Yahoo Tuning Groups Ultimate Backup tuning Poor Man's Power Ranking of Intervals

Thanks Vaisvil,

I am very curious to know how the remaining important pure intervals (ratios
between integers smaller than 10) in the octave fare using your values.
Missing from your list are the following:

7:6 Small Minor Third
6:5 Minor Third
7:5 Tritone
8:5 Minor Sixth
7:4 Harmonic Minor Seventh

Thanks!

Claudio

_____

From: tuning@yahoogroups.com [mailto:tuning@yahoogroups.com] On Behalf Of
vaisvil
Sent: 22 January 2009 00:01
To: tuning@yahoogroups.com
Subject: [tuning] Poor Man's Power Ranking of Intervals

Ok,

Using excel I summed dyads representing the JI ratios of a major scale.
In this scenario a larger number is more consonant.

3/2 = 10.60823809
4/3 = 9.146074713
9/8 = 2.229477297
5/4 = 6.590050018
2/1 = -0.002267916 (an anomaly)
15/8 = 1.081650721
4/3 = 9.146074713
5/3 = 6.964375836

I'm thinking this would follow most people's idea of dyadic tension.

I got to go to an appointment - I'll try triads later.

I used 2 cycles 0.01 to 1 in .01 steps as the generator.

0.01 =(SIN(2*BJ1*3.14)) =(SIN(2*(5/3)*BJ1*3.14)) 0.01 =(BK1+BL1)/2

Chris

🔗djtrancendance@...

A side-note: I have found using two integer ratios going all the way up to about 12 IE 12/11 or 12/5 sound very consonant to my ears...and ones with factors up to about 16 don't sound particularly dissonant either. Over about 16, though (IE 16/15 or 16/13 etc.), there seems to be a distinct problem with too much beating, at least to my ears.

So to make a consonant scale you would think trying to match ALL ratios between all combinations of different notes in general to be summarized into fractions with factors of about 16 or less and, if you "have" to miss the exact ratios for some intervals, try to miss them by about 3-4 cents or less.
_________________________________

Back to the subject; I seriously doubt the historical "myth" that using the factor 5 as a limit (IE 5/4,5/3,4/3,4/2,3/2) is the edge of "consonance", nor is 10....and more like 15 or even 20 depending on your personal tolerance. I am sure it would hold somewhat true with pure sine-wave, but not for complex instruments where at least one harmonic/overtone which is NOT beating can take over as the "good lead" for the ones which are.

--- On Wed, 1/21/09, Claudio Di Veroli <dvc@braybaroque.ie> wrote:

From: Claudio Di Veroli
<dvc@...>
Subject: RE: [tuning] Poor Man's Power Ranking of Intervals
To: tuning@yahoogroups.com
Date: Wednesday, January 21, 2009, 4:11 PM

Thanks Vaisvil,

I am very curious to know how the remaining important
pure intervals (ratios between integers smaller than 10) in the
octave fare using your values. Missing from your
list are the following:

7:6 Small Minor Third
6:5 Minor Third
7:5 Tritone
8:5 Minor Sixth
7:4 Harmonic Minor Seventh

Thanks!

Claudio

From: tuning@yahoogroups. com
[mailto:tuning@ yahoogroups. com] On Behalf Of vaisvil
Sent: 22
January 2009 00:01
To: tuning@yahoogroups. com
Subject:
[tuning] Poor Man's Power Ranking of Intervals

Ok,

Using excel I summed dyads representing the JI ratios of a major
scale.
In this scenario a larger number is more consonant.

3/2 =
10.60823809
4/3 = 9.146074713
9/8 = 2.229477297
5/4 =
6.590050018
2/1 = -0.002267916 (an anomaly)
15/8 = 1.081650721
4/3 =
9.146074713
5/3 = 6.964375836

I'm thinking this would follow most
people's idea of dyadic tension.

I got to go to an appointment - I'll
try triads later.

I used 2 cycles 0.01 to 1 in .01 steps as the
generator.

0.01 =(SIN(2*BJ1* 3.14))
=(SIN(2*(5/3) *BJ1*3.14) ) 0.01 =(BK1+BL1)/2

Chris

🔗massimilianolabardi <labardi@...>

🔗chrisvaisvil@...

Giving you the excel spreadsheet might be clearest.

In short I generated dyads at the ratio specified and added the two signals. Then I summed the resulting wave form.

This is an attempt to follow up on my idea that power is a measure of connsonace. The premise is a pure sine carries the most power. This probably will break when one adds enough signals to emulate classic wave forms like a square wave.

Right now my math is primitive but I had less than an hour to play.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "massimilianolabardi" <labardi@df.unipi.it>

Date: Thu, 22 Jan 2009 10:37:07
To: <tuning@yahoogroups.com>
Subject: [tuning] Re: Poor Man's Power Ranking of Intervals

--- In tuning@yahoogroups.com, "vaisvil" <chrisvaisvil@...> wrote:
>
> Ok,
>
> Using excel I summed dyads representing the JI ratios of a major
scale.
> In this scenario a larger number is more consonant.
>
> 3/2 = 10.60823809
> 4/3 = 9.146074713
> 9/8 = 2.229477297
> 5/4 = 6.590050018
> 2/1 = -0.002267916 (an anomaly)
> 15/8 = 1.081650721
> 4/3 = 9.146074713
> 5/3 = 6.964375836
>
>
> I'm thinking this would follow most people's idea of dyadic
tension.
>
> I got to go to an appointment - I'll try triads later.
>
> I used 2 cycles 0.01 to 1 in .01 steps as the generator.
>
> 0.01 =(SIN(2*BJ1*3.14)) =(SIN(2*(5/3)*BJ1*3.14))
0.01 =(BK1+BL1)/2
>
> Chris
>

Sorry Chris,

I don't quite understand what you have done.... could you explain it
better?

