Yahoo Tuning Groups Ultimate Backup tuning why did Zarlino prefer 2/7-comma meantone?

> From: <manuel.op.de.coul@eon-benelux.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, February 27, 2003 7:44 AM
> Subject: Re: [tuning] why did Zarlino prefer 2/7-comma meantone?
>
>
> You could find out from the source. The whole book is on the
> internet. Follow the link from the bibliography and login with
> "guest" "password". From what I could see quickly you need to
> be in the neighbourhood of the second part, caput 42. But I don't
> understand it without a dictionary.

ah, thanks, Manuel.

> From: <manuel.op.de.coul@eon-benelux.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, February 27, 2003 8:03 AM
> Subject: Re: [tuning] why did Zarlino prefer 2/7-comma meantone?
>
>
> Another aspect is of course that the minor and
> major third are tempered by the same amount, 1/7 comma.

yes, that's a good point, and (without having read
Zarlino yet) it sounds like a reason he would use.

> From: <Afmmjr@aol.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, February 27, 2003 8:00 AM
> Subject: Re: [tuning] why did Zarlino prefer 2/7-comma meantone?
>
>
> Joe, being the mid-point of 1/4 and 1/3 comma
> (and meantone is about midpoints), is the reason
> most given to me for the existence of 1/6 comma
> meantone. 1/6 is in between 1/4 and 1/11 comma
> (or 12 ET/Pythagorean tuning).

yes, that's pretty much the reasoning i was using
when i said that 2/7-comma could be seen as some
type of mediant between 1/3 (= 2/6) and 1/4 (= 2/8).

but i don't quite follow your example: how is
1/6-comma a mid-point between 1/3 and 1/4?

-monz

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

🔗Afmmjr@aol.com

🔗Robert Wendell <rwendell@cangelic.org> <rwendell@cangelic.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

🔗Robert Wendell <rwendell@cangelic.org> <rwendell@cangelic.org>

It seems everyone is ignoring completely the whole reason I brought
up the subject of 2/7-comma meantone in the first place. You're
getting hot, though, with the recognition that the major and minor
thirds are flat by the same amount.

2/7-comma meantone and Paul Erlich's comment mentioning Zarlino's
mathematical description of it came up in the context of my work with
beat synchrony in temperaments. Gene had mentioned that 1/7-comma
meantone has 2.0 beat ratios of minor to major thirds in the close,
root-position triads in all playable keys. I mentioned 2/7-comma
meantone as another example of a synchronous meantone, along with 1/5-
comma.

Remember that major triads with justly tuned perfect fifths will
always have a beat ratio of 3/2 for minor to major thirds. Also
notice that this is a case of the minor third always being exactly as
flat as the major third is sharp, which yields a beat ratio of -1.50
if we include phase relationships by including the sign.

In 2/7-comma, you have a +1.5 beat ratio (opposite sign), since the
major and minor thirds are flat by exactly the same amount. This
accounts for the positive sign indicating phase.

I don't believe these musicians decided favorite temperaments on the
basis of technical considerations. I think they loved how
HARMONIOUSLY SMOOTH AND SATISFYING this temperament is. After all,
this is a MUSICAL criterion resulting from a technical one, to be
sure, but the MUSICALITY of the result is what I believe they found
so convincing.

I mentioned TWO good reasons for this:

1) It is a synchronous temperament, meaning the beat rates are tuned
to synchronize, 3 beats on the minor third for every two on the
major. It is a clear subjective experience for me and some of the top
professional tuners in the Piano Technician's Guild, some of whom
design temperaments themselves, that tuning (synchronizing) the beats
is important, too. It is a kind of second-order coherence that helps
compensate the sacrifice in first-order harmonic coherence that
tempering unavoidably introduces.

2) It is fairly close to Woolhouse's optimization of meantone, which
minimizes the degree to which all three intervals in a triad are
compromised. I believe this, combined with a theoretically perfect
3/2 beat ratio, is what makes it the most harmoniously smooth
meantone temperament, even smoother than Woolhouse's optimization.

In a word, it SOUNDS GOOD! That's why they like it, darn it! It's
MUSICAL! It sounds good! That's it, period, in my view.

Musically yours,

Bob

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> does anyone know *why* Zarlino had a preference
> for 2/7-comma meantone? (which was the first
> meantone to be described with mathematical
> accuracy, 1558)
>
> i suspect that it's because it was a simple
> arithmetical mediant between the other two
> earliest-described meantones, the true meantone
> of 1/4- (= 2/8-)comma), described in 1523 by Aron,
> and the 1/3- (=2/6-)comma of Salinas of 1577.
>
> in fact Salinas's book describes precisely
> these three tunings.
>
> does anyone know of any more substantial reasons,
> and/or are there any arguments as to the plausibility
> of my hypothesis here?
>
>
>
> PS -- i've added a useful graph depicting the
> "chronology of meantone advocacy", halfway down
> the page at
> http://sonic-arts.org/dict/meantone.htm
>
>
>
>
> -monz

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

🔗Robert Wendell <rwendell@cangelic.org> <rwendell@cangelic.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
<rwendell@c...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus
> <wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
> > <rwendell@c...> wrote:
> >
> > > I don't believe these musicians
> >
> > who? which musicians?
>
> Bob:
> Zarlino and any others of the time who liked and used this
> temperament.
> >
> > > decided favorite temperaments on the
> > > basis of technical considerations.

1. we've already heard that zarlino changed his mind . . .

2. your beating criteria are just as "technical" as the error
criteria mentioned . . .

3. to the best of my knowledge, zarlino and tuners of his time had no
way of accurately setting 2/7-comma meantone by ear. using the beat
synchrony would have given them a way of doing so. but tunings in
those times were set primarily by the tuner judging and fudging
*melodic* intervals -- this is documented fact. it is my strong
suspicion that the beat synchrony property of 2/7-comma meantone was
unknown in either theory or practice in the 16th century.

putting this into perspective, i'm willing give terrific odds that
zarlino gave a completely *theoretical* (not merely "technical")
reason for 2/7-comma meantone; that it had nothing to do with
synchronizing beat rates; and (most confidently) that no one in
zarlino's time could set a keyboard tuning accurately enough to
distinguish between 2/7-comma meantone and, say, 7/25-comma meantone
(to pick a not-so-random example), such that anyone could declare one
or the other their favorite on the basis that it *sounded* better.

p.s. why do you assume that monz (and "everyone") is talking about
2/7-comma meantone because *you* brought it up? have the two of you
been talking off-list?

🔗monz <monz@attglobal.net>

hi Bob,

it's good to see you back in action on the
tuning list. however ...

> From: <rwendell@cangelic.org>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, February 27, 2003 3:43 PM
> Subject: [tuning] Re: why did Zarlino prefer 2/7-comma meantone?
>
>
> It seems everyone is ignoring completely the
> whole reason I brought up the subject of 2/7-comma
> meantone in the first place. <snip>
>
> 2/7-comma meantone and Paul Erlich's comment
> mentioning Zarlino's mathematical description of
> it came up in the context of my work with beat
> synchrony in temperaments. Gene had mentioned
> that 1/7-comma meantone has 2.0 beat ratios of
> minor to major thirds in the close, root-position
> triads in all playable keys. I mentioned 2/7-comma
> meantone as another example of a synchronous meantone,
> along with 1/5-comma.

the relevance of your remarks notwithstanding,
my question about 2/7-comma meantone arose as
the result of a telephone conversation i was
having with someone yesterday, and not because
of anything i read here.

