1523)

I was dating this from Zarlino, but I think you're right.
And forgive me if you posted about this recently, but if you
have any more information on these three citations it would
be greatly appreciated!

> > I expect there were a string of theorists noticing non-meantone
> > temperings of the chain of 5ths (Huygens, Newton ... Helmholtz),
> > but the first thorough inventory of linear temperaments with
> > 5th-based generators I know of was due to Bosanquet (1876).
>
> Rightly so! Temperaments of most fifths in non-meantone ways
> (i.e. "irregular" temperaments) were introduced in the late
> 1600's, first as meantone modifications, then as Pythagorean
> modifications, but only in the 1800's were their given the
> correct theoretical context.

Well, here I was not referring to single irregular temperaments
with differing 5ths, but to separate regular temperaments with
different 5ths. Bosanquet looked at where approximations to
higher harmonics (major 3rds and minor 7ths for instance) could
be found in the chain of 5ths as all the 5ths were varied in
size. He classified temperaments this way:

Triply negative 9, 21, 33, 45, 57, 69, 81, 93, 105, 117
Doubly negative 2, 14, 26, 38, 50, 62, 74, 86, 98, 110
Singly negative 7, 19, 31, 43, 55, 67, 79, 91, 103, 115
Zeroly positive 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
Singly positive 5, 17, 29, 41, 53, 65, 77, 89, 101, 113
Doubly positive 10, 22, 34, 46, 58, 70, 82, 94, 106, 118
Triply positive 3, 15, 27, 39, 51, 63, 75, 87, 99, 111

This chart is actually due to Graham Breed, and I don't know
if Bosanquet went all the way out to triply negative and
triply positive, but the idea is the same: to classify
temperaments according to the size of the pythagorean comma,
in steps of the temperament.

By the way, I said that Wilson was the first to rigorously
consider chains of intervals other than the 5th, but this
distinction may belong to Shohe Tanaka, who apparently saw
53-ET as a chain of minor thirds:

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

-Carl

🔗Carl Lumma <carl@...>

🔗Petr Parízek <p.parizek@...>

Hi Carl?

Thanks a lot for your suggestions.

You wrote:

Were you referring to any particular examples when you were saying this? I know next to nothing about Bosanquet.

> Erv Wilson was apparently one of the first to explore
> linear temperaments with non-5th generators.

I'm surprised to hear this; I'd love to know more -- anyone who can suggest some web links?

> Does this answer your questions?

Surely it does.

Petr

🔗Claudio Di Veroli <dvc@...>

Thanks for the detail Carl.

Your comments are most interesting indeed.

>triply positive, but the idea is the same: to classify
>temperaments according to the size of the pythagorean comma,
>in steps of the temperament.

Yes indeed. One unfortunate consequence of the late-19th studies in
temperament was the wrong belief that historical temperaments could be
suitably classified by Pythagorean comma deviations.
Numerically of course you can, but conceptually, so many historical tunings
relate to Syntonic comma (all the meantones and also many irregular circular
ones as well).
The final consequence is that they proposed the "Temperament Unit" (TU) as a
measure instead of the Cent: as most of us know, the TU it is 1/720 of a
Pythagorean comma.
Some modern writers have adopted the TU (e.g. Lehman), and the undesirable
effect is that temperaments based on Pyth.comma fractions look very neat
because their fifths can all be precisely described in integer TU's, while
all the non-Pyth.comma-based temperaments (which are in no way "inferior" or
"ill-defined") look awful in their infinite decimals.

The Cent is much more "objective" (if we forget the ET bias), as it is
1/1200 of the octave, and the octave is the simplest consonant interval,
thus an "absolute" which is not strictly related to any temperament (forget
about octave stretching in pianos which is a very particular case).

Kind regards,

Claudio

http://temper.braybaroque.ie/

🔗Petr Pařízek <p.parizek@...>

🔗Claudio Di Veroli <dvc@...>

> the first to rigorously consider chains of intervals other than the 5th

Strictly speaking, the first goes back to Salinas (1577)!
After Vicentino proposed the ETS 31 by tuning chains of fifths (and checking
for pure thirds), Salinas included in his treatise a few observations on
Vicentino's system.
Salinas showed mathematically that, 31 being prime, the 31-division could be
tuned by 31 successive intervals of any fixed size, provided that the size
was a multiple of the microtone.
(I took this from Barbieri, Enharmonic Instruments, Rome 2008, p.308, much
easier than to read my Salinas's facsimile in Latin!).
Obviously only intervals near to pure ones would be used, because of the
relative ease of gauging their slight temperament.
Thus Salinas's suggestion directly implied that any of the following
intervals, suitably tempered from pure in order to fit an integer number of
microtones, could be used to tune the 31-division by ear:

Pure ratio-name-microtones
3/2-Fifth-18
4/3-Fourth-13
5/4-Major Third-10
6/5-Minor Third-8
5/3-Major Sixth-23

Salinas also found that "the circulation of any of the intervals is
completed after 31 repeats, covering a number of octaves equal to the number
of parts(microtones) making up the interval itself.
This is not immediately obvious to the tuner, though it is mathematically
easy to prove.

