pan-proportionally beating 12-tone temperament

> > http://lumma.org/tuning/A.wav
> > http://lumma.org/tuning/B.wav
> > http://lumma.org/tuning/B2.wav
> > http://lumma.org/tuning/B3.wav

I haven't seen all the discussion on this, and excuse me if this is a silly question (or a repeated question):

In all of these proportionally-beating schemes (which are undeniably *interesting* to listen to), isn't the whole thing predicated on the notion that the attacks of the notes will be *exactly* together? Even if one had (say) a layout where the major 3rd has (say) 2 beats per second, and the 5th has (say) 4 beats per second, if we hit the middle note (or the top note) very slightly late or early, all this "proportional beating" goes out of phase and becomes inconsequential, or develops a "swing" to it. Nicht wahr?

And so, what does any of this have to do with garden-variety music played on keyboards, where even the most scrupulously precise-intending keyboardists might still be hitting the notes slightly separated...especially on non-electronic instruments? Is any of this more than a mental exercise, within a premise that only delivers the desired effects in computer-controlled media?

And as somebody else asked: how do we discern what's "better" or "worse" within this premise, even under such controlled conditions?

From what I've seen in the classic literature so far, "proportional beating" was mainly one of the made-up selling points of Herbert Kellner's temperament, justifying the coincidence that some component of his B major triad beats 6 times as fast as something else; so what? How is that of any use whatsoever in practical music, other than for a tuner-by-ear to get Kellner's own temperament installed precisely, according to a premise that looked important to Kellner? And then it was latched onto by Jorgensen, and promoted even farther beyond its utility, and now here we are.

(And I recall asking rather similar questions, sometime here during the summer of 2004.....)

Brad Lehman

--- In tuning@yahoogroups.com, Brad Lehman <bpl@u...> wrote:
>
> I haven't seen all the discussion on this, and excuse me if this
> is a silly question (or a repeated question):
>
> In all of these proportionally-beating schemes (which are
> undeniably *interesting* to listen to), isn't the whole thing
> predicated on the notion that the attacks of the notes will be
> *exactly* together? Even if one had (say) a layout where the
> major 3rd has(say) 2 beats per second, and the 5th has (say) 4
> beats per second, if we hit the middle note (or the top note) very
> slightly late or early, all this "proportional beating" goes out
> of phase and becomes inconsequential, or develops a "swing" to
> it. Nicht wahr?

That's right, but with diligent practice bad habits can be corrected.

Regards
Charles

> > > http://lumma.org/tuning/A.wav
> > > http://lumma.org/tuning/B.wav
> > > http://lumma.org/tuning/B2.wav
> > > http://lumma.org/tuning/B3.wav
>
> I haven't seen all the discussion on this, and excuse me if
> this is a silly question (or a repeated question):
>
> In all of these proportionally-beating schemes (which are
> undeniably *interesting* to listen to), isn't the whole thing
> predicated on the notion that the attacks of the notes will
> be *exactly* together?

This is the "phase" question we've been discussing. The
jury is still out...

The whole thing definitely is predicated on the relative
loudness of the notes, and on the relative amplitude of the
partials in the timbre. But I think a useful theory can be
had by averaging over these two parameters.

> And as somebody else asked: how do we discern what's "better"
> or "worse" within this premise, even under such controlled
> conditions?

I originally asked for general comments, if anyone
could identify the 1:1:1 triad, and if anyone thought
one sounded better than the other. Seems like a fair
question, and almost everyone agreed that B.wav sounds
better than A.wav (more triads have since been added).

> From what I've seen in the classic literature so
> far, "proportional beating" was mainly one of the
> made-up selling points of Herbert Kellner's temperament,
> justifying the coincidence that some component of his
> B major triad beats 6 times as fast as something else; so
> what?

That's something else altogether.

