Yahoo Tuning Groups Ultimate Backup tuning A question and an update on personal tuning endeavors

Hi, all! I've been away for quite awhile, and still have little time
to participate at the previous level, but I do want to check in.

I've asked this before, but the answer may have changed. I wish to
purchase a keyboard with full midi capability, tuning tables to which
you can download whatever, and ideally, retunable in real time. I'm
going to have to go to the used market, eBay, or whatever, since
funds are severely restricted. I don't know whether it's possible,
but we need to stay at around $200.

Also, despite what some here may have perceived as an obsession with
JI, or very close approximations to it, which remains a strong
interest, I have been venturing into the world of 12-tone temperament
design. This discussion is predicated on harmonic tibmres, or at
least quasi-harmonic (e.g., pianos).

I have especially been fascinated by the prospect of designing well
temperaments with synchronous beating in the triads. The rationale is
simple: If you must have beats, as you clearly must in a 12-tone
temperament, then they should at least have a simple, coherent
relationship to each other, since clean intonation is itself based on
simple, coherent relationships among frequencies.

In word, or rather, phrase, it is a kind of second-order coherence
phenomenon that partially compensates the necessary compromises in
first-order coherence. The subjective result is remarkably superior
to tunings with random beat relationships, and eliminates the usual
jangle one hears on pianos, for example, tuned in standard ET.

After considerable work, always heavily supplemented with empirical
experimentation to ensure a practical development track, I have
concluded that beat ratios between the minor and major thirds of
major triads should be the essential concern, since everything else
falls out nicely from there. Also, the only ratios that seem to have
significant aural value over any old random ratio are 2/1 and 3/2.
That is, the minor third beats twice for every once the major third
beats, or three times for every two. This is, of course, analagous to
the octave and perfect fifth in the intonational aspect.

Mathematically, to achieve the 1.5 (3/2) ratio for a triad, we simply
need to make the fifth just. To achieve a 2/1 ratio, we must temper
the fifth so that it is 1/3rd as flat as the major third is sharp.
Armed with this basic understanding, I began my design work.

In the process, I made an amazing discovery. If we temper six of the
twelve fifths, namely C, G, D, A, F# and Bb, and start with a quarter
syntonic comma on F#, this creates a 2.0 beat ratio (m3/M3) in the
triad on F#. This is the only triad for which only one of the four
fifths from the root to the major third is tempered. Every other
major third forming a major triad with a tempered fifth in the cycle
encompasses at least two tempered fifths.

Working backwards from F# toward C and then Bb, we can calculate the
amount by which each tempered fifth must be altered to make each
triad it encompasses beat in a 2.0 ratio. The idea was that since the
most remote keys are farthest from just, they need the most precisely
synchronized beat ratios to compensate.

The strategy was to start with F# at a perfect 2.0, surrounded by
keys with all just fifths (two before and three after), and work
backwards, tempering each previously designated tempered fifth by an
amount that would yield a perfect 2.0 beat ratio for a triad in root
position on it. Since they all overlap with at least one other
tempered fifth, we could go all the way around to Bb and then total
the amount of tempering on all six fifths to see how much we would
need to fudge in order to null out the Pythagorean comma and come out
with a closed cycle.

Well, when I totaled up the tempering (with all six tempered fifths
sporting perfect 2.0, and the rest with perfect 1.5 ratios), I was
blown away to discover that the sum was 0.0058 cents less than the
Pythagorean comma! I added this difference to the fifth on C, the
major third of which was at only +7.4 cents, so it was the least
needy of any synchronous compensation. This ruined the theoretically
perfect ratio of 2.0 by changing it to 2.0016 (boo-hoo!).

This is being published as the Wendell Natural Synchronous Well. This
work has enable me to create a number of synchronous temperaments,
six of which will be included in the next release of Verituner
(http://www.verituneinc.com), without a doubt the premier electronic
tuning device for professional tuners. They are as follows:

Wendell Well 2002 (an early one, not fully synchronous, but with
other nice properties)

Wendell Synchronous ET Equivalent (varies only slightly from ET, but
fully synchronous, the triad on Ab excepted)

Bold Synchronous Well (M3s from +4.2 on C to +19.6 cents on C# and
Ab)

Very Mild Synchronous Well (M3s from +7.9 on C to +17.2 on C#, +16.8
cents on Ab)

Natural Synchronous Well (M3s from +7.4 on C to +17.6 cents on C# and
Ab)

Synchronous Modified Mean/Well (Designed for Eb through E as playable
keys, but never drops off the cliff, since the M3 on C# is widest at
+21.5 cents. However, the six standard good meantones keys from Bb to
A are significantly nicer than ET, and Eb and E are about like ET.
Its purpose is to enable more historically informed performance than
ET could provide for almost any music intended for meantone, but
without ever having to retune.)

I have to stop now, but will publish ET offsets for the Natural
later. Some of my temperaments are graphically displayed with offests
and other specs at Jason Kanter's Website:

http://www.rollingball.com

All the best,

Bob

🔗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, "Robert Wendell <rwendell@c...>"
> <rwendell@c...> wrote:
>
> > Mathematically, to achieve the 1.5 (3/2) ratio for a triad, we
> simply
> > need to make the fifth just. To achieve a 2/1 ratio, we must
temper
> > the fifth so that it is 1/3rd as flat as the major third is
sharp.
> > Armed with this basic understanding, I began my design work.
>
> This is 1/7-comma meantone, very close to 53/91.

Hi, Gene! Yes, I'm well aware that all the playable triads in 1/7-
comma meantone have perfect 2.0 beat ratios. For precisely that
reason, my modified meantone is a modification of 1/7th comma with an
eye to extending it by a triad on either end of the six normally
playable keys to get 8 playable major and 5 playable minor keys.

Only three minor keys are actually playable in standard meantones,
since good relative minors is not a sufficient condition. Almost all
music written in minor keys requires good parallel major keys because
of the melodic raising of the sixth and seventh scale degrees and
Picardian thirds. So adding two playable triads makes a substantial
contribution by expanding 6 to 8 for major and 3 to 5 for minor keys.

You just bought four more playable keys with only two triads. Good
deal, huh?! That is historically what created the motivation for
modified meantones, which ended up turning into well temperaments
over time. That's exactly what happened to mine without trying. It
demonstrates clearly to me that well temperaments were in the
evolutionary cards for temperament design.

My favorite straight meantone is 1/5th comma. When you have 1/5th
comma left over for the major third after subtracting 4/5ths from the
full comma, you have -1/5th on the fifth and +1/5th on the third.
This gives exactly 3/1 on the beat ratio, so it's also a fully
synchronous temperament in all the playable keys. However, a 3.0 beat
ratio eats up a ton of the total Pyathagorean real estate, so you can
use it only once in a fairly bold well temperament.

Another neat thing about 1/5th-comma is that the minor second is
exactly 15:16, that is, it's just. This is because the fourth and the
third are raised by the same amount, 1/5th comma. This is only 4.3
cents, so neither the third nor the fifth are severely compromised.
The minor third is, of course, at -8.6, but that's about half as
detuned as it is in 12-tET. This meantone is also melodically more
normal sounding to modern ears than the ones on the other side of 1/4-
comma, since it's deviations from just are in the direction of modern
ET instead of away from it.

If you want a telling demonstration as to just how wide the range is
of melodic possibilities that are compatible with meantones closer
to just than 12-tET, try 2/7th-comma meantone. It's also synchronous
at +1.5 ratios instead of the -1.5 on just P5s, and is
hanging out in the general neighborhood of Woolhouse's optimization
for P5s, M3s, and m3s in the triads.

For my taste, it's the most pleasingly harmonious, since the
synchrony more than compensates its relatively short distance from
Woolhouse's 7/26th-comma. But try playing a simple melody on it
without any harmony, especially melodic minor, and it sounds
otherworldly! A good example is the melody from the Coventry
Carol, "Lullay, Lullay". Try it! Weird! Yet if the melody is played
in the context of harmony, it gets "rationalized" to the ear and
sounds fine!

Cheers,

Bob

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

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

Well, as I recall, Woolhouse weighted the three triadic intervals
(root position), the P5, M3, and m3. My understanding of your
definition of "poptimal" is that it gives these intervals equal
weight, so I don't know quite why you say 7/26-comma is poptimal,
unless Woolhouse's weighting was such that it falls within range
anyway.

Further, as I intended to imply rather directly with my previous
recent posts, I feel that given sufficiently small differences in
deviations from just, beat synchrony weighs more than squeaky clean
optimizations.

Nevertheless, the last point in the post to which you are responding
here is different. I think it's fascinating how radically different
from ET, JI, or anything we're used to melodically such a
satisfyingly harmonious temperament as 2/7-comma can be! It is so
weirdly different melodically, absent any harmonic context
to "rationalize" the melody to the ear. All the temperaments that far
south toward the 1/3-comma extreme have very wide minor seconds
(diatonic half steps) and exceedlingly narrow chromatic half steps.

Cheers,

Bob

--- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
<gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
> <rwendell@c...> wrote:
>
> > If you want a telling demonstration as to just how wide the range
> is
> > of melodic possibilities that are compatible with meantones
closer
> > to just than 12-tET, try 2/7th-comma meantone. It's also
> synchronous
> > at +1.5 ratios instead of the -1.5 on just P5s, and is
> > hanging out in the general neighborhood of Woolhouse's
optimization
> > for P5s, M3s, and m3s in the triads.
>
> If you can figure out anything notable about 4/15-comma meantone,
> that one is poptimal. So are both 7/26 and 7/27, as well as 5/19
> and 6/23 comma meantone.

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

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

--- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
<rwendell@c...> wrote:

> Well, as I recall, Woolhouse weighted the three triadic intervals
> (root position), the P5, M3, and m3. My understanding of your
> definition of "poptimal" is that it gives these intervals equal
> weight, so I don't know quite why you say 7/26-comma is poptimal,
> unless Woolhouse's weighting was such that it falls within range
> anyway.

I'm not following--we both use equal weight, so why shouldn't we get
equal answers? By definition, 7/26 is poptimal for exponent p=2.

> Further, as I intended to imply rather directly with my previous
> recent posts, I feel that given sufficiently small differences in
> deviations from just, beat synchrony weighs more than squeaky clean
> optimizations.

Did you read my "Brat" postings? I pointed out you can do both, or
try to. 11/41-comma meantone is poptimal, and has a beat ratio (brat)
of -4; 3/11-comma very nearly is poptimal, at -3.

> Nevertheless, the last point in the post to which you are
responding
> here is different. I think it's fascinating how radically different
> from ET, JI, or anything we're used to melodically such a
> satisfyingly harmonious temperament as 2/7-comma can be!

It does have its own kind of charm, doesn't it?

🔗Jon Szanto <JSZANTO@ADNC.COM> <JSZANTO@ADNC.COM>

Bob,

There is an off-chance something might come *close*, but I actually do believe you are asking the impossible. To specific points:

--- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>" <rwendell@c...> wrote:
> I wish to purchase a keyboard with full midi capability, tuning tables to which you can download whatever

So far, I believe there might be a couple of possibilities, but it may be that you will need a keyboard to drive an external synth box, with the *possible* exceptions of some of the Kurzweil keyboards (and maybe Ensoniq?).

> and ideally, retunable in real time.

Dream on. Seriously, what do *you* mean by "real time" - the ability to shift tunings, across the span of the keyboard, *as* you are performing? I don't believe so, as even if you had an external midi contraption that would send, at given points, a midi tuning file, I would believe all keyboards would have to stop playing to accept a tuning file dump.

> I'm going to have to go to the used market, eBay, or whatever, since funds are severely restricted. I don't know whether it's possible, but we need to stay at around $200.

There isn't anything even close to that price range. What you *might* find in that price range would be one of the external 1 or 2 rack space module - you can even get old TX-802 modules for around that. Even so, you'll need a keyboard to drive it, and I didn't see where you specified what the thing needed to sound like. If you are talking about getting a close approximation of a piano sound, you are going to have to budget a lot more than that.

The only other thing I can think of, and it would enter an entire other world, would be to explore the new realm of software synths, where we are seeing far more development in tuning flexibility. Alas, this also means computers, external keyboards, soundcard latency, and a lot of other cruft.

If you haven't taken a look, you should look at the 4 or so pages of charts that John Loffink maintains on the tuning specs of almost any piece of gear (mostly hard, some soft) that is out there these days. His new url is:

http://www.microtonal-synthesis.com/

I can't guarantee any better results, but you might ask this question over at MMM if no one here has any better ideas...

Cheers,
Jon

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

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

"Did you read my "Brat" postings? I pointed out you can do both, or
try to. 11/41-comma meantone is poptimal, and has a beat ratio (brat)
of -4; 3/11-comma very nearly is poptimal, at -3."

OK, I didn't get it. I don't know much math beyond trig, analytic
geometry, some elementary statistics, and basic concepts
that are the fossil remains of elementary calculus. What beat ratio
are you talking about? I focus on the beat ratio of m3/M3, since the
fifth beats at a relatively slow rate and is therefore not generally
a major producer of the random beating jangle you hear with some
temperaments like 12-tET. Moreover, when there is a simple ratio of
m3/M3, the relationship of both to the beat rate of the P5 is
automatically also simple.

By my reckoning, since the third and fifth are both flat, 11/41-comma
meantone has a beat ratio on m3/M3 of approximately +1.1. On the
other hand, since the M3 in 1/4-comma is pure, this beat ratio is
infinite, or perhaps it's more meaningful to say undefined. The only
beat ratio of significance therefore becomes that of m3/P5, which by
my reckoning is -2.50. Am I off-base here?

In my temperament design work, I use a highly accurate linear
approximation for converting ratios of the cents deviation from just
to beat ratios. The formula is:

R = 1.5(delta P5/delta M3 - 1)

where R is the beat ratio and delta P5 and delta M3 are the
deviations in cents from just (with signs preserved, - is flat, + is
sharp) for the interval under consideration.

I also use half a schisma for the unit, since this is very close to
one cent (0.9775) and allows playing off the syntonic and Pythagorean
commas in terms of the whole numbers 22 and 24 respectively. This
with the simple formula to eliminate the need for log conversions
puts headlights on my design wagon and allows me to run calculations
in my head while walking around, waiting in line at the bank, etc. I
am more of an engineer than a mathematician, although I have no lack
of respect for the power of advanced mathematical knowledge. Wish I
had more.

Cheers,

Bob

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

Hey there, Jon! Great to hear from you again! I just looked at
Justonics page at

http://www.justonic.com/default3.htm

and go this list of supported synths:

Ensoniq MR and ZR series
Kurzweil K2000, K2500 and K2600
Roland - any synthesizer supporting GS mode
Yamaha - any synthesizer supporting XG mode

Justonics' software retunes these synths in real time to produce just
intonation. I suspect none of these keyboards are within reach even
on eBay or whatever as used equipment. Wondering if you know what
range the most economical of these choices might run?

