Proportional stretch function

I do believe the proportional stretch function implies that the pitches of a consonant major triad are to be defined in this way:

if `z` ( i.e. 1.5) becomes `z+n`

then `x` (i.e. 1.25) becomes x+(1 over z/x times n)

For example, if one modifies the pure fifth so that the new value is 732 cents, then to preserve the consonance of the chord, one needs to modify the third by this amount:

386+(30 times 1 over 702/386)= 386+16.5= 403 cents

This yields, if I am not wrong, equal beating of the intervals within a chord, and thus, produces a soothing consonant effect.

Monz, could this be a valuable contribution to your encyclopedia?

Cordially,
Ozan

I can think of no system that incorporates a similar triad for varying size better than the scales of mt . Meru and the your whole family of recurrent sequences.
the one thing they all have in common is what erv calls a fibonacci triplet.
It has the following characteristics
if we symbolize a triad is from bottom to top with A,B,C
A=(2B)-C
B+ A=C/2, 2B= A=C
C=(2B) -A

But even with chords that did not have the fibonacci triplet as the basis of their consonance, it seems that the principle of keeping the relationship of the chord members to it s difference tones a constant quality would be a logical way of stretching.

But one never has equal beating cause it doubles per octave, where as the recurrent sequences produce difference tones result that are members of it own series. When one ups the annie so to speak with the superposition of beating sonorities these themselves reinforces lower level beat. The result is an overall even beating that the more stimuli you add; the more an overall beating rate comes out and is heard as an important quality of the tuning.
i am working on collecting more observations about this family of tunings. Message: 21 Date: Sat, 27 Aug 2005 03:38:08 +0300
From: "Ozan Yarman" <ozanyarman@superonline.com>
Subject: Proportional stretch function

I do believe the proportional stretch function implies that the pitches of a consonant major triad are to be defined in this way:

if `z` ( i.e. 1.5) becomes `z+n`

then `x` (i.e. 1.25) becomes x+(1 over z/x times n)

For example, if one modifies the pure fifth so that the new value is 732 cents, then to preserve the consonance of the chord, one needs to modify the third by this amount:

386+(30 times 1 over 702/386)= 386+16.5= 403 cents

This yields, if I am not wrong, equal beating of the intervals within a chord, and thus, produces a soothing consonant effect.

Monz, could this be a valuable contribution to your encyclopedia?

Cordially,
Ozan

[This message contained attachments]

__________________________________________________________

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Uh Kraig, remember that I'm still a novice and do not grasp the technical lingo yet!

So, what kind of beating exactly does the formula I proposed yields?

Cordially,
Ozan

----- Original Message -----
From: Kraig Grady
To: tuning@yahoogroups.com
Sent: 27 Ağustos 2005 Cumartesi 9:55
Subject: [tuning] Proportional stretch function

I can think of no system that incorporates a similar triad for varying size better than the scales of mt . Meru and the your whole family of recurrent sequences.
the one thing they all have in common is what erv calls a fibonacci triplet.
It has the following characteristics
if we symbolize a triad is from bottom to top with A,B,C
A=(2B)-C
B+ A=C/2, 2B= A=C
C=(2B) -A

But even with chords that did not have the fibonacci triplet as the basis of their consonance, it seems that the principle of keeping the relationship of the chord members to it s difference tones a constant quality would be a logical way of stretching.

But one never has equal beating cause it doubles per octave, where as the recurrent sequences produce difference tones result that are members of it own series. When one ups the annie so to speak with the superposition of beating sonorities these themselves reinforces lower level beat. The result is an overall even beating that the more stimuli you add; the more an overall beating rate comes out and is heard as an important quality of the tuning.
i am working on collecting more observations about this family of tunings.

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> I do believe the proportional stretch function implies that the
pitches of a consonant major triad are to be defined in this way:
>
> if `z` ( i.e. 1.5) becomes `z+n`
>
> then `x` (i.e. 1.25) becomes x+(1 over z/x times n)
>
> For example, if one modifies the pure fifth so that the new value
>is 732 cents, then to preserve the consonance of the chord, one
>needs to modify the third by this amount:
>
> 386+(30 times 1 over 702/386)= 386+16.5= 403 cents

Shall I take this to be 386.31+(30.04*386.31/701.96) = 402.84 cents
as a more exact figure?

