Re: [tuning] Digest Number 3634

I am not sure because one needs to convert it to vibrations per second to really know. cents doesn't contain this type of information

Message: 3 Date: Sat, 27 Aug 2005 14:24:19 +0300
From: "Ozan Yarman" <ozanyarman@superonline.com>
Subject: Re: Proportional stretch function

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 ----- -- 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

Ok, so, how does one convert combined pitches (chords) in cents to vps?

----- Original Message -----
From: Kraig Grady
To: tuning@yahoogroups.com
Sent: 27 Ağustos 2005 Cumartesi 17:24
Subject: Re: [tuning] Digest Number 3634

I am not sure because one needs to convert it to vibrations per second to really know. cents doesn't contain this type of information

Message: 3
Date: Sat, 27 Aug 2005 14:24:19 +0300
From: "Ozan Yarman" <ozanyarman@superonline.com>
Subject: Re: Proportional stretch function

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

Hi Ozan,

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

> > > From: "Ozan Yarman" <ozanyarman@s...>
> > >
> > > 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?
> >
> >
> > From: Kraig Grady
> >
> > I am not sure because one needs to convert it to
> > vibrations per second to really know. cents doesn't
> > contain this type of information
>
>
> Ok, so, how does one convert combined pitches (chords)
> in cents to vps?

First, you need some reference frequency "f".

Then, you measure all cents values in relation to that
reference pitch.

Then, to convert cents to vps (or cps or Hz, if you prefer):

2^(cents/1200) * f

-monz
http://tonalsoft.com
Tonescape microtonal music software

So, if my reference frequency is 260 Hz and my values originally were 386 and 702 cents, then the other two pitches are 260 * 2^(386/1200) Hz and 260 * 2^(702/1200) Hz respectively. This in effect equals:

260*1.25=325 Hz

and

260*1.5=390 Hz.

I already knew these Monz, but how do I calculate the beats from here?

And do you think that the `proportional stretch function` is a viable model for achieving maximum consonance in tempered chords? Do you know if this has been proposed by someone else before?

Cordially,
Ozan

----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 28 Ağustos 2005 Pazar 0:08
Subject: [tuning] convert cents to vps (was: Digest Number 3634)

Hi Ozan,

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

> > > From: "Ozan Yarman" <ozanyarman@s...>
> > >
> > > 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?
> >
> >
> > From: Kraig Grady
> >
> > I am not sure because one needs to convert it to
> > vibrations per second to really know. cents doesn't
> > contain this type of information
>
>
> Ok, so, how does one convert combined pitches (chords)
> in cents to vps?

First, you need some reference frequency "f".

Then, you measure all cents values in relation to that
reference pitch.

Then, to convert cents to vps (or cps or Hz, if you prefer):

2^(cents/1200) * f

-monz
http://tonalsoft.com
Tonescape microtonal music software

Hi Ozan,

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

> So, if my reference frequency is 260 Hz and my values
> originally were 386 and 702 cents, then the other two
> pitches are 260 * 2^(386/1200) Hz and 260 * 2^(702/1200) Hz
> respectively.

That's right.

> This in effect equals:
>
> 260*1.25=325 Hz
>
> and
>
> 260*1.5=390 Hz.
>
> I already knew these Monz, but how do I calculate the beats
> from here?

Well, if you're really talking about JI ratios, then there
will be no beats. The whole point of JI is that the intervals
don't beat.

If you really mean exactly 386 cents, then you will get
a beat rate of approximate 1 beat every 4 seconds.

You derive the beat rate by comparing the tempered interval
with the nearest JI ratio, as follows:

* multiply the higher frequency by the denominator of the ratio

* multiply the lower frequency by the numerator of the ratio

* subtract the results and use the absolute value

Thus, if you really mean exactly 386 cents and not the 5/4
ratio, then you get 2^(386/1200)*260 = ~324.9411127 Hz for
the higher frequency of your major-3rd.