Greatfully,

Bob

--- In tuning@yahoogroups.com, "Jon Szanto <JSZANTO@A...>"
<JSZANTO@A...> wrote:
> Bob,
>
> There is an off-chance something might come *close*, but I actually
do believe you are asking the impossible. To specific points:
>
> --- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
<rwendell@c...> wrote:
> > I wish to purchase a keyboard with full midi capability, tuning
tables to which you can download whatever
>
> So far, I believe there might be a couple of possibilities, but it
may be that you will need a keyboard to drive an external synth box,
with the *possible* exceptions of some of the Kurzweil keyboards (and
maybe Ensoniq?).
>
> > and ideally, retunable in real time.
>
> Dream on. Seriously, what do *you* mean by "real time" - the
ability to shift tunings, across the span of the keyboard, *as* you
are performing? I don't believe so, as even if you had an external
midi contraption that would send, at given points, a midi tuning
file, I would believe all keyboards would have to stop playing to
accept a tuning file dump.
>
> > I'm going to have to go to the used market, eBay, or whatever,
since funds are severely restricted. I don't know whether it's
possible, but we need to stay at around $200.
>
> There isn't anything even close to that price range. What you
*might* find in that price range would be one of the external 1 or 2
rack space module - you can even get old TX-802 modules for around
that. Even so, you'll need a keyboard to drive it, and I didn't see
where you specified what the thing needed to sound like. If you are
talking about getting a close approximation of a piano sound, you are
going to have to budget a lot more than that.
>
> The only other thing I can think of, and it would enter an entire
other world, would be to explore the new realm of software synths,
where we are seeing far more development in tuning flexibility. Alas,
this also means computers, external keyboards, soundcard latency, and
a lot of other cruft.
>
> If you haven't taken a look, you should look at the 4 or so pages
of charts that John Loffink maintains on the tuning specs of almost
any piece of gear (mostly hard, some soft) that is out there these
days. His new url is:
>
> http://www.microtonal-synthesis.com/
>
> I can't guarantee any better results, but you might ask this
question over at MMM if no one here has any better ideas...
>
> Cheers,
> Jon

🔗John Loffink <jloffink@austin.rr.com>

Robert,

You should be able to find a used Roland GS synth in the sub-$800 range.
An XP-10 might run $200-$250.

You may also find Yamaha portable keyboards such as the PSR-530, PSR-540
and PSR-550 are compatible with Justonic, but I would inquire with
Justonic first. These claim XG compatibility but I believe there are
varying levels of it. These are in the $200-$500 range on ebay.

Check out my Microtonal Synthesis web site listed below for a list of
compatible models.

John Loffink
jloffink@austin.rr.com

The Microtonal Synthesis Web Site
http://www.microtonal-synthesis.com/

The Wavemakers Modular and Integrated Synthesizer Web Site
http://www.wavemakers-synth.com/

> Hey there, Jon! Great to hear from you again! I just looked at
> Justonics page at
>
> http://www.justonic.com/default3.htm
>
> and go this list of supported synths:
>
> Ensoniq MR and ZR series
> Kurzweil K2000, K2500 and K2600
> Roland - any synthesizer supporting GS mode
> Yamaha - any synthesizer supporting XG mode
>
> Justonics' software retunes these synths in real time to produce just
> intonation. I suspect none of these keyboards are within reach even
> on eBay or whatever as used equipment. Wondering if you know what
> range the most economical of these choices might run?
>
>
> Greatfully,
>
> Bob
>
> --- In tuning@yahoogroups.com, "Jon Szanto <JSZANTO@A...>"
> <JSZANTO@A...> wrote:
> > Bob,
> >
> > There is an off-chance something might come *close*, but I actually
> do believe you are asking the impossible. To specific points:
> >
> > --- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
> <rwendell@c...> wrote:
> > > I wish to purchase a keyboard with full midi capability, tuning
> tables to which you can download whatever
> >
> > So far, I believe there might be a couple of possibilities, but it
> may be that you will need a keyboard to drive an external synth box,
> with the *possible* exceptions of some of the Kurzweil keyboards (and
> maybe Ensoniq?).
> >
> > > and ideally, retunable in real time.
> >
> > Dream on. Seriously, what do *you* mean by "real time" - the
> ability to shift tunings, across the span of the keyboard, *as* you
> are performing? I don't believe so, as even if you had an external
> midi contraption that would send, at given points, a midi tuning
> file, I would believe all keyboards would have to stop playing to
> accept a tuning file dump.
> >
> > > I'm going to have to go to the used market, eBay, or whatever,
> since funds are severely restricted. I don't know whether it's
> possible, but we need to stay at around $200.
> >
> > There isn't anything even close to that price range. What you
> *might* find in that price range would be one of the external 1 or 2
> rack space module - you can even get old TX-802 modules for around
> that. Even so, you'll need a keyboard to drive it, and I didn't see
> where you specified what the thing needed to sound like. If you are
> talking about getting a close approximation of a piano sound, you are
> going to have to budget a lot more than that.
> >
> > The only other thing I can think of, and it would enter an entire
> other world, would be to explore the new realm of software synths,
> where we are seeing far more development in tuning flexibility. Alas,
> this also means computers, external keyboards, soundcard latency, and
> a lot of other cruft.
> >
> > If you haven't taken a look, you should look at the 4 or so pages
> of charts that John Loffink maintains on the tuning specs of almost
> any piece of gear (mostly hard, some soft) that is out there these
> days. His new url is:
> >
> > http://www.microtonal-synthesis.com/
> >
> > I can't guarantee any better results, but you might ask this
> question over at MMM if no one here has any better ideas...
> >
> > Cheers,
> > Jon
>
>

🔗Jon Szanto <JSZANTO@ADNC.COM> <JSZANTO@ADNC.COM>

Mr. Bob,

--- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>" <rwendell@c...> wrote:
> Hey there, Jon! Great to hear from you again! I just looked at
> Justonics page at
>
> http://www.justonic.com/default3.htm

Pardon me for a moment while I take off my blinders; the callouses from wearing them all the time slows me down. There. That's better...

Seriously, I remember seeing Justonics pages but my own personal bug was that they are operating on only 12 notes per octave and repeating that tuning every octave. However, this may very well suit your particular purpose, and it is poor of me to not remember that everyone might not need (or want) full keyboard microtuning.

One thought that does bother me: most software can be found in demo versions, and it wouldn't hurt for them to do that. That they could retune a midi file for later playing wouldn't be a stretch - there are others that can do that (Scala does, and it's free). But to make tunings on the fly with temporal accuracy is not just a slam dunk.

I've recently been trying soft synths (you may very well go this route at some point) and one problem (that they gloss over just a bit) is latency - the time from when you press a key on the keyboard to the time when you actually hear a sound. Slow computers and slow sound cards can affect this mightily, and I have not yet found a solution that is adequate. But I will!

Rick McGowan has been touting some very nice soft micro synths, and if you end up having to use a computer to use Justonic live, you might as well invesigate some of these. If you still want to have a dedicated hardware synth (to use with Justonic), the Ensoniq MR 76

http://www.sonicstate.com/synth/ensoniq_mr76.cfm

gives you almost a full keyboard with decent sounds, and the above link puts them (used) in around the $800 range. Yes, a bit steep, but... The Kurzweils I have always found superior for reproducing pianos (IIRC, you are in fairly traditional, choral music, right?) which would probably be more important to you than 'spacey' sounds or great drum kits; OTOH, they are going to cost more as well.

John Loffink has already pointed you at the other synths, and after posting this I'm going to download their midi files (Justonic) to try with a Roland JV-1080. Mind you, it's not real-time, but I'm curious.

> Justonics' software retunes these synths in real time to produce
> just intonation.

You know what? I'd write Justonic and see if you can't get software to try - I think before you shell out you ought to know whether it meets your needs. We can all stroll into a music store and try out new instruments.

Cheers,
Jon

🔗Dave Keenan <d.keenan@uq.net.au> <d.keenan@uq.net.au>

--- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
<rwendell@c...> wrote:
> In my temperament design work, I use a highly accurate linear
> approximation for converting ratios of the cents deviation from just
> to beat ratios. The formula is:
>
> R = 1.5(delta P5/delta M3 - 1)
>
> where R is the beat ratio and delta P5 and delta M3 are the
> deviations in cents from just (with signs preserved, - is flat, + is
> sharp) for the interval under consideration.
>
> I also use half a schisma for the unit, since this is very close to
> one cent (0.9775) and allows playing off the syntonic and Pythagorean
> commas in terms of the whole numbers 22 and 24 respectively. This
> with the simple formula to eliminate the need for log conversions
> puts headlights on my design wagon and allows me to run calculations
> in my head while walking around, waiting in line at the bank, etc. I
> am more of an engineer than a mathematician, although I have no lack
> of respect for the power of advanced mathematical knowledge. Wish I
> had more.

Hi Bob,

I love these tricks that let you do the calculations in your head. I'm
also fond of ways of representing them graphically. I suppose I too am
more engineer than mathematician.

I would like to understand the derivation of your formula above,
because it doesn't seem right to me.

It seems to me that the beat rate of the P5 is the frequency
difference between the 3rd harmonic of the low note and the 2nd
harmonic of the high note. And the beat rate of the M3 is 4*f_high -
5*f_low.

In the region of cents deviations we are interested in it is a good
approximation that
delta_frequency ~= frequency * delta_cents/1730

So if the M3 and the P5 share their low note (as in a root position
major triad) then the beat rate of the P5 is
3 * f_low * delta_P5/1730
and the beat rate of the M3 is
5 * f_low * delta_M3/1730
and the ratio of their beat rates is simply
R_M3_P5 = 5/3*delta_M3/delta_P5

Similarly if they share their high note (root position minor triad) I get
R_M3_P5 = 2*delta_M3/delta_P5

The signs of the cents deviations don't matter.

Then for m3 and M3 I get:
In a root position major triad
R_m3_M3 = 3/2*delta_m3/delta_M3
In a root position minor triad
R_m3_M3 = delta_m3/delta_M3

And for m3 and P3 I get:
In a root position major triad
R_m3_P5 = 5/2*delta_m3/delta_P5
In a root position minor triad
R_m3_P5 = 2*delta_m3/delta_P5

Using these formulae and a spreadsheet, I find that 1/4-comma is the
only place on the meantone continuum, where the beat ratios for both
major and minor triad are all either 0:N, 1:N or 2:N (where N is an
integer).

The only other places I can find where the major triad has this
property are at 5/17-comma (equal beating), 5/19-comma and 5/23-comma.

And for the minor triad, 2/7-comma (equal beating), 4/15-comma,
4/17-comma, 2/9-comma, 1/5-comma and 1/6-comma.

Where am I going wrong?

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

--- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
<rwendell@c...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith <gwsmith@s...>"
> <gwsmith@s...> wrote:

> OK, I didn't get it. I don't know much math beyond trig, analytic
> geometry, some elementary statistics, and basic concepts
> that are the fossil remains of elementary calculus. What beat ratio
> are you talking about? I focus on the beat ratio of m3/M3, since
the
> fifth beats at a relatively slow rate and is therefore not
generally
> a major producer of the random beating jangle you hear with some
> temperaments like 12-tET. Moreover, when there is a simple ratio of
> m3/M3, the relationship of both to the beat rate of the P5 is
> automatically also simple.

I'm using *your* beat ratio; in any system of tuning with uniform
sizes for fifths and thirds,
get a "brat", or Robert Wendell beat ratio, of b = 3/2 - 5/4 z, where
z = (P5 - 3/2)/(M3 - 5/4).

The beat ratios are b, 3/5 - 2/5 b, 5b/(3 - 2b) and their
reciprocals, or
3/2 - 5/4 z, z/2, (6 - 5z)/2z and their reciprocals when expressed in
terms of z.

> By my reckoning, since the third and fifth are both flat, 11/41-
comma
> meantone has a beat ratio on m3/M3 of approximately +1.1.

I don't know where this comes from. I get a beat ratio of -3.99334
for 11/41-comma meantone. The value of P5 with a beat ratio of
exactly -4 is the positive real root of
11 P5^4 - 10 P5 - 40, or 1.4950094; in terms of cents that is
696.185532. For a beat ratio of exactly +2, you need the positive
real root of P5^4 + 10 P5 - 20, which is 1.4973363545,
or 698.8780055246 cents.

On the
> other hand, since the M3 in 1/4-comma is pure, this beat ratio is
> infinite, or perhaps it's more meaningful to say undefined. The
only
> beat ratio of significance therefore becomes that of m3/P5, which
by
> my reckoning is -2.50. Am I off-base here?

This is right; if you stick b=infinity into 5b/(3-2b) you get -2.5.

> In my temperament design work, I use a highly accurate linear
> approximation for converting ratios of the cents deviation from
just
> to beat ratios. The formula is:
>
> R = 1.5(delta P5/delta M3 - 1)
>
> where R is the beat ratio and delta P5 and delta M3 are the
> deviations in cents from just (with signs preserved, - is flat, +
is
> sharp) for the interval under consideration.

delta P5 = cents(P5/(3/2)) and delta M3 = cents(M3/(5/4)). Since you
take the ratio, it doesn't
matter which logs you use, so you can use natural logs. If we expand
ln(P5/(3/2)) around 3/2 and ln(M3/(5/4)) around 5/4, we get that the
ratio u = 2/3(P5 - 3/2) + ... / (4/5)(M3 - 5/4) + .
so that u is approximately (2/3)/(4/5) z, or 5/6 z, which gives your
approximation.

> I also use half a schisma for the unit, since this is very close to
> one cent (0.9775) and allows playing off the syntonic and
Pythagorean
> commas in terms of the whole numbers 22 and 24 respectively. This
> with the simple formula to eliminate the need for log conversions
> puts headlights on my design wagon and allows me to run
calculations
> in my head while walking around, waiting in line at the bank, etc.
I
> am more of an engineer than a mathematician, although I have no
lack
> of respect for the power of advanced mathematical knowledge. Wish I
> had more.

I've found a similar system, which measures everything in terms of
the 612 et, works wonderfully for that.

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

🔗Jon Szanto <JSZANTO@ADNC.COM> <JSZANTO@ADNC.COM>

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

--- In tuning@yahoogroups.com, "Jon Szanto <JSZANTO@A...>"
<JSZANTO@A...> wrote:
> Paul,
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus
<wallyesterpaulrus@y...>" <wallyesterpaulrus@y...> wrote:
> > don't forget john delaubenfels' ji relay for real-time adaptive-
ji
> > retuning!!
>
> Could you take on the task of contacting John and getting the most
current link to his software? And maybe he'll let someone host a page
on it and a file for download (I could do it at microtonal.org)?
>
> Cheers,
> Jon

i got an immediate reply from john today!:

Hi, Paul,

Nice to hear from you! I've been away from tuning for awhile now,
working on photos (see http://www.adaptune.com/pix0.htm#enhancement)
and other fun stuff. Every now and then I look at the current digest
on the tuning list, but it seems to have gotten a bit dry lately.

JIRelay has never had a second version. A few months ago I got an
idea for how it might be improved, and I'm not sure this has been
discussed on the list: grounding the tuning of each pitch degree
fairly strongly, but letting the grounding points "ooze" in response
to pressure, so that as one modulated and stayed in a new key, the
chords would become pure over time, without modifying the tuning of a
particular pitch degree so quickly that it was jarring. The same
notion of "floating grounding" (if such a phrase makes sense) could
be applied to retuning that was not real-time as well; there, of
course, a lot more intelligence could be brought to the picture,
since the entire piece is in hand.

How are things going with you? Just before I dropped off the list in
Nov 2001, I'd listened to a piece of yours that was very nice, and I
meant to write a review of it. Don't remember the title, sorry, but
do you have anything new available online?

JdL

🔗Carl Lumma <ekin@lumma.org>

🔗Graham Breed <graham@microtonal.co.uk>

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

My answer follows below these messages, included here to provide
convenient context:

--- In tuning@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
<d.keenan@u...> wrote:
> --- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
> <rwendell@c...> wrote:
> > In my temperament design work, I use a highly accurate linear
> > approximation for converting ratios of the cents deviation from
just
> > to beat ratios. The formula is:
> >
> > R = 1.5(delta P5/delta M3 - 1)
> >
> > where R is the beat ratio and delta P5 and delta M3 are the
> > deviations in cents from just (with signs preserved, - is flat, +
is
> > sharp) for the interval under consideration.
> >
> > I also use half a schisma for the unit, since this is very close
to
> > one cent (0.9775) and allows playing off the syntonic and
Pythagorean
> > commas in terms of the whole numbers 22 and 24 respectively. This
> > with the simple formula to eliminate the need for log conversions
> > puts headlights on my design wagon and allows me to run
calculations
> > in my head while walking around, waiting in line at the bank,
etc. I
> > am more of an engineer than a mathematician, although I have no
lack
> > of respect for the power of advanced mathematical knowledge. Wish
I
> > had more.
>
> Hi Bob,
>
> I love these tricks that let you do the calculations in your head.
I'm
> also fond of ways of representing them graphically. I suppose I too
am
> more engineer than mathematician.
>
> I would like to understand the derivation of your formula above,
> because it doesn't seem right to me...
>
> The signs of the cents deviations don't matter...
>
> Where am I going wrong?