> This yields, if I am not wrong, equal beating of the intervals
>within a chord,

If I take the root to be 440 Hz, the frequencies would be 440,
555.28, and 671.55. The beat rate of the "perfect fifth" is thus

2*671.55 - 3*440 = 23.1 Hz;

the beat rate of the "major third" is

4*555.27 - 5*440 = 21.12 Hz;

and the beat rate of the "minor third" is

5*671.55 - 6*555.27 = 26.13 Hz.

So, if I have understood you correctly, your procedure does not in
fact produce equal beating of any pair of intervals within the chord.
Instead, given a "perfect fifth" of 732 cents, it is a "major third"
of 404.4 cents would cause all three beat rates in the root-position
close-voiced major triad to be equal.

i fail to see how this got some complex, if you know the vps of you intervals, especially the fifth you just divide in half!
if you know the root and third you just add the difference to the third to get your fifth. etc.
-- Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:

> i fail to see how this got some complex, if you know the vps of you
> intervals, especially the fifth you just divide in half!
> if you know the root and third you just add the difference to the
third
> to get your fifth. etc.

I don't know what you mean. Can you elaborate, say with an example? If
you have an easier alternative for the fairly simple computations I
used in this post:

/tuning/topicId_59911.html#59995

which are the same computations Monz proposed, I'm sure we'd both love
to know about it.

The math I used is pretty valid for my purposes and understanding Kraig. Don't you think my formula deserves some tests and `beating`?

Cordially,
Ozan

----- Original Message -----
From: Kraig Grady
To: tuning@yahoogroups.com
Sent: 30 Ağustos 2005 Salı 1:39
Subject: [tuning] Proportional stretch function

i fail to see how this got some complex, if you know the vps of you
intervals, especially the fifth you just divide in half!
if you know the root and third you just add the difference to the third
to get your fifth. etc.

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Dear Paul,
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 30 Ağustos 2005 Salı 0:19
Subject: [tuning] Re: Proportional stretch function

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> I do believe the proportional stretch function implies that the
pitches of a consonant major triad are to be defined in this way:
>
> if `z` ( i.e. 1.5) becomes `z+n`
>
> then `x` (i.e. 1.25) becomes x+(1 over z/x times n)
>
> For example, if one modifies the pure fifth so that the new value
>is 732 cents, then to preserve the consonance of the chord, one
>needs to modify the third by this amount:
>
> 386+(30 times 1 over 702/386)= 386+16.5= 403 cents

Shall I take this to be 386.31+(30.04*386.31/701.96) = 402.84 cents
as a more exact figure?

That is indeed desirable!

> This yields, if I am not wrong, equal beating of the intervals
>within a chord,

If I take the root to be 440 Hz, the frequencies would be 440,
555.28, and 671.55. The beat rate of the "perfect fifth" is thus

2*671.55 - 3*440 = 23.1 Hz;

the beat rate of the "major third" is

4*555.27 - 5*440 = 21.12 Hz;

and the beat rate of the "minor third" is

5*671.55 - 6*555.27 = 26.13 Hz.

So, if I have understood you correctly, your procedure does not in
fact produce equal beating of any pair of intervals within the chord.
Instead, given a "perfect fifth" of 732 cents, it is a "major third"
of 404.4 cents would cause all three beat rates in the root-position
close-voiced major triad to be equal.

Uh, wait a second. Do you mean to say that my stretch function is a `beat-equalizer` formula for consonance? Explain to me how that works please.

Cordially,
Ozan

examples from your numbers 440 and 671.55
add together, divide by two to get your third 555.775

Subject: Re: Proportional stretch function

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:

>> i fail to see how this got some complex, if you know the vps of you >> intervals, especially the fifth you just divide in half!
>> if you know the root and third you just add the difference to the > >
third >> to get your fifth. etc.
> >

I don't know what you mean. Can you elaborate, say with an example? If you have an easier alternative for the fairly simple computations I used in this post:

/tuning/topicId_59911.html#59995

which are the same computations Monz proposed, I'm sure we'd both love to know about it.

