Yaphi spectrum and tuning

Some time ago I happened to be thinking about the role phi could play in
tuning matters. Since I recently
noticed a lot of activity about that on the tuning list, I thought of
sharing these thoughts. I hope I will be
excused for not going through everything that went by here (I do have a
life, you know :-), and beforehand
I make excuses if anything I'm saying has been dealt with already.

As a starting point I think phi, in the context of harmonic spectra, is
mainly interesting to generate maximum
dissonance. Isn't it true that phi can be interpreted as one of the hardest
numbers to approach by a ratio of
small integers? (That's O.K. though. I'm certainly not against creating
dissonance, on the contrary)

However, I next started thinking about adjusting the harmonic spectrum to be
in a better concordance with phi.
I came up with the following construction:
Take the regular harmonic spectrum, designated as (fundamental 1):
1 2 3 4 5 6 ...
then lower all partials by 2-phi (0.38197) (times the fundamental) to get
the series:
0.61803 1.61803 2.61803 ....
For ease of notation of this series here, let's define phiN (N = 0 1 2 ...)
to be phi + N - 1 (so phi1 == phi):
phi0 phi1 phi2 phi3 ...

The first two intervals generated by this series are:
phi1 / phi0 == phi2 == phi^2
phi2 / phi0 == phi^3 (which makes phi2 / phi1 exactly phi)

These nice 'coincidences' are of course completely in the line of phi
properties. So, if we start with phi as basic interval of a scale (which, I
gather has been adopted as 'phitave' in these circles), these intervals are
trivially accounted for.
The next two are:
phi3 / phi0 = 5.8541 which, as an interval is 3.67228 phitave
phi4 / phi0 = 7.47214 ------ 4.17941 phitave

To accommodate these intervals in an ET, we have to find an optimal division
of the phitave to approximate these fractions (.67228 and .17941).
Obviously, 6 is a surprisingly simple and good candidate.
If we express the series as steps in an equal division of the phitave by 6,
we get:

0 12 18 22.034 25.076

Of course, instead of lowering the partials by 2-phi, we can also increase
by (phi-1), giving the same series, but without the first, phi0
1.61803 2.61803 3.61803 ....
With the same basic properties.

I would like everybody to take special note of the fact that we have a
musically correlated sound quality with a related scale construction,
without any reference to regular JI intervals, especially the octave.

Now about ways to obtain that spectrum. FM would be a solution, if you avoid
reflected sidebands. But since I played a lot with FM in the seventies, I've
grown tired of the predictable sound quality.

Another nice way is shifting the partials of any generated sound by
completing the analytic signal it stands for. We have to fabricate the
imaginary part of the signal, which we can do with a Hilbert transform. We
can then shift the complete spectrum by multiplying with fixed frequency
complex signal. This is easily done in Csound.

; suppose p4 is desired fundamental frequency
idfr = - 0.3819660113 * p4
ifr = p4 - idfr
;
; generate any harmonic sound with fundamental frequency ifr into a1
;
areal, aimag hilbert a1
asin1 oscili 1, idfr, 1
acos1 oscili 1, idfr, 1, 0.25
ash1 = areal * acos1 - aimag * asin1
; ash1 is the shifted spectrum with p4 as shifted fundamental

<CsScore>
f 1 0 16384 10 1

That's it for now.

--
Kees van Prooijen
www.kees.cc

That's very interesting. If I'm not wrong, Frequency Shifter used in electronic music of 50's could do similar job in shifting spectra.

I'm glad you've mentioned FM, as I want to go in the same direction. I don't think the result is unpredictable, if one knows something about ratios, modulation index, zero carrier etc. I'm aware of reflecting sidebands. Anytime I can use additive algorithms, and on my SY99 use two-FM-element sound to get 12 harmonics. There's a rarely known and used possibility to combine four sounds in Masterkeyboard mode, to get 48 harmonics. It's a pity my Kawai K5000 can work only with harmonic spectra adn harmonics are fixed...