Bob:

The signs are extremely important. You get very different answers if
you change the signs. They don't matter only at the level of the
resulting beat rate, meaning your ear doesn't care about the phase
relationship, but the equations care a lot!

Here is how I derived it:

Let us use a specific justly tuned triad to illustrate:

G3 6
E3 5
C3 4

The numbers to the right specify the frequency relationships among
these pitches, which are in a ratio of 4:5:6. The common harmonics
for each of the three possible intervals, the perfect fifth, major
third, and minor third, between any two notes of the triad are
illustrated below, including their frequency relationships with
respect to the root position triad.

B5 30 (m3) -
-
-
E5 20 (M3) - - - Common Harmonics
-
-
G4 12 (P5) -

G3 6
E3 5
C3 4

The beat rate on any interval is defined as the frequency difference
in Hz of the first harmonic common to both pitches. To derive a
linear approximation to the logarithmic conversion from cent
deviations to frequency differences corresponding to beats on the
first common harmonic, we could take several approaches. The approach
used here is generalized and does not even need to refer to the
logarithmic equations.

The symbols we will use here are:

F = P5 or Perfect Fifth
M = M3 or Major Third
m = m3 or Minor Third

We will symbolize pitch deviations in cents as delta and the interval
symbols as defined above. So dM will refer to the deviation in cents
from just of the major third (with as flat and + as sharp in
the
substituted values during calculation). In the standard equation for
beat rates in a major triad:

Q = beat rate on the P5
T = " " " " M3
t = " " " " m3

The formula relating these beat rates in the major triad is:

2t + 3T = 5Q

We first assume essentially linear changes across the exceedingly
small deviations in pitch in terms of the width of the interval under
consideration. This directly implies that:

T = 2k*dM

and

t = 3K*dm

where k is some constant of conversion which we do not have to
determine, since it will cancel out in the process of deriving the
approximation formula for the beat ratio of the minor over the major
third, R:

R = t/T = 3k*dm/2k*dM = 1.5(dm/dM)

We know that the sum of the cent deviations on the major and minor
third equals the deviation of the perfect fifth:

dF = dM + dm

dm = dF dM

Substituting right side of this equation for dm in the equation for
R3 yields:

R3 = 1.5 (dF dM) / dM

therefore

R3 = 1.5 (dF/dM 1)

And also dF/dM = R3/1.5 + 1

If you ignore the sign of the beat ratio, you will get terribly
different and wrong results. (That's how I got it, Gene, referring to
my analysis of 11/41-comma meantone.)

🔗Dave Keenan <d.keenan@uq.net.au> <d.keenan@uq.net.au>

--- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
<rwendell@c...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan <d.keenan@u...>"
> <d.keenan@u...> wrote:
> > I would like to understand the derivation of your formula above,
> > because it doesn't seem right to me...
> >
> > The signs of the cents deviations don't matter...
> >
> > Where am I going wrong?
>
> Bob:
>
> The signs are extremely important. You get very different answers if
> you change the signs. They don't matter only at the level of the
> resulting beat rate, meaning your ear doesn't care about the phase
> relationship, but the equations care a lot!
>
> Here is how I derived it:
...

Thanks Bob. My only mistake was to assume that your equation
R = 1.5(delta P5/delta M3 - 1)
was for the beat ratio between the P5 and M3 (because of the deltas it
contains). I agree that this is correct for the beat ratio of m3 to
M3, and yes the signs matter. However, if one derives the beat ratio
between interval x and interval y using only delta x and delta y, then
the signs don't matter.

I agree that, in the case of well temperaments, one can ignore the
other two beat ratios (m3/P5 and M3/P5) however I don't think this is
the case when designing beat-synced meantones.

I find your results fascinating. Have you used Scala to COMPARE your
scales against those in the Scala archive, allowing for say a 1 or 2
cent SET MAXDIFF?

When designing either meantones or well temperaments for synchronised
beats, why do you ignore the minor triads? And why do you ignore other
rotations of the major triads?

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

🔗Carl Lumma <ekin@lumma.org> <ekin@lumma.org>

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

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

? I don't understand why we can ignore the beat ratios
> involving P5 anywhere, let alone only for WT.
>
Dave Keenan:
> >When designing either meantones or well temperaments for
> >synchronised beats, why do you ignore the minor triads? And
> >why do you ignore other rotations of the major triads?
>
> Yeah... Was this answered somewhere but somehow not threaded
> in?
>
> -Carl

Hi, Carl! I explained about the rotations in another reply to Paul
Erlich. All interval inversions do is halve or double the beat rates,
so simple ratios remain simple. On major versus minor, major is
generally more important, and fairly often minor triads in well
temperaments will have a 1/1 ratio between their thirds anyway when
the corresponding majors (e.g., E-G in C major is the bottom third in
E minor, etc.) are in clean ratios.

I don't worry about that though. I let the chips fall where they may.
You can't do it all. You're constrained too much by the automatic
results of everything else. You can't change anything in temperament
design without affecting literally everything else!

However, when I get good synchrony in the major triads all around the
circle of fifths, I'm amazed at how good modern jazz chords in very
chromatic progessions sound, for example. I don't try to explain it,
but you sure can hear it!

On the P5, these beat rates are NOT independent. If they were, then
you'd have to worry about every one of them. If you get the beat
ratio for m3/M3 simple, they will have a simple relationship to P5.
This should be obvious from the formula relating beat rates for all
three intervals. The beat on m3 is t, on M3 is T, and on P5 is Q. So
the formula is:

2t + 3T = 5Q

For example, if t = 2T for a beat ratio of 2.0 relating m3/M3,
for every -1 beats on P5 (flat) you will get -10 beats on m3 (flat)
for every +5 on M3 (sharp) for a negative 2.0 on m3/M3. (Ratio sign
is important only in the formulas and signify phase relationships.)
So why strain your brain on P5? You have no choice there. Further, it
beats at a much slower rate than either m3 or M3 in most practical
temperaments, so it's not where the asynchronous "jangling" is in the
first place. Therefore m3/M3 is the most practical point of focus in
designing temperaments.

My approach is to boil everything down to the simplest possible
parameters, the ones that have the most effect on what you hear, and
at the same time imply the most good stuff about what's happening
automatically with everything else.

🔗Carl Lumma <ekin@lumma.org>

>Hi, Carl! I explained about the rotations in another reply to Paul
>Erlich. All interval inversions do is halve or double the beat rates,
>so simple ratios remain simple. On major versus minor, major is
>generally more important, and fairly often minor triads in well
>temperaments will have a 1/1 ratio between their thirds anyway when
>the corresponding majors (e.g., E-G in C major is the bottom third in
>E minor, etc.) are in clean ratios.
>
>I don't worry about that though. I let the chips fall where they may.
>You can't do it all. You're constrained too much by the automatic
>results of everything else. You can't change anything in temperament
>design without affecting literally everything else!
>
>However, when I get good synchrony in the major triads all around the
>circle of fifths, I'm amazed at how good modern jazz chords in very
>chromatic progessions sound, for example. I don't try to explain it,
>but you sure can hear it!
>
>On the P5, these beat rates are NOT independent. If they were, then
>you'd have to worry about every one of them. If you get the beat
>ratio for m3/M3 simple, they will have a simple relationship to P5.
>This should be obvious from the formula relating beat rates for all
>three intervals. The beat on m3 is t, on M3 is T, and on P5 is Q. So
>the formula is:
>
> 2t + 3T = 5Q
>
>For example, if t = 2T for a beat ratio of 2.0 relating m3/M3,
>for every -1 beats on P5 (flat) you will get -10 beats on m3 (flat)
>for every +5 on M3 (sharp) for a negative 2.0 on m3/M3. (Ratio sign
>is important only in the formulas and signify phase relationships.)
>So why strain your brain on P5? You have no choice there. Further, it
>beats at a much slower rate than either m3 or M3 in most practical
>temperaments, so it's not where the asynchronous "jangling" is in the
>first place. Therefore m3/M3 is the most practical point of focus in
>designing temperaments.
>
>My approach is to boil everything down to the simplest possible
>parameters, the ones that have the most effect on what you hear, and
>at the same time imply the most good stuff about what's happening
>automatically with everything else.

Thanks, I'm grokking it now.

-Carl

🔗manuel.op.de.coul@eon-benelux.com

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

For the two I consider most useful, they are as follows:

Wendell Natural Synchronous Well
(Perfect 2.0 or 1.50 ratios on every triad)
A 0
G# 0.366
G 3.735
F# 1.833
F 2.288
E -2.077
Eb 2.321
D 2.414
C# -1.589
C 4.243
B -0.122
Bb 4.276

Wendell Very Mild Synchronous Well
(Essentially perfect ratios in remote keys, somewhat compromised
elsewhere. Can be substituted for ET without advising the client.)
A 0
G# 1.127
G 4.299
F# 2.265
F 2.686
E -1.6454
D# 3.082
D 3.038
C# -0.828
C 4.241
B 0.310
A# 5.037

NOTE: These temperaments are copyrighted, but can be freely
distributed as long as the names and authorship are respected.

--- In tuning@yahoogroups.com, "Carl Lumma <ekin@l...>" <ekin@l...>
wrote:
> Heya Bob,
>
> > I have to stop now, but will publish ET offsets for the Natural
> > later. Some of my temperaments are graphically displayed with
> > offests and other specs at Jason Kanter's Website:
> >
> > http://www.rollingball.com
>
> This site seems to be down at the moment. Anyway, I'm sure
> we're all awaiting the et offsets for your temperaments...
>
> -Carl

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Robert Wendell <rwendell@cangelic.org>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
> <rwendell@c...> wrote:
>
> > If you want a telling demonstration as to just how wide the range
> is
> > of melodic possibilities that are compatible with meantones
closer
> > to just than 12-tET, try 2/7th-comma meantone. It's also
> synchronous
> > at +1.5 ratios instead of the -1.5 on just P5s, and is
> > hanging out in the general neighborhood of Woolhouse's
optimization
> > for P5s, M3s, and m3s in the triads.
> >
> > For my taste, it's the most pleasingly harmonious, since the
> > synchrony more than compensates its relatively short distance
from
> > Woolhouse's 7/26th-comma.
>
> according to scala, 7/26-comma meantone is (ignoring the usual
> hundredth-of-a-cent error due to the logarithmic approximation) the
> tuning where
>
> Beating of 3/2 = twice 5/4 same.
>
> so am i missing something or is this a seriously synchronous
> temperament?

Bob:
Well, I keep getting questions and feedback that indicate an
obsession with ratios involving the fifth. I repeat that NONE of the
ratios I have cited in describing my work involve the fifth, except
in a couple of places where I was specifically and very explicitly
responding to questions involving ratios with the fifth.

The reasons I have previously cited for this are:

1) The beat rates on the fifths in practical well temperaments are
MUCH slower than the beats on the major and minor thirds, and thus
they are a relatively insignificant contributor to the random janglin
typical of equal temperament.

2) If the ratio of the beat rate on m3 to that on M3 is simple, this
automatically implies a simple relationship of both to the beat rate
on the fifth, since the relationship among the three intervals is
given by the equation 2t + 3T = 5Q, where t, T, and Q are the beat
rates on the minor and major thirds and the perfect fifth
respectively on a given triad in close voiced root position.

Sometimes, in the recent discussions on this topic, I get the
impression that we somehow think we can just use sophisticated
mathematics to bend things to our will and get good beat synchrony,
say 2.0, on every triad. Well, good luck! I am positive that this is
impossible! Everything is highly interactive. This not chess; it's
more like Go, if you're familiar with the Asian game.

You can't change anything in a temperament design without affecting
absolutely EVERYTHING else! It is a very tough business unless you
think very holistically. The linear, fragmented, object-rather-than-
context-oriented focus of the typical western scientific mind has
trouble with the kind of thinking this requires. You have to see how
the WHOLE BALLGAME is responding to every little thing you do.

For example, if you start with a quarter comma on a fifth followed by
three just fifths, you will get a majro third of one syntonic comma S
minus 1/4 comma, leaving it 3/4 comma sharp. This cent ratio of -1/3
on the P5/M3 ratio yields a beat ratio on m3/M3 of 2.0. Please note
that I make NO mention here of a beat ratio involving the fifth. I
use the CENT ratio of the tempering on the fifth and major third to
calculate simple and directly the BEAT ratio of the minor to major
third.

If you then move to the fifth just previous to the 1/4-comma tempered
fifth, you can calculate the amount by which you must temper that to
get another 2.0 ratio on that fifth. The equation is f = (S - F)/4,
where f is the amount by which you must temper the fifth, S is the
syntonic comma, and F is the total amount of tempering placed on the
other three fifths. I won't derive that one for you here, but it's
very simple algebra from my linear approximation formula.

If, starting from the 1/4 comma, you iterate this process backwards
around the circle of fifths, you will converge relatively quickly at
first toward 1/7-comma, then oscillate slightly above and below it
for awhile, converging on it ever more tightly from either side. The
total tempering when you get to the twelfth fifth will be over 40
cents! So you're not very close to a closed temperament, are you?

If you start with 1/7 comma, you will stay there all the way around
and end up with 12/7 comma total for an open temperament called 1/7-
comma meantone. If you go back to the previous strategy, starting
with 1/4 comma, and iterate back to your starting pitch, you will
have a feedback loop that eventually ends up with an oscillating
asymptotic approach to 1/7-comma meantone.

If you leave a just fifth between the end and beginning of this
cycle, you will create a feedback loop that converges toward an
irregular temperament that is an odd, anomalous well temperament, if
you could call it that. If you open the feedback loop so there is no
overlap, as I did with the natural, but leave all the other fifths
tempered, you will end up with a huge wolf at the break point.

If you insert untempered fifths at strategic locations that work to
make C the most just key and imitate the historical key color pattern
of well temperaments since Werkmeister, you will end up with my
Natural Synchronous Well.

Hope this clarifies things a bit.

Cheers,

Bob

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Robert Wendell" <rwendell@c...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, "Robert Wendell <rwendell@c...>"
> > <rwendell@c...> wrote:
> >
> > > If you want a telling demonstration as to just how wide the
range
> > is
> > > of melodic possibilities that are compatible with meantones
> closer
> > > to just than 12-tET, try 2/7th-comma meantone. It's also
> > synchronous
> > > at +1.5 ratios instead of the -1.5 on just P5s, and is
> > > hanging out in the general neighborhood of Woolhouse's
> optimization
> > > for P5s, M3s, and m3s in the triads.
> > >
> > > For my taste, it's the most pleasingly harmonious, since the
> > > synchrony more than compensates its relatively short distance
> from
> > > Woolhouse's 7/26th-comma.
> >
> > according to scala, 7/26-comma meantone is (ignoring the usual
> > hundredth-of-a-cent error due to the logarithmic approximation)
the
> > tuning where
> >
> > Beating of 3/2 = twice 5/4 same.
> >
> > so am i missing something or is this a seriously synchronous
> > temperament?
>
> Bob:
> Well, I keep getting questions and feedback that indicate an
> obsession with ratios involving the fifth. I repeat that NONE of
the
> ratios I have cited in describing my work involve the fifth, except
> in a couple of places where I was specifically and very explicitly
> responding to questions involving ratios with the fifth.
>
> The reasons I have previously cited for this are:
>
> 1) The beat rates on the fifths in practical well temperaments are
> MUCH slower than the beats on the major and minor thirds, and thus
> they are a relatively insignificant contributor to the random
janglin
> typical of equal temperament.

but bob, *you* were the one who brought up 2/7-comma meantone in this
context -- and this is *not* an example of a "practical well
temperament" -- or one where the beats of the fifths are slow! so why
can't i bring up 7/26-comma and note that it, too, has notable beat
synchrony? you're really tying me up in knots, bob!