Thanks,

Max

🔗Charles Lucy <lucy@...>

Strangely enough, I disagree with Carl Lumma, and all others on this most dissonant 12edo interval issue.

with unique reasoning, of course;-)

Anyway here is a "different" perspective on the question, and the values for LucyTuned intervals, for various common intervals from 12edo values.

from notenames C, A, difference in cents, scale position, edo semitones, mean difference for 12 edo, plus Large and small intervals. values

C A 0.0 I 0 0
C# A# -31.5 #I 1 27.0 L-s
Db Bb 22.5 bII 1 27.0 s
D B - 9.0 II 2 9 L
D# B# -40.5 #II 3 27.0 2L-s
Eb C 13.5 bIII 3 27.0 L+s
E C# 18.0 III 4 27.0 2L
Fb Db 36.0 bIV 4 27.0 L+2s
E# Cx -49.6 #III 5 27.0 3L-s
F D 4.5 IV 5 27.0 2L+s
F# D# -23.0 #IV 6 25.0 3L
Gb Eb 27.0 bV 6 25.0 2L+2s
G E -4.5 V 7 4.5 3L+s
G# E# -36.0 #V 8 27.0 4L
Ab F 18.0 bVI 8 27.0 3L+2s
A F# -13.5 VI 9 13.5 4L+s
A# Fx 45.0 #VI 10 42.8 5L
Bb Gb 40.6 bVII 10 42.8 3L+3s
B G 9.0 VII 11 20.3 4L+2s
Cb Ab 31.5 bVIII 11 20.3 4L+3s
B# G# -22.5 #VII 12 11.3 5L+s

which suggests that the most distant 12edo interval is the #V1th or bVIIth.
C-A# and C-Bb.
You could of course interpret these figures in dozens of different ways, which I suppose does expose the thinking methods of the diverse tunaniks, yet become pretty meaningless for all practical purposes.

(Riding in top deck - back seat of London red bus, with mobile internet, is my current excuse for any typos or calculation errors)
On 22 Jan 2009, at 12:48, chrisvaisvil@... wrote:

> Giving you the excel spreadsheet might be clearest.
>
> In short I generated dyads at the ratio specified and added the two > signals. Then I summed the resulting wave form.
>
> This is an attempt to follow up on my idea that power is a measure > of connsonace. The premise is a pure sine carries the most power. > This probably will break when one adds enough signals to emulate > classic wave forms like a square wave.
>
> Right now my math is primitive but I had less than an hour to play.
>
> Sent via BlackBerry from T-Mobile
>
>
> From: "massimilianolabardi"
> Date: Thu, 22 Jan 2009 10:37:07 -0000
> To: <tuning@yahoogroups.com>
> Subject: [tuning] Re: Poor Man's Power Ranking of Intervals
>
> --- In tuning@yahoogroups.com, "vaisvil" <chrisvaisvil@...> wrote:
> >
> > Ok,
> >
> > Using excel I summed dyads representing the JI ratios of a major
> scale.
> > In this scenario a larger number is more consonant.
> >
> > 3/2 = 10.60823809
> > 4/3 = 9.146074713
> > 9/8 = 2.229477297
> > 5/4 = 6.590050018
> > 2/1 = -0.002267916 (an anomaly)
> > 15/8 = 1.081650721
> > 4/3 = 9.146074713
> > 5/3 = 6.964375836
> >
> >
> > I'm thinking this would follow most people's idea of dyadic
> tension.
> >
> > I got to go to an appointment - I'll try triads later.
> >
> > I used 2 cycles 0.01 to 1 in .01 steps as the generator.
> >
> > 0.01 =(SIN(2*BJ1*3.14)) =(SIN(2*(5/3)*BJ1*3.14))
> 0.01 =(BK1+BL1)/2
> >
> > Chris
> >
>
> Sorry Chris,
>
> I don't quite understand what you have done.... could you explain it
> better?
>
> Thanks,
>
> Max
>
>
>
>
Charles Lucy
lucy@...

- Promoting global harmony through LucyTuning -

for information on LucyTuning go to:
http://www.lucytune.com

For LucyTuned Lullabies go to:
http://www.lullabies.co.uk

🔗massimilianolabardi <labardi@...>

Dear Chris,

I have tried myself on the track of operations you put into Excel
and your explanation. What you are doing is calculating a numerical
integral over one period of oscillation of a sine wave at f1 summed
with another sine wave at r*f1 with r a JI ratio (3:2, 5/4 etc.). I
have obtained practically the same result, so I think my
interpretation is correct. But:

If you do this, what you get depends on where you stop the integral.
If you integrate over many cycles, the result is zero. If you just
make the summation as you do, the result depends on the phase delay
of one sinusoid with respect to the other one at the point where you
stop the summation. You could try for instance to sum between 0.01
and 2 instead than 1, your result would change for each of the
dyads, and there will be no rule for the ranking of result (they
will change more or less randomly if you increase the summation
limit to 3, 4 or any number, even not integer). Also, in the 2:1
case the phase at the end of a cycle happens to be 0 for both
sinewaves, therefore their summation will integrate to zero. That's
not an anomaly.

If you wanted to quantify the "power" as you call it, you should at
least integrate the absolute value of the sum of the two sine waves;
that would have a nonzero average value and gives a value of
amplitude of the summed waves. But also here you would have the same
phase effect, therefore your result would depend on the choice of
upper limit of your summation.

On general grounds (energy conservation), the acoustic power of two
sine waves of equal amplitude summed to one another is the same
regardless their frequency ratios. Interference makes sometimes
their peak amplitude to double, sometimes to get to zero, but the
average amplitude is always the sum of two amplitudes. The case of
two equal frequencies is a special one, they are a linear
combination of each other and therefore they do not represent a good
example to discuss........ I mean, they can be written as a single
sine function with different amplitude and phase.

Max

🔗Chris Vaisvil <chrisvaisvil@...>

I'm not sure I understand your point.

Yes I am doing discrete integration. The was the core of my idea - the
integral is related to the power of the wave form.

That it worked on integration of + and - surprised me.

However, empirically, it appears to be predictive so far, so I guess I don't
understand what you are saying.
If it breaks in other models, like more periods or phase relations, with all
due respect is immaterial in my opinion.
I'm only presenting my one model.

Remember all I had was about an hour of free time between work and my
daughter's appointment.
I'm not trying to represent this as anything more than a curio now. When I
try Caleb's ratios it may go to hell in a hand basket.