Have you used your spectra in some compositions? Is it possible to hear it somewhere?

Thank you.

Daniel Forro

P.S.: Nice art you have done, and you like Antheil, me too. Beautiful pictures from Kyoto... I had the same Shinto wedding ceremony as your son :-)

On 9 May 2009, at 9:48 AM, Kees van Prooijen wrote:
> Some time ago I happened to be thinking about the role phi could > play in tuning matters. Since I recently
> noticed a lot of activity about that on the tuning list, I thought > of sharing these thoughts. I hope I will be
> excused for not going through everything that went by here (I do > have a life, you know :-), and beforehand
> I make excuses if anything I'm saying has been dealt with already.
>
> As a starting point I think phi, in the context of harmonic > spectra, is mainly interesting to generate maximum
> dissonance. Isn't it true that phi can be interpreted as one of > the hardest numbers to approach by a ratio of
> small integers? (That's O.K. though. I'm certainly not against > creating dissonance, on the contrary)
>
> However, I next started thinking about adjusting the harmonic > spectrum to be in a better concordance with phi.
> I came up with the following construction:
> Take the regular harmonic spectrum, designated as (fundamental 1):
> 1 2 3 4 5 6 ...
> then lower all partials by 2-phi (0.38197) (times the fundamental) > to get the series:
> 0.61803 1.61803 2.61803 ....
> For ease of notation of this series here, let's define phiN (N = 0 > 1 2 ...) to be phi + N - 1 (so phi1 == phi):
> phi0 phi1 phi2 phi3 ...
>
> The first two intervals generated by this series are:
> phi1 / phi0 == phi2 == phi^2
> phi2 / phi0 == phi^3 (which makes phi2 / phi1 exactly phi)
>
> These nice 'coincidences' are of course completely in the line of > phi properties. So, if we start with phi as basic interval of a > scale (which, I gather has been adopted as 'phitave' in these > circles), these intervals are trivially accounted for.
> The next two are:
> phi3 / phi0 = 5.8541 which, as an interval is 3.67228 phitave
> phi4 / phi0 = 7.47214 ------ 4.17941 phitave
>
> To accommodate these intervals in an ET, we have to find an optimal > division of the phitave to approximate these fractions (.67228 and .> 17941). Obviously, 6 is a surprisingly simple and good candidate.
> If we express the series as steps in an equal division of the > phitave by 6, we get:
>
> 0 12 18 22.034 25.076
>
> Of course, instead of lowering the partials by 2-phi, we can also > increase by (phi-1), giving the same series, but without the first, > phi0
> 1.61803 2.61803 3.61803 ....
> With the same basic properties.
>
> I would like everybody to take special note of the fact that we > have a musically correlated sound quality with a related scale > construction, without any reference to regular JI intervals, > especially the octave.
>
> Now about ways to obtain that spectrum. FM would be a solution, if > you avoid reflected sidebands. But since I played a lot with FM in > the seventies, I've grown tired of the predictable sound quality.
>
> Another nice way is shifting the partials of any generated sound by > completing the analytic signal it stands for. We have to fabricate > the imaginary part of the signal, which we can do with a Hilbert > transform. We can then shift the complete spectrum by multiplying > with fixed frequency complex signal. This is easily done in Csound.
>
> ; suppose p4 is desired fundamental frequency
> idfr = - 0.3819660113 * p4
> ifr = p4 - idfr
> ;
> ; generate any harmonic sound with fundamental frequency ifr into a1
> ;
> areal, aimag hilbert a1
> asin1 oscili 1, idfr, 1
> acos1 oscili 1, idfr, 1, 0.25
> ash1 = areal * acos1 - aimag * asin1
> ; ash1 is the shifted spectrum with p4 as shifted fundamental
>
> <CsScore>
> f 1 0 16384 10 1
>
> That's it for now.
>
> --> Kees van Prooijen
> www.kees.cc

On Fri, May 8, 2009 at 8:30 PM, Daniel Forro <dan.for@...> wrote:

> That's very interesting. If I'm not wrong, Frequency Shifter used in
> electronic music of 50's could do similar job in shifting spectra.