>
> 2) If the ratio of the beat rate on m3 to that on M3 is simple,
this
> automatically implies a simple relationship of both to the beat
rate
> on the fifth, since the relationship among the three intervals is
> given by the equation 2t + 3T = 5Q, where t, T, and Q are the beat
> rates on the minor and major thirds and the perfect fifth
> respectively on a given triad in close voiced root position.

so what does this have to do with 7/26-comma meantone? they're all
pretty simple, right?

>
> Sometimes, in the recent discussions on this topic, I get the
> impression that we somehow think we can just use sophisticated
> mathematics to bend things to our will and get good beat synchrony,
> say 2.0, on every triad. Well, good luck!

gene, on tuning-math, just came up with over 100 examples where the
beat rate ratios are all either 2.0 or 1.5 -- some have six and six,
and some have seven and five.

> I am positive that this is
> impossible! Everything is highly interactive.

this is where mathematics shines.

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> > > according to scala, 7/26-comma meantone is (ignoring the usual
> > > hundredth-of-a-cent error due to the logarithmic approximation)
> the
> > > tuning where
> > >
> > > Beating of 3/2 = twice 5/4 same.
> > >
> > > so am i missing something or is this a seriously synchronous
> > > temperament?

I get that 7/26-comma has a brat of -15/4, and major beat ratios

-15/4, 10/21, -14/25

minor beat ratios

-5/2, 7/10, -4/7

which doesn't seem all that great. Better would be 3/11-comma.

> so what does this have to do with 7/26-comma meantone? they're all
> pretty simple, right?

See above.

Thanks for the comments on my big parade of temperaments; I'd like to
sort them out somehow. I could comapre the tuning, but I'd like to
know what Bob (or anyone else who has an opinion) thinks are the
desiderata of a good well-temperament.

Bob's point that this isn't always easy is correct, I find. The brat
for 12-equal is between 1.5 and 2, which probably explains why it
works for these. I've been fooling around with b=infinity without
getting much that looks useful. I'd like to sort it out better before
moving on to well-temperaments of 7 notes with 25/24 comma, or the like.

🔗manuel.op.de.coul@eon-benelux.com

🔗Afmmjr@aol.com

🔗monz <monz@attglobal.net>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Afmmjr@aol.com

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗David Beardsley <davidbeardsley@biink.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, David Beardsley <davidbeardsley@b...>
wrote:
> ----- Original Message -----
> From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
>
> > johnny, the problem is that string lengths on a monochord do not
> > translate exactly to inverse frequencies, which they would have
to in
> > order for the tunings in question to exhibit the properties under
> > debate here. even if one could stop the string with infinite
> > accuracy, there will be a slight nonlinear correction due to the
> > *bending* of the string down from its relaxed position to the
> > position where it makes contact with the markings on the
monochord.
> > if you've ever tried this yourself, you'll know that pressing
down on
> > the midpoint of the string gets you quite a few cents sharp of
the
> > octave, due to the added tension of pressing down on it.
>
> Wouldn't you compensate for this when you calculate your markings?

we're talking about what would have happened in the 16th century. no
one then had any clue that there was anything to compensate, since
the physical basis of the ratios was still placed in string lengths
(the physics of strings would be worked out beginning in the
following century) and not yet in vibration frequencies (which were
just beginning to be uncovered in the first place). for example, in
those days a minor triad was 4:5:6, and a major triad was 10:12:15,
since these numbers were referring to string lengths and no
physically more relevant correlate of the numbers was yet widely
known . . . and the relationships between string deflection, tension,
and pitch were centuries away . . .

remember, dave, we're talking about the monochords for
*temperaments*, so even if just intervals could be corrected by ear,
there would still be no way to correct the positions of the
*tempered* markings on the monochord . . .

🔗Afmmjr@aol.com

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Robert Wendell <rwendell@cangelic.org>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Robert Wendell" <rwendell@c...>
wrote:
>
> > Sometimes, in the recent discussions on this topic, I get the
> > impression that we somehow think we can just use sophisticated
> > mathematics to bend things to our will and get good beat
synchrony,
> > say 2.0, on every triad. Well, good luck! I am positive that this
is
> > impossible!
>
> It's impossible for a well-temperament, since it's 1/7 (or 2/7 comma
> for a -2 brat) comma meantone. That means you could use the 91 et,
> which is not something you tune a piano to, but which makes perfect
> sense in other contexts.
>
> Everything is highly interactive. This not chess; it's
> > more like Go, if you're familiar with the Asian game.
>
> I'm aware that it's not as easy as one might think, since I've been
> trying it a little. However, did you check the list of 114
> temperaments I posted on tuning-math? I'd like feedback on that,
and I
> suggest you not assume things must converge on Wendell Well until
> we've actually looked into it.

Bob:
I have been careful to specify from the outset that I am strictly
referring to close, 12-tone temperament design. I am aware that 1/7-
comma meantone is equivalent to 91-tone ET, if you don't take the
closest approximations to the just intervals, but restrict yourself
exclusively to intervals equivalent to those available 1/7th comma.
meantone. That was one of the first things I checked out after
realizing that 1/7-comma had 2.0 beat ratios for m3/M3 on all the
playable triads.

There may be other possibilities for perfect 2.00 and 1.50 ratios,
but the question is how many options do you have that will parallel
the historical key color pattern. I've already looked at other
possibilities that work for perfect ratios, but yield squirrelly
intonation sequences. Not saying no one will ever find one, but I
think the probability of that is rather low.

Sincerely,

Bob

🔗Robert Wendell <rwendell@cangelic.org>

Thank you, Gene. I have answers to your "desiderata" question further
below.

Everyone, I would like to request of all involved in this thread that
my comments here be understood as referring by default to closed 12-
tone temperaments only. Any reference to other kinds of temperaments
have been in this context for illustrative purposes only. For
example, I referred to 2/7-comma meantone as a meantone example of
beat synchrony between m3/M3, but in the larger context of 12-tone
well temperament.

I did this as a side note in reference to the main theme and
seemingly deluded myself that I'd made that very clear. When Monz
brought up 2/7-comma meantone, I mistakenly took it as a response to
something I had said and said so later.

The desiderata I perceive as ideal for an historically based well are
as follows:

1) C is the most consonant (i.e., most nearly just) key, and we pay
the piper for that most severely around the F#-C#-G# neighborhood.

2) Almost all historical temperaments are biased toward the sharp
side as implied in the previous point. C# is often the axis around
which the widest thirds occur. This dates all the way back to the
traditional convention of tuning meantones for playable keys between
from Bb major to A major (2 flats to 3 sharps).

3) The width of the major thirds should becomes progressively wider
as we move away from C in either direction.

4) The flat side consequently must move more rapidly toward wider
thirds than the sharp side, so Bb should normally have a wider third
on it than D, and so forth. (The Natural has some minor anomalies in
this regard, dictated by teh goal of perfect synchrony. I
have "fixed" some of them in the Very Mild version, but these
anomalies are mild enough to be virtually inaudible inthe first
place.)

Hope this helps.

Cheers,

Bob

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > > > according to scala, 7/26-comma meantone is (ignoring the
usual
> > > > hundredth-of-a-cent error due to the logarithmic
approximation)
> > the
> > > > tuning where
> > > >
> > > > Beating of 3/2 = twice 5/4 same.
> > > >
> > > > so am i missing something or is this a seriously synchronous
> > > > temperament?
>
> I get that 7/26-comma has a brat of -15/4, and major beat ratios
>
> -15/4, 10/21, -14/25
>
> minor beat ratios
>
> -5/2, 7/10, -4/7
>
> which doesn't seem all that great. Better would be 3/11-comma.
>
> > so what does this have to do with 7/26-comma meantone? they're
all
> > pretty simple, right?
>
> See above.
>
> Thanks for the comments on my big parade of temperaments; I'd like
to
> sort them out somehow. I could comapre the tuning, but I'd like to
> know what Bob (or anyone else who has an opinion) thinks are the
> desiderata of a good well-temperament.
>
> Bob's point that this isn't always easy is correct, I find. The brat
> for 12-equal is between 1.5 and 2, which probably explains why it
> works for these. I've been fooling around with b=infinity without
> getting much that looks useful. I'd like to sort it out better
before
> moving on to well-temperaments of 7 notes with 25/24 comma, or the
like.

🔗monz <monz@attglobal.net>

hi paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, March 06, 2003 6:58 AM
> Subject: [tuning] Re: A question and an update on personal tuning
endeavors
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> >
> > From: <Afmmjr@a...>
> > To: <tuning@yahoogroups.com>
> > Sent: Thursday, March 06, 2003 5:35 AM
> > Subject: Re: [tuning] Re: A question and an update on
> > personal tuning endeavors
> >
> >
> > > Paul, are you sure that Zarlino did not use a
> > > monochord to calculate 2/7ths comma meantone tuning?
> > > Johnny
> >
> >
> >
> > Zarlino *DID* use a monochord to calculate
> > 2/7-comma meantone!!! see my previous post
> > stating this.
> >
> > ... paul is saying that the difficulty comes in trying
> > to tune a harpsichord *to* a monochord. but personally
> > i don't see that as so difficult.
>
> i didn't say it was difficult to do -- just that it's difficult to
> end up with the desired tuning actually tuned up with accuracy of a
> small fraction of a cent (which would be required here).

OK, my bad. i agree totally with what you're saying.

> > i've often used
> > an electronic keyboard to check the pitch of pianos
> > that customers have had me tune in 12edo.
>
> what does this have to do with monochords?

i was just making an analogy with the tuning procedure:
tuning keyboard strings by ear to another external sound
source which is already in the desired tuning.

-monz

🔗Leonardo Perretti <dombedos@tiscalinet.it>

wallyesterpaulrus wrote:

>johnny, the problem is that string lengths on a monochord do not
>translate exactly to inverse frequencies, which they would have to in
>order for the tunings in question to exhibit the properties under
>debate here. even if one could stop the string with infinite
>accuracy, there will be a slight nonlinear correction due to the
>*bending* of the string down from its relaxed position to the
>position where it makes contact with the markings on the monochord.
>if you've ever tried this yourself, you'll know that pressing down on
>the midpoint of the string gets you quite a few cents sharp of the
>octave, due to the added tension of pressing down on it.

This is not true for Zarlino. He used a plain table to set the proportional markings, and put the strings at some height above it; then, he used a mobile tool, that he calls "scannello", a sort of triangular prism set on one of its faces, so that one of its corners touches the string from the bottom (see the image in Zarlino's Part 2, Cap.18). The tool is moved on the table, aligned to the markings, to set the notes. This is similar to the method described by the ancient writers, as, for example, Ptolemy. Zarlino remarks that the mobile "scannello" should be exactly as high as the two stationary "scannello's" that limit the string at its edges, so it should not add any tension.

Regards
Leonardo Perretti

🔗Afmmjr@aol.com

🔗Robert Wendell <rwendell@cangelic.org>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> ----- Original Message -----
> From: <Afmmjr@a...>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, March 06, 2003 5:35 AM
> Subject: Re: [tuning] Re: A question and an update on personal
tuning
> endeavors
>
>
> > Paul, are you sure that Zarlino did not use a
> > monochord to calculate 2/7ths comma meantone tuning?
> > Johnny
>
>
>
> Zarlino *DID* use a monochord to calculate
> 2/7-comma meantone!!! see my previous post
> stating this.
>
> ... paul is saying that the difficulty comes in trying
> to tune a harpsichord *to* a monochord. but personally
> i don't see that as so difficult. i've often used
> an electronic keyboard to check the pitch of pianos
> that customers have had me tune in 12edo.
>
> i certainly agree that people in Zarlino's time
> could tune a harpsichord by listening to the beating
> of the mistuned ratios, without knowing theoretically
> what was going on. a musician with a highly sensitive
> ear knows when something sounds good ... whatever
> that means exactly, it certainly does mean something.
>
> -monz

Bob:
Thank you both! This is what I've been trying to say all along! I
have choir members who don't understand or hear the physical basis
behind the mathematically designed and AMAZINGLY effective tuning
exercises I use to warm them up and train their ears, but they "get
it" intuitively well enough to be able to DO IT!!! Sometimes these
discussions get so academically constipated that I can't seem to go
anywhere with any really practical considerations.

Convince me that people tuning unisons never heard beats, no matter
how they conceived or named them! The fundamentals in this case are
totally, blatantly obvious in their alternating constructive and
destructive interference in totally regular periodicity! That they
couldn't hear that well enough to respond to it intuitively is not
credible, no matter what academic "evidence" pretends to support it.
And if they hear beats in unisons and they could tell whether octaves
were in tune, then they could respond to accurately beats on the
second harmonic, no matter how they described or failed to describe
them. The rest is a matter of degree and not principle, right?!

Using modern pianists to "prove" that people back then didn't hear
beats is completely without any basis. I am a choral director, and I
know only too much about most other choral directors. Most of them
come from a vocal and/or keyboard background steeped in "ET" that is
not even worth THAT name more than about once or twice a year or so.
They never tuned anything themselves, not even a flute to a single
pitch, and in terms of accurate intonation, typically have the most
poorly trained ears of anyone involved seriously in the performance
of music. Just listen to some of the so-called "word-class" singers
who perform at the Met!

And these "musicians" being used as evidence that some of the
greatest musician/composer/theorists that who ever lived and who
often tuned entire keyboard instruments more than once a day couldn't
hear what THESE modern pianists don't?!?!?!?!?!? Some of these latter
could retune in record time to accomodate more remote keys in
meantone. The singers in my choirs who have only played piano or sung
in other choirs are typically the worst at both intonation and
following a conductor and generally adapting well to an ensenble
situation. They are usually NOT well-rounded musically.

Some probably think I'm chauvinistically assuming that what we hear
today applies to what they could hear then. Well, how chauvinistic is
it to assume that only we wonderful moderns with all our scientific
understanding and mathematics could hear beats? They may have taken
however they perceived or intuited their presence so much for granted
they didn't even feel it was worth mentioning, or didn't have any
conceptual basis that would have allowed any verbal communication of
the experience. There are, after all, other musical attributes that
fit nicely into that category even today, aren't there?!

And there are other means besides a monochord to mathematically
construct accurate tuning models, such as columns of air and lengths
of long, thin metallic objects, etc. This would require only unisons
and octaves to set a temperament accurately. And monchords would at
the very least familiarize musicians with the physical property of
string harmonics, with plainly visible nodes you could touch to HEAR
them.

Cheers,

Bob

🔗Jon Szanto <JSZANTO@ADNC.COM>

🔗Robert Wendell <rwendell@cangelic.org>

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

>...no one then had any clue that there was anything to compensate...

Bob:
How can anyone propose this seriously? Paul, you just pointed out
yourself that when you push the string down at the halfway point, the
pressure mistunes the octave. Is this not a truly big, fat clue?! Or
are we being so chauvinistic in our viewpoint that we really believe
no one earlier than a couple of hundred years ago could tell when
anything, especially a simple octave on a harmonic string, was
mistuned? We had to wait for science for that, I suppose.

As I pointed out in another recent post, you don't even have to push
the string down on a monochord. You can plainly SEE the nodes and
just touch one lightly, as when playing harmonics on bowed
instruments.