On Thu, Jan 22, 2009 at 11:03 AM, massimilianolabardi
<labardi@...>wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "vaisvil"
> <chrisvaisvil@...> wrote:
> >
> > Ok,
> >
> > Using excel I summed dyads representing the JI ratios of a major
> scale.
> > In this scenario a larger number is more consonant.
> >
> > 3/2 = 10.60823809
> > 4/3 = 9.146074713
> > 9/8 = 2.229477297
> > 5/4 = 6.590050018
> > 2/1 = -0.002267916 (an anomaly)
> > 15/8 = 1.081650721
> > 4/3 = 9.146074713
> > 5/3 = 6.964375836
> >
> >
> > I'm thinking this would follow most people's idea of dyadic
> tension.
> >
> > I got to go to an appointment - I'll try triads later.
> >
> > I used 2 cycles 0.01 to 1 in .01 steps as the generator.
> >
> > 0.01 =(SIN(2*BJ1*3.14)) =(SIN(2*(5/3)*BJ1*3.14))
> 0.01 =(BK1+BL1)/2
> >
> > Chris
> >
>
> Dear Chris,
>
> I have tried myself on the track of operations you put into Excel
> and your explanation. What you are doing is calculating a numerical
> integral over one period of oscillation of a sine wave at f1 summed
> with another sine wave at r*f1 with r a JI ratio (3:2, 5/4 etc.). I
> have obtained practically the same result, so I think my
> interpretation is correct. But:
>
> If you do this, what you get depends on where you stop the integral.
> If you integrate over many cycles, the result is zero. If you just
> make the summation as you do, the result depends on the phase delay
> of one sinusoid with respect to the other one at the point where you
> stop the summation. You could try for instance to sum between 0.01
> and 2 instead than 1, your result would change for each of the
> dyads, and there will be no rule for the ranking of result (they
> will change more or less randomly if you increase the summation
> limit to 3, 4 or any number, even not integer). Also, in the 2:1
> case the phase at the end of a cycle happens to be 0 for both
> sinewaves, therefore their summation will integrate to zero. That's
> not an anomaly.
>
> If you wanted to quantify the "power" as you call it, you should at
> least integrate the absolute value of the sum of the two sine waves;
> that would have a nonzero average value and gives a value of
> amplitude of the summed waves. But also here you would have the same
> phase effect, therefore your result would depend on the choice of
> upper limit of your summation.
>
> On general grounds (energy conservation), the acoustic power of two
> sine waves of equal amplitude summed to one another is the same
> regardless their frequency ratios. Interference makes sometimes
> their peak amplitude to double, sometimes to get to zero, but the
> average amplitude is always the sum of two amplitudes. The case of
> two equal frequencies is a special one, they are a linear
> combination of each other and therefore they do not represent a good
> example to discuss........ I mean, they can be written as a single
> sine function with different amplitude and phase.
>
> Max
>
>
>

🔗massimilianolabardi <labardi@...>

🔗Chris Vaisvil <chrisvaisvil@...>

That's interesting.

What I'd like is a function in excel that would discard the negative values
of a waveform.
That is what I was aiming for but couldn't find it easy enough within my
time constraints.
I guess you could play games with the abs() function and do it.

If you have the time could you try the other intervals caleb suggested?
I'll be home from work in a few hours and I can try them then.

Thanks,

Chris
On Thu, Jan 22, 2009 at 12:33 PM, massimilianolabardi
<labardi@...>wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...>
> wrote:
>
> Hi Chris,
>
> I wanted to say that the definite integral of a sinusoidal waveform
> over a time period that is an integer number of periods is zero
> because it is symmetric around zero, that is, the area of the
> positive part is equal to the area of the negative part. This is
> true if you integrate full periods. If you truncate the integration
> at some fraction of period, the value of the integral will be
> different depending on where you set your integration limits.
>
> > However, empirically, it appears to be predictive so far, so I
> guess I don't
> > understand what you are saying.
>
> I have tried with the tritone. It sums up close to the most
> consonant interval (the fifth): 10.4 instead of 10.6.
>
> Max
>
>
>

🔗Carl Lumma <carl@...>

🔗Chris Vaisvil <chrisvaisvil@...>

You know.... I spent an hour on a simple experiement.
Not two years on a doctorial thesis.

Considering it was your idea to actually attempt to demonstrate the proposal
I made from a mental picture I am disappointed.

I'm a scientist - I don't rig anything.

I go by what the data says.

If a system works, be it by accident or not, it works.

If you are going to speak intelligently on the subject of what I am doing
then you should understand the constraints of the test. It is obvious from
your reply you do not.

Chris

On Thu, Jan 22, 2009 at 1:16 PM, Carl Lumma <carl@...> wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> >
> > However, empirically, it appears to be predictive so far, so I
> > guess I don't understand what you are saying.
>
> The scoring you posted look OK to me, but you only posted
> a few (and only 5-limit ratios) so the chances of failure
> were limited. Moreover, there are many ways to compute a
> ranking on JI intervals that comes out right -- one must also
> ask if the method used has any justification, or if it is
> less complicated than another method which produces equivalent
> results. Your method is based on sines -- does it work for
> arbitrary waveforms? Your method also ignores the negative
> part of the waveform -- how do you justify that? Finally,
> any measure of consonance must be able to handle irrational
> intervals -- you should try some.
>
> > If it breaks in other models, like more periods or phase
> > relations, with all due respect is immaterial in my opinion.
>
> So if it only works when you rig it to work, that's all
> that counts?
>
> -Carl
>
>
>

🔗massimilianolabardi <labardi@...>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> That's interesting.
>
> What I'd like is a function in excel that would discard the
negative values
> of a waveform.
> That is what I was aiming for but couldn't find it easy enough
within my
> time constraints.
> I guess you could play games with the abs() function and do it.
>
> If you have the time could you try the other intervals caleb
suggested?
> I'll be home from work in a few hours and I can try them then.
>
> Thanks,
>
> Chris

Hi Chris,

surely we can easily calculate the rest of the intervals. Before I
would like to point out the following: if you integrate as you did
over the two sine functions, being the integration a linear operator,
its result is the same than doing the same operation on each of the
waveforms and then summing the two results. Now, the integral over
the first waveform is always zero, since the summation is done
exactly over one full period of the sine. Then the result is only the
second integral. You can calculate it as

integral between 0 and 1 of: sin(2 Pi r t) dt, that equals - cos(2 Pi
r t)/2 Pi r calculated between 0 and 1 (r is the frequency ratio).
Therefore the result is

(1 - cos(2 Pi r))/4 Pi r.