No, sorry. That's something completely different. That's a relative
frequency shift that keeps all the partial relations the same. It would
result in a multiplication of the spectral elements with a constant factor,
instead of adding a constant value.

> I'm glad you've mentioned FM, as I want to go in the same direction.
> I don't think the result is unpredictable,

I said the result of FM was predictable :-)
I meant, it always sounds like ... well, FM.

> if one knows something
> about ratios, modulation index, zero carrier etc. I'm aware of
> reflecting sidebands. Anytime I can use additive algorithms, and on
> my SY99 use two-FM-element sound to get 12 harmonics. There's a
> rarely known and used possibility to combine four sounds in
> Masterkeyboard mode, to get 48 harmonics. It's a pity my Kawai K5000
> can work only with harmonic spectra adn harmonics are fixed...

> Have you used your spectra in some compositions? Is it possible to
> hear it somewhere?

Working on it.

> Thank you.

> Daniel Forro

> P.S.: Nice art you have done, and you like Antheil, me too.
> Beautiful pictures from Kyoto... I had the same Shinto wedding
> ceremony as your son :-)

Thanks

Kees

To make it absolutely clear, my suggested scale is:

! phi_6.scl
!
Phi equal division by 6
6
!
138.84838
277.69677
416.54515
555.39353
694.24191
833.09030

To be used with sound spectrum (recalculated for fundamental 1.0):
1.0 - 2.618 - 4.2361 - 5.8541 - 7.4721 - 9.0902 - ....
or
1.0 - 1.618 - 2.2361 - 2.841 - 3.4721 - ....

Kees wrote:

> No, sorry. That's something completely different. That's a relative
> frequency shift that keeps all the partial relations the same. It would
> result in a multiplication of the spectral elements with a constant factor,
> instead of adding a constant value.

As far as I can remember, there WERE some gadgets who could do linear frequency shifts but certainly not in the 50s -- or, at least, I can't imagine at all how something like that could have been possibleas early as that.

Petr

--- In tuning@yahoogroups.com, Kees van Prooijen <keesvp@...> wrote:
>
> On Fri, May 8, 2009 at 8:30 PM, Daniel Forro <dan.for@...> wrote:
>
> > That's very interesting. If I'm not wrong, Frequency Shifter >used in
> > electronic music of 50's could do similar job in shifting >spectra.
>
> No, sorry. That's something completely different. That's a relative
> frequency shift that keeps all the partial relations the same. It >would
> result in a multiplication of the spectral elements with a constant >factor,
> instead of adding a constant value.

The Bode frequency shifter from the mid-late 60s shifts the partials by a constant Hz amount, and that's the one Daniel is thinking of.

>
> > I'm glad you've mentioned FM, as I want to go in the same >direction.
> > I don't think the result is unpredictable,
>
> I said the result of FM was predictable :-)
> I meant, it always sounds like ... well, FM.

FM is completely predictable, and easy as far as what partials are going to appear, but calculating the resulting amplitudes uses Bessel functions and is beyond me. However you can get a very good feel with practice. One curious thing about FM is the energy distribution- basically when you increase the modulation amount, you're spreading the energy into more sidebands, and that's part of the "FM sound". It gets weedy sounding because when you're making more high partials, you're robbing energy from the low to do so, if that makes sense.

A simple cheat- a 1 or 2 pole pitch-tracking LPF and make-up gain. :-)
>

On 10 May 2009, at 1:42 AM, Kees van Prooijen wrote:

> > I'm glad you've mentioned FM, as I want to go in the same > direction.
> > I don't think the result is unpredictable,
>
> I said the result of FM was predictable :-)
>

Oops, my mistake, you wrote it. So we have the same experience with it.