The 31 division saw many instruments (not only keyboards) being built to it
in the 17th century.
Most of the sources show that they were tuned by either tempered fifths or
else using a monochord divided into 31 equal microtones.

Kind regards,

Claudio

🔗Graham Breed <gbreed@...>

Petr Pařízek wrote:
> > Graham wrote:
> >> You could maybe treat Vicentino as an exception. Though he
>> didn't describe it this way, his enharmonic system is really
>> rank 2 with a neutral third generator.
> > This sounds fascinating to me. Though I’m not sure what you’re talking about, I > assume this means mapping 3/2 to 2 generators but I don’t know how the other > primes would be mapped there -- or, more precisely, which intervals should be > tempered out in this system.

It's a 5-limit system in practice. He talked about other ratios, including 11:9 for the neutral third, but he didn't use them as consonances and he (or his printer) didn't always get them right.

It was implemented with two manuals tuned a "diesis" apart. If the standard tuning was roughly quarter comma meantone, the result would be close to 31-equal and he sometimes used equivalences from it.

The enharmonic scale has 24 notes. You can get that by taking a 12 note scale and transposing it by a diesis. The result is that chromatic semitones get split into two equal parts, but diatonic semitones are split into a diesis and a chromatic semitone (which he called a different kind of diesis I think).

You can get the same scale as a single chain of neutral thirds. Usually, a fifth divides into a major and minor third. The difference between them is a chromatic semitone, which is divided into equal parts to give the neutral third.

If you were to tune the fifth to 12-equal, the neutral third would require 24-equal. And the enharmonic scale becomes an equal quartertone scale. If you wrote music with strict spelling it could be played in a range of temperaments, much like a linear temperament class. But strictly speaking it isn't a temperament because not every interval is an approximation of a 5-limit ratio.

There's also a maximally even interpretation. The enharmonic is 24 notes taken from 31.

Graham

🔗Aaron Andrew Hunt <aaronhunt@...>

Hi Petr.

Here is a summary chart of the historical path you describe:

<http://www.h-pi.com/theory/huntsystem2.html#3>

In the text I clearly state that I am not an historian. Claudio,
if you are reading, perhaps you can critique the dates I use
in this table?

Cheers,
Aaron
=====

--- In tuning@yahoogroups.com, Petr PaÅÃzek <p.parizek@...> wrote:
>
> Hi there.
>
> A few days ago, I was wondering about the progress of the idea of 2D (or
> rank two) tunings over time. The oldest ever record of a 2D tuning in use,
> AFAIK, is the use of Pythagorean tuning. And then, many centuries later,
> there were meantone temperaments which are also 2D tunings. And then, some
> centuries later again, there's now the huge set of 2D temperaments, some of
> which we're still experimenting with. The thing is that I have a bit of
> chaos in my knowledge. Most importantly, I'm not sure how the idea of 2D
> tunings evolved over time and I would be quite surprised if it had "frozen"
> in the meantone form for such a long time and then turned into such an
> "outburst" of 2D temperaments at "one particular point in time", simply
> said. What's more, as early as in the 18th century, there were suggestions
> for 3D JI models (like the Euler's monochord scale) along with that.
> What do you think?
>
> Petr
>

🔗Claudio Di Veroli <dvc@...>

> Here is a summary chart of the historical path you describe:
<http://www.h- <http://www.h-pi.com/theory/huntsystem2.html#3>
pi.com/theory/huntsystem2.html#3>

> In the text I clearly state that I am not an historian.
> Claudio, if you are reading, perhaps you can critique the dates I use
> in this table?

With pleasure Aaron, first because your "Traditional System" table is very
similar to one published in my first book of 1978, second because it is an
excellent table.
These are my suggested Approximate Years:

1- OK

2 - 1500-1850: better 1500-1750.
By 1750 meantone had disappeared from the musical scene, with two MINOR
exceptions:
- English-speaking countries, producing few important musicians w.r.t.
the rest of Europe
- Church organs, an instrument now largely out of fashion (which is why
they were not retuned)

3 - 1600-1900: better 1650-1800.
Even if Schlick and Mersenne described two Well-tempered systems before
1650, they had scarcely any followers.
After 1800 everybody was tuning Equal Temperament except
English-speaking or Spanish/Italian organs.

4- 1850-present: better 1750-present.
Important musicians proven to have advocated/used Equal Temperament
c.1750-1800: Rameau, C.P.E. Bach, Haydn, Mozart.
Important musicians proven to have kept Unequal Temperaments after 1750:
Tartini, Quantz, Telemann. After 1770: none.
A recent study shows that even in the exceptional English-speaking
countries, by 1810 most pianos were tuned to Equal Temperament.

All the best,

Claudio

http://temper.braybaroque.ie/

🔗Chris Vaisvil <chrisvaisvil@...>

ok - this is clear. thanks!

"D" represents the number of generators for a tuning.