> How is that of any use whatsoever in practical music, other
> than for a tuner-by-ear to get Kellner's own temperament
> installed precisely, according to a premise that looked
> important to Kellner? And then it was latched onto by
> Jorgensen, and promoted even farther beyond its utility,
> and now here we are.

Lots of bearing plans are given this way (it certainly didn't
start with Kellner) and it's a good way to tune.

But beat ratios and equal-beating are different things. We're
currently trying to figure out if the former mean anything.
The latter, I dunno... here's one gloss about how it could go:

() Aside from its total mistuning from JI, the percieved beat
rate of an interval or chord is a 2ndary perceptual property
of it.

() If equal mistuning is used across an instrument, equal
beat rates will not be achieved, and vice versa.

() But since equal beat rates are much more volitile with
respect to tuning than is percieved mistuning error, equal
beat rates can be usually achieved without changing the
overall character of a tuning.

() If equal temperament is the goal, a slightly unequal
version with equal beat rates might sound more equal.

() If unequal temperament is the goal, but one cannot
tolerate too much error, beat rates could be purposely
varied over the scale.

-Carl

These are very interesting speculations that require deeper investigation. But I was assuming that phase shift is an altogether different attribute next to beat rate. It is important to check whether the phase shift of proportional beating triads has any noticable effect on consonance.

----- Original Message -----
From: Carl Lumma
To: tuning@yahoogroups.com
Sent: 04 Kasım 2005 Cuma 0:07
Subject: [tuning] Re: pan-proportionally beating 12-tone temperament

SNIP!

here's one gloss about how it could go:

() Aside from its total mistuning from JI, the percieved beat
rate of an interval or chord is a 2ndary perceptual property
of it.

() If equal mistuning is used across an instrument, equal
beat rates will not be achieved, and vice versa.

() But since equal beat rates are much more volitile with
respect to tuning than is percieved mistuning error, equal
beat rates can be usually achieved without changing the
overall character of a tuning.

() If equal temperament is the goal, a slightly unequal
version with equal beat rates might sound more equal.

() If unequal temperament is the goal, but one cannot
tolerate too much error, beat rates could be purposely
varied over the scale.

-Carl

--- In tuning@yahoogroups.com, Brad Lehman <bpl@u...> wrote:
>
> > > http://lumma.org/tuning/A.wav
> > > http://lumma.org/tuning/B.wav
> > > http://lumma.org/tuning/B2.wav
> > > http://lumma.org/tuning/B3.wav
>
> I haven't seen all the discussion on this, and excuse me if this is
a
> silly question (or a repeated question):
>
> In all of these proportionally-beating schemes (which are
undeniably
> *interesting* to listen to), isn't the whole thing predicated on
the
> notion that the attacks of the notes will be *exactly*
> together? Even if one had (say) a layout where the major 3rd has
> (say) 2 beats per second, and the 5th has (say) 4 beats per second,
> if we hit the middle note (or the top note) very slightly late or
> early, all this "proportional beating" goes out of phase and
becomes
> inconsequential, or develops a "swing" to it. Nicht wahr?

Why would one necessarily conclude that beats out of phase with one
another make the equal beating inconsequential? What's wrong
with "swing"?

> And as somebody else asked: how do we discern what's "better" or
> "worse" within this premise, even under such controlled conditions?
>
> From what I've seen in the classic literature so
far, "proportional
> beating" was mainly one of the made-up selling points of Herbert
> Kellner's temperament,

Proportional beating has come up in far wider contexts that the mere
consideration of Bach tunings by Kellner!

> justifying the coincidence that some component
> of his B major triad beats 6 times as fast as something else; so
> what? How is that of any use whatsoever in practical music,

If it sounds good, it's of use -- only listening can really determine
if it sounds good, and only to that particular listener.

>And then it was latched onto by Jorgensen, and promoted
> even farther beyond its utility, and now here we are.

You are mistaken. Jorgenson talks about tunings with equal beats
among different instances of the same interval; here we are talking
about equal beats among *different* intervals within a single chord.
Apples and oranges.