> I meant, it always sounds like ... well, FM.
>

Therefore I like Yamaha algorithms on DX/TX series where on later models different basic spectra are used, not only sine wave, and FM is combined with additive approach. Also RCM synthesis on SY/TG77 and SY99 goes more far with 45 algorithms and free definable algorithm by user, digital filters, freely definable feedback, looping envelopes and possibility to include samples. Nothing to say about the last FM generation in FS1r where they used 8 tone + 8 noise operators, 88 algorithms and combination with formant synthesis.

> Working on it.
>

Looking forward to it.

Daniel Forro

On 10 May 2009, at 5:47 AM, Cameron Bobro wrote:

> The Bode frequency shifter from the mid-late 60s shifts the > partials by a constant Hz amount, and that's the one Daniel is > thinking of.
>
>

You've got it, Cameron, thanks for mentioning.

> FM is completely predictable, and easy as far as what partials are > going to appear, but calculating the resulting amplitudes uses > Bessel functions and is beyond me.
>

I have some tables, but...

> However you can get a very good feel with practice.
>

... exactly. The ear is judge here.

> One curious thing about FM is the energy distribution- basically > when you increase the modulation amount, you're spreading the > energy into more sidebands, and that's part of the "FM sound". It > gets weedy sounding because when you're making more high partials, > you're robbing energy from the low to do so, if that makes sense.
>
> A simple cheat- a 1 or 2 pole pitch-tracking LPF and make-up gain. :-)
> >
>

Yes, exactly this was missing in DX/TX series, that's what Yamaha added in later FM generations on their synths - digital filters.

Daniel Forro

--- In tuning@yahoogroups.com, Daniel Forro <dan.for@...> wrote:

> >
>
> I have some tables, but...
>
> > However you can get a very good feel with practice.
> >
>
> ... exactly. The ear is judge here.

If I recall correctly, even one of the FM pioneers- Vercoe? maybe said something about how it's better just to go through it listening. After all, even if you can predict the numbers it doesn't mean you can predict the effect of the sound, especially since FM can do some pretty "unnatural" things. Even the inharmonic spectra are really too orderly to actually be like a struck piece of metal for example, so their metallic sound is like an abstraction.

Hello Kees,
This would be extremely easy to do using IFFT synthesis in MATLAB or
with C++. I don't know how csound works but I'm sure it might be
doable in there as well. Once I'm done with finals here, I can come up
with a program to do it if people would find it useful.

Since this signal won't be periodic, creating a sample of a period to
get this happening simply won't work. A trivially easy way to do this
would be to simply come up with some code that makes a second or two
of the signal and then to simply loop that sample, possibly
crossfading it into adjacent samples. I would hope most decent
commercial sampling engines do this.

As for your FM synth comment - I don't think generating the output
spectrum by means of FM vs generating the output spectrum by layering
successive sine waves on top of each other vs generating the output
spectrum by frequency shifting a harmonic series would make any
difference :) FM synthesis is simply a component of signal processing
- it doesn't have to be related to 80's power ballad keyboard sounds
:)

But the next time I take a study break I'll post a 1s sample of
precisely this spectrum, unless it's been done already - would be
interesting.

-Mike

Petr Par�zek wrote:
> > > Kees wrote:
> >> No, sorry. That's something completely different. That's a relative
>> frequency shift that keeps all the partial relations the same. It would
>> result in a multiplication of the spectral elements with a constant > factor,
>> instead of adding a constant value.
> > As far as I can remember, there WERE some gadgets who could do linear > frequency shifts but certainly not in the 50s -- or, at least, I can�t > imagine at all how something like that could have been possibleas early > as that.
> > Petr

Single-sideband modulation existed in the 50's and was used in radio transmission. If the receiver is slightly mistuned, the frequency spectrum will be shifted. Whether anyone had the idea to use radio equipment for musical purposes is another question, but it wouldn't surprise me if someone did.