... it just seemed that i should mention that. :)

-monz

🔗Robert Wendell <rwendell@cangelic.org> <rwendell@cangelic.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

🔗monz <monz@attglobal.net>

hi paul,

> From: <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, February 27, 2003 4:22 PM
> Subject: [tuning] Re: why did Zarlino prefer 2/7-comma meantone?
>
>
> 3. to the best of my knowledge, zarlino and tuners of
> his time had no way of accurately setting 2/7-comma
> meantone by ear. using the beat synchrony would have
> given them a way of doing so. but tunings in those
> times were set primarily by the tuner judging and
> fudging *melodic* intervals -- this is documented fact.
> it is my strong suspicion that the beat synchrony
> property of 2/7-comma meantone was unknown in either
> theory or practice in the 16th century.

in fact Zarlino specified 2/7-comma meantone as measured
divisions of a string-length on a monochord.

so most likely, if he himself actually did tune any
keyboards to that temperament, he probably would have
done it simply by matching the sound of the harpsichord
strings to those of the monochord.

i agree with you that at this early stage of the game
(1558), Zarlino was probably unaware of beat synchrony.
this was still several decades before the recognition
of the overtone series, which forms the basis of
calculating beats.

> putting this into perspective, i'm willing give
> terrific odds that zarlino gave a completely
> *theoretical* (not merely "technical") reason for
> 2/7-comma meantone; that it had nothing to do with
> synchronizing beat rates;

well ...

> From: "Leonardo Perretti" <dombedos@tiscalinet.it>
> To: <tuning@yahoogroups.com>
> Sent: Monday, March 03, 2003 3:51 AM
> Subject: [tuning] Re.: new Tuning Dictionary page: 2/7-comma meantone
>
>
> Hi monz,
>
> if you like, I can translate the chapters of Zarlino's
> treatise you mention. I have also some practice with
> Renaissance Italian :-)

Leonardo, i'll take your offer!

so, paul, hopefully we'll find out soon exactly what
Zarlino himself had to say about it. but don't
look for a wager from me.

> and (most confidently) that no one in zarlino's time
> could set a keyboard tuning accurately enough to
> distinguish between 2/7-comma meantone and, say,
> 7/25-comma meantone (to pick a not-so-random example),
> such that anyone could declare one or the other their
> favorite on the basis that it *sounded* better.

well, you *certainly* won't be able to get *me* to bet
against you on that one!

[ 3/25 -3/25 7/25 ] = 7/25-comma meantone "5th"
- [ 1/7 -1/7 2/7 ] = 2/7-comma meantone "5th"
-------------------------
[-4/175 4/175 -1/175] = difference between the two

= ~0.122893083 cent = ~1/8 cent

which is very close to 3 jots and 5 cawapus

... certainly not an audible difference. it's
highly doubtful to me that this particular difference
could ever be accomplished before the advent of
digital electronic tuning aids ... and i'd bet
that even with the help of them, it's hard to
distinguish between the two.

-monz

🔗Gene Ward Smith <gwsmith@svpal.org> <gwsmith@svpal.org>

🔗Robert Wendell <rwendell@cangelic.org> <rwendell@cangelic.org>

--- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > i agree with you that at this early stage of the game
> > (1558), Zarlino was probably unaware of beat synchrony.
> > this was still several decades before the recognition
> > of the overtone series, which forms the basis of
> > calculating beats.
>
> Robert's theory as I understand it is not that Zarlino noticed
> synchronized beating, but that he noticed the result sounded good.
If
> synchronized beating was part of the reason why, it would serve as a
> reason for both 2/7 and 1/4 comma advocacy.

Bob:
Precisely, Gene! Thank you. That's exactly my hypothesis, and what I
meant earlier by the reasons not being technical. A preference may
have a basis in some technical aspect underlying the good sound, but
liking something because it sounds good, whatever the underlying
technical reasons for the perception may be, is not in itself a
technical reason for the preference. We may like the smooth, mellow
sound of a French horn because of the lovely, balanced regularity of
its harmonics, but this reaction may occur despite a total lack of
unawareness of this, even though it has everything to do with it.

Also, although I don't think a valid argument for it sounding better
is dependent on hearing beats, I don't believe that it is necessary
to understand analytically the existence of harmonics to hear beats.
In answer to one of Paul Erlich's questions, "tuning wrench in hand",
1/4-comma meantone has an interesting characteristic that makes it
easier to tune and check for accuracy:

The major thirds are completely just, so a fourth above the top tone
(e.g., A above a M3, C-E) will beat with the top note (E) at exactly
the same rate as it does with the bottom not a major sixth below it
(C). This is because both notes in the major third have a common
harmonic with A, as well as with each other. The common harmonic with
each other has no beats on it, since the third is just, so they both
beat at the same rate with the common harmonic on the A.

This means you can tune the third justly not simply by eliminating
beats between the common harmonic on the two notes in the third, but
even more accurately by making the beating with the fourth above the
top note exactly the same rate. This is actually easier and more
accurate on piano strings than trying to eliminate beats between the
two notes of the third itself.

The beats on the latter are the fifth and fourth harmonics of the
bottom and top notes respectively. The beats of either with the
fourth above the top note is lower in the harmonic series, on the
stronger third harmonic of the fourth above. This beating is much
easier to hear. It makes a wonderful cross-check as well as a means
to move quickly to remote parts of the cycle of fifths to forestall
the effect of miniscule but cumulative errors, the universal bane of
even the finest ears among tuners. Further, it helps compensate
inharmonicity on piano strings to tune 1/4-comma this way.

It is difficult to imagine that people who tuned their own
instruments every day before playing, and even multiple times per day
in some circumstances, and who often had a choice of several commonly
used temperaments in their era that were not so grossly different
from each other, would never notice this beating even if they never
once thought in terms of harmonics. The other arts were highly
refined, with a deep, if not always scientifically understood,
mastery of technical considerations. I find it difficult to think,
all things considered, that the art of music represented some kind of
contrast in this regard.

I find it much more likely that is was subjectively refined beyond
the imagination of most musicians living in an age in which tuning
temperaments is left to professional technicians, and who are never
confronted with the fine pitch distinctions that inevitably manifest
when you set any kind of temperament, and which mandate tempering in
the first place. It is important to remind ourselves that mathematics
conjured up in the whimsical imaginations of theorists such as
Riemann and Lobachevsky, end up being used to describe large-scale
reality in Einstein's General Relativity.

This illustrates that there are qualities in the human mind that
reflect the "external" realities in nature around us. Not everything
that is written about the external completey explains
the "knowingness" that we possess within ourselves and
our "subjective" reality. So some things may be considered too
commonplace or "natural" or "subjective" or subtle or even obvious
but conceptually elusive to bother to write down.