I think the theoretical work of George Secor and Erv Wilson, to name
two, has had much more to do with the interest in this question here
than Jorgenson or Kellner.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > > http://lumma.org/tuning/A.wav
> > > > http://lumma.org/tuning/B.wav
> > > > http://lumma.org/tuning/B2.wav
> > > > http://lumma.org/tuning/B3.wav
> >
> > I haven't seen all the discussion on this, and excuse me if
> > this is a silly question (or a repeated question):
> >
> > In all of these proportionally-beating schemes (which are
> > undeniably *interesting* to listen to), isn't the whole thing
> > predicated on the notion that the attacks of the notes will
> > be *exactly* together?
>
> This is the "phase" question we've been discussing. The
> jury is still out...
>
> The whole thing definitely is predicated on the relative
> loudness of the notes, and on the relative amplitude of the
> partials in the timbre. But I think a useful theory can be
> had by averaging over these two parameters.

Huh? Multiplying them gives you the amplitudes of the potentially
beating partials; when they're the same, the corresponding beating
can go all the way to zero amplitude each cycle.

> () If equal temperament is the goal, a slightly unequal
> version with equal beat rates might sound more equal.

You don't really mean "more equal", just "better" here, right?

> > > > > http://lumma.org/tuning/A.wav
> > > > > http://lumma.org/tuning/B.wav
> > > > > http://lumma.org/tuning/B2.wav
> > > > > http://lumma.org/tuning/B3.wav
> > >
> > > I haven't seen all the discussion on this, and excuse me if
> > > this is a silly question (or a repeated question):
> > >
> > > In all of these proportionally-beating schemes (which are
> > > undeniably *interesting* to listen to), isn't the whole thing
> > > predicated on the notion that the attacks of the notes will
> > > be *exactly* together?
> >
> > This is the "phase" question we've been discussing. The
> > jury is still out...
> >
> > The whole thing definitely is predicated on the relative
> > loudness of the notes, and on the relative amplitude of the
> > partials in the timbre. But I think a useful theory can be
> > had by averaging over these two parameters.
>
> Huh? Multiplying them gives you the amplitudes of the potentially
> beating partials; when they're the same, the corresponding beating
> can go all the way to zero amplitude each cycle.

Multiplying what? Are you saying that making all the
amplitudes the same is not a good place to observe from?
What place would be better?

> > () If equal temperament is the goal, a slightly unequal
> > version with equal beat rates might sound more equal.
>
> You don't really mean "more equal", just "better" here, right?

I mean more equal (since beat rates are a quality of the
chord) and therefore better (since that was the "goal" in
this case.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > > > > http://lumma.org/tuning/A.wav
> > > > > > http://lumma.org/tuning/B.wav
> > > > > > http://lumma.org/tuning/B2.wav
> > > > > > http://lumma.org/tuning/B3.wav
> > > >
> > > > I haven't seen all the discussion on this, and excuse me if
> > > > this is a silly question (or a repeated question):
> > > >
> > > > In all of these proportionally-beating schemes (which are
> > > > undeniably *interesting* to listen to), isn't the whole thing
> > > > predicated on the notion that the attacks of the notes will
> > > > be *exactly* together?
> > >
> > > This is the "phase" question we've been discussing. The
> > > jury is still out...
> > >
> > > The whole thing definitely is predicated on the relative
> > > loudness of the notes, and on the relative amplitude of the
> > > partials in the timbre. But I think a useful theory can be
> > > had by averaging over these two parameters.
> >
> > Huh? Multiplying them gives you the amplitudes of the potentially
> > beating partials; when they're the same, the corresponding
beating
> > can go all the way to zero amplitude each cycle.
>
> Multiplying what?

The relative amplitude of the notes times the amplitude of the
partial in question relative to the amplitude of the fundamental.

> Are you saying that making all the
> amplitudes the same is not a good place to observe from?
> What place would be better?