On Sat, Mar 28, 2009 at 11:48 PM, Carl Lumma <carl@...> wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> >
> > Could it be placed into the xenharmonic wiki?
> >
> > I have read that fairly well.
>
> Paul's paper is a PDF... I suppose it could be uploaded as a
> file. I should visit more to the xenharmonic wiki, but every
> time I stop by it looks like it's in good hands.
>
> I'll try a very brief answer to your question to see where
> that gets us: the number of generators a tuning requires is
> its dimensionality. So Pythagorean tuning only requires
> 3/2 and 2/1. All pitches can be expressed as some number
> of 3/2s and some number of 2/1s. That makes it a 2-D tuning,
> or more formally, a rank 2 tuning. 7-limit JI would be
> rank 3. Temperaments work the same way. 12-ET is rank 1
> because all you need is the semitone.
>
> -Carl
>
> _
>

🔗Carl Lumma <carl@...>

Hi Petr,

> > I expect there were a string of theorists noticing non-meantone
> > temperings of the chain of 5ths (Huygens, Newton ... Helmholtz),
> > but the first thorough inventory of linear temperaments with
> > 5th-based generators I know of was due to Bosanquet (1876).
>
> Were you referring to any particular examples when you were
> saying this? I know next to nothing about Bosanquet.

As Graham mentioned, he built a schismatic (53) reed organ.
But I was referring to his systematic classification of
5th-based linear systems and their associated keyboards
layouts (see my subsequent post). You might consider
getting his book:
http://diapason.xentonic.org/ttl/ttl04.html

> > Erv Wilson was apparently one of the first to explore
> > linear temperaments with non-5th generators.
>
> I'm surprised to hear this; I'd love to know more -- anyone
> who can suggest some web links?

Graham and I mentioned Shohe Tanaka, who may have been the
first to consider a chain of minor 3rds. I don't have access
to his original work, so I'm relying on Wikipedia and on
information presented on this list in the past, but it doesn't
look like he extended Bosanquet's program to arbitrary
generators. Wilson did that. The web link is of course
the Anaphorian Wilson archive. For starters,

http://www.anaphoria.com/xen2.PDF
http://www.anaphoria.com/xen3a.PDF
http://www.anaphoria.com/xen3b.PDF

> > Does this answer your questions?
>
> Surely it does.

I noticed I made some errors when giving the dimensionality of
some of those systems. I corrected them when I added my text
to my tuning FAQ. Did I mention I'm writing a tuning FAQ?

I've been collecting my responses to questions here for the
past several months, and heavily edited them. :) Here's a
preview:

http://lumma.org/music/theory/TuningFAQ.txt

-Carl

🔗Carl Lumma <carl@...>

Hi Claudio,

> Yes indeed. One unfortunate consequence of the late-19th studies
> in temperament was the wrong belief that historical temperaments
> could be suitably classified by Pythagorean comma deviations.
> Numerically of course you can, but conceptually, so many
> historical tunings relate to Syntonic comma (all the meantones
> and also many irregular circular ones as well).

Bosanquet did it this way because his generalized keyboard had
12 columns on it, and he was trying to show which systems would
well fit on the keyboard and which would not. You can see a
schematic of his keyboard on the lower portion of this page:

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

> http://temper.braybaroque.ie/

By the way, several weeks ago already I perused your site and
downloaded some of the audio files. Very nice! It's a pleasure
to have a musician of your caliber here on the list and
interested in such matters.

-Carl

🔗Carl Lumma <carl@...>

🔗Aaron Andrew Hunt <aaronhunt@...>

Hi Claudio.

--- In tuning@yahoogroups.com, "Claudio Di Veroli" <dvc@...> wrote:
> <http://www.h-pi.com/theory/huntsystem2.html#3>
> your "Traditional System" table is very
> similar to one published in my first book of 1978

Great! I had written that webpage about a year
before reading UT 1978. But in fact I have your book
checked out of the library at the moment, so I just
looked for the table you mention but couldn't find it.
I guess if it is not there explicitly it is certainly all there
in your text. (aside: for anyone else reading, Claudio's
book is the very best on the subject of unequal
temperaments, and on tuning in general.) Obviously,
Claudio, I still need to get the updated eBook version
of your text.

About the suggested changes, first thanks for
responding, and especially for getting exactly the
gist of what I'm trying to do and maintaining the
idea of century to mid century, even though obviously
history does not usually work this way! I am trying
to keep the dates focused on what we know to be
actual practice, and I wonder if there could be possible
compromises between what I have there now and what
you have suggested. Please let me try to explain ...

> These are my suggested Approximate Years:
>
> 1- OK
>
> 2 - 1500-1850: better 1500-1750.
> By 1750 meantone had disappeared from the
> musical scene, with two MINOR exceptions:
> - English-speaking countries, producing few
> important musicians w.r.t. the rest of Europe
> - Church organs, an instrument now largely out
> of fashion (which is why they were not retuned)

Could I use 1500 - 1800? I wanted to include organs
as an important vestige of meantone especially because
as I understand it subsemitones persisted on these
instruments at least in Italy and maybe Germany for
quite some time - I have yet to study Ibo Ortgies's work
enough to be able to substantiate this so I could be
wrong. The death of Bach is always a handy reference
for any historical table, and it is tempting to use it just
for that reason, because it is already memorable.