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
> Dear Paul,
> ----- Original Message -----
> From: wallyesterpaulrus
> To: tuning@yahoogroups.com
> Sent: 30 Aðustos 2005 Salý 0:19
> Subject: [tuning] Re: Proportional stretch function
>
>
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...>
wrote:
>
> > I do believe the proportional stretch function implies that the
> pitches of a consonant major triad are to be defined in this way:
> >
> > if `z` ( i.e. 1.5) becomes `z+n`
> >
> > then `x` (i.e. 1.25) becomes x+(1 over z/x times n)
> >
> > For example, if one modifies the pure fifth so that the new
value
> >is 732 cents, then to preserve the consonance of the chord, one
> >needs to modify the third by this amount:
> >
> > 386+(30 times 1 over 702/386)= 386+16.5= 403 cents
>
> Shall I take this to be 386.31+(30.04*386.31/701.96) = 402.84
cents
> as a more exact figure?
>
>
> That is indeed desirable!
>
>
>
>
> > This yields, if I am not wrong, equal beating of the intervals
> >within a chord,
>
> If I take the root to be 440 Hz, the frequencies would be 440,
> 555.28, and 671.55. The beat rate of the "perfect fifth" is thus
>
> 2*671.55 - 3*440 = 23.1 Hz;
>
> the beat rate of the "major third" is
>
> 4*555.27 - 5*440 = 21.12 Hz;
>
> and the beat rate of the "minor third" is
>
> 5*671.55 - 6*555.27 = 26.13 Hz.
>
> So, if I have understood you correctly, your procedure does not
in
> fact produce equal beating of any pair of intervals within the
chord.
> Instead, given a "perfect fifth" of 732 cents, it is a "major
third"
> of 404.4 cents would cause all three beat rates in the root-
position
> close-voiced major triad to be equal.
>
>
>
>
> Uh, wait a second. Do you mean to say that my stretch function is a
>`beat-equalizer` formula for consonance?

I don't know what that means. For consonance??

> Explain to me how that works please.

I'm sorry we appear to have lost each other. What I showed above is
that your scheme of proportional stretching does not produce equal-
beating triads, contradicting your assertion that it does. Instead of
the major third of 402.84 cents that you derived, it is instead a
major third of 404.4 cents that would be needed in order to give an
equal-beating triad when the context is a root-position close-voiced
major triad.

Best,
Paul

Oh, that's what you meant! Yes, Kraig, that's indeed the easiest way
to get the third that results in an equal-beating triad. But as I did
not show how I got *that* third, I don't know on what basis you're
assuming I *didn't* use this method. I thought you were objecting to
the calculations that are actually shown in

/tuning/topicId_59911.html#59995

and the similar ones Monz did and had another way of doing that. I
guess you aren't?

But you are right in that Ozan could probably have learned more if
presented with this fact right away (counterpoised to
his "proportional-stretch" proposal) rather than being shown that his
proposal was wrong and then being given the right answer without
explanation.

So you deserve thanks, Kraig!

In other words, we can sum up the whole topic by saying that starting
with a JI triad n:n+1:n+2, equal beating results if the triad is
stretched "proportionally", but not in cents space; rather, in
frequency (cps or vps or Hz) space. Ozan really had the right idea,
in a sense . . .

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> examples from your numbers 440 and 671.55
> add together, divide by two to get your third 555.775
>
>
> Subject: Re: Proportional stretch function
>
> --- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
>
>
> >> i fail to see how this got some complex, if you know the vps of
you
> >> intervals, especially the fifth you just divide in half!
> >> if you know the root and third you just add the difference to
the
> >
> >
> third
>
> >> to get your fifth. etc.
> >
> >
>
> I don't know what you mean. Can you elaborate, say with an example?
If
> you have an easier alternative for the fairly simple computations I
> used in this post:
>
> /tuning/topicId_59911.html#59995
>
> which are the same computations Monz proposed, I'm sure we'd both
love
> to know about it.
>
> --
> Kraig Grady
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
> The Wandering Medicine Show
> KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

Paul,

>
> Uh, wait a second. Do you mean to say that my stretch function is a
>`beat-equalizer` formula for consonance?

I don't know what that means. For consonance??

Aie, I give up trying to use funny words, and retract all my statements in that regard.

> Explain to me how that works please.

I'm sorry we appear to have lost each other. What I showed above is
that your scheme of proportional stretching does not produce equal-
beating triads, contradicting your assertion that it does.

That was my previous assertion which I corrected immediately. However the results are very close, and that's what deceived me at first.

Instead of
the major third of 402.84 cents that you derived, it is instead a
major third of 404.4 cents that would be needed in order to give an
equal-beating triad when the context is a root-position close-voiced major triad.

I have no objections to that! But I still do believe you have to explain why the results are so close in both cases.