🔗manuel.op.de.coul@eon-benelux.com

I hadn't written yet that it's possible to print out the cross
checks that Robert mentioned with Scala. The command is
SHOW/TEST BEATS. They are sometimes called Nix or Kröber tests.

The output looks like this for the Wendell Natural Synchronous Well:

Base frequency : 261.6256 Hertz
Equal beating tests of 6/5 5/4 3/2 5/3
-10 -3 D.-1 A.-1 -1.1091 Hz. = -5 2 G.-1 D.0 -1.1113 Hz.
0 7 C.0 G.0 -1.1158 Hz. = -5 2 G.-1 D.0 -1.1113 Hz.
2 9 D.0 A.0 -2.2182 Hz. = 7 14 G.0 D.1 -2.2226 Hz.
7 14 G.0 D.1 -2.2226 Hz. = 12 19 C.1 G.1 -2.2317 Hz.
-10 -6 D.-1 F#.-1 5.5727 Hz. = -10 -1 D.-1 B.-1 5.5727 Hz.
0 4 C.0 E.0 5.5779 Hz. = -10 -1 D.-1 B.-1 5.5727 Hz.
0 4 C.0 E.0 5.5779 Hz. = -5 -1 G.-1 B.-1 5.5782 Hz.
-5 -1 G.-1 B.-1 5.5782 Hz. = -8 -5 E.-1 G.-1 -5.5782 Hz.
-5 -1 G.-1 B.-1 5.5782 Hz. = -5 4 G.-1 E.0 5.5782 Hz.
-8 -4 E.-1 G#.-1 7.6853 Hz. = -8 1 E.-1 C#.0 7.6853 Hz.
-8 -4 E.-1 G#.-1 7.6853 Hz. = -11 -8 C#.-1 E.-1 -7.6853 Hz.
-8 1 E.-1 C#.0 7.6853 Hz. = -3 1 A.-1 C#.0 7.6945 Hz.
-11 -8 C#.-1 E.-1 -7.6853 Hz. = -3 1 A.-1 C#.0 7.6945 Hz.
-12 -9 C.-1 Eb.-1 -7.9223 Hz. = -7 2 F.-1 D.0 7.9788 Hz.
-9 0 Eb.-1 C.0 7.9223 Hz. = -7 2 F.-1 D.0 7.9788 Hz.
-9 0 Eb.-1 C.0 7.9223 Hz. = -10 -7 D.-1 F.-1 -7.9788 Hz.
-7 2 F.-1 D.0 7.9788 Hz. = -2 2 Bb.-1 D.0 7.9871 Hz.
-10 -7 D.-1 F.-1 -7.9788 Hz. = -2 2 Bb.-1 D.0 7.9871 Hz.
0 9 C.0 A.0 8.6410 Hz. = -11 -2 C#.-1 Bb.-1 8.6328 Hz.
-3 0 A.-1 C.0 -8.6410 Hz. = -6 -2 F#.-1 Bb.-1 8.6460 Hz.
0 9 C.0 A.0 8.6410 Hz. = -6 -2 F#.-1 Bb.-1 8.6460 Hz.
-6 -2 F#.-1 Bb.-1 8.6460 Hz. = -6 3 F#.-1 Eb.0 8.6460 Hz.
-6 -2 F#.-1 Bb.-1 8.6460 Hz. = -9 -6 Eb.-1 F#.-1 -8.6460 Hz.
-4 0 G#.-1 C.0 10.5630 Hz.= -4 5 G#.-1 F.0 10.5630 Hz.
-4 0 G#.-1 C.0 10.5630 Hz.= -7 -4 F.-1 G#.-1 -10.5630 Hz.
2 6 D.0 F#.0 11.1455 Hz.= -3 6 A.-1 F#.0 11.1318 Hz.
2 6 D.0 F#.0 11.1455 Hz.= -6 -3 F#.-1 A.-1 -11.1318 Hz.
2 6 D.0 F#.0 11.1455 Hz.= -1 2 B.-1 D.0 -11.1455 Hz.
2 6 D.0 F#.0 11.1455 Hz.= 2 11 D.0 B.0 11.1455 Hz.
12 16 C.1 E.1 11.1557 Hz.= -1 2 B.-1 D.0 -11.1455 Hz.
2 11 D.0 B.0 11.1455 Hz.= 12 16 C.1 E.1 11.1557 Hz.
12 16 C.1 E.1 11.1557 Hz.= 7 11 G.0 B.0 11.1564 Hz.
7 11 G.0 B.0 11.1564 Hz.= 7 16 G.0 E.1 11.1564 Hz.
7 11 G.0 B.0 11.1564 Hz.= 4 7 E.0 G.0 -11.1564 Hz.
5 9 F.0 A.0 11.5213 Hz.= -1 3 B.-1 Eb.0 11.5280 Hz.
-1 3 B.-1 Eb.0 11.5280 Hz.= -4 -1 G#.-1 B.-1 -11.5280 Hz.
-1 3 B.-1 Eb.0 11.5280 Hz.= -1 8 B.-1 G#.0 11.5280 Hz.
4 8 E.0 G#.0 15.3706 Hz.= 4 13 E.0 C#.1 15.3706 Hz.
4 8 E.0 G#.0 15.3706 Hz.= 1 4 C#.0 E.0 -15.3706 Hz.
4 13 E.0 C#.1 15.3706 Hz.= 9 13 A.0 C#.1 15.3889 Hz.
1 4 C#.0 E.0 -15.3706 Hz.= 9 13 A.0 C#.1 15.3889 Hz.
3 12 Eb.0 C.1 15.8446 Hz.= 2 5 D.0 F.0 -15.9577 Hz.
0 3 C.0 Eb.0 -15.8446 Hz.= 2 5 D.0 F.0 -15.9577 Hz.
0 3 C.0 Eb.0 -15.8446 Hz.= 5 14 F.0 D.1 15.9577 Hz.
2 5 D.0 F.0 -15.9577 Hz.= 10 14 Bb.0 D.1 15.9742 Hz.
5 14 F.0 D.1 15.9577 Hz.= 10 14 Bb.0 D.1 15.9742 Hz.

This scale is quite fascinating. I wonder if it's possible to make
the beat rate quotients for the triads with tempered fifths all exactly
the same. This gives a set of equations that's probably too hard to
solve analytically so it must be done numerically. Any takers?
The closest scale in the archive is Neidhardt 2 by the way, although
not a very close match.