> 3 - 1600-1900: better 1650-1800.
> Even if Schlick and Mersenne described two Well-
> tempered systems before 1650, they had scarcely
> any followers. After 1800 everybody was tuning
> Equal Temperament except English-speaking or
> Spanish/Italian organs.

> 4- 1850-present: better 1750-present.
> Important musicians proven to have advocated/used
> Equal Temperament c.1750-1800: Rameau, C.P.E. Bach,
> Haydn, Mozart.
> Important musicians proven to have kept Unequal
> Temperaments after 1750:
> Tartini, Quantz, Telemann. After 1770: none.
> A recent study shows that even in the exceptional
> English-speaking countries, by 1810 most pianos were
> tuned to Equal Temperament.

1750 as both the cutoff for MT and start of ET makes
sense, but as I wrote above, I wonder if what was called
ET at that time would have actually been WT? Or is it
impossible to know?

Thanks so much!
Aaron
=====

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "Claudio Di Veroli" <dvc@...> wrote:

> 4- 1850-present: better 1750-present.
> Important musicians proven to have advocated/used Equal
> Temperament c.1750-1800: Rameau, C.P.E. Bach, Haydn, Mozart.
[snip]
> A recent study shows that even in the exceptional English-speaking
> countries, by 1810 most pianos were tuned to Equal Temperament.

There has been a great deal of debate here about the birth of
equal temperament. The date usually given for a factory piano
tuning that was properly equal is 1900. I understand there
are factory tunings extant from the 1850s that are very nearly
equal, and I tend to consider them to be, for all intents and
purposes. However their bearing plans would strictly speaking
lead to mild circulating temperament.

I'm told that some theorists (e.g. Werckmeister and Neihardt)
did discuss 1/12-comma pythagorean temperament, but that they
lacked any means for accurately tuning it.

I would welcome a fresh perspective on this.

-Carl

🔗Claudio Di Veroli <dvc@...>

Thanks for your kind works Aaron

> 2 - 1500-1850: better 1500-1750.

Could I use 1500 - 1800? I wanted to include organs
as an important vestige of meantone
You're welcome, do not need my permission ... :-)

as I understand it subsemitones persisted on these
instruments at least in Italy and maybe Germany for
quite some time - I have yet to study Ibo Ortgies's work
enough to be able to substantiate this so I could be
wrong.
Read Ibo's tables: he found no organs built with subsemitones from 1700 on
(i.e. none at all until the present-day revival) in German-speaking places.
A few survived of course.
And instruments with split sharps were still being made in English-speaking
countries throughout the 18th century.
So from this point of view -1800 is OK.

OK, how about 1650 - 1850? I used 1900 there because
I thought even though many important folks were
proponents of ET by 1800, actual tuning practice would
mostly still have been WT, or quite inaccurate ET, until
the 20th century when the technology of instrument
building and tuning practices started making more
precise ET possible. But I don't know!
I do: the above is a myth by a few modern writers, debunked independently by
different modern researchers which I quote in my book.
Reasonably good ET was practised in Mozart's time. Accurate ET in Schumann's
time.
There is not a shred of doubt, and plenty of evidence.
By 1830 the precision had almost reached modern standards.
And when it was not, it erred randomly, not in the WT direction.
As for WT, users of those systems in 1800-1850 were a ridiculous minority
(though they wrote, like Young).

1750 as both the cutoff for MT and start of ET makes
sense, but as I wrote above, I wonder if what was called
ET at that time would have actually been WT? Or is it
impossible to know?
It is well known: it was ET, with a few fifths randomly about 1 cent off.
This did NOT concentrate in the diatonic fifths (which would have produced a
WT).
Leading musicians of the time clarified the matter in very minute detail:
"identical tuning in all the tonalities": CPE Bach, Tartini.
The result was not unlike that of an average-not-outstanding modern tuner
without the recent electronic devices.
Unequal comebacks were proposed, claimed, discussed, but they mostly never
happened and when they did it was statistically insignificant, or else the
departure from E.T. was insignificant (De Lorenzi from 1830 on).

All the best!

Claudio

🔗Claudio Di Veroli <dvc@...>

Hi Carl again! :-)
my post of minutes ago guess is the answer to your comments: find below a
few more details anyway.

> There has been a great deal of debate here about the birth of
> equal temperament.
Indeed this matter has been researched continuously ever since Ellis's
times.

I'm told that some theorists (e.g. Werckmeister and Neihardt)
did discuss 1/12-comma pythagorean temperament, but that they
lacked any means for accurately tuning it.

Maybe they lacked those means, c. (1690-1730), but the next generation
certainly did not.
Even without beat-rates (KNOWN but not practised since 1749) the following
procedures were widely known around 1750 and guaranteed a quite accurate ET
tuning:
1. Use Sabattini's method (17th c.): start by splitting an octave in three
equally-tempered major thirds: this is easy to do with good precision.
2. Subdivide each major third in four fifths, each one audibly but slightly
tempered.
(A novice tuner can use Fogliano's method: tune the outer two fifths pure,
then you need to tune only one note to split the second into two equally
tempered fifths. Once this is done, retune the two pure fifths etc.)
3. (After 1800) Add lots of checks with other intervals: Montal's book has
been experimentally verified to reach modern standards back in 1830.