Best,
Paul

Cordially,
Ozan

Drat, I really thought I came up with something interesting this time. So, is every one allied against the idea that consonance of a triad in root position requires equal beating?
----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 30 Ağustos 2005 Salı 3:11
Subject: [tuning] Re: Proportional stretch function

Oh, that's what you meant! Yes, Kraig, that's indeed the easiest way
to get the third that results in an equal-beating triad. But as I did
not show how I got *that* third, I don't know on what basis you're
assuming I *didn't* use this method. I thought you were objecting to
the calculations that are actually shown in

/tuning/topicId_59911.html#59995

and the similar ones Monz did and had another way of doing that. I
guess you aren't?

But you are right in that Ozan could probably have learned more if
presented with this fact right away (counterpoised to
his "proportional-stretch" proposal) rather than being shown that his
proposal was wrong and then being given the right answer without
explanation.

So you deserve thanks, Kraig!

In other words, we can sum up the whole topic by saying that starting
with a JI triad n:n+1:n+2, equal beating results if the triad is
stretched "proportionally", but not in cents space; rather, in
frequency (cps or vps or Hz) space. Ozan really had the right idea,
in a sense . . .

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> > Explain to me how that works please.
>
> I'm sorry we appear to have lost each other. What I showed above is
> that your scheme of proportional stretching does not produce equal-
> beating triads, contradicting your assertion that it does.
>
>
>
> That was my previous assertion which I corrected immediately.
>However the results are very close, and that's what deceived me at
>first.

There's close and then there's close! The closeness here is
absolutely distant compared with, say, the closeness of certain n/d-
comma meantones to having exact-integer beat rates (as mentioned in
some current posts here) -- there we are often talking about
thousandths of a cent, if that.

>> Instead of
>> the major third of 402.84 cents that you derived, it is instead a
>> major third of 404.4 cents that would be needed in order to give
an
>> equal-beating triad when the context is a root-position close-
>>voiced major triad.
>
>
>
> I have no objections to that! But I still do believe you have to
>explain why the results are so close in both cases.

As Kraig pointed out, you would have gotten exactly the right results
if you had been "proportionally stretching" the frequencies (Hz or
cps or vps) instead of the logarithmic intervals (cents). As long as
the stretches are small ones, these two procedures will produce very
close results, as the logarithmic function is approximately linear
within a small enough range.

Regards,
Paul

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 30 Ağustos 2005 Salı 3:46
Subject: [tuning] Re: Proportional stretch function

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> > Explain to me how that works please.
>
> I'm sorry we appear to have lost each other. What I showed above is
> that your scheme of proportional stretching does not produce equal-
> beating triads, contradicting your assertion that it does.
>
>
>
> That was my previous assertion which I corrected immediately.
>However the results are very close, and that's what deceived me at
>first.

There's close and then there's close! The closeness here is
absolutely distant compared with, say, the closeness of certain n/d-
comma meantones to having exact-integer beat rates (as mentioned in
some current posts here) -- there we are often talking about
thousandths of a cent, if that.

Do you claim to hear the difference of 2 cents by the beats of a pulsating major triad as compared to equal beating?

>> Instead of
>> the major third of 402.84 cents that you derived, it is instead a
>> major third of 404.4 cents that would be needed in order to give
an
>> equal-beating triad when the context is a root-position close-
>>voiced major triad.
>
>
>
> I have no objections to that! But I still do believe you have to
>explain why the results are so close in both cases.

As Kraig pointed out, you would have gotten exactly the right results
if you had been "proportionally stretching" the frequencies (Hz or
cps or vps) instead of the logarithmic intervals (cents). As long as
the stretches are small ones, these two procedures will produce very
close results, as the logarithmic function is approximately linear
within a small enough range.

Ah, so, you deem that the logarithmic stretch has no bearing whatsoever for the consonance of a major triad? You are sure that the stretch must produce exact equal beating to sound consonant? This was the main concern really, you do remember of course.

Regards,
Paul

Cordially,
Ozan

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> There's close and then there's close! The closeness here is
> absolutely distant compared with, say, the closeness of certain
n/d-
> comma meantones to having exact-integer beat rates (as mentioned
in
> some current posts here) -- there we are often talking about
> thousandths of a cent, if that.
>
>
> Do you claim to hear the difference of 2 cents by the beats of a
>pulsating major triad as compared to equal beating?