Then if you put in for example r=3/2 you get 1/3 Pi, that is about
0.106. Since you made integration by summing 100 intervals, you
should multiply this by 100 to obtain your result 10.6 for the
perfect fifth.

What intervals from Caleb do you mean? 9th min, 2nd min, and tritone?
I know 2nd min in 12-tet being 1.059, this ranks 0.51 in your units.
9th min ranks 0.99. tritone ranks 10.45.

Max

> On Thu, Jan 22, 2009 at 12:33 PM, massimilianolabardi
> <labardi@...>wrote:
>
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris
Vaisvil
> > <chrisvaisvil@>
> > wrote:
> >
> > Hi Chris,
> >
> > I wanted to say that the definite integral of a sinusoidal
waveform
> > over a time period that is an integer number of periods is zero
> > because it is symmetric around zero, that is, the area of the
> > positive part is equal to the area of the negative part. This is
> > true if you integrate full periods. If you truncate the
integration
> > at some fraction of period, the value of the integral will be
> > different depending on where you set your integration limits.
> >
> > > However, empirically, it appears to be predictive so far, so I
> > guess I don't
> > > understand what you are saying.
> >
> > I have tried with the tritone. It sums up close to the most
> > consonant interval (the fifth): 10.4 instead of 10.6.
> >
> > Max
> >
> >
> >
>

🔗chrisvaisvil@...

Thanks for the calculus. I rarely use in my job and then often just concepts.

Caleb had two minor 3rds and some other examples I can't remember. If I can get my berry to be a tethered modem I can find them quickly.

Is the solution for three tones similar?

Also, my original thought took only the positive portion of the sum of the intervals to avoid the negative cancelling out the positive problem

Ahhh I wonder if half a wave is sufficient - it would avoid the cancellation I'm thinking.
Sent via BlackBerry from T-Mobile

-----Original Message-----
From: "massimilianolabardi" <labardi@df.unipi.it>

Date: Thu, 22 Jan 2009 19:50:59
To: <tuning@yahoogroups.com>
Subject: [tuning] Re: Poor Man's Power Ranking of Intervals

Hi Chris,

integral between 0 and 1 of: sin(2 Pi r t) dt, that equals - cos(2 Pi
r t)/2 Pi r calculated between 0 and 1 (r is the frequency ratio).
Therefore the result is

(1 - cos(2 Pi r))/4 Pi r.

What intervals from Caleb do you mean? 9th min, 2nd min, and tritone?
I know 2nd min in 12-tet being 1.059, this ranks 0.51 in your units.
9th min ranks 0.99. tritone ranks 10.45.

Max

🔗Chris Vaisvil <chrisvaisvil@...>

my bad - it was Claudio:

I will be home in about 2 hrs now fo rwhat its worth.

Thanks Vaisvil,

I am very curious to know how the remaining important pure intervals (ratios
between integers smaller than 10) in the octave fare using your values.
Missing from your list are the following:

7:6 Small Minor Third
6:5 Minor Third
7:5 Tritone
8:5 Minor Sixth
7:4 Harmonic Minor Seventh

Thanks!

Claudio

On Thu, Jan 22, 2009 at 3:05 PM, <chrisvaisvil@gmail.com> wrote:

> Thanks for the calculus. I rarely use in my job and then often just
> concepts.
>
> Caleb had two minor 3rds and some other examples I can't remember. If I can
> get my berry to be a tethered modem I can find them quickly.
>
> Is the solution for three tones similar?
>
> Also, my original thought took only the positive portion of the sum of the
> intervals to avoid the negative cancelling out the positive problem
>
> Ahhh I wonder if half a wave is sufficient - it would avoid the
> cancellation I'm thinking.
>
> Sent via BlackBerry from T-Mobile
>
> ------------------------------
> *From*: "massimilianolabardi"
> *Date*: Thu, 22 Jan 2009 19:50:59 -0000
> *To*: <tuning@yahoogroups.com>
> *Subject*: [tuning] Re: Poor Man's Power Ranking of Intervals
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> >
> > That's interesting.
> >
> > What I'd like is a function in excel that would discard the
> negative values
> > of a waveform.
> > That is what I was aiming for but couldn't find it easy enough
> within my
> > time constraints.
> > I guess you could play games with the abs() function and do it.
> >
> > If you have the time could you try the other intervals caleb
> suggested?
> > I'll be home from work in a few hours and I can try them then.
> >
> > Thanks,
> >
> > Chris
>
> Hi Chris,
>
> surely we can easily calculate the rest of the intervals. Before I
> would like to point out the following: if you integrate as you did
> over the two sine functions, being the integration a linear operator,
> its result is the same than doing the same operation on each of the
> waveforms and then summing the two results. Now, the integral over
> the first waveform is always zero, since the summation is done
> exactly over one full period of the sine. Then the result is only the
> second integral. You can calculate it as
>
> integral between 0 and 1 of: sin(2 Pi r t) dt, that equals - cos(2 Pi
> r t)/2 Pi r calculated between 0 and 1 (r is the frequency ratio).
> Therefore the result is
>
> (1 - cos(2 Pi r))/4 Pi r.
>
> Then if you put in for example r=3/2 you get 1/3 Pi, that is about
> 0.106. Since you made integration by summing 100 intervals, you
> should multiply this by 100 to obtain your result 10.6 for the
> perfect fifth.
>
> What intervals from Caleb do you mean? 9th min, 2nd min, and tritone?
> I know 2nd min in 12-tet being 1.059, this ranks 0.51 in your units.
> 9th min ranks 0.99. tritone ranks 10.45.
>
> Max
>
> > On Thu, Jan 22, 2009 at 12:33 PM, massimilianolabardi
> > <labardi@...>wrote:
> >
> > > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com> <tuning%
> 40yahoogroups.com>, Chris
> Vaisvil
> > > <chrisvaisvil@>
> > > wrote:
> > >
> > > Hi Chris,
> > >
> > > I wanted to say that the definite integral of a sinusoidal
> waveform
> > > over a time period that is an integer number of periods is zero
> > > because it is symmetric around zero, that is, the area of the
> > > positive part is equal to the area of the negative part. This is
> > > true if you integrate full periods. If you truncate the
> integration
> > > at some fraction of period, the value of the integral will be
> > > different depending on where you set your integration limits.
> > >
> > > > However, empirically, it appears to be predictive so far, so I
> > > guess I don't
> > > > understand what you are saying.
> > >
> > > I have tried with the tritone. It sums up close to the most
> > > consonant interval (the fifth): 10.4 instead of 10.6.
> > >
> > > Max
> > >
> > >
> > >
> >
>
>
>