I have devoted some study to the matter, reading both ancient sources and
modern studies.
The progress of TUNING PRECISION of E.T. from c1750 up to nowadays is
described in my book in a long section: 9.10.
The history of the UNIVERSAL ADOPTION OF E.T. is scrutinised at length in my
sections 11.11 and 11.12 on temperament from Classical times up to nowadays.
I quote all the necessary ancient sources and/or modern scholars and their
evidence.
Indeed quite a few scholars have scrutinised musical writings c.1800-1850 on
the tuning methods: the evidence for a precise ET is absolutely complete.
The suggestion that old-times ET was not an ET or a rought ET but actually a
circulating temperament is absolutely groundless and not difficult to
disprove: this is done in my book as well.

Kind regards!

Claudio

🔗Aaron Andrew Hunt <aaronhunt@...>

OK, thank you Claudio. I decided to go with your
recommendations and update the table accordingly:

<http://www.h-pi.com/theory/huntsystem2.html#3>

I hope you don't mind, I also mentioned your assistance
and linked to your work. This seems especially appropriate
because your book covers exactly the material which I had
previously acknowledged as missing from my website
(that is, Stages 2 and 3 of the table).

Cheers,
Aaron
=====

--- In tuning@yahoogroups.com, "Claudio Di Veroli" <dvc@...> wrote:
>
> Thanks for your kind works Aaron
>
> > 2 - 1500-1850: better 1500-1750.
>
> Could I use 1500 - 1800? I wanted to include organs
> as an important vestige of meantone
> You're welcome, do not need my permission ... :-)
>
> as I understand it subsemitones persisted on these
> instruments at least in Italy and maybe Germany for
> quite some time - I have yet to study Ibo Ortgies's work
> enough to be able to substantiate this so I could be
> wrong.
> Read Ibo's tables: he found no organs built with subsemitones from 1700 on
> (i.e. none at all until the present-day revival) in German-speaking places.
> A few survived of course.
> And instruments with split sharps were still being made in English-speaking
> countries throughout the 18th century.
> So from this point of view -1800 is OK.
>
> > 3 - 1600-1900: better 1650-1800.
> > Even if Schlick and Mersenne described two Well-
> > tempered systems before 1650, they had scarcely
> > any followers. After 1800 everybody was tuning
> > Equal Temperament except English-speaking or
> > Spanish/Italian organs.
>
> OK, how about 1650 - 1850? I used 1900 there because
> I thought even though many important folks were
> proponents of ET by 1800, actual tuning practice would
> mostly still have been WT, or quite inaccurate ET, until
> the 20th century when the technology of instrument
> building and tuning practices started making more
> precise ET possible. But I don't know!
> I do: the above is a myth by a few modern writers, debunked independently by
> different modern researchers which I quote in my book.
> Reasonably good ET was practised in Mozart's time. Accurate ET in Schumann's
> time.
> There is not a shred of doubt, and plenty of evidence.
> By 1830 the precision had almost reached modern standards.
> And when it was not, it erred randomly, not in the WT direction.
> As for WT, users of those systems in 1800-1850 were a ridiculous minority
> (though they wrote, like Young).
>
> > 4- 1850-present: better 1750-present.
> > Important musicians proven to have advocated/used
> > Equal Temperament c.1750-1800: Rameau, C.P.E. Bach,
> > Haydn, Mozart.
> > Important musicians proven to have kept Unequal
> > Temperaments after 1750:
> > Tartini, Quantz, Telemann. After 1770: none.
> > A recent study shows that even in the exceptional
> > English-speaking countries, by 1810 most pianos were
> > tuned to Equal Temperament.
>
> 1750 as both the cutoff for MT and start of ET makes
> sense, but as I wrote above, I wonder if what was called
> ET at that time would have actually been WT? Or is it
> impossible to know?
> It is well known: it was ET, with a few fifths randomly about 1 cent off.
> This did NOT concentrate in the diatonic fifths (which would have produced a
> WT).
> Leading musicians of the time clarified the matter in very minute detail:
> "identical tuning in all the tonalities": CPE Bach, Tartini.
> The result was not unlike that of an average-not-outstanding modern tuner
> without the recent electronic devices.
> Unequal comebacks were proposed, claimed, discussed, but they mostly never
> happened and when they did it was statistically insignificant, or else the
> departure from E.T. was insignificant (De Lorenzi from 1830 on).
>
> All the best!
>
> Claudio
>

🔗Claudio Di Veroli <dvc@...>

🔗Carl Lumma <carl@...>

🔗Graham Breed <gbreed@...>

Carl Lumma wrote:

> Graham and I mentioned Shohe Tanaka, who may have been the
> first to consider a chain of minor 3rds. I don't have access
> to his original work, so I'm relying on Wikipedia and on
> information presented on this list in the past, but it doesn't
> look like he extended Bosanquet's program to arbitrary
> generators. Wilson did that. The web link is of course
> the Anaphorian Wilson archive. For starters,

I think this is the relevant Tanaka file:

http://anaphoria.com/Shohe.PDF

It's in German, and I've never puzzled out exactly what he was saying. But he knows that tempering out a kleisma gives a chain of minor thirds.