Manuel

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
<rwendell@c...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
> <gwsmith@s...> wrote:
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > > i agree with you that at this early stage of the game
> > > (1558), Zarlino was probably unaware of beat synchrony.
> > > this was still several decades before the recognition
> > > of the overtone series, which forms the basis of
> > > calculating beats.
> >
> > Robert's theory as I understand it is not that Zarlino noticed
> > synchronized beating, but that he noticed the result sounded
good.
> If
> > synchronized beating was part of the reason why, it would serve
as a
> > reason for both 2/7 and 1/4 comma advocacy.
>
> Bob:
> Precisely, Gene! Thank you.

dear bob,

please read monz's reply to what i wrote again.

beats and overtones were unknown in zarlino's day.

there was no way to accurately tune, IN THE FIRST PLACE 2/7-comma
meantone as opposed to, say, 7/25-comma (also "optimal", though by
different criteria), in those days.

so what sounded good and what didn't sound good could have been 2/7-
comma and 7/25-comma, or the reverse, or none of the above.

no one had any way of knowing in those days. the "result" gene
postulates above simply had no way of coming about, with any
accuracy, in the first place -- as opposed to some "result" in which
the beat synchrony did not appear.

we take accurate tuning for granted today with our electronic tuners
and computers. but before beats and overtones were even suspected of
existing (and logarithms hadn't been invented yet), these small
fractions of a comma were completely *theoretical*, in the realm of
ideals rather than tangible musical entities.

> It is difficult to imagine that people who tuned their own
> instruments every day before playing, and even multiple times per
day
> in some circumstances, and who often had a choice of several
commonly
> used temperaments in their era that were not so grossly different
> from each other, would never notice this beating even if they never
> once thought in terms of harmonics.

the problem is that this "choice" was not simply "given" the way it
is with today's electronic gear. until the 17th century, there were
absolutely no aural guides for setting temperaments -- neither
eliminating beats nor getting their rates to match. so even if
they "preferred" one temperament over another, they wouldn't know
what they were comparing to what! until mersenne came around, setting
different varieties of meantones was more guessing game than science.

as to noticing beats, the vast majority of pianists whom i've talked
to about tuning are unable to hear them even with a great deal of
prodding. i walk over to the piano, play a major third, and say "hear
the beats"? if we're using a synth, i'll play a just major third for
comparison. "hear how that *doesn't* beat?" silence. i go back to the
tempered major third and sing "wawawawa" in time with the beating
common overtone. "hear that?" almost no pianist does. it's no wonder
beating escaped everyone's attention until the 17th century.

> The other arts were highly
> refined, with a deep, if not always scientifically understood,
> mastery of technical considerations. I find it difficult to think,
> all things considered, that the art of music represented some kind
of
> contrast in this regard.

everything i've said is well-documented. hopefully you'll be able, at
least, to get a library to provide you with a copy of the jorgenson
book. also, if possible, read up on mersenne's _harmonie
universelle_, the state of tuning technique before it, and the state
of tuning technique after it. you'll find some good info in the new
grove dictionary. the "scientific revolution" happened a bit later in
the musical arts than in the other arts in our civilization.

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com> <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, manuel.op.de.coul@e... wrote:
> I hadn't written yet that it's possible to print out the cross
> checks that Robert mentioned with Scala. The command is
> SHOW/TEST BEATS. They are sometimes called Nix or Kröber tests.
>
> The output looks like this for the Wendell Natural Synchronous Well:
>
> Base frequency : 261.6256 Hertz
> Equal beating tests of 6/5 5/4 3/2 5/3
> -10 -3 D.-1 A.-1 -1.1091 Hz. = -5 2 G.-1 D.0 -1.1113
Hz.
> 0 7 C.0 G.0 -1.1158 Hz. = -5 2 G.-1 D.0 -1.1113
Hz.
> 2 9 D.0 A.0 -2.2182 Hz. = 7 14 G.0 D.1 -2.2226
Hz.
> 7 14 G.0 D.1 -2.2226 Hz. = 12 19 C.1 G.1 -2.2317
Hz.
> -10 -6 D.-1 F#.-1 5.5727 Hz. = -10 -1 D.-1 B.-1 5.5727
Hz.
> 0 4 C.0 E.0 5.5779 Hz. = -10 -1 D.-1 B.-1 5.5727
Hz.
> 0 4 C.0 E.0 5.5779 Hz. = -5 -1 G.-1 B.-1 5.5782
Hz.
> -5 -1 G.-1 B.-1 5.5782 Hz. = -8 -5 E.-1 G.-1 -5.5782
Hz.
> -5 -1 G.-1 B.-1 5.5782 Hz. = -5 4 G.-1 E.0 5.5782
Hz.
> -8 -4 E.-1 G#.-1 7.6853 Hz. = -8 1 E.-1 C#.0 7.6853
Hz.
> -8 -4 E.-1 G#.-1 7.6853 Hz. = -11 -8 C#.-1 E.-1 -7.6853
Hz.
> -8 1 E.-1 C#.0 7.6853 Hz. = -3 1 A.-1 C#.0 7.6945
Hz.
> -11 -8 C#.-1 E.-1 -7.6853 Hz. = -3 1 A.-1 C#.0 7.6945
Hz.
> -12 -9 C.-1 Eb.-1 -7.9223 Hz. = -7 2 F.-1 D.0 7.9788
Hz.
> -9 0 Eb.-1 C.0 7.9223 Hz. = -7 2 F.-1 D.0 7.9788
Hz.
> -9 0 Eb.-1 C.0 7.9223 Hz. = -10 -7 D.-1 F.-1 -7.9788
Hz.
> -7 2 F.-1 D.0 7.9788 Hz. = -2 2 Bb.-1 D.0 7.9871
Hz.
> -10 -7 D.-1 F.-1 -7.9788 Hz. = -2 2 Bb.-1 D.0 7.9871
Hz.
> 0 9 C.0 A.0 8.6410 Hz. = -11 -2 C#.-1 Bb.-1 8.6328
Hz.
> -3 0 A.-1 C.0 -8.6410 Hz. = -6 -2 F#.-1 Bb.-1 8.6460
Hz.
> 0 9 C.0 A.0 8.6410 Hz. = -6 -2 F#.-1 Bb.-1 8.6460
Hz.
> -6 -2 F#.-1 Bb.-1 8.6460 Hz. = -6 3 F#.-1 Eb.0 8.6460
Hz.
> -6 -2 F#.-1 Bb.-1 8.6460 Hz. = -9 -6 Eb.-1 F#.-1 -8.6460
Hz.
> -4 0 G#.-1 C.0 10.5630 Hz.= -4 5 G#.-1 F.0
10.5630 Hz.
> -4 0 G#.-1 C.0 10.5630 Hz.= -7 -4 F.-1 G#.-1 -
10.5630 Hz.
> 2 6 D.0 F#.0 11.1455 Hz.= -3 6 A.-1 F#.0
11.1318 Hz.
> 2 6 D.0 F#.0 11.1455 Hz.= -6 -3 F#.-1 A.-1 -
11.1318 Hz.
> 2 6 D.0 F#.0 11.1455 Hz.= -1 2 B.-1 D.0 -
11.1455 Hz.
> 2 6 D.0 F#.0 11.1455 Hz.= 2 11 D.0 B.0
11.1455 Hz.
> 12 16 C.1 E.1 11.1557 Hz.= -1 2 B.-1 D.0 -
11.1455 Hz.
> 2 11 D.0 B.0 11.1455 Hz.= 12 16 C.1 E.1
11.1557 Hz.
> 12 16 C.1 E.1 11.1557 Hz.= 7 11 G.0 B.0
11.1564 Hz.
> 7 11 G.0 B.0 11.1564 Hz.= 7 16 G.0 E.1
11.1564 Hz.
> 7 11 G.0 B.0 11.1564 Hz.= 4 7 E.0 G.0 -
11.1564 Hz.
> 5 9 F.0 A.0 11.5213 Hz.= -1 3 B.-1 Eb.0
11.5280 Hz.
> -1 3 B.-1 Eb.0 11.5280 Hz.= -4 -1 G#.-1 B.-1 -
11.5280 Hz.
> -1 3 B.-1 Eb.0 11.5280 Hz.= -1 8 B.-1 G#.0
11.5280 Hz.
> 4 8 E.0 G#.0 15.3706 Hz.= 4 13 E.0 C#.1
15.3706 Hz.
> 4 8 E.0 G#.0 15.3706 Hz.= 1 4 C#.0 E.0 -
15.3706 Hz.
> 4 13 E.0 C#.1 15.3706 Hz.= 9 13 A.0 C#.1
15.3889 Hz.
> 1 4 C#.0 E.0 -15.3706 Hz.= 9 13 A.0 C#.1
15.3889 Hz.
> 3 12 Eb.0 C.1 15.8446 Hz.= 2 5 D.0 F.0 -
15.9577 Hz.
> 0 3 C.0 Eb.0 -15.8446 Hz.= 2 5 D.0 F.0 -
15.9577 Hz.
> 0 3 C.0 Eb.0 -15.8446 Hz.= 5 14 F.0 D.1
15.9577 Hz.
> 2 5 D.0 F.0 -15.9577 Hz.= 10 14 Bb.0 D.1
15.9742 Hz.
> 5 14 F.0 D.1 15.9577 Hz.= 10 14 Bb.0 D.1
15.9742 Hz.
>
> This scale is quite fascinating. I wonder if it's possible to make
> the beat rate quotients for the triads with tempered fifths all
exactly
> the same. This gives a set of equations that's probably too hard to
> solve analytically so it must be done numerically. Any takers?
> The closest scale in the archive is Neidhardt 2 by the way, although
> not a very close match.
>
> Manuel