🔗caleb morgan <calebmrgn@...>

hey, anything Claudio can do, I can do better, except higher math and psychoacoustics, of course.

I can do more pushups than Milton Babbit !

I've made more money from film-scoring than Harry Partch ever did!

I can pick faster than Wes Montgomery!

caleb

On Jan 22, 2009, at 3:09 PM, Chris Vaisvil wrote:

>
> my bad - it was Claudio:
>
> I will be home in about 2 hrs now fo rwhat its worth.
>
> Thanks Vaisvil,
>
> I am very curious to know how the remaining important pure intervals > (ratios between integers smaller than 10) in the octave fare using > your values. Missing from your list are the following:
>
> 7:6 Small Minor Third
> 6:5 Minor Third
> 7:5 Tritone
> 8:5 Minor Sixth
> 7:4 Harmonic Minor Seventh
>
> Thanks!
>
> Claudio
>
> On Thu, Jan 22, 2009 at 3:05 PM, <chrisvaisvil@...> wrote:
> Thanks for the calculus. I rarely use in my job and then often just > concepts.
>
> Caleb had two minor 3rds and some other examples I can't remember. > If I can get my berry to be a tethered modem I can find them quickly.
>
> Is the solution for three tones similar?
>
> Also, my original thought took only the positive portion of the sum > of the intervals to avoid the negative cancelling out the positive > problem
>
> Ahhh I wonder if half a wave is sufficient - it would avoid the > cancellation I'm thinking.
> Sent via BlackBerry from T-Mobile
>
>
> From: "massimilianolabardi"
> Date: Thu, 22 Jan 2009 19:50:59 -0000
>
> To: <tuning@yahoogroups.com>
> Subject: [tuning] Re: Poor Man's Power Ranking of Intervals
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
> >
> > That's interesting.
> >
> > What I'd like is a function in excel that would discard the
> negative values
> > of a waveform.
> > That is what I was aiming for but couldn't find it easy enough
> within my
> > time constraints.
> > I guess you could play games with the abs() function and do it.
> >
> > If you have the time could you try the other intervals caleb
> suggested?
> > I'll be home from work in a few hours and I can try them then.
> >
> > Thanks,
> >
> > Chris
>
> Hi Chris,
>
> surely we can easily calculate the rest of the intervals. Before I
> would like to point out the following: if you integrate as you did
> over the two sine functions, being the integration a linear operator,
> its result is the same than doing the same operation on each of the
> waveforms and then summing the two results. Now, the integral over
> the first waveform is always zero, since the summation is done
> exactly over one full period of the sine. Then the result is only the
> second integral. You can calculate it as
>
> integral between 0 and 1 of: sin(2 Pi r t) dt, that equals - cos(2 Pi
> r t)/2 Pi r calculated between 0 and 1 (r is the frequency ratio).
> Therefore the result is
>
> (1 - cos(2 Pi r))/4 Pi r.
>
> Then if you put in for example r=3/2 you get 1/3 Pi, that is about
> 0.106. Since you made integration by summing 100 intervals, you
> should multiply this by 100 to obtain your result 10.6 for the
> perfect fifth.
>
> What intervals from Caleb do you mean? 9th min, 2nd min, and tritone?
> I know 2nd min in 12-tet being 1.059, this ranks 0.51 in your units.
> 9th min ranks 0.99. tritone ranks 10.45.
>
> Max
>
> > On Thu, Jan 22, 2009 at 12:33 PM, massimilianolabardi
> > <labardi@...>wrote:
> >
> > > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris
> Vaisvil
> > > <chrisvaisvil@>
> > > wrote:
> > >
> > > Hi Chris,
> > >
> > > I wanted to say that the definite integral of a sinusoidal
> waveform
> > > over a time period that is an integer number of periods is zero
> > > because it is symmetric around zero, that is, the area of the
> > > positive part is equal to the area of the negative part. This is
> > > true if you integrate full periods. If you truncate the
> integration
> > > at some fraction of period, the value of the integral will be
> > > different depending on where you set your integration limits.
> > >
> > > > However, empirically, it appears to be predictive so far, so I
> > > guess I don't
> > > > understand what you are saying.
> > >
> > > I have tried with the tritone. It sums up close to the most
> > > consonant interval (the fifth): 10.4 instead of 10.6.
> > >
> > > Max
> > >
> > >
> > >
> >
>
>
>
>
>
>

🔗massimilianolabardi <labardi@...>

Hi Chris,

I have calculated ranking of the intervals you asked by means of
your criterium:

7/6 3.41
6/5 4.58
7/5 10.28
8/5 8.99
7/4 4.55

By the way, I have uploaded (on my dir Max in this list) a plot with
the result of your summation as a function of the interval frequency
ratio (filename chris.jpg).

A note: if you choose as upper limit of integration the denominator
of your ratio (i.e. in the case of 3/2 you choose 2 instead of 1)
the result of the integral will be always zero. This is because
after 2 cycles of the first sine there have been 3 cycles of the
second sine, therefore their phase will be the same at the upper
limit of integration, the integral is performed on exactly full
periods, thereby the result is null.

I also did the absolute value of the waveform sum. The result was

1 63.64
9/8 59.62
5/4 55.19
4/3 51.37
3/2 40.22
5/3 36.26
15/8 38.65
2/1 39.74

Also in this case, in my opinion, the result depends only on the
phase difference between the two summed wave at the chosen upper
limit for the integral. If you choose the upper limit to be the
denominator, you should get always a constant power value,
regardless of the interval chosen.