> http://www.anaphoria.com/xen2.PDF
> http://www.anaphoria.com/xen3a.PDF
> http://www.anaphoria.com/xen3b.PDF

I don't remember Erv using arbitrary generators in Xenharmonikon but I don't have time to check these now. Some of the later Scale Tree diagrams will have anything you like.

Graham

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> I think this is the relevant Tanaka file:
>
> http://anaphoria.com/Shohe.PDF
>
> It's in German, and I've never puzzled out exactly what he
> was saying. But he knows that tempering out a kleisma gives
> a chain of minor thirds.

Thanks for the link.

> > http://www.anaphoria.com/xen2.PDF
> > http://www.anaphoria.com/xen3a.PDF
> > http://www.anaphoria.com/xen3b.PDF
>
> I don't remember Erv using arbitrary generators in
> Xenharmonikon but I don't have time to check these now.
> Some of the later Scale Tree diagrams will have anything you
> like.

Not in those, no. Actually the earliest thing I have
here is more recent than I thought -- Keyboard Schemata
from the Scale Tree, 1991.

-Carl

🔗Carl Lumma <carl@...>

🔗Graham Breed <gbreed@...>

Carl Lumma wrote:

>>> http://www.anaphoria.com/xen2.PDF
>>> http://www.anaphoria.com/xen3a.PDF
>>> http://www.anaphoria.com/xen3b.PDF
>> I don't remember Erv using arbitrary generators in >> Xenharmonikon but I don't have time to check these now. >> Some of the later Scale Tree diagrams will have anything you >> like.

Sorry, I got my timetable wrong. I have two hours too look at this :-)

> Not in those, no. Actually the earliest thing I have
> here is more recent than I thought -- Keyboard Schemata
> from the Scale Tree, 1991.

Right. So Larry Hanson has the record.

www.anaphoria.com/hanson.PDF

The keyboard goes back to 1942. But "Erv ... pointed out that my layout _could_ be considered to _have been_ generated by the interval of a minor third, and that the 19 tone subset _did_ have the properties of a Wilson MOS scale." This was in 1978.

Graham

🔗Claudio Di Veroli <dvc@...>

Thanks Carl!

No the book is not protected, the file has no password.
Do not know where they say it, but they do not allow protected PDF files.
(Incidentally, I am not aware of what is the problem with protected PDF).

Please let me know: all problems have a solution! :-)

Kind regards,

Claudio

_____

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of
Carl Lumma
Sent: 30 March 2009 00:22
To: tuning@yahoogroups.com
Subject: [tuning] Re: 2D tunings in use? - Historical dates

--- In tuning@yahoogroups. <mailto:tuning%40yahoogroups.com> com, "Claudio
Di Veroli" <dvc@...> wrote:
>
> Hi Carl again! :-)
> my post of minutes ago guess is the answer to your comments:
> find below a few more details anyway.

I'm convinced. ;)

I've got your book in my cart at Lulu, but I won't purchase
it if the PDF is "protected" in some way. Do you happen to
know if it is? I've searched the Lulu FAQs and can't find
an answer to this.

Thx,

-Carl

🔗Graham Breed <gbreed@...>

Carl Lumma wrote:

> Not in those, no. Actually the earliest thing I have
> here is more recent than I thought -- Keyboard Schemata
> from the Scale Tree, 1991.

The Letter to John Chalmers (1975) that defines "MOS" has examples of each MOS from 17-equal. So at this point
he's caught up with Salinas: scales but not temperaments.

http://www.anaphoria.com/mos.PDF

George Secor's paper on what we now call miracle temperament was published in 1975. His more recent paper makes it clear he was looking at rank 2 temperaments with a range of generators independently of Erv. Bottom of page 6 of

http://www.anaphoria.com/SecorMiracle.pdf

This would be some time between 1963 and 1974. Both of them can comment on who thought of what when, if they care.

Graham

🔗Carl Lumma <carl@...>

🔗Claudio Di Veroli <dvc@...>

Hi Carl,

The main reason why you cannot copy from some PDF books is that they are
assemblies of scans: they are bitmap graphics, not vector graphics.
Thus there is no text there, just photographic dots!
In my main PC I can copy from the my PDF book
(and I do NOT have Adobe full installed: I make the conversions in another
PC).
So I guess everybody can!.
I give instructions in the very first page on how to copy from a PDF text.
If you had difficulties, I can always provide you with a Word text, do not
worry.