Where the partials which beat are of equal amplitude. If all the
notes have the same relative amplitudes of their partials relative to
the fundamental (and assuming the partials aren't of equal amplitude
in each note), you'd need different amplitudes for the notes
themselves to make this happen.

> > > () If equal temperament is the goal, a slightly unequal
> > > version with equal beat rates might sound more equal.
> >
> > You don't really mean "more equal", just "better" here, right?
>
> I mean more equal (since beat rates are a quality of the
> chord)

I don't get it. What's "the chord" here?

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

> But beat ratios and equal-beating are different things. We're
> currently trying to figure out if the former mean anything.

No we aren't. Obviously, it means something, the question is exactly what?

> () But since equal beat rates are much more volitile with
> respect to tuning than is percieved mistuning error, equal
> beat rates can be usually achieved without changing the
> overall character of a tuning.

One cannot pick just any "good" brat, however.

> () If equal temperament is the goal, a slightly unequal
> version with equal beat rates might sound more equal.

Where does this come from?

--- In tuning@yahoogroups.com, Brad Lehman <bpl@u...> wrote:
> I haven't seen all the discussion on this, and excuse me if this
is a
> silly question (or a repeated question):

No. A good question.

> In all of these proportionally-beating schemes (which are
undeniably
> *interesting* to listen to), isn't the whole thing predicated on
the
> notion that the attacks of the notes will be *exactly*
> together?

Not necessarily. See below.

> Even if one had (say) a layout where the major 3rd has
> (say) 2 beats per second, and the 5th has (say) 4 beats per
second,
> if we hit the middle note (or the top note) very slightly late or
> early, all this "proportional beating" goes out of phase and
becomes
> inconsequential, or develops a "swing" to it. Nicht wahr?
>
> And so, what does any of this have to do with garden-variety music
> played on keyboards, where even the most scrupulously
> precise-intending keyboardists might still be hitting the notes
> slightly separated...especially on non-electronic instruments? Is
> any of this more than a mental exercise, within a premise that
only
> delivers the desired effects in computer-controlled media?

Good point. It probably has nothing to do with music produced on
garden variety instruments. One could however have an electronic
instrument where all the oscillators were always running and so
always in the same phase relationship, and were just gated to the
output as needed, and so relative timing of attacks would be
irrelevant. But it doesn't seem worth the trouble, to me.

-- Dave Keenan

> > Are you saying that making all the
> > amplitudes the same is not a good place to observe from?
> > What place would be better?
>
> Where the partials which beat are of equal amplitude. If all
> the notes have the same relative amplitudes of their partials
> relative to the fundamental (and assuming the partials aren't
> of equal amplitude in each note), you'd need different
> amplitudes for the notes themselves to make this happen.

I understand, and this is doable in Cool Edit. It will
take a lot more time to generate the demos this way, though.
:(

> > > > () If equal temperament is the goal, a slightly unequal
> > > > version with equal beat rates might sound more equal.
> > >
> > > You don't really mean "more equal", just "better" here,
> > > right?
> >
> > I mean more equal (since beat rates are a quality of the
> > chord)
>
> I don't get it. What's "the chord" here?

Any chord that's used to judge a temperament, to put it
crudely.
If beat rates are an independent quality, and the goal
is to make all these chords sound the same, then clearly
equal beating is something to consider, especially if
it can be had without making the mistunings of these
chords (the other quality defined) very unequal.

-Carl

> > () But since equal beat rates are much more volitile with
> > respect to tuning than is percieved mistuning error, equal
> > beat rates can be usually achieved without changing the
> > overall character of a tuning.
>
> One cannot pick just any "good" brat, however.

Here I was referring to equal-beating, not brats.

> > () If equal temperament is the goal, a slightly unequal
> > version with equal beat rates might sound more equal.
>
> Where does this come from?