I suppose no one ever noticed this until the seventeenth century. And
I suppose the 16th-century tunings of 31 ET demonstrate a major
tuning disability, especially the ones using two such temperaments
offset by a quarter comma. I suppose all this demonstrates a marked
inferiority or at least similarity, to the hearing of those modern-
day pianists unable hear beats even when they're pointed out to them
and contrasted with beatless intervals. :)

> there would still be no way to correct the positions of the
> *tempered* markings on the monochord . . .

Bob:
Pretty bold statement. On what basis? There should be about a hundred
ways to refute that one in simple, empirical, and not very
analytically sophisticated practice.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, David Beardsley
<davidbeardsley@b...>
> wrote:
> > ----- Original Message -----
> > From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> >
> > > johnny, the problem is that string lengths on a monochord do
not
> > > translate exactly to inverse frequencies, which they would have
> to in
> > > order for the tunings in question to exhibit the properties
under
> > > debate here. even if one could stop the string with infinite
> > > accuracy, there will be a slight nonlinear correction due to
the
> > > *bending* of the string down from its relaxed position to the
> > > position where it makes contact with the markings on the
> monochord.
> > > if you've ever tried this yourself, you'll know that pressing
> down on
> > > the midpoint of the string gets you quite a few cents sharp of
> the
> > > octave, due to the added tension of pressing down on it.
> >
> > Wouldn't you compensate for this when you calculate your markings?
>
> we're talking about what would have happened in the 16th century.
no
> one then had any clue that there was anything to compensate, since
> the physical basis of the ratios was still placed in string lengths
> (the physics of strings would be worked out beginning in the
> following century) and not yet in vibration frequencies (which were
> just beginning to be uncovered in the first place). for example, in
> those days a minor triad was 4:5:6, and a major triad was 10:12:15,
> since these numbers were referring to string lengths and no
> physically more relevant correlate of the numbers was yet widely
> known . . . and the relationships between string deflection,
tension,
> and pitch were centuries away . . .
>
> remember, dave, we're talking about the monochords for
> *temperaments*, so even if just intervals could be corrected by
ear,
> there would still be no way to correct the positions of the
> *tempered* markings on the monochord . . .

🔗Robert Wendell <rwendell@cangelic.org>

--- In tuning@yahoogroups.com, Leonardo Perretti <dombedos@t...>
wrote:
> wallyesterpaulrus wrote:
>
> >johnny, the problem is that string lengths on a monochord do not
> >translate exactly to inverse frequencies, which they would have to
in
> >order for the tunings in question to exhibit the properties under
> >debate here. even if one could stop the string with infinite
> >accuracy, there will be a slight nonlinear correction due to the
> >*bending* of the string down from its relaxed position to the
> >position where it makes contact with the markings on the monochord.
> >if you've ever tried this yourself, you'll know that pressing down
on
> >the midpoint of the string gets you quite a few cents sharp of the
> >octave, due to the added tension of pressing down on it.
>
> This is not true for Zarlino. He used a plain table to set the
> proportional markings, and put the strings at some height above it;
> then, he used a mobile tool, that he calls "scannello", a sort of
> triangular prism set on one of its faces, so that one of its
corners
> touches the string from the bottom (see the image in Zarlino's Part
> 2, Cap.18). The tool is moved on the table, aligned to the
markings,
> to set the notes. This is similar to the method described by the
> ancient writers, as, for example, Ptolemy. Zarlino remarks that the
> mobile "scannello" should be exactly as high as the two stationary
> "scannello's" that limit the string at its edges, so it should not
> add any tension.
>
> Regards
> Leonardo Perretti

Bob:
Beautiful! Exactly the kind of resourcefulness I would have imagined
existed without any specific academic knowledge to draw from on this
particular subject. Nonetheless, it would be very
uncharacteristically clutzy for the period in general to be so easily
stymied by such a simple mechanical/musical problem.

I can't help but wonder at the very narrow sense of superiority
twentieth century science seems to impose on its interpretation of
past cultures and their level of understanding on a practical level.
I think its very ironically chauvinistic, considering the degree to
which that perspective seems to project itself on those who would
disagree.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "Robert Wendell" <rwendell@c...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>

> Or
> are we being so chauvinistic in our viewpoint that we really
believe
> no one earlier than a couple of hundred years ago could tell when
> anything, especially a simple octave on a harmonic string, was
> mistuned?

bob, you are obviously not taking the time to read my posts
carefully. this is quite a distortion . . .

> As I pointed out in another recent post, you don't even have to
push
> the string down on a monochord. You can plainly SEE the nodes and
> just touch one lightly, as when playing harmonics on bowed
> instruments.

and yet this observation does not exist in the recorded literature
before 16blah blah blah. we're going around in circles. all i know is
that plenty of people fail to notice such things for their whole
lives, and at some point, civilization as a whole had yet to notice.
we can argue forever when that point was, i'm just trying to point
out that certain pronouncements made here make certain assumptions
about this which are quite speculative.

>
> I suppose no one ever noticed this until the seventeenth century.
And
> I suppose the 16th-century tunings of 31 ET demonstrate a major
> tuning disability, especially the ones using two such temperaments
> offset by a quarter comma. I suppose all this demonstrates a marked
> inferiority or at least similarity, to the hearing of those modern-
> day pianists unable hear beats even when they're pointed out to
them
> and contrasted with beatless intervals. :)

i'm glad the smiley is there because you know i don't feel this way.

🔗Afmmjr@aol.com

🔗Robert Wendell <rwendell@cangelic.org>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Robert Wendell" <rwendell@c...>
wrote:
> > --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> > <wallyesterpaulrus@y...> wrote:
> >
>
> > Or
> > are we being so chauvinistic in our viewpoint that we really
> believe
> > no one earlier than a couple of hundred years ago could tell when
> > anything, especially a simple octave on a harmonic string, was
> > mistuned?
>
> bob, you are obviously not taking the time to read my posts
> carefully. this is quite a distortion . . .
>
> > As I pointed out in another recent post, you don't even have to
> push
> > the string down on a monochord. You can plainly SEE the nodes and
> > just touch one lightly, as when playing harmonics on bowed
> > instruments.
>
> and yet this observation does not exist in the recorded literature
> before 16blah blah blah. we're going around in circles. all i know
is
> that plenty of people fail to notice such things for their whole
> lives, and at some point, civilization as a whole had yet to
notice.
> we can argue forever when that point was, i'm just trying to point
> out that certain pronouncements made here make certain assumptions
> about this which are quite speculative.
>
> >
> > I suppose no one ever noticed this until the seventeenth century.
> And
> > I suppose the 16th-century tunings of 31 ET demonstrate a major
> > tuning disability, especially the ones using two such
temperaments
> > offset by a quarter comma. I suppose all this demonstrates a
marked
> > inferiority or at least similarity, to the hearing of those
modern-
> > day pianists unable hear beats even when they're pointed out to
> them
> > and contrasted with beatless intervals. :)
>
> i'm glad the smiley is there because you know i don't feel this way.

Bob:
I would think not, judging from the many very astute posts I have
read from you in the past, Paul. However, your citing of your
experiments with present-day pianists puzzles me in this context, for
reasons I've already outlined at some length in another recent post.

I wish to concede an important point here, though. I just ran some
calculations on the 2/7- versus 7/26-comma meantones. A difference of
only 0.35 cents in the tempering on the fifths causes the +1.5 beat
ratio in 2/7-comma to change radically to over 3 as I recall (here
away from my HP scientific, +3.75 I think). I agree that this would
make it unlikely that beat synchrony constitutes a compelling
rationale for any preference for that tuning.

There is one thing I feel we need to maintain clearly in our
awareness, however: The tempering on the fifths for any temperament
is often very small, and varies from one temperament to another by
fractions of a cent in any temperaments anywhere near each other
along the continuum from 1/3- to 1/11-comma meantones. I find it
highly speculative to assume that all the careful weighing and
choosing implied in the documentation often cited on this list from
the long-lived meantone era was strictly theoretical. This is
especially true concerning the plethora of temperaments referred to
as being in use at many junctures along this centuries-long road.

Even modern equal temperament uses tempering just shy of two cents on
each fifth. The slightest deviation from this by fractions of a cent
in a sequence of several fifths will cause gross cumulative tuning
errors on other intervals, most notably the thirds. This is not a new
problem, of course. Nonetheless we have to deal with it today as much
as ever if we tune aurally. We do this with cross checks and
iterative tuning techniques if we're conscientious and neither tuning
electronically nor using unisons against an external physical
standard of some sort.

I don't pretend to know how they tuned hundreds of years ago.
However, it is difficult to believe they couldn't hear beats on
unisons, no matter how they interpreted them subjectively, or tune
octaves by ear accurately on reasonably harmonic strings. Or that
with all their advancements in other areas as early as the
Renaissance they were not resourceful enough or musically refined
enough with respect to intonation that they failed to discriminate in
a practically meaningful way among all the different tunings on which
they wrote their discourses. All the evidence seems to me to point in
the opposite direction: that they were much more sophisticated,
refined, and discriminating with regard to intonation than musicians
typically are today.

I've already enumerated many of the reasons for our slack standards
in this regard in other posts from the time I joine this list. Even
so, for the sake of convenience I will posit a few in summary here:

1) Tuning of keyboards relegated almost exclusively to professional
technicians who are often notmusicians themselves

2) Consequent elimination of the daily need to confront finer
(microtonal) pitch discrepancies commonly commonplace for musicians
in previous eras

3) Consequent elimination from the awareness of a huge portion of
modern musicians that such discrepancies even exist

4) Grossly infrequent tuning of keyboard instruments, most notably
pianos, resulting in the "training" of modern ears to tolerate
incredibly poor intonation on a daily basis

Cheers,

Bob :)

🔗Kraig Grady <kraiggrady@anaphoria.com>

Obviously people have been tuning quite accurately for thousands of years without science . half the world still does and those culture which relies most on science seems to have the worst ears in relation to pitch. Amiya dasqupta could hear a two cents differenace yet didn't know what a cent was. Possibly in history there was less noise and hence the ears might have been
more sensitive.

>
>
> From: "Robert Wendell" <rwendell@cangelic.org>
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> >...no one then had any clue that there was anything to compensate...
>
> Bob:
> How can anyone propose this seriously? Paul, you just pointed out
> yourself that when you push the string down at the halfway point, the
> pressure mistunes the octave. Is this not a truly big, fat clue?! Or
> are we being so chauvinistic in our viewpoint that we really believe
> no one earlier than a couple of hundred years ago could tell when
> anything, especially a simple octave on a harmonic string, was
> mistuned? We had to wait for science for that, I suppose.
>
> As I pointed out in another recent post, you don't even have to push
> the string down on a monochord. You can plainly SEE the nodes and
> just touch one lightly, as when playing harmonics on bowed
> instruments.
>
> I suppose no one ever noticed this until the seventeenth century. And
> I suppose the 16th-century tunings of 31 ET demonstrate a major
> tuning disability, especially the ones using two such temperaments
> offset by a quarter comma. I suppose all this demonstrates a marked
> inferiority or at least similarity, to the hearing of those modern-
> day pianists unable hear beats even when they're pointed out to them
> and contrasted with beatless intervals. :)
>
> > there would still be no way to correct the positions of the
> > *tempered* markings on the monochord . . .
>
> Bob:
> Pretty bold statement. On what basis? There should be about a hundred
> ways to refute that one in simple, empirical, and not very
> analytically sophisticated practice.
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> > --- In tuning@yahoogroups.com, David Beardsley
> <davidbeardsley@b...>
> > wrote:
> > > ----- Original Message -----
> > > From: "wallyesterpaulrus" <wallyesterpaulrus@y...>
> > >
> > > > johnny, the problem is that string lengths on a monochord do
> not
> > > > translate exactly to inverse frequencies, which they would have
> > to in
> > > > order for the tunings in question to exhibit the properties
> under
> > > > debate here. even if one could stop the string with infinite
> > > > accuracy, there will be a slight nonlinear correction due to
> the
> > > > *bending* of the string down from its relaxed position to the
> > > > position where it makes contact with the markings on the
> > monochord.
> > > > if you've ever tried this yourself, you'll know that pressing
> > down on
> > > > the midpoint of the string gets you quite a few cents sharp of
> > the
> > > > octave, due to the added tension of pressing down on it.
> > >
> > > Wouldn't you compensate for this when you calculate your markings?
> >
> > we're talking about what would have happened in the 16th century.
> no
> > one then had any clue that there was anything to compensate, since
> > the physical basis of the ratios was still placed in string lengths
> > (the physics of strings would be worked out beginning in the
> > following century) and not yet in vibration frequencies (which were
> > just beginning to be uncovered in the first place). for example, in
> > those days a minor triad was 4:5:6, and a major triad was 10:12:15,
> > since these numbers were referring to string lengths and no
> > physically more relevant correlate of the numbers was yet widely
> > known . . . and the relationships between string deflection,
> tension,
> > and pitch were centuries away . . .
> >
> > remember, dave, we're talking about the monochords for
> > *temperaments*, so even if just intervals could be corrected by
> ear,
> > there would still be no way to correct the positions of the
> > *tempered* markings on the monochord . . .
>
> ________________________________________________________________________
>

-- -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>

--- In tuning@yahoogroups.com, "Robert Wendell" <rwendell@c...> wrote:
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"

>
> I wish to concede an important point here, though. I just ran some
> calculations on the 2/7- versus 7/26-comma meantones. A difference
of
> only 0.35 cents in the tempering on the fifths causes the +1.5 beat
> ratio in 2/7-comma to change radically to over 3 as I recall (here
> away from my HP scientific, +3.75 I think). I agree that this would
> make it unlikely that beat synchrony constitutes a compelling
> rationale for any preference for that tuning.

thank you robert. that's all i was arguing. and, obliquely, this was
the point i was trying to make by mentioning 7/25-comma meantone,
which is so close to 2/7, and also "optimal" according to certain
criteria, but whose beat rate ratios already depart markedly from
those of 2/7-comma meantone.

>
> There is one thing I feel we need to maintain clearly in our
> awareness, however: The tempering on the fifths for any temperament
> is often very small, and varies from one temperament to another by
> fractions of a cent in any temperaments anywhere near each other
> along the continuum from 1/3- to 1/11-comma meantones. I find it
> highly speculative to assume that all the careful weighing and
> choosing implied in the documentation often cited on this list from
> the long-lived meantone era was strictly theoretical. This is
> especially true concerning the plethora of temperaments referred to
> as being in use at many junctures along this centuries-long road.

after mersenne, these things exploded into the world of accurate
verifiability and testability (if that's a word). that's where its
useful to have a historical roadmap like jorgenson to look at.
zarlino's was the *first* theorist to put forward a meantone
temperament in methematically precise form. explicit knowledge of
beats, harmonics, and aural tuning rules came later, but still early
in the centuries-long meantone era.

>
> Even modern equal temperament uses tempering just shy of two cents
on
> each fifth. The slightest deviation from this by fractions of a
cent
> in a sequence of several fifths will cause gross cumulative tuning
> errors on other intervals, most notably the thirds. This is not a
new
> problem, of course. Nonetheless we have to deal with it today as
much
> as ever if we tune aurally. We do this with cross checks and
> iterative tuning techniques if we're conscientious and neither
tuning
> electronically nor using unisons against an external physical
> standard of some sort.

jorgenson describes all these checks in painful detail for hundreds
of historical temperaments. that's what makes the book so big and
weighty.

> I don't pretend to know how they tuned hundreds of years ago.
> However, it is difficult to believe they couldn't hear beats on
> unisons, no matter how they interpreted them subjectively, or tune
> octaves by ear accurately on reasonably harmonic strings.

agreed.