I'm not sure what you're asking, exactly. But to be on the safe side,
I'll say no, I don't claim to hear the difference.

> Ah, so, you deem that the logarithmic stretch has no bearing
>whatsoever for the consonance of a major triad?

If you're asking whether 'proportional stretch' is optimal when one
of the tempered intervals is given and the others are a free choice,
it seems like a pretty good strategy but not necessarily the best one.

>You are sure that the stretch must produce exact equal beating to
>sound consonant?

I suggested no such thing -- the triad I suggested as possibly most
consonant for a given stretched fifth was not an equal-beating one at
all, and I only addressed the subject of equal beating *after* it was
explicitly brought up by others.

Paul,

>
>
> Do you claim to hear the difference of 2 cents by the beats of a
>pulsating major triad as compared to equal beating?

I'm not sure what you're asking, exactly. But to be on the safe side, I'll say no, I don't claim to hear the difference.

Ah, that is why, in my reply to Monz, I have excused myself since the difference between my stretched triad and the equal-beating triad was quite unnoticable. My ear assumed previously that my formula produced equal-beating intervals. Then I said that the results were not exact, but very close, and that's what deceived me. Hopefully, the test will show that many people, if not all, are incapable of discerning the difference.

> Ah, so, you deem that the logarithmic stretch has no bearing
>whatsoever for the consonance of a major triad?

If you're asking whether 'proportional stretch' is optimal when one of the tempered intervals is given and the others are a free choice, it seems like a pretty good strategy but not necessarily the best one.

Thank you for acknowledging that! But, what is the best strategy according to you?

>You are sure that the stretch must produce exact equal beating to
>sound consonant?

I suggested no such thing -- the triad I suggested as possibly most consonant for a given stretched fifth was not an equal-beating one at all, and I only addressed the subject of equal beating *after* it was explicitly brought up by others.

You are aware, of course, that the main reason we started this conversation was to find a way to explain why 5/4 sounds horrible in a triad whose fifth is stretched as much as 20 cents. It appears both the equal-beating formula, and my `proportional stretch` explains the phenomenon adequately. Or do you dismiss my approach to be inappropriate for defining the consonance of a chord?

Cordially,
Ozan

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

>
>
>> > Ah, so, you deem that the logarithmic stretch has no bearing
>> >whatsoever for the consonance of a major triad?
>
>> If you're asking whether 'proportional stretch' is optimal when
>>one of the tempered intervals is given and the others are a free
>>choice, it seems like a pretty good strategy but not necessarily
the >>best one.
>
>
>
> Thank you for acknowledging that! But, what is the best strategy
>according to you?

I tried to illustrate one possible answer to this here:

/tuning/topicId_59689.html#59994

>> >You are sure that the stretch must produce exact equal beating
to
>> >sound consonant?
>
>> I suggested no such thing -- the triad I suggested as possibly
>>most consonant for a given stretched fifth was not an equal-beating
>>one at all, and I only addressed the subject of equal beating
>>*after* it was explicitly brought up by others.

Again, the text below didn't appear until I hit "Reply". Could one of
the moderators look into this? I may have been missing large amounts
of information in my reading of this list for . . . how long?

> You are aware, of course, that the main reason we started this
>conversation was to find a way to explain why 5/4 sounds horrible in
>a triad whose fifth is stretched as much as 20 cents. It appears
>both the equal-beating formula, and my `proportional stretch`
>explains the phenomenon adequately.

To repeat my earlier statements, I believe that neither of these is
an 'explanation' of any sort. Rather, I believe it's the fact that
the 6:5 ratio between the upper two notes in the chord is sensitive
to mistuning which provides the real explanation as to why a pure 5:4
will not do between the lowest two notes of a close-voiced root-
position major triad with a highly stretched fifth.

>Or do you dismiss my approach to be inappropriate for defining the
>consonance of a chord?

I'm not sure, but I'd stress that just because a chord can be made
slightly more concordant through slight adjustments, doesn't mean
that the chord wasn't concordant to begin with. This is true when the
slight adjustments lead to JI and certainly true when they lead to
synchronized beating, let alone a proportional stretch in cents. I
only say this because of the language you've been using, such
as "defining" above. There is a continuum of chords and I believe
that concordance is a continuous function of all the intervals in the
chord, with local maxima at simple JI chords (for most timbres). It's
not a black or white kind of deal.