For example, the notion that this scale (which is probably
pretty close to what Norman Henry has been tuning on his
customers' pianos for 50 years) sounds more equal than
equal temperament:

!
Equal-beating version of 12-tET.
12
!
100.03402
199.51879
299.79965
399.51612
500.01742
599.94076
699.32161
799.50359
899.12725
999.54025
1099.38074
2/1

-Carl

> > But beat ratios and equal-beating are different things. We're
> > currently trying to figure out if the former mean anything.
>
> No we aren't. Obviously, it means something, the question is
> exactly what?

And the answer is: almost certainly very little.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > () But since equal beat rates are much more volitile with
> > > respect to tuning than is percieved mistuning error, equal
> > > beat rates can be usually achieved without changing the
> > > overall character of a tuning.
> >
> > One cannot pick just any "good" brat, however.
>
> Here I was referring to equal-beating, not brats.
>
> > > () If equal temperament is the goal, a slightly unequal
> > > version with equal beat rates might sound more equal.
> >
> > Where does this come from?
>
> For example, the notion that this scale (which is probably
> pretty close to what Norman Henry has been tuning on his
> customers' pianos for 50 years) sounds more equal than
> equal temperament:
>
> !
> Equal-beating version of 12-tET.
> 12
> !
> 100.03402
> 199.51879
> 299.79965
> 399.51612
> 500.01742
> 599.94076
> 699.32161
> 799.50359
> 899.12725
> 999.54025
> 1099.38074
> 2/1
>
> -Carl

I'd love to see an analysis that shows how this is equal-beating.

> > For example, the notion that this scale (which is probably
> > pretty close to what Norman Henry has been tuning on his
> > customers' pianos for 50 years) sounds more equal than
> > equal temperament:
> >
> > !
> > Equal-beating version of 12-tET.
> > 12
> > !
> > 100.03402
> > 199.51879
> > 299.79965
> > 399.51612
> > 500.01742
> > 599.94076
> > 699.32161
> > 799.50359
> > 899.12725
> > 999.54025
> > 1099.38074
> > 2/1
> >
> > -Carl
>
> I'd love to see an analysis that shows how this is equal-beating.

Get ready to love...

Beat frequencies of 3/2
0: 0.000: -1.1930
1: 100.034: -1.1930
2: 199.519: -1.1930
3: 299.800: -1.1930
4: 399.516: -1.1930
5: 500.017: -1.1930
6: 599.941: -1.1930
7: 699.322: -1.1930
8: 799.504: -1.1930
9: 899.127: -1.1930
10: 999.540: -1.1930
11: 1099.381: -1.1930
12: 1200.000: -2.3859

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > But beat ratios and equal-beating are different things. We're
> > > currently trying to figure out if the former mean anything.
> >
> > No we aren't. Obviously, it means something, the question is
> > exactly what?
>
> And the answer is: almost certainly very little.

It's clear from your own examples that it means the chords sound
different; they have a regular, pulsing rythem.

> > > > But beat ratios and equal-beating are different things. We're
> > > > currently trying to figure out if the former mean anything.
> > >
> > > No we aren't. Obviously, it means something, the question is
> > > exactly what?
> >
> > And the answer is: almost certainly very little.
>
> It's clear from your own examples that it means the chords sound
> different; they have a regular, pulsing rythem.

They all seemed to have regular rhythms.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > > > But beat ratios and equal-beating are different things. We're
> > > > > currently trying to figure out if the former mean anything.
> > > >
> > > > No we aren't. Obviously, it means something, the question is
> > > > exactly what?
> > >
> > > And the answer is: almost certainly very little.
> >
> > It's clear from your own examples that it means the chords sound
> > different; they have a regular, pulsing rythem.
>
> They all seemed to have regular rhythms.