> Or that
> with all their advancements in other areas as early as the
> Renaissance they were not resourceful enough or musically refined
> enough with respect to intonation that they failed to discriminate
in
> a practically meaningful way among all the different tunings on
which
> they wrote their discourses.

zarlino's 2/7-comma proposal was the first ever made with
mathematical precision. that's what we were arguing about, and it
happened in 1558. by 1571(?), another possibility was in the air --
1/4-comma meantone -- and zarlino changed his preference to this.
there were no other alteratives yet defined at this time, except 1/3-
comma meantone by salinas. 1/5-comma and 1/6-comma came a bit later.

> All the evidence seems to me to point in
> the opposite direction: that they were much more sophisticated,
> refined, and discriminating with regard to intonation than
musicians
> typically are today.

by 1640 or so, this was undoubtedly true.
>
> I've already enumerated many of the reasons for our slack standards
> in this regard in other posts from the time I joine this list. Even
> so, for the sake of convenience I will posit a few in summary here:
>
> 1) Tuning of keyboards relegated almost exclusively to professional
> technicians who are often notmusicians themselves
>
> 2) Consequent elimination of the daily need to confront finer
> (microtonal) pitch discrepancies commonly commonplace for musicians
> in previous eras
>
> 3) Consequent elimination from the awareness of a huge portion of
> modern musicians that such discrepancies even exist
>
> 4) Grossly infrequent tuning of keyboard instruments, most notably
> pianos, resulting in the "training" of modern ears to tolerate
> incredibly poor intonation on a daily basis
>
> Cheers,
>
> Bob :)

bob, i'm in complete agreement, except on the last point -- many
composers are known to favor out-of-tune pianos, kind of like some
artists would prefer to start with an irregular lump of stone than a
perfect cube of marble . . .

🔗Robert Wendell <rwendell@cangelic.org>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Robert Wendell" <rwendell@c...>
wrote:

> > 4) Grossly infrequent tuning of keyboard instruments, most
notably
> > pianos, resulting in the "training" of modern ears to tolerate
> > incredibly poor intonation on a daily basis
> >
Paul:
> bob, i'm in complete agreement, except on the last point -- many
> composers are known to favor out-of-tune pianos, kind of like some
> artists would prefer to start with an irregular lump of stone than
a
> perfect cube of marble . . .

Bob:
So this predisposes you to disagree that people who grow surrounded
by grossly out of tune pianos don't get a kind of negative ear
training that cultivates a tolerance for very bad intonation? In that
regard, I studied under a world-class vocal pedagogue who didn't seem
to mind playing on a horribly out-of-tune piano at New England
Conservatory.

Now you would think they would have their pianos tuned regularly, but
there were grossly out of tune pianos in many studios there in the
early seventies and vocal students had to sing with them. I don't
think that's a great help, and I think we hear the results in lously
intonation on the part of a lot of vocalists. And I have to deal with
it in my choir when new people come in. Their ears are used to badly
out of tune choirs and pianos.

Also, there is nothing in my experience that leads me to believe that
composers necessarily have any better-trained ears intonationally
than many other musicians these days.

🔗Graham Breed <graham@microtonal.co.uk>

wallyesterpaulrus wrote:

> zarlino's 2/7-comma proposal was the first ever made with > mathematical precision. that's what we were arguing about, and it > happened in 1558. by 1571(?), another possibility was in the air -- > 1/4-comma meantone -- and zarlino changed his preference to this. > there were no other alteratives yet defined at this time, except 1/3-
> comma meantone by salinas. 1/5-comma and 1/6-comma came a bit later.

1/4-comma meantone must have been in the air at least by 1555, when Vicentino wrote his treatise. He describes just thirds, but doesn't specify how the fifths be tempered, and uses 31-equal enharmonies that wouldn't work so well in 2/7-comma. He says this is the usual way of tuning a keyboard. So wouldn't Zarlino have known about it? From these potted biographies

http://www.hoasm.org/IVO/Vicentino.html
http://www.hoasm.org/IVN/Zarlino.html

they were both in Venice until 1546. That gives Vicentino 9 years to adopt the Roman practice of 1/4-comma meantone, without the mathematical precision. If Zarlino was capable of deriving 2/7-comma meantone as the temperament with equal errors for the major and minor thirds, he should have been able to use the same method for 1/4-comma meantone as the temperament with pure thirds. Possibly this method wasn't available in 1546, or it was and Vicentino didn't understand it.

In this new translation of Zarlino we have

http://sonic-arts.org/monzo/zarlino/1558/cap42-43.txt

the second part, starting "Demonstration from which one can understand..." considers some alternative temperament. Can anybody make sense of it? He says two intervals are mistuned by half a comma, which could be the two whole tones of 1/4-comma meantone. But he also says the other intervals should be pure. So is he mishearing a description of 1/4-comma meantone, or is this Vicentino's second tuning of the archicembalo which requires two manuals to make sense? Or something else?

In 1563, Vicentino moved to Vicenza, which seems to be near Venice. That gave him plenty of time to convince Zarlino of the merits of 1/4-comma meantone by 1571.

Graham

🔗Leonardo Perretti <dombedos@tiscalinet.it>

Graham Breed wrote:

>In this new translation of Zarlino we have
>
>http://sonic-arts.org/monzo/zarlino/1558/cap42-43.txt
>
>the second part, starting "Demonstration from which one can
>understand..." considers some alternative temperament. Can anybody make
>sense of it? He says two intervals are mistuned by half a comma, which
>could be the two whole tones of 1/4-comma meantone. But he also says
>the other intervals should be pure. So is he mishearing a description
>of 1/4-comma meantone, or is this Vicentino's second tuning of the
>archicembalo which requires two manuals to make sense? Or something else?

This is an interesting point.

In the chapters preceding the 42, Zarlino goes extensively through the ancient musical theory, mostly as described by Ptolemy and Boethius, and at the end he concludes that the best musical system for the "modern" music is the Greater Perfect System joined with the Lesser Perfect System, built by the "Syntonic Diatonic" tetrachord, also defined as the "Tense Diatonic" (16/15 x 9/8 x 10/9) by other writers. As known, the two mentioned systems were made with a total of five tetrachords, the classical Hypaton, Meson, Diezeugmenon and Hyperbolaion to build the GPS, with the addition of the Synemmenon joint to the Meson to complete the LPS. All this is illustrated in the figure of the divided monochord in the cap 40.
See the diagrams in Monzo's web page with the translated chapters.

In short, by this construction Zarlino individuates the comma as a difference between the Nete Synemmenon and the Paranete Diezeugmenon, that both should correspond to the second D of the scale (points R and M). In cap. 40 Zarlino clearly states that the Comma, "while is not useable in any genus, it is not born without any use, as by it many consonances could be gained", and "in the artificial instruments it is divided by mean of its distribution, that is made between many intervals, between eight strings, as we will see elsewhere".
In the following chapters Zarlino describes his temperament starting from the previously built scale, where the only evident comma is the above mentioned, so when he says that there is someone who uses to simply split comma (which in the monochord is placed between the points R and M) between the two adjacent intervals he refers to *that* comma, it means that the two strings R and M were unified in a unique string placed half-way (harmonically), so that the comma is divided between the intervals N-RM and RM-L, the remaining scale remaining untempered. I don't know if this was a general use, or just an erudite discussion between scholars.

Another point.
The main reason why Zarlino used 2/7 - and here we turn to Monz's original question - seems to be the necessity of obtaining all the intervals of the same kind equal one another, as he clearly and repeatedly states through the cap. 42 and 43.

Why 2/7?
The most logical line of thought seems the following. As seen before, Zarlino takes into consideration only the Diatonic "syntonic" genus, i.e. the diatonic scale; this is made of eight strings and, consequently, seven intervals, so he tries by dividing the comma in seven parts. In doing that, he not only distributes the comma, but also reorganizes the intervals, and makes it in such a way, that all the intervals of the same kind are equalized.
The actual procedure he uses is "clearly" explained in the cap. 43, and the figure is explicative, particularly in the upper band.

It is interesting to note that Zarlino almost certainly was not aware in 1558 of the 1/4 temperament; in Cap. 47 he describes the utility of the addition of the strings of the Enarmonic thetrachord, and concludes that it is useful in that it offers a way to obtain the pure thirds, settling them on additional red keys. This would have no sense with 1/4.
I don't know the evolution of his theory in the next edition of his treatise.

Comments?

Leonardo Perretti

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, Leonardo Perretti <dombedos@t...>
wrote:

> so when he says that there is someone who uses to
> simply split comma (which in the monochord is placed between the
> points R and M) between the two adjacent intervals he refers to
> *that* comma, it means that the two strings R and M were unified in
a
> unique string placed half-way (harmonically), so that the comma is
> divided between the intervals N-RM and RM-L, the remaining scale
> remaining untempered. I don't know if this was a general use, or
just
> an erudite discussion between scholars.

this may well be a reference to fogliano. the latter has some
association with 1/2-comma meantone that is obscure to me; judging
from fogliano's monochords, it may well have been a construction just
like this.

> Another point.
> The main reason why Zarlino used 2/7 - and here we turn to Monz's
> original question - seems to be the necessity of obtaining all the
> intervals of the same kind equal one another, as he clearly and
> repeatedly states through the cap. 42 and 43.

terrific. i anticipated an explanation like this, either here or in
an off-list message to monz.

> The actual procedure he uses is "clearly" explained in the cap. 43,
> and the figure is explicative, particularly in the upper band.

i'll have to take a look at it later.

> It is interesting to note that Zarlino almost certainly was not
aware
> in 1558 of the 1/4 temperament;

wow.

> in Cap. 47 he describes the utility
> of the addition of the strings of the Enarmonic thetrachord, and
> concludes that it is useful in that it offers a way to obtain the
> pure thirds, settling them on additional red keys. This would have
no
> sense with 1/4.

hmm . . . yeah i'll read that too, and let you know what i think . . .

> Comments?
>
> Leonardo Perretti

cheers,
paul

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "wallyesterpaulrus"

/tuning/topicId_42480.html#42846

<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
> >
> > /tuning/topicId_42480.html#42743
> >
> > >
> > > i was just making an analogy with the tuning procedure:
> > > tuning keyboard strings by ear to another external sound
> > > source which is already in the desired tuning.
> > >
> > >
> > >
> > > -monz
> >
> > ***Actually, while we're on the subject of "tuning..." (this *is*
> the
> > *Tuning List* after all... :) using a differently-timbred
external
> > sound source is not always as easy as one might thing.
> >
> > Johnny Reinhard and I purchased a nice, and rather pricey, Korg
> tuner
> > some time ago, and I was thinking of using it to tune a piano,
but
> > because of the difference in timbre, it was almost impossible to
> > aurally "line it up" for that purpose for whatever reason.... :(
> >
> > J. Pehrson
>
> joseph, i'm pretty sure that what you're up against here is the
> *inharmonicity* of the piano (guitar would also fall into this
> category), versus the harmonicity of sustaining acoustic
instruments and periodic electronically-generated signals.

***Thanks, Paul. That would make some sense. It was quite
startling, though, since there was nothing to "hang on too..." It
seemed like the proverbial "apples and oranges..." funny... :(

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗monz <monz@attglobal.net>

thanks, guys.

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, March 11, 2003 7:52 PM
> Subject: [tuning] Re: Zarlino's 2/7 (was: A question and an update...)
>
>
> --- In tuning@yahoogroups.com, "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
>
> > my comment to monzo is
> > now at the bottom of this page:
> >
> > http://sonic-arts.org/dict/2-7cmt.htm
>
> monz's last four words are "the four 5-limit ratios". strictly
> speaking, it should really say "the four ratios of 5", since there
> are *seven* 5-odd-limit ratios (as the 5-limit Tonality Diamond
> shows), and there are *an infinite number* of 5-prime-limit ratios.

i've fixed that, and also made some new corrections
and additions near the bottom of the page.

the link in the 2/7-comma page also now points to the
webpage i made of Leonardo Perretti's translation
of cap. 24, 25, 42, 43 of part 2 of _le institutione
harmoniche_, Zarlino 1558.

that webpage now has Zarlino's text broken up into
much easier-to-read paragraphs, and includes his
diagrams and several of my own, which show precisely
how Zarlino tempered certain JI intervals (those
of the "Perfect Immutable System" tuned in Ptolemy's
"Syntonic Diatonic" genus), reducing the 9 distinct
8ve-equivalent pitches of Ptolemy's scale into an
8-tone (8ve-equivalent) 2/7-comma meantone.

http://sonic-arts.org/monzo/zarlino/1558/zarlino1558-2.htm

Zarlino actually describes the tuning of 3 pairs
of 8ves in the 2/7-comma temperament, so that he
actually reduces a 12-tone set of JI pitches to
an 11-tone meantone.

-monz

🔗Graham Breed <graham@microtonal.co.uk>

🔗monz <monz@attglobal.net>

> From: "Graham Breed" <graham@microtonal.co.uk>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, March 12, 2003 12:37 AM
> Subject: [tuning] Re: Zarlino's 2/7 (was: A question and an update...)
>
>
> Leonardo Perretti wrote:
>
> > It is interesting to note that Zarlino almost certainly was not aware
> > in 1558 of the 1/4 temperament; in Cap. 47 he describes the utility
> > of the addition of the strings of the Enarmonic thetrachord, and
> > concludes that it is useful in that it offers a way to obtain the
> > pure thirds, settling them on additional red keys. This would have no
> > sense with 1/4.
>
> I don't think that's tenable. I've done a bit more research -- 1/4
> comma meantone was fist given by Pietro Aaron in 1529 in Venice. While
> I don't know how he specified it, all sources agree it was 1/4 comma.
> Surely Zarlino must have known about this, either from Willaert or
> because it was the standard tuning in Venice.

without knowing any more about the circumstances myself,
i'd go along with Graham on this.

while by Zarlino's lifetime Venice was no longer the
most powerful city in Europe, at that time it was
perhaps still the richest, most opulent, most
culturally diverse and most technologically advanced.
this era is known as its period of decadent decline.

if 1/4-comma meantone was introduced in Venice in 1529
by Aron, then my guess is that it would have become
wildly popular as "the next new thing" in music.

ever since the obliteration from the cultural fund of
knowledge of the richness of Greek tuning theory, with
the German invasions of the western Roman Empire in the
late 400s, a simple Pythagorean diatonic scale had been
assumed as the normal tuning in Europe for about 1000 years.

while tantalizing hints of 5-limit use in practice
are noted by English writers as early as the 1200s,
5-limit JI had only been first detailed in a theory
treatise in 1480 by Ramos. by c.1600 theorists were
commenting on "comma drift" problems.

if 1/4-comma meantone developed shortly after that,
as Schlick's work (1511) seems to indicate, it certainly
would have been viewed as a radically modern idea,
and most likely would have spread like wildfire in
Venetian musical circles. (yes, i know, wild speculations
produce wild adjectives ...)

it seems to me that Zarlino was probably familiar with
1/4-comma meantone in 1558, and would have understood
exactly how its error from 5-limit JI is distributed.

(... as depicted in my applet graph at)
http://sonic-arts.org/dict/meantone-error/meantone-error.htm

he obviously was well aware of this error distribution
in 2/7-comma meantone, since he describes it in detail.
so to me, it's just as obvious that its even error
distribution of the "ratios of 5" was the main reason
he chose it, and as Leonardo pointed out, Zarlino states
this himself several times.

Leonardo has already told me that he'll undertake the
translation of cap. 46 and 47, so i'm eager to see
what it says.

... and for those who might be wondering, i'm trying
to talk Leonardo into producing a complete English
translation of part 2 of Zarlino's book, which i will
publish online alongside the webpage already existing.
my feeling is that the importance of Zarlino's work
to all subsequent western theorizing merits the labor
involved.

-monz

🔗monz <monz@attglobal.net>

oops, my bad ...