i'd love to tackle any numerical problems you guys might care to put
forward -- i'm equipped with matlab! so what is it exactly that you
were asking here? can you write the set of equations down? maybe we
can even discover some new temperaments that verituner would be
interested in . . .

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗manuel.op.de.coul@eon-benelux.com

🔗Kraig Grady <kraiggrady@anaphoria.com>

Paul!

I am not sure of the dates but they did have organ pipes that beat and ttrue as many of the physical laws
were not known, it never stopped organ builders. The other question is whether beats can be "sensed"
without being percieved. the possibility of this is high when you consider most of music is things still
more "sensed" than percieved in a way that we can analytical desipher!

>
> From: "wallyesterpaulrus <wallyesterpaulrus@yahoo.com>" <wallyesterpaulrus@yahoo.com>
>
>
>
> dear bob,
>
> please read monz's reply to what i wrote again.
>
> beats and overtones were unknown in zarlino's day.
>
> there was no way to accurately tune, IN THE FIRST PLACE 2/7-comma
> meantone as opposed to, say, 7/25-comma (also "optimal", though by
> different criteria), in those days.
>
> so what sounded good and what didn't sound good could have been 2/7-
> comma and 7/25-comma, or the reverse, or none of the above.
>
> no one had any way of knowing in those days. the "result" gene
> postulates above simply had no way of coming about, with any
> accuracy, in the first place -- as opposed to some "result" in which
> the beat synchrony did not appear.
>
> we take accurate tuning for granted today with our electronic tuners
> and computers. but before beats and overtones were even suspected of
> existing (and logarithms hadn't been invented yet), these small
> fractions of a comma were completely *theoretical*, in the realm of
> ideals rather than tangible musical entities.
>
> > It is difficult to imagine that people who tuned their own
> > instruments every day before playing, and even multiple times per
> day
> > in some circumstances, and who often had a choice of several
> commonly
> > used temperaments in their era that were not so grossly different
> > from each other, would never notice this beating even if they never
> > once thought in terms of harmonics.
>
> the problem is that this "choice" was not simply "given" the way it
> is with today's electronic gear. until the 17th century, there were
> absolutely no aural guides for setting temperaments -- neither
> eliminating beats nor getting their rates to match. so even if
> they "preferred" one temperament over another, they wouldn't know
> what they were comparing to what! until mersenne came around, setting
> different varieties of meantones was more guessing game than science.
>
> as to noticing beats, the vast majority of pianists whom i've talked
> to about tuning are unable to hear them even with a great deal of
> prodding. i walk over to the piano, play a major third, and say "hear
> the beats"? if we're using a synth, i'll play a just major third for
> comparison. "hear how that *doesn't* beat?" silence. i go back to the
> tempered major third and sing "wawawawa" in time with the beating
> common overtone. "hear that?" almost no pianist does. it's no wonder
> beating escaped everyone's attention until the 17th century.
>
> > The other arts were highly
> > refined, with a deep, if not always scientifically understood,
> > mastery of technical considerations. I find it difficult to think,
> > all things considered, that the art of music represented some kind
> of
> > contrast in this regard.
>
> everything i've said is well-documented. hopefully you'll be able, at
> least, to get a library to provide you with a copy of the jorgenson
> book. also, if possible, read up on mersenne's _harmonie
> universelle_, the state of tuning technique before it, and the state
> of tuning technique after it. you'll find some good info in the new
> grove dictionary. the "scientific revolution" happened a bit later in
> the musical arts than in the other arts in our civilization.
>
>

-- -Kraig Grady
North American Embassy of Anaphoria Island
http://www.anaphoria.com
The Wandering Medicine Show
KXLU 88.9 FM WED 8-9PM PST

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Jon Szanto <JSZANTO@ADNC.COM>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

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

> oops, how do i copy output from the scala window? ctrl-c doesn't
>work.

figured it out. here is the result from my last tuning, the one based
on 90 (which i multiplied by 3 to make it 270, a "C"):