I hope this could be useful.

Cheers

Max

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...>
wrote:
>
> my bad - it was Claudio:
>
> I will be home in about 2 hrs now fo rwhat its worth.
>
> Thanks Vaisvil,
>
> I am very curious to know how the remaining important pure
intervals (ratios
> between integers smaller than 10) in the octave fare using your
values.
> Missing from your list are the following:
>
> 7:6 Small Minor Third
> 6:5 Minor Third
> 7:5 Tritone
> 8:5 Minor Sixth
> 7:4 Harmonic Minor Seventh
>
> Thanks!
>
> Claudio
>
> On Thu, Jan 22, 2009 at 3:05 PM, <chrisvaisvil@...> wrote:
>
> > Thanks for the calculus. I rarely use in my job and then often
just
> > concepts.
> >
> > Caleb had two minor 3rds and some other examples I can't
remember. If I can
> > get my berry to be a tethered modem I can find them quickly.
> >
> > Is the solution for three tones similar?
> >
> > Also, my original thought took only the positive portion of the
sum of the
> > intervals to avoid the negative cancelling out the positive
problem
> >
> > Ahhh I wonder if half a wave is sufficient - it would avoid the
> > cancellation I'm thinking.
> >
> > Sent via BlackBerry from T-Mobile
> >
> > ------------------------------
> > *From*: "massimilianolabardi"
> > *Date*: Thu, 22 Jan 2009 19:50:59 -0000
> > *To*: <tuning@yahoogroups.com>
> > *Subject*: [tuning] Re: Poor Man's Power Ranking of Intervals
> >
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>,
Chris Vaisvil
> > <chrisvaisvil@> wrote:
> > >
> > > That's interesting.
> > >
> > > What I'd like is a function in excel that would discard the
> > negative values
> > > of a waveform.
> > > That is what I was aiming for but couldn't find it easy enough
> > within my
> > > time constraints.
> > > I guess you could play games with the abs() function and do it.
> > >
> > > If you have the time could you try the other intervals caleb
> > suggested?
> > > I'll be home from work in a few hours and I can try them then.
> > >
> > > Thanks,
> > >
> > > Chris
> >
> > Hi Chris,
> >
> > surely we can easily calculate the rest of the intervals. Before
I
> > would like to point out the following: if you integrate as you
did
> > over the two sine functions, being the integration a linear
operator,
> > its result is the same than doing the same operation on each of
the
> > waveforms and then summing the two results. Now, the integral
over
> > the first waveform is always zero, since the summation is done
> > exactly over one full period of the sine. Then the result is
only the
> > second integral. You can calculate it as
> >
> > integral between 0 and 1 of: sin(2 Pi r t) dt, that equals - cos
(2 Pi
> > r t)/2 Pi r calculated between 0 and 1 (r is the frequency
ratio).
> > Therefore the result is
> >
> > (1 - cos(2 Pi r))/4 Pi r.
> >
> > Then if you put in for example r=3/2 you get 1/3 Pi, that is
about
> > 0.106. Since you made integration by summing 100 intervals, you
> > should multiply this by 100 to obtain your result 10.6 for the
> > perfect fifth.
> >
> > What intervals from Caleb do you mean? 9th min, 2nd min, and
tritone?
> > I know 2nd min in 12-tet being 1.059, this ranks 0.51 in your
units.
> > 9th min ranks 0.99. tritone ranks 10.45.
> >
> > Max
> >
> > > On Thu, Jan 22, 2009 at 12:33 PM, massimilianolabardi
> > > <labardi@>wrote:
> > >
> > > > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>
<tuning%
> > 40yahoogroups.com>, Chris
> > Vaisvil
> > > > <chrisvaisvil@>
> > > > wrote:
> > > >
> > > > Hi Chris,
> > > >
> > > > I wanted to say that the definite integral of a sinusoidal
> > waveform
> > > > over a time period that is an integer number of periods is
zero
> > > > because it is symmetric around zero, that is, the area of the
> > > > positive part is equal to the area of the negative part.
This is
> > > > true if you integrate full periods. If you truncate the
> > integration
> > > > at some fraction of period, the value of the integral will be
> > > > different depending on where you set your integration limits.
> > > >
> > > > > However, empirically, it appears to be predictive so far,
so I
> > > > guess I don't
> > > > > understand what you are saying.
> > > >
> > > > I have tried with the tritone. It sums up close to the most
> > > > consonant interval (the fifth): 10.4 instead of 10.6.
> > > >
> > > > Max
> > > >
> > > >
> > > >
> > >
> >
> >
> >
>

🔗Carl Lumma <carl@...>

🔗Chris Vaisvil <chrisvaisvil@...>

7:6 Small Minor Third
6:5 Minor Third
7:5 Tritone
8:5 Minor Sixth
7:4 Harmonic Minor Seventh

7/6 3.41
6/5 4.58
7/5 10.28
8/5 8.99
7/4 4.55

Thank you very much Max - I was quite busy yesterday and today. Especially
today - fights between bosses that I was trying to keep myself out of and
trying to fix mistakes some others did. (its my job to help fix them).

Ok,

I'm not at all familiar with some of these intervals. I need to tabulate
them at some point for the ranking.
Next up is making a spreadsheet that is more flexible - I just copied over
columns and made the appropriate fromula changes - I want to use variables -
and them look at triads. However, inspection only goes so far. If this
approach still seems reasonable a proof would really be needed.

I agree about the phase difference. I think if we are looking at the ranking
of intervals and not frequency + timbres sines only and starting at zero
makes sense.

"A note: if you choose as upper limit of integration the denominator
of your ratio (i.e. in the case of 3/2 you choose 2 instead of 1)
the result of the integral will be always zero."

So, if I understand correctly it is good to avoid a full cycle.

Thanks Max,

I appreciate it! - I'll go look at the jpeg you posted.