> From: "monz" <monz@attglobal.net>
> To: <tuning@yahoogroups.com>
> Sent: Wednesday, March 12, 2003 1:05 AM
> Subject: Re: [tuning] Re: Zarlino's 2/7 (was: A question and an update...)
>

>
> while tantalizing hints of 5-limit use in practice
> are noted by English writers as early as the 1200s,
> 5-limit JI had only been first detailed in a theory
> treatise in 1480 by Ramos. by c.1600 theorists were
> commenting on "comma drift" problems.

that should say:

by c.1500 theorists were commenting on "comma drift" problems.

i also could have mentioned the "fifth-tones" of
Marchetto (1318), whatever they're supposed to mean,
as early examples of intonational adventure.

see my Marchetto webpage:
http://sonic-arts.org/monzo/marchet/marchet.htm

Europeans returning home from the Crusades during
the 1100-1300s brought back many Greek books, which
is a large part of what caused the Renaissance in
the first place.

many late-medieval and Renaissance theorists described
(or at least attempted to describe) the ancient
Greek _genera_, some with admirable mathematical
accuracy. Zarlino was one of them.

it was inevitable that contemporary theorists and
composers would want to try these interesting new
sounds in their work.

after a couple of centuries of this revival
of ancient Greek knowledge of JI, the invention
of temperament would be seen as very modern.

-monz

🔗Afmmjr@aol.com

🔗Robert Wendell <rwendell@cangelic.org>

One very important point:

In the meantones "south" of 1/4-comma toward 1/3-comma on the
opposite side of 1/4-comma from ET, where the beat ratios of m3/M3
are positive, the ratios change radically with very small tuning
shifts. On the other side toward ET (e.g., 1/5-comma, 1/7-comma,
etc.), the beat synchrony is much more robust.

As I observed in a previous post acceding this point to Paul, the
+1.5 beat ratio shifted to +3.75 with only a 0.35 cent shift in
intonation! On a triad with a just fifth, where the beat ratio is
always at -1.50, with a third at +11 cents you can change the fifth
by 0.4 cents and the ratio changes relatively little to -1.55. I had
overlooked this important fact in my exchanges with Paul on this
issue.

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, "Joseph Pehrson" <jpehrson@r...>
wrote:
> > --- In tuning@yahoogroups.com, "Robert Wendell" <rwendell@c...>
> wrote:
> >
> > /tuning/topicId_42480.html#42746
> >
> > > Well, how chauvinistic is
> > > it to assume that only we wonderful moderns with all our
> scientific
> > > understanding and mathematics could hear beats?
> >
> > ***Now here's a good one. Bob says they *did* and Paul says they
> > *don't...*
>
> not really. thankfully, bob and i seem to have settled this thread
> and moved on . . . i hope you'll "catch up" . . .

🔗Robert Wendell <rwendell@cangelic.org>

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Robert Wendell <rwendell@cangelic.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > From: "Graham Breed" <graham@m...>
> > To: <tuning@yahoogroups.com>
> > Sent: Wednesday, March 12, 2003 12:37 AM
> > Subject: [tuning] Re: Zarlino's 2/7 (was: A question and an
update...)
> >
> >
> > Leonardo Perretti wrote:
> >
> > > It is interesting to note that Zarlino almost certainly was not
aware
> > > in 1558 of the 1/4 temperament; in Cap. 47 he describes the
utility
> > > of the addition of the strings of the Enarmonic thetrachord,
and
> > > concludes that it is useful in that it offers a way to obtain
the
> > > pure thirds, settling them on additional red keys. This would
have no
> > > sense with 1/4.
> >
> > I don't think that's tenable. I've done a bit more research --
1/4
> > comma meantone was fist given by Pietro Aaron in 1529 in Venice.
While
> > I don't know how he specified it, all sources agree it was 1/4
comma.
> > Surely Zarlino must have known about this, either from Willaert
or
> > because it was the standard tuning in Venice.
>
>
>
> without knowing any more about the circumstances myself,
> i'd go along with Graham on this.
>
> while by Zarlino's lifetime Venice was no longer the
> most powerful city in Europe, at that time it was
> perhaps still the richest, most opulent, most
> culturally diverse and most technologically advanced.
> this era is known as its period of decadent decline.
>
> if 1/4-comma meantone was introduced in Venice in 1529
> by Aron, then my guess is that it would have become
> wildly popular as "the next new thing" in music.

aron was merely describing already existing tuning practices. margo,
quoting lindley, has posted an estimate of something like 1480 for
when meantone became prevalent.

but even aron didn't give a mathematically precise description, as
monz and margo were quick to point out way back when.

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Joseph Pehrson <jpehrson@rcn.com>

🔗Leonardo Perretti <dombedos@tiscalinet.it>

Graham Breed wrote:

>> It is interesting to note that Zarlino almost certainly was not aware
>> in 1558 of the 1/4 temperament; in Cap. 47 he describes the utility
>> of the addition of the strings of the Enarmonic thetrachord, and
>> concludes that it is useful in that it offers a way to obtain the
>> pure thirds, settling them on additional red keys. This would have no
>> sense with 1/4.
>
>I don't think that's tenable. I've done a bit more research -- 1/4
>comma meantone was fist given by Pietro Aaron in 1529 in Venice. While
>I don't know how he specified it, all sources agree it was 1/4 comma.
>Surely Zarlino must have known about this, either from Willaert or
>because it was the standard tuning in Venice.

and Johnny Reinhard wrote:

>I believe Mark Lindley proved that Aaron did not specifically give >quarter-comma, but >was close. Aaron to meantone is like Schlick is >to well-temperament, a bit obscure >and pioneering.

I read "at first hand" Aaron's relevant part of the "Toscanello", and it actually seems to describe something fairly different from 1/4 meantone.
Does anyone (perhaps Margo ?) know by what arguments Aaron's treatise was interpreted as a description of 1/4 meantone? Maybe I am misunderstanding it.

Regards
Leonardo Perretti

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

--- In tuning@yahoogroups.com, Leonardo Perretti <dombedos@t...>
wrote:

> I read "at first hand" Aaron's relevant part of the "Toscanello",
and
> it actually seems to describe something fairly different from 1/4
> meantone.
> Does anyone (perhaps Margo ?) know by what arguments Aaron's
treatise
> was interpreted as a description of 1/4 meantone? Maybe I am
> misunderstanding it.
>
> Regards
> Leonardo Perretti

in
/tuning/topicId_8553.html#8553

margo writes:

'Aaron, writing an introduction to music in Italian (_Toscanello_ is a
title honoring his native Tuscany), advises the not necessarily
experienced reader to start by making the octave C-C "just," and then
the major third C-E "sonorous and just, as united as possible." While
Aaron does not specify a pure 5:4 for the major third, this seems to
me a natural reading of "sonorous and just." Then the fifths C-G-D-A-E
are tuned so that each is slightly "flat" or "lacking," and each by
the same amount. For example, A should be the same "distance" from D
as from E -- in other words, the fifths D-A and A-E should be tempered
from "perfection" by the same quantity.

As far as I'm concerned, this is enough to justify listing Aaron's
instructions as a description the procedure for obtaining 1/4-comma
meantone -- as various authors have done, and Paul does in his table.
Further, Aaron adds after his C-G-D-A-E series of fifths that F should
be tuned by a similar but opposite procedure, making F in the fifth
F-C "a little high, passing a bit beyond perfection." If one assumes
symmetry, then 1/4-comma meantone indeed results.

Correctly pointing out that Aaron's instructions do not give a
mathematical description of 1/4-comma meantone, or explicitly direct
that all major thirds be made pure, Lindley raises the question of
what Aaron means by his statement that "thirds and sixths are blunted
or diminished" in this temperament.

One interpretation which occurs to me is that Aaron is comparing
meantone _major_ thirds and sixths with the same intervals in a
Pythagorean tuning with pure fifths -- where they are indeed somewhat
larger. Lindley emphasizes such ambiguous statements to argue that
Aaron's instructions do specify 1/4-comma for the first notes
C-G-D-A-E, but might lend themselves to various slightly irregular
tunings for the other fifths and thirds.

Another point made by Lindley is that Aaron uses the adjective
_giusta_ ("just") to refer not only to his initial octave and major
third (where "pure" is an attractive reading), but also for the
temperament as a whole -- _participatione & acordo giusto & buono_, "a
just and good temperament and tuning." However, Lindley himself is
ready to accept Aaron's opening C-G-D-A-E with its "sonorous and just"
major third C-E as a description of 1/4-comma. His statement about the
octave C-C and major third C-E being as sonorous or "united" as
possible would support this conclusion even we assume that "just" may
mean simply "euphonious" or "pleasing."

In his instructions for the final stage of the temperament, the tuning
of the sharps, Aaron directs that C#, tuned in relation to the fifth
A-E, should be a major third from A and a minor third from E, and
likewise with F# in relation to the fifth D-A, etc.

From one viewpoint, this language may simply be reminding the student
of the locations of major and minor thirds involving accidentals.
Lindley, however, pursuing his argument that a regular 1/4-comma
temperament is not _necessarily_ implied, argues that one _could_ read
this language to suggest something like Zarlino's 2/7-comma meantone
for the temperament of the sharps, with major and minor thirds
compromised by about the same amount from pure.

In arguing that Aaron's tuning is not _necessarily_ a regular
1/4-comma meantone, Lindley may have two motivations. First, he wishes
to emphasize that Aaron's instructions are not mathematically precise;
and indeed, I would agree that Zarlino (1571) and Salinas (1577) may
be the first known theorists to give such mathematical definitions of
1/4-comma and other meantone temperaments.

Secondly, Lindley wants to correct the view that 1/4-comma was a
universal standard in the 16th century. Since Aaron's instructions are
often taken as the paradigm case of this tuning, showing that they are
actually open to more than one interpretation would fit with his
larger campaign against "1/4-comma hegemony."

However, I find it noteworthy that without invoking any complex
mathematical concepts or even defining a syntonic comma, Aaron has
described in what I find beautiful as well as musicianly terms the
idea of tuning a pure major third and then dividing it into four
equally tempered fifths. Aaron's remaining instructions, including his
suggestion of a similar but opposite tempering of fifths in the flat
direction (F-C, and then Bb-F and Eb-Bb), permit a regular 1/4-comma
temperament, even if they do not compel it or define it in
mathematical terms.

If I were making a table like Paul's, I might list Aaron for 1/4-comma
and add a footnote or annotation like this:

"Aaron evidently describes a temperament with a pure major third C-E
and equally narrowed fifths for his first five notes C-G-D-A-E, with
further instructions permitting but not explicitly specifying that
other major thirds are pure; Zarlino (1571) and Salinas (1577) give
mathematically precise descriptions of a regular 1/4-comma tuning."'

monz might want to revise some statements on his 1/4-comma meantone
page after rereading the above.

🔗monz <monz@attglobal.net>

my take on this was that when Zarlino described
what was apparently a 5-limit JI diatonic scale
with a literal meantone subsituted for the both
9/8 and 10/9, he was referring to Aron's _Toscanello_.

Leonardo, would you agree or disagree with this?
you've read Aron, and i haven't.

( ... well, i've read the Italian original text,
and don't understand much of it ... )

-monz

----- Original Message -----
From: "Leonardo Perretti" <dombedos@tiscalinet.it>
To: <tuning@yahoogroups.com>
Sent: Thursday, March 13, 2003 2:21 PM
Subject: Re: [tuning] Re: Zarlino's 2/7 (was: A question and an update...)

> Graham Breed wrote:
>
> >> It is interesting to note that Zarlino almost certainly was not aware
> >> in 1558 of the 1/4 temperament; in Cap. 47 he describes the utility
> >> of the addition of the strings of the Enarmonic thetrachord, and
> >> concludes that it is useful in that it offers a way to obtain the
> >> pure thirds, settling them on additional red keys. This would have no
> >> sense with 1/4.
> >
> >I don't think that's tenable. I've done a bit more research -- 1/4
> >comma meantone was fist given by Pietro Aaron in 1529 in Venice. While
> >I don't know how he specified it, all sources agree it was 1/4 comma.
> >Surely Zarlino must have known about this, either from Willaert or
> >because it was the standard tuning in Venice.
>
> and Johnny Reinhard wrote:
>
> >I believe Mark Lindley proved that Aaron did not specifically give
> >quarter-comma, but >was close. Aaron to meantone is like Schlick is
> >to well-temperament, a bit obscure >and pioneering.
>
> I read "at first hand" Aaron's relevant part of the "Toscanello", and
> it actually seems to describe something fairly different from 1/4
> meantone.
> Does anyone (perhaps Margo ?) know by what arguments Aaron's treatise
> was interpreted as a description of 1/4 meantone? Maybe I am
> misunderstanding it.
>
> Regards
> Leonardo Perretti

🔗monz <monz@attglobal.net>

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Thursday, March 13, 2003 2:28 PM
> Subject: [tuning] Re: Zarlino's 2/7 (was: A question and an update...)
>

>
> in
> /tuning/topicId_8553.html#8553
>
> margo writes:
>
> 'Aaron, writing an introduction to music in Italian (_Toscanello_ is a
> title honoring his native Tuscany), advises the not necessarily
> experienced reader to start by making the octave C-C "just," and then
> the major third C-E "sonorous and just, as united as possible." While
> Aaron does not specify a pure 5:4 for the major third, this seems to
> me a natural reading of "sonorous and just." Then the fifths C-G-D-A-E
> are tuned so that each is slightly "flat" or "lacking," and each by
> the same amount. For example, A should be the same "distance" from D
> as from E -- in other words, the fifths D-A and A-E should be tempered
> from "perfection" by the same quantity.
>
> < etc. ...snip>

thanks, paul. the most expedient way to do it
was to simply add the quote from Margo at the
end of the page, which i've done.

http://sonic-arts.org/dict/1-4cmt.htm

-monz

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

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

> thanks, paul. the most expedient way to do it
> was to simply add the quote from Margo at the
> end of the page, which i've done.
>
> http://sonic-arts.org/dict/1-4cmt.htm

> -monz

the beginning of the page, unfortunately, contains some misleading
statements (if you'll allow the criticism) . . .

'the paradigms postulated by "common-practice" music-theory to a
great extent depend on . . . the fact that "flats" are higher in
pitch than "sharps"'

not true, since the 'expressive intonation' paradigm introduced
c.1800 fits just as snugly with common-practice paradigms even though
it has the opposite tendency

'1/4-comma meantone was assumed as the standard tuning in most of
Europe from approximately 1500 to 1700'

this is the statement i hoped margo's post would lead you to weaken --
i certainly see no reason to "assume" that all of the other meantone
tunings described by salinas and rossi were "non-standard" during
this time.

.
.
.

and now, since most people didn't seem to understand it before, let's
look at this optimal meantone graph again:

/tuning/files/perlich/woptimal.gif

zarlino's optimal meantone (at least in 1558) is based on minimizing
the maximum absolute error of the 5-limit consonances (hence we're on
the red line) while weighting the ratios of 3 insignificantly
relative to the ratios of 5 (hence we're way over on the left side of
the graph). as the graph shows, 2/7-comma meantone is optimal under
these criteria. speculating wildly, zarlino may have later decided
that all the 5-limit consonances should be weighted equally (on the
graph, exactly in the center between left and right, at w=1), in
which case staying on the red line (that is, continuing to minimize
the maximum absolute error) would have taken him to 1/4-comma
meantone.

and again, woolhouse's optimal meantone is based on minimizing the
sum-of-squares errors of the 5-limit consonances (hence we're on the
green line) while weighting all the 5-limit consonances equally
(hence we're exactly in the center between left and right, at w=1).
as the graph shows, 7/26-comma meantone is optimal under these
criteria.

finally, the blue line shows that as long as the ratios of 3 are
weighted anywhere from 0 to 7 times as much as the ratios of 5, the
lowest sum of absolute errors of the consonances occurs with 1/4-
comma meantone. hence 1/4-comma meantone arises as optimal under the
widest variety of assumptions, but anything from 2/7-comma to 3/14-
comma meantone (and beyond) can be justified by plausible optimality
criteria, and would probably not have sounded out-of-place for the
body of western music from 1500 to 1700.

is anyone still not understanding this graph?