~324.9411127 * 4 = ~1299.764451
260 * 5 = 1300

Absolute value of the difference = ~0.235549362 beats
per second, which is the same as 1 beat every ~4.245394646
seconds.

By comparison, the 12-edo major-3rd is much more active:

2^(400/1200) * 4 = ~327.579473 Hz.

~327.579473 * = ~1310.317892
260 * 5 = 1300

Absolute value of difference = ~10.31789189 beats per second.

> And do you think that the `proportional stretch function`
> is a viable model for achieving maximum consonance in
> tempered chords? Do you know if this has been proposed
> by someone else before?

I wasn't really following your discussion, and only jumped
in here where i saw that i could answer a question you had.

The concordance of intervals in a proportional stretch tuning
is highly dependent on the harmonic spectrum of the timbres
involved. The reason why a stretch tuning works well on a
piano is because the piano strings are under such high tension
that their harmonics (overtones) are higher in pitch than
the integer-values of the overtones would suggest. So stretching
the actual tuning of the notes makes them fit the stretched
harmonics better, giving an overall smoother sound.

Not really sure which "proportional stretch function"
you mean. I posted a graphic of the stretch tuning data
given in the manual for the Rhodes electric piano, which
i posted here:

/tuning/files/monz/rhodes.jpg

and discussed here:

/tuning/topicId_13830.html#13830

Allan Myhara provided a formula which described this curve:

/tuning/topicId_13830.html#13907

Hope that helps.

-monz
http://tonalsoft.com
Tonescape microtonal music software

Dear Monz,

Thank you for showing me how to calculate the beat rates!

So, a 392 cents major third from the base frequence of 260 Hz has the beat rate:

2^ (392/1200) * 260 = 326.07 Hz

(326.07 * 4) - (260 * 5) = 1304.28-1300 bps

which makes 4.28 bps or

1 over 4.28 = one beat every 0.233645 seconds.

In order to achieve the same beat rate for the minor third in a major triad, I have to temper the fifth so that the result is:

(f*2) - (260 * 3) = 4.28 bps

thence,

2f - 780 = 4.28

2f = 780 + 4.28

f= 784.28 / 2

f= 392.14 Hz

Thus the relative frequency of the fifth from the base is:

392.14 / 260 Hz = 1.5082307692307692307692307692308

The size in cents is:

1200 / (log 2) * log 1.5082307692307692307692307692308 =

3986.3137 * 0.1784678 =

~711.429 cents

Thus the equal beating tempered triad has the frequencies:

260 Hz
312.68 Hz
392.14 Hz

and the cent values are:

0
392
711.429

Is this correct so far?

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

Assuming that z>x,

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

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

Conversely, if `x` becomed `x+n`

then `z` becomes z+(z/x times n).

For example, if one modifies the pure third so that the new value is 392 cents (+5.686 cents), then to preserve the consonance of the chord, one needs to modify the fifth by this amount:

701.955+(5.686 times 701.955 / 386.314)= 701.955+10.332= 712.288 cents

Notice how close the result is between this fifth and the equal-beating one calculated above. The difference is less than a cent.

I had said previously that `this yields, if I am not wrong, equal beating of the intervals within a chord, and thus, produces a soothing consonant effect.`

I should now say that the results yield very close values to `equal-beating consonant chords`.

Monz, could this be a valuable contribution to your encyclopedia, since I didn't see any article on `beats`?

Cordially,
Ozan

----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 28 Ağustos 2005 Pazar 20:49
Subject: [tuning] Re: convert cents to vps (was: Digest Number 3634)

Hi Ozan,

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

> So, if my reference frequency is 260 Hz and my values
> originally were 386 and 702 cents, then the other two
> pitches are 260 * 2^(386/1200) Hz and 260 * 2^(702/1200) Hz
> respectively.

That's right.

> This in effect equals:
>
> 260*1.25=325 Hz
>
> and
>
> 260*1.5=390 Hz.
>
> I already knew these Monz, but how do I calculate the beats
> from here?

Well, if you're really talking about JI ratios, then there
will be no beats. The whole point of JI is that the intervals
don't beat.