Chris

On Fri, Jan 23, 2009 at 4:07 AM, massimilianolabardi <labardi@...>wrote:

> Hi Chris,
>
> I have calculated ranking of the intervals you asked by means of
> your criterium:
>
> 7/6 3.41
> 6/5 4.58
> 7/5 10.28
> 8/5 8.99
> 7/4 4.55
>
> By the way, I have uploaded (on my dir Max in this list) a plot with
> the result of your summation as a function of the interval frequency
> ratio (filename chris.jpg).
>
> A note: if you choose as upper limit of integration the denominator
> of your ratio (i.e. in the case of 3/2 you choose 2 instead of 1)
> the result of the integral will be always zero. This is because
> after 2 cycles of the first sine there have been 3 cycles of the
> second sine, therefore their phase will be the same at the upper
> limit of integration, the integral is performed on exactly full
> periods, thereby the result is null.
>
> I also did the absolute value of the waveform sum. The result was
>
> 1 63.64
> 9/8 59.62
> 5/4 55.19
> 4/3 51.37
> 3/2 40.22
> 5/3 36.26
> 15/8 38.65
> 2/1 39.74
>
> Also in this case, in my opinion, the result depends only on the
> phase difference between the two summed wave at the chosen upper
> limit for the integral. If you choose the upper limit to be the
> denominator, you should get always a constant power value,
> regardless of the interval chosen.
>
> I hope this could be useful.
>
> Cheers
>
> Max
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...>
> wrote:
> >
> > my bad - it was Claudio:
> >
> > I will be home in about 2 hrs now fo rwhat its worth.
> >
> > Thanks Vaisvil,
> >
> > I am very curious to know how the remaining important pure
> intervals (ratios
> > between integers smaller than 10) in the octave fare using your
> values.
> > Missing from your list are the following:
> >
> > 7:6 Small Minor Third
> > 6:5 Minor Third
> > 7:5 Tritone
> > 8:5 Minor Sixth
> > 7:4 Harmonic Minor Seventh
> >
> > Thanks!
> >
> > Claudio
> >
> > On Thu, Jan 22, 2009 at 3:05 PM, <chrisvaisvil@...> wrote:
> >
> > > Thanks for the calculus. I rarely use in my job and then often
> just
> > > concepts.
> > >
> > > Caleb had two minor 3rds and some other examples I can't
> remember. If I can
> > > get my berry to be a tethered modem I can find them quickly.
> > >
> > > Is the solution for three tones similar?
> > >
> > > Also, my original thought took only the positive portion of the
> sum of the
> > > intervals to avoid the negative cancelling out the positive
> problem
> > >
> > > Ahhh I wonder if half a wave is sufficient - it would avoid the
> > > cancellation I'm thinking.
> > >
> > > Sent via BlackBerry from T-Mobile
> > >
> > > ------------------------------
> > > *From*: "massimilianolabardi"
> > > *Date*: Thu, 22 Jan 2009 19:50:59 -0000
> > > *To*: <tuning@yahoogroups.com <tuning%40yahoogroups.com>>
> > > *Subject*: [tuning] Re: Poor Man's Power Ranking of Intervals
> > >
> > > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com> <tuning%
> 40yahoogroups.com>,
> Chris Vaisvil
> > > <chrisvaisvil@> wrote:
> > > >
> > > > That's interesting.
> > > >
> > > > What I'd like is a function in excel that would discard the
> > > negative values
> > > > of a waveform.
> > > > That is what I was aiming for but couldn't find it easy enough
> > > within my
> > > > time constraints.
> > > > I guess you could play games with the abs() function and do it.
> > > >
> > > > If you have the time could you try the other intervals caleb
> > > suggested?
> > > > I'll be home from work in a few hours and I can try them then.
> > > >
> > > > Thanks,
> > > >
> > > > Chris
> > >
> > > Hi Chris,
> > >
> > > surely we can easily calculate the rest of the intervals. Before
> I
> > > would like to point out the following: if you integrate as you
> did
> > > over the two sine functions, being the integration a linear
> operator,
> > > its result is the same than doing the same operation on each of
> the
> > > waveforms and then summing the two results. Now, the integral
> over
> > > the first waveform is always zero, since the summation is done
> > > exactly over one full period of the sine. Then the result is
> only the
> > > second integral. You can calculate it as
> > >
> > > integral between 0 and 1 of: sin(2 Pi r t) dt, that equals - cos
> (2 Pi
> > > r t)/2 Pi r calculated between 0 and 1 (r is the frequency
> ratio).
> > > Therefore the result is
> > >
> > > (1 - cos(2 Pi r))/4 Pi r.
> > >
> > > Then if you put in for example r=3/2 you get 1/3 Pi, that is
> about
> > > 0.106. Since you made integration by summing 100 intervals, you
> > > should multiply this by 100 to obtain your result 10.6 for the
> > > perfect fifth.
> > >
> > > What intervals from Caleb do you mean? 9th min, 2nd min, and
> tritone?
> > > I know 2nd min in 12-tet being 1.059, this ranks 0.51 in your
> units.
> > > 9th min ranks 0.99. tritone ranks 10.45.
> > >
> > > Max
> > >
> > > > On Thu, Jan 22, 2009 at 12:33 PM, massimilianolabardi
> > > > <labardi@>wrote:
> > > >
> > > > > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com> <tuning%
> 40yahoogroups.com>
> <tuning%
> > > 40yahoogroups.com>, Chris
> > > Vaisvil
> > > > > <chrisvaisvil@>
> > > > > wrote:
> > > > >
> > > > > Hi Chris,
> > > > >
> > > > > I wanted to say that the definite integral of a sinusoidal
> > > waveform
> > > > > over a time period that is an integer number of periods is
> zero
> > > > > because it is symmetric around zero, that is, the area of the
> > > > > positive part is equal to the area of the negative part.
> This is
> > > > > true if you integrate full periods. If you truncate the
> > > integration
> > > > > at some fraction of period, the value of the integral will be
> > > > > different depending on where you set your integration limits.
> > > > >
> > > > > > However, empirically, it appears to be predictive so far,
> so I
> > > > > guess I don't
> > > > > > understand what you are saying.
> > > > >
> > > > > I have tried with the tritone. It sums up close to the most
> > > > > consonant interval (the fifth): 10.4 instead of 10.6.
> > > > >
> > > > > Max
> > > > >
> > > > >
> > > > >
> > > >
> > >
> > >
> > >
> >
>
>
>

🔗Chris Vaisvil <chrisvaisvil@...>

Hi Max,

I've looked at your plot. It intuitively makes sense in the fashion that the
further away from tonic you are the less dissonant. The exception is the
tritone...