Not to the same extent. Compare A.wav to B2.wav. A.wav goes
WaWaWaWaWaWaWa, and B2.wav goes WaaaaaaWhWhWhWaaaaaaWhWhWhWaaaaaa,
but in fact the sound is more complex than I can transcribe it, with
more stuff going on.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > For example, the notion that this scale (which is probably
> > > pretty close to what Norman Henry has been tuning on his
> > > customers' pianos for 50 years) sounds more equal than
> > > equal temperament:
> > >
> > > !
> > > Equal-beating version of 12-tET.
> > > 12
> > > !
> > > 100.03402
> > > 199.51879
> > > 299.79965
> > > 399.51612
> > > 500.01742
> > > 599.94076
> > > 699.32161
> > > 799.50359
> > > 899.12725
> > > 999.54025
> > > 1099.38074
> > > 2/1
> > >
> > > -Carl
> >
> > I'd love to see an analysis that shows how this is equal-beating.
>
> Get ready to love...
>
> Beat frequencies of 3/2
> 0: 0.000: -1.1930
> 1: 100.034: -1.1930
> 2: 199.519: -1.1930
> 3: 299.800: -1.1930
> 4: 399.516: -1.1930
> 5: 500.017: -1.1930
> 6: 599.941: -1.1930
> 7: 699.322: -1.1930
> 8: 799.504: -1.1930
> 9: 899.127: -1.1930
> 10: 999.540: -1.1930
> 11: 1099.381: -1.1930
> 12: 1200.000: -2.3859

So it's "equal minus one." :) How do you decide which note to start
this on?

> > Beat frequencies of 3/2
> > 0: 0.000: -1.1930
> > 1: 100.034: -1.1930
> > 2: 199.519: -1.1930
> > 3: 299.800: -1.1930
> > 4: 399.516: -1.1930
> > 5: 500.017: -1.1930
> > 6: 599.941: -1.1930
> > 7: 699.322: -1.1930
> > 8: 799.504: -1.1930
> > 9: 899.127: -1.1930
> > 10: 999.540: -1.1930
> > 11: 1099.381: -1.1930
> > 12: 1200.000: -2.3859
>
> So it's "equal minus one." :)

Do you mean -2.3859? The beat rates double every
octave... I think that's pretty standard for these
scales.

> How do you decide which note to start this on?

Not sure what the question is...

-Carl

On Tuesday 08 November 2005 12:26 pm, wallyesterpaulrus wrote:
> --- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
> > > > For example, the notion that this scale (which is probably
> > > > pretty close to what Norman Henry has been tuning on his
> > > > customers' pianos for 50 years) sounds more equal than
> > > > equal temperament:
> > > >
> > > > !
> > > > Equal-beating version of 12-tET.
> > > > 12
> > > > !
> > > > 100.03402
> > > > 199.51879
> > > > 299.79965
> > > > 399.51612
> > > > 500.01742
> > > > 599.94076
> > > > 699.32161
> > > > 799.50359
> > > > 899.12725
> > > > 999.54025
> > > > 1099.38074
> > > > 2/1
> > > >
> > > > -Carl
> > >
> > > I'd love to see an analysis that shows how this is equal-beating.
> >
> > Get ready to love...
> >
> > Beat frequencies of 3/2
> > 0: 0.000: -1.1930
> > 1: 100.034: -1.1930
> > 2: 199.519: -1.1930
> > 3: 299.800: -1.1930
> > 4: 399.516: -1.1930
> > 5: 500.017: -1.1930
> > 6: 599.941: -1.1930
> > 7: 699.322: -1.1930
> > 8: 799.504: -1.1930
> > 9: 899.127: -1.1930
> > 10: 999.540: -1.1930
> > 11: 1099.381: -1.1930
> > 12: 1200.000: -2.3859
>
> So it's "equal minus one." :) How do you decide which note to start
> this on?

No. The next new octave naturally will beat at twice the rate....

-Aaron.

