Hello All You, out there in PD World.

This being my #1 Post, I want to say: Thanks for having me! Here, amidst the Pure Data community. I love open source software, and the principal of sharing knowledge generally.

I only recently discovered PD--last week! I am currently on vacation from obligation, and so I am taking my time to do some intellectual callibration, and I sure do appreciate all the help, guys. already! Specifically, Maelstorm & katjav have been super helpful and enlightening. katjav: I freaking love your website (I do not understand your website, yet). But I believe vortex-based mathematics is the key to free energy and harmonics (Though, I must say: I do not know why, exactly). And DSP is so very fascinating. But I have only the most rudimentary understanding of PD, thus far. Thanks to cheetomoskeeto and his youtube page, for the very helpful video tutorials.

As my first patch, merely an adolescent, I would like to share my Pythagorean tuning wheel.

Pythagoras' tuning system, generated by whole-number ratios of perfect fifths and octaves, is more ancient than our current international standard of tuning, the "equal temperament" of our pianos and MIDI keyboards. And if we examine the pitches and their relationships mathematically, we find that the ancient Pythagorean system is more elegant and beautiful (e.g., the standard tuning of Middle 'C' is 261.625565 hertz, Vs. the Pythagorean 'C' @ 256 | or, standard 'D' @ 293.67, Vs. Py's 'D' @ 288. This follows, because the Py scale is *generated by pure ratios: 3/2 (the Perfect Fifth) & 2/1 (the Octave, but of course). In contradistinction, my fellows, our MIDI standard is *Tempered--the ratios are not pure. This is not merely the unfolding of some conspiratory plot. It is for very good reason, I'm sure (I just . . not sure what the reason, here is. . what IS the hold up? Why the glitch? What is this infamous comma anywho?). It is a story, unfolding.

A digital signal is a stream of numbers. Pythagoras was the father of numerology. Give this a try if you want.

Feedback is greatly appreciated.
Peace~

http://www.pdpatchrepo.info/hurleur/Pythagorean_Octaves.pd

the comma is, to quote wikipedia, ' the difference between twelve just perfect fifths and seven octaves'
to see this use your ratios for fifth and octave 12 and 7 times respectively
there is also some information on the wikipedia page http://en.wikipedia.org/wiki/Pythagorean_comma
as well as a star graphic showing it similar to your own
another way to say this is that when you go around the circle of fifths to reach the same note it is not the same as the note that many octaves up, and because of this it becomes necessary to decide how, if you mean to derive all 12 tones from the circle of fifths, you divide the pitch set. Do you simply keep going up the circle of fifths until you reach the beginning pitch? or do you start going down from the fundamental when you reach the tritone?

edit: I see now that the definition of Pythagorean tuning http://en.wikipedia.org/wiki/Pythagorean_tuning uses the latter

in addition weird intervals can occur between the notes of a scale generated in this way, again to quote wikipedia from this page
http://en.wikipedia.org/wiki/Wolf_interval

In Pythagorean tuning, there are eleven justly tuned fifths sharper than 700 cents by about 1.955 cents (or exactly one twelfth of a Pythagorean comma), and hence one fifth will be flatter by twelve times that, which is 23.460 cents (one Pythagorean comma) flatter than a just fifth. A fifth this flat can also be regarded as howling like a wolf. There are also now eight sharp and four flat major thirds.

more stuff from the first page:

In equal temperament, pairs of enharmonic notes such as A♭ and G♯ are thought of as being exactly the same note — however, as the above table indicates, in Pythagorean tuning they have different ratios with respect to D, which means they are at a different frequency. This discrepancy, of about 23.46 cents, or nearly one quarter of a semitone, is known as a Pythagorean comma.

To get around this problem, Pythagorean tuning ignores A♭, and uses only the 12 notes from E♭ to G♯. This, as shown above, implies that only eleven just fifths are used to build the entire chromatic scale. The remaining fifth (from G♯ to E♭) is left badly out-of-tune, meaning that any music which combines those two notes is unplayable in this tuning. A very out-of-tune interval such as this one is known as a wolf interval. In the case of Pythagorean tuning, all the fifths are 701.96 cents wide, in the exact ratio 3:2, except the wolf fifth, which is only 678.49 cents wide, nearly a quarter of a semitone flatter.

If the notes G♯ and E♭ need to be sounded together, the position of the wolf fifth can be changed. For example, a C-based Pythagorean tuning would produce a stack of fifths running from D♭ to F♯, making F♯-D♭ the wolf interval. However, there will always be one wolf fifth in Pythagorean tuning, making it impossible to play in all keys in tune.

however for completely diatonic songs in one key it can sound very good

hope it helps!

Howling like a wolf! Thanks very much for the reply, sebfusmaster.

This is something I'm still very much grappling with understanding. This sort of 'comma' seems to be the main quirk inherent to music theory which has motivated the many variations in tone and complexity of music systems.