🔗alternativetuning <alternativetuning@yahoo.com>

🔗wallyesterpaulrus <wallyesterpaulrus@yahoo.com>

🔗Leonardo Perretti <dombedos@tiscalinet.it>

monz wrote:

>my take on this was that when Zarlino described
>what was apparently a 5-limit JI diatonic scale
>with a literal meantone subsituted for the both
>9/8 and 10/9, he was referring to Aron's _Toscanello_.
>
>Leonardo, would you agree or disagree with this?
>you've read Aron, and i haven't.
>
>( ... well, i've read the Italian original text,
>and don't understand much of it ... )

Disagree.
In cap. 40 Aaron describes his monochord, in which he uses only one kind of tone, divided in a major and a minor semitone. The diatonic semitones are minor. He does not specify the sizes of the intervals, but all the construction seems to lead to a Pythagorean scale starting from Bb, with the Pythagorean comma set on D#-Eb (of course). He clearly affirms that the temperament he describes in the following chapter is referred to this monochord.
Aaron does not build his monochord stepping by fifths, but by the degrees of the diatonic scale, using both the Greek, and usual names. The black keys are described as long as they appear between the diatonic degrees. I will provide a translation, perhaps tomorrow.

About the paper by Margo, I believe I have not the authority to discuss or criticize her affirmations, nor the ones of Mark Lindley; anyway, here are some comments about Aaron's cap. 41.

Aaron starts saying that the temperament has to be made in three parts.
The first part essentially describes how to temper the diatonic notes, excluding F, with pure thirds; then the fifths, that are prescribed to be equal, should be 1/4, as universally agreed. No doubts about that.

The second part describes how to temper F, Bb and Eb, that must be three equal fifths, tuned by descending.
Apropos of this Margo wrote:

>Further, Aaron adds after his C-G-D-A-E series of fifths that F should
>be tuned by a similar but opposite procedure, making F in the fifth
>F-C "a little high, passing a bit beyond perfection." If one assumes
>symmetry, then 1/4-comma meantone indeed results.
>
>Correctly pointing out that Aaron's instructions do not give a
>mathematical description of 1/4-comma meantone, or explicitly direct
>that all major thirds be made pure, Lindley raises the question of
>what Aaron means by his statement that "thirds and sixths are blunted
>or diminished" in this temperament.

The expression: "thirds and sixths are blunted or diminished", seems to contradict the previous statement that the thirds have to be "sonorous and just.". Actually, it becomes coherent when one considers that this is comprised in the second section of the description of the temperament; then, it refer to the notes involved there only, that are F, Bb and Eb, as well as the previous statement refers to notes of the first section. The three parts of the temperament are very well separated in the original text.
In view of that, here is my opinion.
As known, the third become diminished if made with fifths accumulating a diminution of more than (-)1 sc, and the sixth with negative corrections greater then 4/3 sc.. Then, probably Aaron prescribed corrections equal or greater then 1/3 for these fifths. The thirds and sixths become diminished and all returns coherent with the text.

The third part specifies how to temper the "major semitones", that have to be tempered "between their thirds". The "major semitones" are the black keys that were built on the monochord with the major semitone in the low side of the tone that comprises it. They are C#, F# and G#. The C# has be tempered against A and E, "so much that it remains in the middle major third with A, and minor with E". As for Lindley in Margo paper, it should mean something like Zarlino's 2/7. The same is for F#.
G# is not mentioned but it could be inferred that it follows the same prescriptions.

The hole that opens between G# and Eb is not really bigger then in some variants of the regular middletones. In the description of the monochord, anyway, in cap. 40 Aaron talks of splitting the Eb, so that the convenient intervals (before temperament) could be obtained.

Is this hypothesis teneable? Is it musically likely? I don't feel entitled to make those evaluations. I call upon you wizards to speak.

Regards
Leonardo Perretti

🔗Leonardo Perretti <dombedos@tiscalinet.it>

🔗Robert Wendell <rwendell@cangelic.org>

Oh, on why negative for the beat ratios, Gene, in my formulas I
conceptually plug in negative values for flat tempering and positive
for sharpness in the thirds. However, sometimes I bend the formulas
by changing the sign and plug in positive values when I'm actually
using negative tempering. I do this to save myself the hassle of
repeatedly having to enter negative values into my HP programmable
scientific and when talking to others it's easier than explaining
what adding negative values means to the mathematically
disadvantaged, among which, in your company, I include myself.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
>
> The beat ratio will be very sensitive to tuning near 1/4-comma,
since
> that is a beat ratio of infinity, and the same is true for the
inverse
> of the beat ratio near 1/3-comma.

My point was simply this:

+1.50 (by the definition posited above) occurs on just fifths, which
are nowhere near 1/4-comma. By contrast, -1.50 occurs on 2/7-comma.
If you look at the two in light of the very accurate linear
approximation formula I posted originally, you can easily see the
contrast.

R = 1.5(deltaP5/deltaM3 - 1), where the deta values are in cents.

For just P5:

R = 1.5(0 - 1) = -1.50

The thirds don't effect the ratio as long as the fifth is just and
the fifth can change quite a bit without unduly upsetting the ratio.

For -2/7 comma P5, the thirds are 1 + 4*(-2/7) = 1 + (-8/7) = -1/7
comma:

R = 1.5[(-2/7)/(-1/7) - 1 = 1.5(2 - 1) = 1.50

You can see that the slightest mistuning will throw this off pretty
fast.

That's all I meant. :)

Cheers,

Bob

🔗Robert Wendell <rwendell@cangelic.org>

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

Graph:
> /tuning/files/perlich/woptimal.gif
>
Some explanation of the graph:
> zarlino's optimal meantone (at least in 1558) is based on
minimizing
> the maximum absolute error of the 5-limit consonances (hence we're
on
> the red line) while weighting the ratios of 3 insignificantly
> relative to the ratios of 5 (hence we're way over on the left side
of
> the graph). as the graph shows, 2/7-comma meantone is optimal under
> these criteria. speculating wildly, zarlino may have later decided
> that all the 5-limit consonances should be weighted equally (on the
> graph, exactly in the center between left and right, at w=1), in
> which case staying on the red line (that is, continuing to minimize
> the maximum absolute error) would have taken him to 1/4-comma
> meantone.
>
> and again, woolhouse's optimal meantone is based on minimizing the
> sum-of-squares errors of the 5-limit consonances (hence we're on
the
> green line) while weighting all the 5-limit consonances equally
> (hence we're exactly in the center between left and right, at w=1).
> as the graph shows, 7/26-comma meantone is optimal under these
> criteria.
>
> finally, the blue line shows that as long as the ratios of 3 are
> weighted anywhere from 0 to 7 times as much as the ratios of 5, the
> lowest sum of absolute errors of the consonances occurs with 1/4-
> comma meantone. hence 1/4-comma meantone arises as optimal under
the
> widest variety of assumptions, but anything from 2/7-comma to 3/14-
> comma meantone (and beyond) can be justified by plausible
optimality
> criteria, and would probably not have sounded out-of-place for the
> body of western music from 1500 to 1700.
>
> is anyone still not understanding this graph?

Bob:
Lovely graph, Paul! Very nice. It explains a great deal very
economically.

🔗Robert Wendell <rwendell@cangelic.org>

I want to congratulate Margo for her wonderfully balanced and
insightful input on this issue. :)

Cheers,

Bob

--- In tuning@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> --- In tuning@yahoogroups.com, Leonardo Perretti <dombedos@t...>
> wrote:
>
> > I read "at first hand" Aaron's relevant part of the "Toscanello",
> and
> > it actually seems to describe something fairly different from 1/4
> > meantone.
> > Does anyone (perhaps Margo ?) know by what arguments Aaron's
> treatise
> > was interpreted as a description of 1/4 meantone? Maybe I am
> > misunderstanding it.
> >
> > Regards
> > Leonardo Perretti
>
>
> in
> /tuning/topicId_8553.html#8553
>
> margo writes:
>
> 'Aaron, writing an introduction to music in Italian (_Toscanello_
is a
> title honoring his native Tuscany), advises the not necessarily
> experienced reader to start by making the octave C-C "just," and
then
> the major third C-E "sonorous and just, as united as possible."
While
> Aaron does not specify a pure 5:4 for the major third, this seems to
> me a natural reading of "sonorous and just." Then the fifths C-G-D-
A-E
> are tuned so that each is slightly "flat" or "lacking," and each by
> the same amount. For example, A should be the same "distance" from D
> as from E -- in other words, the fifths D-A and A-E should be
tempered
> from "perfection" by the same quantity.
>
> As far as I'm concerned, this is enough to justify listing Aaron's
> instructions as a description the procedure for obtaining 1/4-comma
> meantone -- as various authors have done, and Paul does in his
table.
> Further, Aaron adds after his C-G-D-A-E series of fifths that F
should
> be tuned by a similar but opposite procedure, making F in the fifth
> F-C "a little high, passing a bit beyond perfection." If one assumes
> symmetry, then 1/4-comma meantone indeed results.
>
> Correctly pointing out that Aaron's instructions do not give a
> mathematical description of 1/4-comma meantone, or explicitly direct
> that all major thirds be made pure, Lindley raises the question of
> what Aaron means by his statement that "thirds and sixths are
blunted
> or diminished" in this temperament.
>
> One interpretation which occurs to me is that Aaron is comparing
> meantone _major_ thirds and sixths with the same intervals in a
> Pythagorean tuning with pure fifths -- where they are indeed
somewhat
> larger. Lindley emphasizes such ambiguous statements to argue that
> Aaron's instructions do specify 1/4-comma for the first notes
> C-G-D-A-E, but might lend themselves to various slightly irregular
> tunings for the other fifths and thirds.
>
> Another point made by Lindley is that Aaron uses the adjective
> _giusta_ ("just") to refer not only to his initial octave and major
> third (where "pure" is an attractive reading), but also for the
> temperament as a whole -- _participatione & acordo giusto &
buono_, "a
> just and good temperament and tuning." However, Lindley himself is
> ready to accept Aaron's opening C-G-D-A-E with its "sonorous and
just"
> major third C-E as a description of 1/4-comma. His statement about
the
> octave C-C and major third C-E being as sonorous or "united" as
> possible would support this conclusion even we assume that "just"
may
> mean simply "euphonious" or "pleasing."
>
> In his instructions for the final stage of the temperament, the
tuning
> of the sharps, Aaron directs that C#, tuned in relation to the fifth
> A-E, should be a major third from A and a minor third from E, and
> likewise with F# in relation to the fifth D-A, etc.
>
> From one viewpoint, this language may simply be reminding the
student
> of the locations of major and minor thirds involving accidentals.
> Lindley, however, pursuing his argument that a regular 1/4-comma
> temperament is not _necessarily_ implied, argues that one _could_
read
> this language to suggest something like Zarlino's 2/7-comma meantone
> for the temperament of the sharps, with major and minor thirds
> compromised by about the same amount from pure.
>
> In arguing that Aaron's tuning is not _necessarily_ a regular
> 1/4-comma meantone, Lindley may have two motivations. First, he
wishes
> to emphasize that Aaron's instructions are not mathematically
precise;
> and indeed, I would agree that Zarlino (1571) and Salinas (1577) may
> be the first known theorists to give such mathematical definitions
of
> 1/4-comma and other meantone temperaments.
>
> Secondly, Lindley wants to correct the view that 1/4-comma was a
> universal standard in the 16th century. Since Aaron's instructions
are
> often taken as the paradigm case of this tuning, showing that they
are
> actually open to more than one interpretation would fit with his
> larger campaign against "1/4-comma hegemony."
>
> However, I find it noteworthy that without invoking any complex
> mathematical concepts or even defining a syntonic comma, Aaron has
> described in what I find beautiful as well as musicianly terms the
> idea of tuning a pure major third and then dividing it into four
> equally tempered fifths. Aaron's remaining instructions, including
his
> suggestion of a similar but opposite tempering of fifths in the flat
> direction (F-C, and then Bb-F and Eb-Bb), permit a regular 1/4-comma
> temperament, even if they do not compel it or define it in
> mathematical terms.
>
> If I were making a table like Paul's, I might list Aaron for 1/4-
comma
> and add a footnote or annotation like this:
>
> "Aaron evidently describes a temperament with a pure major third C-E
> and equally narrowed fifths for his first five notes C-G-D-A-E, with
> further instructions permitting but not explicitly specifying that
> other major thirds are pure; Zarlino (1571) and Salinas (1577) give
> mathematically precise descriptions of a regular 1/4-comma tuning."'
>
>
> monz might want to revise some statements on his 1/4-comma meantone
> page after rereading the above.

🔗monz <monz@attglobal.net>

hi paul,

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning@yahoogroups.com>
> Sent: Friday, March 14, 2003 5:18 AM
> Subject: [tuning] Re: Zarlino's 2/7 (was: A question and an update...)
>
>
> --- In tuning@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > thanks, paul. the most expedient way to do it
> > was to simply add the quote from Margo at the
> > end of the page, which i've done.
> >
> > http://sonic-arts.org/dict/1-4cmt.htm
>
> > -monz
>
> the beginning of the page, unfortunately, contains some misleading
> statements (if you'll allow the criticism) . . .
>
> 'the paradigms postulated by "common-practice" music-theory to a
> great extent depend on . . . the fact that "flats" are higher in
> pitch than "sharps"'
>
> not true, since the 'expressive intonation' paradigm introduced
> c.1800 fits just as snugly with common-practice paradigms even though
> it has the opposite tendency
>
> '1/4-comma meantone was assumed as the standard tuning in most of
> Europe from approximately 1500 to 1700'
>
> this is the statement i hoped margo's post would lead you to weaken --
> i certainly see no reason to "assume" that all of the other meantone
> tunings described by salinas and rossi were "non-standard" during
> this time.

how's this? (i've change the beginning of the webpage) :

>> 1/4-comma meantone is historically very important
>> to the development of Western music, as the paradigms
>> postulated by "common-practice" music-theory to a great
>> extent depend on the elimination or tempering-out of the
>> syntonic comma, which is probably the most prominent
>> feature of all meantones.
>>
>> Note that while in the familiar 12edo (which also belongs
>> to the meantone family) there is a complete set of
>> enharmonic equivalents, such that all 7 notes which have
>> "flats" can also be "spelled" as 7 different notes which
>> have "sharps", in all other meantones the "flats" are
>> higher in pitch than supposedly enharmonically-equivalent
>> "sharps", which is the opposite of the case in the much
>> older Pythagorean tuning, and also the opposite of the
>> case in the "expressive intonation" which has been widely
>> taught in Eurocentric schools of playing since Mozart's
>> time (late 1700s).
>>
>> Based on my research, I would say that 1/4-comma meantone
>> was the closest thing to a "standard" tuning in most of
>> Europe from approximately 1500 to 1700, and was still
>> commonly found on keyboards (especially organs) until
>> about 1850. It would probably be fair to say that most
>> instrumental music of the Renaissance and Baroque periods
>> was intended to be played in 1/4-comma meantone or a
>> close relative, and even after the growth in popularity
>> of well-temperaments for keyboards after 1700, some form
>> of meantone (generally more like 1/6-comma or 55edo) was
>> usually still intended for orchestral music.

> <snip>
>
> ... anything from 2/7-comma to 3/14-
> comma meantone (and beyond) can be justified by plausible optimality
> criteria, and would probably not have sounded out-of-place for the
> body of western music from 1500 to 1700.

i've added some commentary here:

http://sonic-arts.org/dict/meantone-error/meantone-error.htm

which explains some of this.

-monz