Linear evenness and partial periodicity

Hi tuners.

Although I wasn't here for some time, I kept on improving my understanding about spectra and periodicity and all that stuff -- I mean, purely in terms of the waveforms themselves regardless of if we're talking about motion of acoustic pressure, heat, or anything else. Now I think I'm starting to give even more significance to chords made of "linearly equidistant pitches" than I was in the past -- and maybe some of you will do so too after reading what I've just finished writing.

Because there was some discussion of this topic some time ago, I decided to summarize some of my conclusions in a short document and demonstrate them by a couple of examples, which is exactly what I'm going to do now.

One sound example was made simply by repeating an ordinary unit impulse 4 times per second in such a way that the phase relationships in the entire frequency range were shifted 90 degrees ahead in each consecutive impulse. The other example was made by mixing cosine waves (of equal volumes) whose frequencies were 4Hz, 10, 16, 22, 28, Etc.

You can observe that the former, although it was made using phase shifting, is almost the same as mixing cosine waves of 3:7:11:15 and so on. Likewise, the latter, although it was made by mixing cosine waves, is almost the same as phase-shifting every consecutive impulse by -120° (the same as +240°).

The explanation for all this is found in the document.

Here's the link:
/tuning/files/PetrParizek/PP_Phase_Shift.rar

I'm thinking about the idea of possibly writing a "part 2" in the future; but this is what I have so far.

FYI: Since English is not my native language, there are a few spots where I'm not sure about the proper choice of words; so sorry for any inconvenience of this kind.

Comments are appreciated.

Petr

I wrote:

> You can observe that the former, although it was made using phase > shifting, is almost the same as mixing cosine waves of 3:7:11:15 and so > on.

Hell, I meant "1,5, 9, 13 etc.", of course.

Petr

Hi Petr,

These chords are neat. It looks like you're only focusing on JI for the
moment, or with rational chords (and hence not things like 1:phi:2phi-1).
If so, I think these are the same thing as George Secor's "isoharmonic"
chords.

I've been considering chords for a while now which don't have to be JI, and
also which don't have to be separated by a constant frequency difference -
so something like 1:phi:3phi-2 would be an example - it's a subset of the
chain 1:phi:2phi-1:3phi-2. I've since termed these chords "metaharmonic,"
to avoid conflict with the "isoharmonic" term.

They're fairly useful for periodicity buzz and proportional beating chords,
although I find the effect is stronger for chords that have smaller dyads
in general - 8:9:10 generates the effect more than 1:3:5, in other words.

-Mike

2011/11/25 Petr Pařízek <petrparizek2000@...>

> **
>
>
> Hi tuners.
>
> Although I wasn't here for some time, I kept on improving my understanding
> about spectra and periodicity and all that stuff -- I mean, purely in
> terms
> of the waveforms themselves regardless of if we're talking about motion of
> acoustic pressure, heat, or anything else. Now I think I'm starting to
> give
> even more significance to chords made of "linearly equidistant pitches"
> than
> I was in the past -- and maybe some of you will do so too after reading
> what
> I've just finished writing.
>
> Because there was some discussion of this topic some time ago, I decided
> to
> summarize some of my conclusions in a short document and demonstrate them
> by
> a couple of examples, which is exactly what I'm going to do now.
>
> One sound example was made simply by repeating an ordinary unit impulse 4
> times per second in such a way that the phase relationships in the entire
> frequency range were shifted 90 degrees ahead in each consecutive impulse.
> The other example was made by mixing cosine waves (of equal volumes) whose
> frequencies were 4Hz, 10, 16, 22, 28, Etc.
>
> You can observe that the former, although it was made using phase
> shifting,
> is almost the same as mixing cosine waves of 3:7:11:15 and so on.
> Likewise,
> the latter, although it was made by mixing cosine waves, is almost the
> same
> as phase-shifting every consecutive impulse by -120° (the same as +240°).
>
> The explanation for all this is found in the document.
>
> Here's the link:
> /tuning/files/PetrParizek/PP_Phase_Shift.rar
>
> I'm thinking about the idea of possibly writing a "part 2" in the future;
> but this is what I have so far.
>
> FYI: Since English is not my native language, there are a few spots where
> I'm not sure about the proper choice of words; so sorry for any
> inconvenience of this kind.
>
> Comments are appreciated.
>
> Petr
>
>
>

Mike wrote:

> If so, I think these are the same thing as George Secor's "isoharmonic"
> chords.

Exactly. This is what I had in mind. My aim was to possibly find new temperaments with a little help of the knowledge about these phenomena. For example, if we treat a concord like 1:2:3:4 or 2:3:4:5 as if it were 2:4:6:8 or 4:6:8:10, then we can make a concord which is acoustically "symmetric" by turning it into 3:5:7:9. And if 2:4:6:8 is the origin of Pythagorean intonation and 4:6:8:10 is the origin of 5-limit (or Didymean) intonation, then 3:5:7:9 (or 1:3:5:7) suggests a couple of nice non-octave temperaments, one of them being the equal-tempered version of BP. Similarly, something like 3:6:9:12:15 can be turned into 2:5:8:11:14 or 4:7:10:13:16, the latter of which includes the 4/1 and therefore also makes it possible to play 1:4:7:10:13:16. This "triharmonic" series of tones also suggests a few possible temperaments and the one I like the most of all can be found as "parizek_triharmonic.scl" in Manuel's archive. As to the 2:5:8:11:14 thing, I haven't explored it a lot yet.

> I've been considering chords for a while now which don't have to be JI, > and
> also which don't have to be separated by a constant frequency difference -
> so something like 1:phi:3phi-2 would be an example - it's a subset of the
> chain 1:phi:2phi-1:3phi-2. I've since termed these chords "metaharmonic,"
> to avoid conflict with the "isoharmonic" term.

I see.

> They're fairly useful for periodicity buzz and proportional beating > chords,

Agreed.

> although I find the effect is stronger for chords that have smaller dyads
> in general - 8:9:10 generates the effect more than 1:3:5, in other words.

And in the situations when the periodicity may be better perceivable, the actual frequencies are below our hearing range; try comparing 105:115:125 to 5:15:25.

Petr