Claudio

_____

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of
Carl Lumma
Sent: 30 March 2009 03:46
To: tuning@yahoogroups.com
Subject: [tuning] Re: 2D tunings in use? - Historical dates

--- In tuning@yahoogroups. <mailto:tuning%40yahoogroups.com> com, "Claudio
Di Veroli" <dvc@...> wrote:
>
> Thanks Carl!
>
> No the book is not protected, the file has no password.
> Do not know where they say it, but they do not allow protected
> PDF files. (Incidentally, I am not aware of what is the problem
> with protected PDF).

It's possible for a publisher to restrict certain capabilities
in a PDF without a password. You can lock things like the ability
to copy text from the PDF to the clipboard. That's an onerous
one for those of us who might need to cite material -- you have
to retype it by hand. Other examples abound.

-Carl

🔗Carl Lumma <carl@...>

🔗monz <joemonz@...>

Hi Chris,

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> so the "D" refers to the dimensions in the figures
> on the diagrams of note relationships?
>
> if so - that seems simple.
>
> On Sat, Mar 28, 2009 at 9:53 PM, Graham Breed <gbreed@...> wrote:
>
> > Chris Vaisvil wrote:
> > > I'll read it - thanks.
> > >
> > > At this moment I don't know that the "D" stands for
> >
> > dimensional

The number of dimensions is the number of generators
from which the tuning is constructed.

In a JI system, the generators are either the prime-factors
or the odd-factors, depending on whether one is considering
prime-limit or odd-limit as the basis of the tuning.

For temperaments, the generators are usually tempered
approximations to the prime-factors, but they don't
necessarily have to be. Gene Ward Smith constructed
many temperaments from generators whose size is based
on other mathematical properties.

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

🔗monz <joemonz@...>

Hi Carl and Claudio,

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> > > In Europe, the first extant description of a systematic
> > > tempering of these 5ths dates from the late 1500s;
> >
> > You mean early 1500s (Schlick 1511, Aron 1516 and 1523)
>
> I was dating this from Zarlino, but I think you're right.
> And forgive me if you posted about this recently, but if you
> have any more information on these three citations it would
> be greatly appreciated!

A few years ago i put up this English translation of
Aron's seminal work:

http://tonalsoft.com/monzo/aron/toscanello/aron_toscanello.htm

> By the way, I said that Wilson was the first to rigorously
> consider chains of intervals other than the 5th, but this
> distinction may belong to Shohe Tanaka, who apparently saw
> 53-ET as a chain of minor thirds:
>
> http://en.wikipedia.org/wiki/Shohe_Tanaka

My webpage on "kleisma" shows it visually
on Tonescape lattices:

http://tonalsoft.com/enc/k/kleisma.aspx

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

🔗monz <joemonz@...>

Hi Carl and Chris,

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> ... 12-ET is rank 1
> because all you need is the semitone.

Or the tempered perfect-4th.

12-edo is a particularly interesting tuning precisely
because it can be generated from either the semitone
or perfect-4th (or of course from the inversions of
those: the major-7th or perfect-5th), but not from
any other intervals in the tuning.

Most of the other interesting low-cardinality EDOs
have a cardinality which is a prime-number (i.e.,
19, 31, 41, 53, etc.), which means that cycling
all the way thru any of the intervals in the
tuning will eventually produce the whole tuning
(assuming octave-equivalence).

In 12-edo this doesn't work, because 12 can be divided
so many ways. Thus, in 12-edo:

* a cycle of major-2nds produces only whole-tone scale
* a cycle of minor-3rds produces only the diminished-7th tetrad
* a cycle of major-3rds produces only the augmented triad
* a cycle of augmented-4ths produces only the tritone

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

🔗Claudio Di Veroli <dvc@...>

Thanks Joe.

Was not aware of your version of Aron: it is indeed very useful for
non-Italian speakers to have Aron's discussion on Tones and Temperament
translated into English.
[Question: Aron inconsistently entitled his chapter 40 XXXX and the chapter
41 XLI . Is for consistency that you translated the latter as XXXXI or are
we reading from different originals?)

My U.T. Book only translates the temperament chapter XLI: I rendered it into
modern English usage (including the names of notes and accidentals), and
included a few comments so that the average modern musician and tuner can
easily understand it.
E.g. "le quali son tutte quinte che non si tirano al segno de la perfettione
mancando dal canto di sopra".
I translated it as: "which are all of them fifths that do not reach purity
as they are lacking in the upper part (i.e. they are flat)."
You rendered it as: "that are all fifths, which aren't drawn to the
perfection, being lacking on the upper edge."
Your translation is of course better for the scholar as it is very strictly
literal.

Kind regards,
Claudio

http://temper.braybaroque.ie/

_____

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of
monz
Sent: 31 March 2009 13:21
To: tuning@yahoogroups.com
Subject: [tuning] Re: 2D tunings in use?