The pitches that I have solved for above were derived from D @ 288Hz. Knowing that, would you be able to tell me explicitly, where this wolf fifth might be hiding? I'm sure with enough focused analysis, it will become apparent, but I certainly do appreciate the expertise.

I am interested in the possibility of visualizing the sound w/ an oscilliscope, or an hilber scope like in this video:

The program is written for macOS, which I do not use. I want to be able to do something like this with PD. I have experimented with: coreyker's patch with kinda poor yet somewhat promising results. I see also that there may be other possibilities offered by the [hilbert~] object: http://puredata.hurleur.com/sujet-6428-hilbert, Fascinating! I will definitely be looking into that, because visualizing the sounds will help to reveal properties of intervals (including if they are in tune or not!) and, perhaps, may reveal many other properties and geometries. This sort of functionaliy would greatly sweeten my little suite of tuning patches.

As a final note, I'm interested in these ladies' work:

. Although I cannot find their source, they claim to have written software capable of dynamically re-tuning intervals during play. This seems like a fairly simple operation once the musical intervals are perfectly understood.

I have a lot of work to do. Please, do share any insights.

For this next patch, I saved the note frequencies to a file that is transfered to an array when the patch loads. The array is queried as midi notes are received into pd, this in turn is handled, as according to Maelstorm's "Polyphonic voice management" tutorial, by the poly function to allow for multiple voices.

So basically, its a polyphonic synth controlled by midi input, tuned according to the text file. It's another method to experiment with alternate midi tunings.

http://www.pdpatchrepo.info/hurleur/Tuned_Polyphonic_Synth.zip

Here's another instance of tuning functionality that am interested in creating in pd, much like the FlexTune system mentioned above: Unleash the beauty of pure harmony!

I'm definitely not interested in releasing any pay-to-play windows software.

i'll try to derive the wolf fifth: D=288
* 3/2 = 432 (A)
* 3/2 * 1/2 = 324 (E)
* 3/2 = 486 (B)
* 3/2 * 1/2 = 364 1/2 (F#)
* 3/2 = 546 3/4 (C#)
* 3/2 * 1/2 = 410 1/16 (G#)

now, for pythagorean tuning the circle of fifths are traversed downward from the fundamental at this point instead of continuing from the tritone so that the wolf interval is as far removed harmonically from the fundamental as possible (and not between the fourth and fundamental!):
288 (D)
* 2/3 * 2 = 384 (G)
* 2/3 * 2 = 512 (C)
* 2/3 = 341 1/3 (F)
* 2/3 * 2 = 455 1/9 (Bb)
* 2/3 = 303 11/27 (Eb)
* 2/3 * 2 = 404 44/81 (Ab)

well it is immediately apparent that Ab is not the same as G#, and for this reason the Ab is omitted from the pythagorean tuning system.
but the wolf fifth is still present:

G# - Eb (transposed up an octave) - 606 22/27 hz.
the relationship between fifths is of course supposed to be 3/2, let's see the difference in the ratio:
take this Eb/G# in this pythagorean system just derived in order to obtain the ratio of their frequencies:
(606 22/27) / (410 1/16) = 1 84997/177147 which does not equal 3/2. instead of being 1.5, it's like 1.47981055....
a perfect pythagorean fifth (ratio of 3:2) is 701.955...etc cents. The pythagorean comma is also evident from this number's deviation from 700.
but the wolf 5th is 678.494...etc cents (the formula to get cents is 1200 * log base 2 of pitch ratio, in this case 1.47981055 for the wolf and 1.5 for the perfect fifth). so the difference is basically a whole quarter tone, the wolf 5th is a quarter tone flat!
edit: whoops, not a quarter tone, rather a quarter of a half-step

I've been kind of an insomniac the past few days, and I am not in the proper state of mind to give your response full attention at this moment. But I will.

I just wanted to post a pair of patches that I am working on. The core of my 'just intonation' generator patch is all up and running--I might add some file option functionality. You input a midi key # and the frequency you desire it to be, and it uses the key as a fundamental to generate all notes based on the ratios (that I have found) of Just Intonation, storing them into a text file, which can be opened 'Tuned Synth.pd'

I will make it easier to switch between tunings with the synth controller open, using lists or perhaps a knob. But that is for another day . . tomorrow

. .
I am going to research scales and chords . . and music theory in general in the next few weeks.

http://www.pdpatchrepo.info/hurleur/Just_Intonation_-_Pure_Data.zip

I'm pretty much following you, but I need to do the math on paper. Or, w/ a brain. I will try again tom, I'm burned.

G'night

stumbled upon this just now https://github.com/fabb/Rational-Piano

Btw-- it seemed like a fun exercise so I ported that fifths diagram to Pd-l2ork:

www.jonathanwilkes.net/wheel.webm

It's built using some new features of data structures seen in the subpatches of the video. (I don't have a text object yet but you get the idea.)