Thanks,

Off to triads!

Chris

On Fri, Jan 23, 2009 at 4:07 AM, massimilianolabardi <labardi@...>wrote:

🔗Chris Vaisvil <chrisvaisvil@...>

Carl,

Did you try the square root method against irrationals?

Also, the reason I called it a "poor man's" is because I used Excel instead
of purchasing Mathmatica.
-> also I wanted to avoid the learning curve - and I have just 15 days on
the trial.

It seems to me that intervals are understood fairly well (harmonic
entropy???) but the devil is introduced with 3 or more tones... correct?

Chris

On Fri, Jan 23, 2009 at 3:36 PM, Carl Lumma <carl@...> wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>,
> "massimilianolabardi" <labardi@...> wrote:
> >
> > Hi Chris,
> >
> > I have calculated ranking of the intervals you asked by means of
> > your criterium:
> >
> > 7/6 3.41
> > 6/5 4.58
> > 7/5 10.28
> > 8/5 8.99
> > 7/4 4.55
>
> I don't think 7/6 is more consonant than 7/4.
>
> By the way, I'd hardly call this a "poor man's" ranking,
> since it requires calculus. sqrt(n*d) gives the following
> ranking
>
> 5/3 3.87
> 5/4 4.47
> 7/4 5.29
> 6/5 5.48
> 7/5 5.92
> 8/5 6.32
> 7/6 6.48
>
> which I think agrees very well with experience.
> Indeed, the only contentious bit over the years
> has been 7/5 vs. 8/5. Here, I think cultural
> conditioning clouds the issue. 7/5 is actually
> quite consonant, but it close to a tritone, which
> is a functional dissonance in Western tonal music,
> owing to its place in the diatonic scale more
> than its sensory nature.
>
> -Carl
>
>
>

🔗Carl Lumma <carl@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Carl Lumma <carl@...>

🔗Chris Vaisvil <chrisvaisvil@...>

Not to belabor the point.

If you can use the relationship in music then you can at least closely
approximate the result as a ratio of frequencies.

4/3 from C = (262*4/3)/(262) = 349.3333... / 262 = 1.33333....

So 1.33333... = (262*1.33333)/(262)

and 3.1415926535 = (262 * 3.1415)/(262)

From a musical perspective a few decimals deep is certainly equivalent to
the ear to the real irrational.

Unless I'm missing something here.... I would think you can evaluate these
intervals.

Chris

On Sun, Jan 25, 2009 at 2:58 AM, Carl Lumma <carl@...> wrote:

> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
>
> > > > Did you try the square root method against irrationals?
> > >
> > > Can't be done, of course, since it's defined in terms of
> > > numerator and denominator. However harmonic entropy, which
> > > agrees with sqrt(n*d) for dyads, does work for irrationals,
> > > and has been tried and tested many times over the years.
> >
> > the irrationals come from some ratio so even a frequency ratio
> > will do.
>
> Yes, a ratio, but not a rational number (thing with a numerator
> and denominator). One can express irrational numbers as
> continued fractions, and there's been much speculation on this
> list about Noble numbers
> http://mathworld.wolfram.com/NobleNumber.html
> in particular, their relationship to harmonic entropy maxima.
> But to my knowledge, nobody's ever proposed a dissonance
> measure based on, say, the number of terms before the
> partial quotients stop changing
> http://mathworld.wolfram.com/PartialQuotient.html
> (though that would be interesting)
>
> -Carl
>
>
>

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:

> > Yes, a ratio, but not a rational number (thing with a numerator
> > and denominator). One can express irrational numbers as
> > continued fractions, and there's been much speculation on this
> > list about Noble numbers
> > http://mathworld.wolfram.com/NobleNumber.html
> > in particular, their relationship to harmonic entropy maxima.
> > But to my knowledge, nobody's ever proposed a dissonance
> > measure based on, say, the number of terms before the
> > partial quotients stop changing
> > http://mathworld.wolfram.com/PartialQuotient.html
> > (though that would be interesting)
>
> Not to belabor the point.
> If you can use the relationship in music then you can at
> least closely approximate the result as a ratio of frequencies.
> 4/3 from C = (262*4/3)/(262) = 349.3333... / 262 = 1.33333....

I think you mean 262*4/3 = 349.333...

> So 1.33333... = (262*1.33333)/(262)
> and 3.1415926535 = (262 * 3.1415)/(262)

Trivially.

> From a musical perspective a few decimals deep is certainly
> equivalent to the ear to the real irrational.
> Unless I'm missing something here.... I would think you can
> evaluate these intervals.

One can certainly evaluate irrational intervals based on
the sqrt(n*d) of a nearby rational plus a mistuning factor.
However, one needs a method for finding nearby rationals.
Since the rationals are dense, and small movements result
in wild changes in the numerator and denominator, it's not
clear this makes sense. If you have a method please share
your results.

When evaluating temperaments, we have no trouble saying
396 cents is a better 5:4 than 400 cents, but we don't
usually compare an approximation of one rational to an
approximation of another, e.g. which is more consonant,
396 cents or 696 cents?

-Carl

Poor Man's Power Ranking of Intervals

🔗vaisvil <chrisvaisvil@...>

🔗Claudio Di Veroli <dvc@...>

🔗djtrancendance@...

🔗massimilianolabardi <labardi@...>

🔗chrisvaisvil@...

🔗Charles Lucy <lucy@...>

🔗massimilianolabardi <labardi@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗massimilianolabardi <labardi@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Carl Lumma <carl@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗massimilianolabardi <labardi@...>

🔗chrisvaisvil@...

🔗Chris Vaisvil <chrisvaisvil@...>

🔗caleb morgan <calebmrgn@...>

🔗massimilianolabardi <labardi@...>

🔗Carl Lumma <carl@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Carl Lumma <carl@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Carl Lumma <carl@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Carl Lumma <carl@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Carl Lumma <carl@...>