Paul,

>You are aware, of course, that the main reason we started this
>conversation was to find a way to explain why 5/4 sounds horrible >in a triad whose fifth is stretched as much as 20 cents. It appears >both the equal-beating formula, and my `proportional stretch`
>explains the phenomenon adequately.

To repeat my earlier statements, I believe that neither of these is
an 'explanation' of any sort. Rather, I believe it's the fact that
the 6:5 ratio between the upper two notes in the chord is sensitive
to mistuning which provides the real explanation as to why a pure 5:4 will not do between the lowest two notes of a close-voiced root-
position major triad with a highly stretched fifth.

Agreed! You succinctly described the phenomenon which I was trying to put into words.

>Or do you dismiss my approach to be inappropriate for defining the
>consonance of a chord?

I'm not sure, but I'd stress that just because a chord can be made
slightly more concordant through slight adjustments, doesn't mean
that the chord wasn't concordant to begin with. This is true when the slight adjustments lead to JI and certainly true when they lead to synchronized beating, let alone a proportional stretch in cents. I only say this because of the language you've been using, such
as "defining" above. There is a continuum of chords and I believe
that concordance is a continuous function of all the intervals in the chord, with local maxima at simple JI chords (for most timbres). It's not a black or white kind of deal.

Agreed again.

Cordially,
Ozan

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Agreed again.

Unfortunately, this latter agreement was snipped by Yahoo and is
invisible in the archives. It only appears when I hit *reply*. Again, I
wonder what other crucial bits of conversation I may have missed or be
missing :(

Best,
Paul

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> To repeat my earlier statements, I believe that neither of these is
> an 'explanation' of any sort. Rather, I believe it's the fact that
> the 6:5 ratio between the upper two notes in the chord is sensitive
> to mistuning which provides the real explanation as to why a pure
5:4 will not do between the lowest two notes of a close-voiced root-
> position major triad with a highly stretched fifth.

This suggests that chord concordance might be linear--that is, it
might simply be a sum of correctly defined interval concordance
functions. If true, it would make life much simpler, but raises the
question of what those interval functions should be.

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:
>
> > To repeat my earlier statements, I believe that neither of these is
> > an 'explanation' of any sort. Rather, I believe it's the fact that
> > the 6:5 ratio between the upper two notes in the chord is sensitive
> > to mistuning which provides the real explanation as to why a pure
> 5:4 will not do between the lowest two notes of a close-voiced root-
> > position major triad with a highly stretched fifth.
>
> This suggests that chord concordance might be linear--that is, it
> might simply be a sum of correctly defined interval concordance
> functions. If true, it would make life much simpler, but raises the
> question of what those interval functions should be.

After investigating this with quite a few listeners on the Harmonic
Entropy list, it seemed a pretty clear conclusion that chord
concordance is not even a *function*, let alone a sum, of any interval
concordance functions.

Essentially, I think you are a valuable objective eye in the tuning list and possess the expertise to choose the correct phrasing as compared to most. So, I agree with you concerning the latest topics, `identity interval`, `proportional stretch`, `midi tuning sensitivity`, etc...

Cordially,
Ozan

----- Original Message -----
From: wallyesterpaulrus
To: tuning@yahoogroups.com
Sent: 06 Eylül 2005 Salı 22:48
Subject: [tuning] Re: Proportional stretch function

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> Agreed again.

Unfortunately, this latter agreement was snipped by Yahoo and is
invisible in the archives. It only appears when I hit *reply*. Again, I
wonder what other crucial bits of conversation I may have missed or be
missing :(

Best,
Paul

Simple integer ratios perhaps Gene?

----- Original Message -----
From: Gene Ward Smith
To: tuning@yahoogroups.com
Sent: 06 Eylül 2005 Salı 22:59
Subject: [tuning] Re: Proportional stretch function

--- In tuning@yahoogroups.com, "Ozan Yarman" <ozanyarman@s...> wrote:

> To repeat my earlier statements, I believe that neither of these is
> an 'explanation' of any sort. Rather, I believe it's the fact that
> the 6:5 ratio between the upper two notes in the chord is sensitive
> to mistuning which provides the real explanation as to why a pure
5:4 will not do between the lowest two notes of a close-voiced root-
> position major triad with a highly stretched fifth.

This suggests that chord concordance might be linear--that is, it
might simply be a sum of correctly defined interval concordance
functions. If true, it would make life much simpler, but raises the
question of what those interval functions should be.