Base frequency : 270.0000 Hertz
Equal beating tests of 6/5 5/4 3/2 5/3
-8 -1 E.-1 B.-1 1.4999 Hz. = -4 3 G#.-1 Eb.0 -1.4999
Hz.
-4 3 G#.-1 Eb.0 -1.4999 Hz. = -10 -3 D.-1 A.-1 -1.5000
Hz.
-10 -3 D.-1 A.-1 -1.5000 Hz. = -5 2 G.-1 D.0 -1.5000
Hz.
-5 2 G.-1 D.0 -1.5000 Hz. = -3 4 A.-1 E.0 -1.5000
Hz.
-3 4 A.-1 E.0 -1.5000 Hz. = -11 -4 C#.-1 G#.-1 1.5000
Hz.
-11 -4 C#.-1 G#.-1 1.5000 Hz. = -6 1 F#.-1 C#.0 -1.5000
Hz.
-6 1 F#.-1 C#.0 -1.5000 Hz. = -9 -2 Eb.-1 Bb.-1 -1.5001
Hz.
4 11 E.0 B.0 2.9998 Hz. = -1 6 B.-1 F#.0 -2.9998
Hz.
4 11 E.0 B.0 2.9998 Hz. = 8 15 G#.0 Eb.1 -2.9998
Hz.
8 15 G#.0 Eb.1 -2.9998 Hz. = 2 9 D.0 A.0 -3.0000
Hz.
2 9 D.0 A.0 -3.0000 Hz. = -12 -8 C.-1 E.-1 3.0000
Hz.
7 14 G.0 D.1 -3.0000 Hz. = -12 -8 C.-1 E.-1 3.0000
Hz.
7 14 G.0 D.1 -3.0000 Hz. = 9 16 A.0 E.1 -3.0001
Hz.
9 16 A.0 E.1 -3.0001 Hz. = 1 8 C#.0 G#.0 3.0001
Hz.
1 8 C#.0 G#.0 3.0001 Hz. = 6 13 F#.0 C#.1 -3.0001
Hz.
6 13 F#.0 C#.1 -3.0001 Hz. = 3 10 Eb.0 Bb.0 -3.0002
Hz.
-8 -5 E.-1 G.-1 -4.5000 Hz. = -12 -3 C.-1 A.-1 4.5000
Hz.
-5 4 G.-1 E.0 4.5000 Hz. = -12 -3 C.-1 A.-1 4.5000
Hz.
-12 -3 C.-1 A.-1 4.5000 Hz. = -10 -6 D.-1 F#.-1 4.5001
Hz.
11 18 B.0 F#.1 -5.9996 Hz. = 16 23 E.1 B.1 5.9996
Hz.
16 23 E.1 B.1 5.9996 Hz. = 14 21 D.1 A.1 -5.9999
Hz.
0 4 C.0 E.0 6.0000 Hz. = 14 21 D.1 A.1 -5.9999
Hz.
0 4 C.0 E.0 6.0000 Hz. = -7 -3 F.-1 A.-1 6.0000
Hz.
-7 -3 F.-1 A.-1 6.0000 Hz. = 13 20 C#.1 G#.1 6.0001
Hz.
13 20 C#.1 G#.1 6.0001 Hz. = 15 22 Eb.1 Bb.1 -6.0004
Hz.
-5 -1 G.-1 B.-1 7.4998 Hz. = -9 -5 Eb.-1 G.-1 7.4998
Hz.
-9 -5 Eb.-1 G.-1 7.4998 Hz. = -12 -9 C.-1 Eb.-1 -7.4998
Hz.
-9 -5 Eb.-1 G.-1 7.4998 Hz. = -9 0 Eb.-1 C.0 7.4998
Hz.
-12 -9 C.-1 Eb.-1 -7.4998 Hz. = -10 -1 D.-1 B.-1 7.4998
Hz.
-9 0 Eb.-1 C.0 7.4998 Hz. = -10 -1 D.-1 B.-1 7.4998
Hz.
-10 -1 D.-1 B.-1 7.4998 Hz. = -4 0 G#.-1 C.0 7.4999
Hz.
-4 0 G#.-1 C.0 7.4999 Hz. = -7 -4 F.-1 G#.-1 -7.4999
Hz.
-4 0 G#.-1 C.0 7.4999 Hz. = -4 5 G#.-1 F.0 7.4999
Hz.
-7 -4 F.-1 G#.-1 -7.4999 Hz. = -6 -2 F#.-1 Bb.-1 7.5000
Hz.
-4 5 G#.-1 F.0 7.4999 Hz. = -6 -2 F#.-1 Bb.-1 7.5000
Hz.
-6 -2 F#.-1 Bb.-1 7.5000 Hz. = -3 1 A.-1 C#.0 7.5000
Hz.
-3 1 A.-1 C#.0 7.5000 Hz. = -11 -7 C#.-1 F.-1 7.5000
Hz.
-11 -7 C#.-1 F.-1 7.5000 Hz. = -11 -2 C#.-1 Bb.-1 7.5000
Hz.
-11 -7 C#.-1 F.-1 7.5000 Hz. = -11 -8 C#.-1 E.-1 -7.5000
Hz.
-11 -2 C#.-1 Bb.-1 7.5000 Hz. = -8 1 E.-1 C#.0 7.5000
Hz.
-11 -2 C#.-1 Bb.-1 7.5000 Hz. = -8 1 E.-1 C#.0 7.5000
Hz.
7 16 G.0 E.1 9.0000 Hz. = -7 2 F.-1 D.0 9.0000
Hz.
7 16 G.0 E.1 9.0000 Hz. = -7 2 F.-1 D.0 9.0000
Hz.
4 7 E.0 G.0 -9.0000 Hz. = -10 -7 D.-1 F.-1 -9.0000
Hz.
4 7 E.0 G.0 -9.0000 Hz. = -10 -7 D.-1 F.-1 -9.0000
Hz.
0 9 C.0 A.0 9.0000 Hz. = -7 2 F.-1 D.0 9.0000
Hz.
0 9 C.0 A.0 9.0000 Hz. = -7 2 F.-1 D.0 9.0000
Hz.
-10 -7 D.-1 F.-1 -9.0000 Hz. = -3 0 A.-1 C.0 -9.0000
Hz.
-10 -7 D.-1 F.-1 -9.0000 Hz. = -3 0 A.-1 C.0 -9.0000
Hz.
0 9 C.0 A.0 9.0000 Hz. = 2 6 D.0 F#.0 9.0001
Hz.
2 6 D.0 F#.0 9.0001 Hz. = -3 0 A.-1 C.0 -9.0000
Hz.
2 6 D.0 F#.0 9.0001 Hz. = -1 3 B.-1 Eb.0 9.0005
Hz.
-3 6 A.-1 F#.0 10.5000 Hz.= -8 -4 E.-1 G#.-1 10.5001
Hz.
-6 -3 F#.-1 A.-1 -10.5000 Hz.= -8 -4 E.-1 G#.-1 10.5001
Hz.
-8 -4 E.-1 G#.-1 10.5001 Hz.= -6 3 F#.-1 Eb.0 10.5001
Hz.
-8 -4 E.-1 G#.-1 10.5001 Hz.= -9 -6 Eb.-1 F#.-1 -10.5001
Hz.
12 16 C.1 E.1 11.9999 Hz.= -2 2 Bb.-1 D.0 12.0000
Hz.
5 9 F.0 A.0 12.0000 Hz.= -2 2 Bb.-1 D.0 12.0000
Hz.
5 9 F.0 A.0 12.0000 Hz.= -4 -1 G#.-1 B.-1 -12.0004
Hz.
5 9 F.0 A.0 12.0000 Hz.= -1 8 B.-1 G#.0 12.0004
Hz.
7 11 G.0 B.0 14.9995 Hz.= 3 7 Eb.0 G.0 14.9997
Hz.
3 7 Eb.0 G.0 14.9997 Hz.= 3 12 Eb.0 C.1 14.9997
Hz.
3 7 Eb.0 G.0 14.9997 Hz.= 0 3 C.0 Eb.0 -14.9997
Hz.
3 12 Eb.0 C.1 14.9997 Hz.= -1 2 B.-1 D.0 -14.9997
Hz.
0 3 C.0 Eb.0 -14.9997 Hz.= -1 2 B.-1 D.0 -14.9997
Hz.
0 3 C.0 Eb.0 -14.9997 Hz.= 2 11 D.0 B.0 14.9997
Hz.
8 12 G#.0 C.1 14.9999 Hz.= -1 2 B.-1 D.0 -14.9997
Hz.
2 11 D.0 B.0 14.9997 Hz.= 8 12 G#.0 C.1 14.9999
Hz.
8 12 G#.0 C.1 14.9999 Hz.= 5 8 F.0 G#.0 -14.9999
Hz.
8 12 G#.0 C.1 14.9999 Hz.= 8 17 G#.0 F.1 14.9999
Hz.
5 8 F.0 G#.0 -14.9999 Hz.= 6 10 F#.0 Bb.0 14.9999
Hz.
8 17 G#.0 F.1 14.9999 Hz.= 6 10 F#.0 Bb.0 14.9999
Hz.
6 10 F#.0 Bb.0 14.9999 Hz.= 9 13 A.0 C#.1 14.9999
Hz.
9 13 A.0 C#.1 14.9999 Hz.= -2 7 Bb.-1 G.0 15.0000
Hz.
9 13 A.0 C#.1 14.9999 Hz.= -5 -2 G.-1 Bb.-1 -15.0000
Hz.
1 5 C#.0 F.0 15.0000 Hz.= -2 7 Bb.-1 G.0 15.0000
Hz.
1 5 C#.0 F.0 15.0000 Hz.= -5 -2 G.-1 Bb.-1 -15.0000
Hz.
1 5 C#.0 F.0 15.0000 Hz.= -2 1 Bb.-1 C#.0 -15.0000
Hz.
1 5 C#.0 F.0 15.0000 Hz.= 1 10 C#.0 Bb.0 15.0000
Hz.
1 4 C#.0 E.0 -15.0000 Hz.= -2 1 Bb.-1 C#.0 -15.0000
Hz.
1 4 C#.0 E.0 -15.0000 Hz.= -2 1 Bb.-1 C#.0 -15.0000
Hz.
1 10 C#.0 Bb.0 15.0000 Hz.= 4 13 E.0 C#.1 15.0000
Hz.
1 10 C#.0 Bb.0 15.0000 Hz.= 4 13 E.0 C#.1 15.0000
Hz.