Just little bit more points on this:

As far as I know SSB Modulators (called also Frequency Shifters) were
known in electronic music of 50's. It's necessary to distinguish
Frequency Shifter (shifting all harmonics/non harmonics, that means
changing timbre) and Pitch Shifter (shifting the pitch, of course
also here is some timbre change by formant shifting, Mickey Mouse
Effect).

Another processors:
Studio für elektronische Musik in Köln used Klangumwandler (which
combined two ring modulators and filters) since 1955. Some info is here:

www.aes.org/e-lib/browse.cfm?elib=155

(The principle was improved by Harald Bode in the beginning of 70's).

Check also this:

cgi.ebay.com/ws/eBayISAPI.dll?
ViewItem&item=220395709855&_trksid=p2762.l1259#ebayphotohosting

and

matrixsynth.blogspot.com/2009/04/360-systems-2020-frequency-shifter-
bode.html

and

www.wendycarlos.com/surround/surround4.html

Studio für elektronische Musik in München used Frequenzumsetzer since
about beginning of 60's.

Digital version:
Virsyn Prism

Daniel Forro

On 11 May 2009, at 11:17 AM, Herman Miller wrote:

> Petr Parízek wrote:
>>
>>
>> Kees wrote:
>>
>>> No, sorry. That's something completely different. That's a relative
>>> frequency shift that keeps all the partial relations the same.
>>> It would
>>> result in a multiplication of the spectral elements with a constant
>> factor,
>>> instead of adding a constant value.
>>
>> As far as I can remember, there WERE some gadgets who could do linear
>> frequency shifts but certainly not in the 50s -- or, at least, I
>> can’t
>> imagine at all how something like that could have been possibleas
>> early
>> as that.
>>
>> Petr
>
> Single-sideband modulation existed in the 50's and was used in radio
> transmission. If the receiver is slightly mistuned, the frequency
> spectrum will be shifted. Whether anyone had the idea to use radio
> equipment for musical purposes is another question, but it wouldn't
> surprise me if someone did.

I've done much experimenting with alternative tuning and creating my own scales and such since 2007. I've created songs using phi. A simple phi chord would be to use phi and inverse phi. I've actually used phi in my binaural beats as well.

A phitave scale consists of the ratio PHI:1, 833.0902963567408 cents.
An octave scale consists of the ratio 2:1, 1200.0000000000000 cents
A tritave scale consists of the ratio 3:1, 1901.9550008653877 cents.
A pitave (hahaha) scale consists of the ratio PI:1, 1981.7953553667823 cents.

You've discovered the equal-tempered PHI-6 scale, except the way you explained it is difficult to follow. You've only scratched the surface. I created a file that outputs all frequencies for all Equal-Tempered scales from PHI-2 to 3-13 haha.

But, Equal-Tempered scales aren't where you create a name for yourself. It's actually in the use of it... But more in the use of a Justly Tuned scale. Take the Bohlen-Pierce for example (3-13).

An Equal-Tempered thirteen step scale for example has zero intervals with twelve 146.3 cent steps. The BP Justly Tuned scale has thirteen steps, approximately three intervals which include 133, 169, 133, 148, 154, 147, 134, 147, 154, 148, 133, 169, 133 cent steps.

Personally, the scale sounds awful.

What you want to work with is the step beyond Justly Tuned, which I've also created scales for ^_^. It actually gets far more complex in the next step, but far more accurate and aesthetically pleasing.

Sorry it took this long for me to reply. I would have replied in May if I knew this existed. Today I got bored and typed in PHITAVE into Google and found this. It's a term I've used, but haven't really spread because no one really understands what I'm talking about.

I'll elaborate later on the next step if you don't understand where I'm going. It actually involves way more math.