Hi Carl and Claudio,

--- In tuning@yahoogroups. <mailto:tuning%40yahoogroups.com> com, "Carl
Lumma" <carl@...> wrote:

A few years ago i put up this English translation of
Aron's seminal work:

http://tonalsoft.
<http://tonalsoft.com/monzo/aron/toscanello/aron_toscanello.htm>
com/monzo/aron/toscanello/aron_toscanello.htm

My webpage on "kleisma" shows it visually
on Tonescape lattices:

http://tonalsoft. <http://tonalsoft.com/enc/k/kleisma.aspx>
com/enc/k/kleisma.aspx

-monz
http://tonalsoft. <http://tonalsoft.com/tonescape.aspx> com/tonescape.aspx
Tonescape microtonal music software

🔗monz <joemonz@...>

Hi Claudio,

Thanks for all the kind words about my work.

However, please note that i did not make the
English translation of Aron -- that was done
by Leonardo Perretti.

Leonardo made the translation at my request,
and i simply turned his plain text into a webpage.

I actually had to do a search to find this page,
because i never integrated it properly into my
Encyclopedia. I'll say more on this soon.

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

--- In tuning@yahoogroups.com, "Claudio Di Veroli" <dvc@...> wrote:
>
> Thanks Joe.
>
> Was not aware of your version of Aron: it is indeed very useful for
> non-Italian speakers to have Aron's discussion on Tones and Temperament
> translated into English.
> [Question: Aron inconsistently entitled his chapter 40 XXXX and the chapter
> 41 XLI . Is for consistency that you translated the latter as XXXXI or are
> we reading from different originals?)
>
> My U.T. Book only translates the temperament chapter XLI: I rendered it into
> modern English usage (including the names of notes and accidentals), and
> included a few comments so that the average modern musician and tuner can
> easily understand it.
> E.g. "le quali son tutte quinte che non si tirano al segno de la perfettione
> mancando dal canto di sopra".
> I translated it as: "which are all of them fifths that do not reach purity
> as they are lacking in the upper part (i.e. they are flat)."
> You rendered it as: "that are all fifths, which aren't drawn to the
> perfection, being lacking on the upper edge."
> Your translation is of course better for the scholar as it is very strictly
> literal.
>
> Kind regards,
> Claudio
>
> http://temper.braybaroque.ie/

🔗Carl Lumma <carl@...>

🔗Leonardo Perretti <dombedos@...>

Hi Claudio and Joe.

Starting from your last question.

Claudio Di Veroli wrote:
>My U.T. Book only translates the temperament chapter XLI: I rendered >it into modern English usage (including the names of notes and >accidentals), and included a few comments so that the average >modern musician and tuner can easily understand it.
>E.g. "le quali son tutte quinte che non si tirano al segno de la >perfettione mancando dal canto di sopra".
>I translated it as: "which are all of them fifths that do not reach >purity as they are lacking in the upper part (i.e. they are flat)."
>You rendered it as: "that are all fifths, which aren't drawn to the >perfection, being lacking on the upper edge."
>Your translation is of course better for the scholar as it is very >strictly literal.

I made this translation, and the other from Zarino, with the aim to present the antique text to English-speaking readers as close to the original as possible, in an attempt to show, together with the actual content of the text, the "taste" or "color" of the antique language. I don't know if I succeeded in doing it; someone commented that it was an "exercise of style", and this left me really surprised, as I have quite no ambition to appear as a man of letters, let alone in English language; the "style" here is just that of Aaron and Zarlino.

>Question: Aron inconsistently entitled his chapter 40 XXXX and the >chapter 41 XLI . Is for consistency that you translated the latter >as XXXXI or are we reading from different originals?

For this translation I used a transcription from the Thesaurus Musicarum Italicarum site; the relevant page is here:
http://euromusicology.cs.uu.nl:6334/dynaweb/tmiweb/a/aartos/@Generic__BookView;cs=default;ts=default;lang=it
It appears the text is taken from the 1523 and 1529 editions and reports "XXXXI" for the chap 41 of the second book. I have no access to the original books for those editions. I have got reproductions of the 1531 and 1539 editions, where the chapters are numbered as you report. There is no error anyway as both forms represent the 41 exactly
Anyway, I think the use of "XXXX" for 40, and "XLI" for 41 is not a real inconsistence for Aaron's part . While the common modern use for Roman numeration is the subtractive form , in late Middle-Ages and Renaissance the extended form was still in use as well, nor were they so meticulous with numbers. Remaining on Aaron's "Toscanello" (1531 ed.), you will find:
-all 4 are in extended form (IIII, XIIII, XXIIII etc.)
-all 9 are in subtractive form (IX, XIX, XXIX etc), but the chap. 29 of the first book is numbered XXVIIII. Some chapters are numbered wrongly: in the first book, chap. 9 is VIII, chap 39 is XXXVIII; in the second book, chap 19 is XVIII. Chap 40 of the first book is in the form XL. The 1539 edition shows even more errors.
It appears such errors, probably due to inaccuracy by the typographer, were common and tolerated at that time.

The text Joe published had been written as a draft in plain text, so there was no way to represent accidentals properly; also, there are other small errors needing to be amended. Of course, all comments are welcome for my part.

Finally, I wish to seize the opportunity to thank Claudio Di Veroli for his writings. His first book on UT was my first source of knowledge about temperaments when I had my apprenticeship in early 80's.

Best regards

Leonardo