everything's a multiple of 1.5 Hz.

🔗Carl Lumma <ekin@lumma.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Carl Lumma <ekin@lumma.org>

>> They obviously knew how to set 1/4 comma meantone.
>
>obviously?

What was the standard tuning of the day, do you think?
Pythagorean?

>to what degree of accuracy?

You can tune a just 5:4 to a high degree of accuracy, and
then estimate the fifths.

>> Why not 2/7 comma?
>
>how? what procedure would be used? try tuning a harpsichord to a
>monochord sometime and tell me what accuracy you can get.

I've tuned harpsichords in Kirnberger, using beating tests. Is
it impossible that these existed in Zarlino's day? One doesn't
need knowledge of overtones to hear beating.

Norman Henry tunes pianos quite well without knowing any explicit
tests. He listens to beating, tuning by 4ths and 5ths, trying
to get them all "about the same", but follows no set procedure.
He's been tuning since he was a kid and he just figured out how to
do it somehow. Equal temperament is the tuning of the day and he
can set it.

-Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗akjmicro <akjmicro@comcast.net>

Hey guys,

I know this is an old thread, but I've been taking an interest in the 2/7 comma
meantone lately. I found this thread in a Google search.

I wanted to make a point that wallyesterpaulrus is missing something in his
argument here. I will state my objection in the form of a question:

If beating was unknown or unpercievable in Zarlino's time, what was the motivation
to temper at all, if not the sensation that the supertonic D-A in C just tuning was
harsh and wobbly, for example? Are we actually willing to semantically brush aside
the idea that these musicians of the past could hear beating? Could they not
recognize a just interval?

The absurdity of thinking otherwise make me side with Robert Wendell on this one.

P.S. Just because many modern musicians can't percieve beating, doesn't
validate that the ancients couldn't, or that certain moderns can. It's all perception
and perception of beating can be trained!!! After all, those of us whoo can hear
beating at one point had to be introduced to the concept, or discovered it!

Any thoughts?

Best,
Aaron.

--- In tuning@yahoogroups.com, "wallyesterpaulrus <wallyesterpaulrus@y...>"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
> <rwendell@c...> wrote:
> > --- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
> > <gwsmith@s...> wrote:
> > > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > >
> > > > i agree with you that at this early stage of the game
> > > > (1558), Zarlino was probably unaware of beat synchrony.
> > > > this was still several decades before the recognition
> > > > of the overtone series, which forms the basis of
> > > > calculating beats.
> > >
> > > Robert's theory as I understand it is not that Zarlino noticed
> > > synchronized beating, but that he noticed the result sounded
> good.
> > If
> > > synchronized beating was part of the reason why, it would serve
> as a
> > > reason for both 2/7 and 1/4 comma advocacy.
> >
> > Bob:
> > Precisely, Gene! Thank you.
>
> dear bob,
>
> please read monz's reply to what i wrote again.
>
> beats and overtones were unknown in zarlino's day.
>
> there was no way to accurately tune, IN THE FIRST PLACE 2/7-comma
> meantone as opposed to, say, 7/25-comma (also "optimal", though by
> different criteria), in those days.
>
> so what sounded good and what didn't sound good could have been 2/7-
> comma and 7/25-comma, or the reverse, or none of the above.
>
> no one had any way of knowing in those days. the "result" gene
> postulates above simply had no way of coming about, with any
> accuracy, in the first place -- as opposed to some "result" in which
> the beat synchrony did not appear.
>
> we take accurate tuning for granted today with our electronic tuners
> and computers. but before beats and overtones were even suspected of
> existing (and logarithms hadn't been invented yet), these small
> fractions of a comma were completely *theoretical*, in the realm of
> ideals rather than tangible musical entities.
>
> > It is difficult to imagine that people who tuned their own
> > instruments every day before playing, and even multiple times per
> day
> > in some circumstances, and who often had a choice of several
> commonly
> > used temperaments in their era that were not so grossly different
> > from each other, would never notice this beating even if they never
> > once thought in terms of harmonics.
>
> the problem is that this "choice" was not simply "given" the way it
> is with today's electronic gear. until the 17th century, there were
> absolutely no aural guides for setting temperaments -- neither
> eliminating beats nor getting their rates to match. so even if
> they "preferred" one temperament over another, they wouldn't know
> what they were comparing to what! until mersenne came around, setting
> different varieties of meantones was more guessing game than science.
>
> as to noticing beats, the vast majority of pianists whom i've talked
> to about tuning are unable to hear them even with a great deal of
> prodding. i walk over to the piano, play a major third, and say "hear
> the beats"? if we're using a synth, i'll play a just major third for
> comparison. "hear how that *doesn't* beat?" silence. i go back to the
> tempered major third and sing "wawawawa" in time with the beating
> common overtone. "hear that?" almost no pianist does. it's no wonder
> beating escaped everyone's attention until the 17th century.
>
> > The other arts were highly
> > refined, with a deep, if not always scientifically understood,
> > mastery of technical considerations. I find it difficult to think,
> > all things considered, that the art of music represented some kind
> of
> > contrast in this regard.
>
> everything i've said is well-documented. hopefully you'll be able, at
> least, to get a library to provide you with a copy of the jorgenson
> book. also, if possible, read up on mersenne's _harmonie
> universelle_, the state of tuning technique before it, and the state
> of tuning technique after it. you'll find some good info in the new
> grove dictionary. the "scientific revolution" happened a bit later in
> the musical arts than in the other arts in our civilization.