If you really mean exactly 386 cents, then you will get
a beat rate of approximate 1 beat every 4 seconds.

You derive the beat rate by comparing the tempered interval
with the nearest JI ratio, as follows:

* multiply the higher frequency by the denominator of the ratio

* multiply the lower frequency by the numerator of the ratio

* subtract the results and use the absolute value

Thus, if you really mean exactly 386 cents and not the 5/4
ratio, then you get 2^(386/1200)*260 = ~324.9411127 Hz for
the higher frequency of your major-3rd.

~324.9411127 * 4 = ~1299.764451
260 * 5 = 1300

Absolute value of the difference = ~0.235549362 beats
per second, which is the same as 1 beat every ~4.245394646
seconds.

By comparison, the 12-edo major-3rd is much more active:

2^(400/1200) * 4 = ~327.579473 Hz.

~327.579473 * = ~1310.317892
260 * 5 = 1300

Absolute value of difference = ~10.31789189 beats per second.

> And do you think that the `proportional stretch function`
> is a viable model for achieving maximum consonance in
> tempered chords? Do you know if this has been proposed
> by someone else before?

I wasn't really following your discussion, and only jumped
in here where i saw that i could answer a question you had.

The concordance of intervals in a proportional stretch tuning
is highly dependent on the harmonic spectrum of the timbres
involved. The reason why a stretch tuning works well on a
piano is because the piano strings are under such high tension
that their harmonics (overtones) are higher in pitch than
the integer-values of the overtones would suggest. So stretching
the actual tuning of the notes makes them fit the stretched
harmonics better, giving an overall smoother sound.

Not really sure which "proportional stretch function"
you mean. I posted a graphic of the stretch tuning data
given in the manual for the Rhodes electric piano, which
i posted here:

/tuning/files/monz/rhodes.jpg

and discussed here:

/tuning/topicId_13830.html#13830

Allan Myhara provided a formula which described this curve:

/tuning/topicId_13830.html#13907

Hope that helps.

-monz
http://tonalsoft.com
Tonescape microtonal music software

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Hi Ozan,

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

> Dear Monz,
>
> Thank you for showing me how to calculate the beat rates!

Glad that i could be of help!

I don't have time to give a proper response to your post
right now ... but here's one thing:

> <big snip>
>
> I had said previously that `this yields, if I am not wrong,
> equal beating of the intervals within a chord, and thus,
> produces a soothing consonant effect.`
>
> I should now say that the results yield very close values
> to `equal-beating consonant chords`.
>
> Monz, could this be a valuable contribution to your
> encyclopedia, since I didn't see any article on `beats`?

Absolutely ... one of the glaring ommissions in the
Encyclopedia right now is the lack of a page about "beats".
I really do need to include something on that ASAP.

Please note my previous criticism of equal-beating
temperaments: personally, i do not like the pulsating
effect, unless the musical style specifically makes
us of it as part of the piece's rhythmic structure,
in which case it can be put to very interesting use.

Hopefully i'll have time to address the rest of your post.

-monz
http://tonalsoft.com
Tonescape microtonal music software

----- Original Message -----
From: monz
To: tuning@yahoogroups.com
Sent: 29 Ağustos 2005 Pazartesi 21:35
Subject: [tuning] Re: convert cents to vps (was: Digest Number 3634)

Hi Ozan,

>
> Monz, could this be a valuable contribution to your
> encyclopedia, since I didn't see any article on `beats`?

Absolutely ... one of the glaring ommissions in the
Encyclopedia right now is the lack of a page about "beats".
I really do need to include something on that ASAP.

Now that I am smug about beats, maybe I can assist you with the text?

Please note my previous criticism of equal-beating
temperaments: personally, i do not like the pulsating
effect, unless the musical style specifically makes
us of it as part of the piece's rhythmic structure,
in which case it can be put to very interesting use.

Hopefully i'll have time to address the rest of your post.

Anytime!

-monz
http://tonalsoft.com
Tonescape microtonal music software

Oz.