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > Beat frequencies of 3/2
> > > 0: 0.000: -1.1930
> > > 1: 100.034: -1.1930
> > > 2: 199.519: -1.1930
> > > 3: 299.800: -1.1930
> > > 4: 399.516: -1.1930
> > > 5: 500.017: -1.1930
> > > 6: 599.941: -1.1930
> > > 7: 699.322: -1.1930
> > > 8: 799.504: -1.1930
> > > 9: 899.127: -1.1930
> > > 10: 999.540: -1.1930
> > > 11: 1099.381: -1.1930
> > > 12: 1200.000: -2.3859
> >
> > So it's "equal minus one." :)
>
> Do you mean -2.3859? The beat rates double every
> octave... I think that's pretty standard for these
> scales.

For all scales. But here, there's a "discontinuity" at a single
semitone, and I was just asking you which one below:

> > How do you decide which note to start this on?
>
> Not sure what the question is...

You could start on A, you could start on C, (...)

> > > > Beat frequencies of 3/2
> > > > 0: 0.000: -1.1930
> > > > 1: 100.034: -1.1930
> > > > 2: 199.519: -1.1930
> > > > 3: 299.800: -1.1930
> > > > 4: 399.516: -1.1930
> > > > 5: 500.017: -1.1930
> > > > 6: 599.941: -1.1930
> > > > 7: 699.322: -1.1930
> > > > 8: 799.504: -1.1930
> > > > 9: 899.127: -1.1930
> > > > 10: 999.540: -1.1930
> > > > 11: 1099.381: -1.1930
> > > > 12: 1200.000: -2.3859
> > >
> > > So it's "equal minus one." :)
> >
> > Do you mean -2.3859? The beat rates double every
> > octave... I think that's pretty standard for these
> > scales.
>
> For all scales. But here, there's a "discontinuity" at a
> single semitone,

I don't see it.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > > > Beat frequencies of 3/2
> > > > > 0: 0.000: -1.1930
> > > > > 1: 100.034: -1.1930
> > > > > 2: 199.519: -1.1930
> > > > > 3: 299.800: -1.1930
> > > > > 4: 399.516: -1.1930
> > > > > 5: 500.017: -1.1930
> > > > > 6: 599.941: -1.1930
> > > > > 7: 699.322: -1.1930
> > > > > 8: 799.504: -1.1930
> > > > > 9: 899.127: -1.1930
> > > > > 10: 999.540: -1.1930
> > > > > 11: 1099.381: -1.1930
> > > > > 12: 1200.000: -2.3859
> > > >
> > > > So it's "equal minus one." :)
> > >
> > > Do you mean -2.3859? The beat rates double every
> > > octave... I think that's pretty standard for these
> > > scales.
> >
> > For all scales. But here, there's a "discontinuity" at a
> > single semitone,
>
> I don't see it.

Play the perfect fifths up the keyboard, ascending one semitone at a
time. Most of the time the beat rate will stay the same. But 1 time
out of every 12, it will double. This is a discontinuity, and in the
part of my post which I think you snipped, I asked where you like to
put this discontinuity.

> > > > > > Beat frequencies of 3/2
> > > > > > 0: 0.000: -1.1930
> > > > > > 1: 100.034: -1.1930
> > > > > > 2: 199.519: -1.1930
> > > > > > 3: 299.800: -1.1930
> > > > > > 4: 399.516: -1.1930
> > > > > > 5: 500.017: -1.1930
> > > > > > 6: 599.941: -1.1930
> > > > > > 7: 699.322: -1.1930
> > > > > > 8: 799.504: -1.1930
> > > > > > 9: 899.127: -1.1930
> > > > > > 10: 999.540: -1.1930
> > > > > > 11: 1099.381: -1.1930
> > > > > > 12: 1200.000: -2.3859
> > > > >
> > > > > So it's "equal minus one." :)
> > > >
> > > > Do you mean -2.3859? The beat rates double every
> > > > octave... I think that's pretty standard for these
> > > > scales.
> > >
> > > For all scales. But here, there's a "discontinuity" at
> > > a single semitone,
> >
> > I don't see it.
>
> Play the perfect fifths up the keyboard, ascending one semitone
> at a time. Most of the time the beat rate will stay the same.
> But 1 time out of every 12, it will double. This is a
> discontinuity, and in the part of my post which I think you
> snipped, I asked where you like to put this discontinuity.