🔗Paul Erlich <paul@stretch-music.com>

--- In tuning@yahoogroups.com, "akjmicro" <akjmicro@c...> wrote:
> Hey guys,
>
> I know this is an old thread, but I've been taking an interest in
the 2/7 comma
> meantone lately. I found this thread in a Google search.
>
> I wanted to make a point that wallyesterpaulrus is missing
something in his
> argument here. I will state my objection in the form of a question:
>
> If beating was unknown or unpercievable in Zarlino's time,

not exactly what i said, if you follow the rest of the thread.

> what was the motivation
> to temper at all, if not the sensation that the supertonic D-A in C
just tuning was
> harsh and wobbly, for example?

because it was harsh and wobbly! there are a lot of other factors
influencing consonance besides beating. but yes, beating was
certainly one of the factors that made this interval undesirable at
the time.

> Could they not
> recognize a just interval?

of course they could.

> The absurdity of thinking otherwise make me side with Robert
>Wendell on this one.

if you continue through the thread, you'll find that robert wendell
and i ended up in complete agreement.

🔗Paul Erlich <paul@stretch-music.com>

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

> --- In tuning@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
> > <rwendell@c...> wrote:
> > > --- In tuning@yahoogroups.com, "Gene Ward Smith
<gwsmith@s...>"
> > > <gwsmith@s...> wrote:
> > > > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > > >
> > > > > i agree with you that at this early stage of the game
> > > > > (1558), Zarlino was probably unaware of beat synchrony.
> > > > > this was still several decades before the recognition
> > > > > of the overtone series, which forms the basis of
> > > > > calculating beats.
> > > >
> > > > Robert's theory as I understand it is not that Zarlino
noticed
> > > > synchronized beating, but that he noticed the result sounded
> > good.
> > > If
> > > > synchronized beating was part of the reason why, it would
serve
> > as a
> > > > reason for both 2/7 and 1/4 comma advocacy.
> > >
> > > Bob:
> > > Precisely, Gene! Thank you.

aaron, my point in this thread was not that lack of beating wouldn't
have been noticeable in the 16th century. my point wasn't even that
synchronized beating wouldn't have sounded good in the 16th century
(although that's more of a stretch). my point was that zarlino
couldn't have preferred 2/7-comma meantone on the basis of its
synchronized beating, because the technology and knowledge didn't
exist, at the time, to produce 2/7-comma meantone with such accuracy
that synchronized beating occured -- as bob wendell ultimately
agreed -- for comparison against some other, similar tuning, without
synchronized beating, which presumably would have been "unpreferred".
and, thanks to leonardo peretti (i think), we got to read zarlino's
writings, in which there's every indication that he derived 2/7-comma
meantone (and his preference for it) as paper theory before ever
trying to tune it up, and there's no mention of the existence of any
type of meantone that would be considered "non-synchronous" today and
could form a valid basis for the presumed comparison, and thus
preference, on that (unconscious) basis. and, needless to say
(because it's made irrelevant by the above), there's no mention of
beating, let along synchronized beating.

hope i've made my remarks clearer to you -- i had quite a rough time
clarifying them to the parties involved at the time, but with a lot
of effort, i succeeded.

🔗Aaron K. Johnson <akjmicro@comcast.net>

Paul,

Thanks for the crystal clarification !!!

Yes, I agree that beat synchrony was definately NOT the reason for 2/7 comma
meantone. I still think it's an open question though, whether they could hear
beat synchrony, although if the literature of the time doesn't speak of it in
those terms, it appears doubtful.

BTW, I didn't know "wallyyesterpaulrus" was you! I like your new handle, 'Paul
Erlich', better.....

All best,
Aaron.

On Monday 06 October 2003 10:27 pm, Paul Erlich wrote:
> --- In tuning@yahoogroups.com, "akjmicro" <akjmicro@c...> wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus
>
> <wallyesterpaulrus@y...>"
>
> > <wallyesterpaulrus@y...> wrote:
> > > --- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
> > >
> > > <rwendell@c...> wrote:
> > > > --- In tuning@yahoogroups.com, "Gene Ward Smith
>
> <gwsmith@s...>"
>
> > > > <gwsmith@s...> wrote:
> > > > > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> > > > > > i agree with you that at this early stage of the game
> > > > > > (1558), Zarlino was probably unaware of beat synchrony.
> > > > > > this was still several decades before the recognition
> > > > > > of the overtone series, which forms the basis of
> > > > > > calculating beats.
> > > > >
> > > > > Robert's theory as I understand it is not that Zarlino
>
> noticed
>
> > > > > synchronized beating, but that he noticed the result sounded
> > >
> > > good.
> > >
> > > > If
> > > >
> > > > > synchronized beating was part of the reason why, it would
>
> serve
>
> > > as a
> > >
> > > > > reason for both 2/7 and 1/4 comma advocacy.
> > > >
> > > > Bob:
> > > > Precisely, Gene! Thank you.
>
> aaron, my point in this thread was not that lack of beating wouldn't
> have been noticeable in the 16th century. my point wasn't even that
> synchronized beating wouldn't have sounded good in the 16th century
> (although that's more of a stretch). my point was that zarlino
> couldn't have preferred 2/7-comma meantone on the basis of its
> synchronized beating, because the technology and knowledge didn't
> exist, at the time, to produce 2/7-comma meantone with such accuracy
> that synchronized beating occured -- as bob wendell ultimately
> agreed -- for comparison against some other, similar tuning, without
> synchronized beating, which presumably would have been "unpreferred".
> and, thanks to leonardo peretti (i think), we got to read zarlino's
> writings, in which there's every indication that he derived 2/7-comma
> meantone (and his preference for it) as paper theory before ever
> trying to tune it up, and there's no mention of the existence of any
> type of meantone that would be considered "non-synchronous" today and
> could form a valid basis for the presumed comparison, and thus
> preference, on that (unconscious) basis. and, needless to say
> (because it's made irrelevant by the above), there's no mention of
> beating, let along synchronized beating.
>
> hope i've made my remarks clearer to you -- i had quite a rough time
> clarifying them to the parties involved at the time, but with a lot
> of effort, i succeeded.
>
>
>
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