My piano's severely out of tune at the moment, and Norman just
moved to Denver (heads up, Neil!). But it looks to me like I
can play fifths degrees 0 - 11 (12 in all) without experiencing
a change in the beat rate.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <clumma@y...> wrote:
>
> > > > > > > Beat frequencies of 3/2
> > > > > > > 0: 0.000: -1.1930
> > > > > > > 1: 100.034: -1.1930
> > > > > > > 2: 199.519: -1.1930
> > > > > > > 3: 299.800: -1.1930
> > > > > > > 4: 399.516: -1.1930
> > > > > > > 5: 500.017: -1.1930
> > > > > > > 6: 599.941: -1.1930
> > > > > > > 7: 699.322: -1.1930
> > > > > > > 8: 799.504: -1.1930
> > > > > > > 9: 899.127: -1.1930
> > > > > > > 10: 999.540: -1.1930
> > > > > > > 11: 1099.381: -1.1930
> > > > > > > 12: 1200.000: -2.3859
> > > > > >
> > > > > > So it's "equal minus one." :)
> > > > >
> > > > > Do you mean -2.3859? The beat rates double every
> > > > > octave... I think that's pretty standard for these
> > > > > scales.
> > > >
> > > > For all scales. But here, there's a "discontinuity" at
> > > > a single semitone,
> > >
> > > I don't see it.
> >
> > Play the perfect fifths up the keyboard, ascending one semitone
> > at a time. Most of the time the beat rate will stay the same.
> > But 1 time out of every 12, it will double. This is a
> > discontinuity, and in the part of my post which I think you
> > snipped, I asked where you like to put this discontinuity.
>
> My piano's severely out of tune at the moment, and Norman just
> moved to Denver (heads up, Neil!). But it looks to me like I
> can play fifths degrees 0 - 11 (12 in all) without experiencing
> a change in the beat rate.

I meant *all the way up the keyboard*. How about the 13th fifth?

> > > > > > > > Beat frequencies of 3/2
> > > > > > > > 0: 0.000: -1.1930
> > > > > > > > 1: 100.034: -1.1930
> > > > > > > > 2: 199.519: -1.1930
> > > > > > > > 3: 299.800: -1.1930
> > > > > > > > 4: 399.516: -1.1930
> > > > > > > > 5: 500.017: -1.1930
> > > > > > > > 6: 599.941: -1.1930
> > > > > > > > 7: 699.322: -1.1930
> > > > > > > > 8: 799.504: -1.1930
> > > > > > > > 9: 899.127: -1.1930
> > > > > > > > 10: 999.540: -1.1930
> > > > > > > > 11: 1099.381: -1.1930
> > > > > > > > 12: 1200.000: -2.3859
> > > > > > >
> > > > > > > So it's "equal minus one." :)
> > > > > >
> > > > > > Do you mean -2.3859? The beat rates double every
> > > > > > octave... I think that's pretty standard for these
> > > > > > scales.
> > > > >
> > > > > For all scales. But here, there's a "discontinuity" at
> > > > > a single semitone,
> > > >
> > > > I don't see it.
> > >
> > > Play the perfect fifths up the keyboard, ascending one semitone
> > > at a time. Most of the time the beat rate will stay the same.
> > > But 1 time out of every 12, it will double. This is a
> > > discontinuity, and in the part of my post which I think you
> > > snipped, I asked where you like to put this discontinuity.
> >
> > My piano's severely out of tune at the moment, and Norman just
> > moved to Denver (heads up, Neil!). But it looks to me like I
> > can play fifths degrees 0 - 11 (12 in all) without experiencing
> > a change in the beat rate.
>
> I meant *all the way up the keyboard*. How about the 13th fifth?

I suppose I'd but it between B and C.

-Carl