A more perfect 12 tone octave?

First off, I'm curious to know what software programs can be used to play non 12tet tunings and temperaments, and whether or not there are plugins for Logic Pro or Pro Tools that I could use.

Now, on to the more experimental stuff...

Recently, I examined the absolute frequencies of all the pitches in a 12 tet octave, and came to the realization that the Perfect 4th and the Perfect 5th are located at ~33% (1/3rd) and ~50% (1/2) of the total octave, respectively. The Flat 7th scale degree is at ~75% (3/4ths) of the total octave. If you were to tune these pitches to the exact fraction, 1/3rd, 1/2, 3/4ths and so on, would that be a better approximation of how these pitches relate with the root note of whatever scale you play? For instance, would a Perfect 5th, tuned to the mid point of the octave (exactly 50%), blend more consonantly to the root?? How could I experiment with this today, needing no extra hardware besides a computer with a keyboard?

-- In tuning@yahoogroups.com, "zacks" <forgiste@...> wrote:
> First off, I'm curious to know what software programs can be used to play non 12tet tunings and temperaments, and whether or not there are plugins for Logic Pro or Pro Tools that I could use.

I don't know about Logic Pro (I am sure others will chime in) but I use Scala software:
http://www.huygens-fokker.org/scala/

With this I can create text files to specify experimental intervals and sequences and play them back through MIDI. I know that much more is possible but have not done it myself.

> ... the Perfect 4th and the Perfect 5th are located at ~33% (1/3rd) and ~50% (1/2) of the total octave, respectively. The Flat 7th scale degree is at ~75% (3/4ths) of the total octave. ... would a Perfect 5th, tuned to the mid point of the octave (exactly 50%), blend more consonantly to the root??

Yes. You are talking in terms of adding frequencies; it is better to think in terms of multiplying frequencies, e.g. perfect 4th is 4/3 times the root, the perfect 5th is 3/2 times the root, flat 7th is 7/4 times the root (and octave is 2/1 times the root). You missed major 3rd is "1/4 of the octave" or better to say 5/4 times the root.

Better still to use ratio notation and express the above as 3:4, 2:3, 4:7, 1:2 and 4:5. Then you can extend to triads, as in the other thread going on at the moment, about 5:6:7, 7:9:11, 10:13:15, 14:18:21 etc.

I am sure I should be referring you to a FAQ about Just Intonation at this point.

HTH, and welcome to the list.

Steve M.

I'll attempt an answer.

Logic Pro does indeed let you adjust the tuning, with the limitation that you
can only have 12 pitches. When you adjust the C#, for example, it affects all
the C#'s (or Db's.)

http://www.sengpielaudio.com/calculator-centsratio.htm

The fifth (700 cents) is only flat by nearly 2 cents, so making the fifth
"perfect" at around 701.955 cents will produce a subtle difference, mostly in
the rate of beating.

2 cents is a very small interval and doesn't produce a big difference in quality
or feel.

So too with the fourth (500 cents)(ca. 4/3) and the "flat seventh" (1000 cents)
(ca.16/9).

So, these intervals -- generated by ratios with highest prime 3, are pretty
accurate in 12-tone equal temperament.

However, the *other* kinds of intervals -- thirds, the natural 7th, the 11th and
13th partial -- are not so accurate in 12-tone ET (or "EDO" as it is somewhat
more carefully referred to, a minor point.)

If you were to continue the process of generating a scale from exact 3/2 ratios
@ 701.955 cents, you'd be making a "Pythagorean" scale. As the chain of 5ths
continues, the deviation from 12EDO grows larger.

The range of tolerable 5ths might be +- around 6 cents -- that's partly a matter
of opinion. Maybe others can give a more accurate number.

People *do* design scales where very fine distinctions are important for the
effects on "beating" -- or the oscillation in amplitude, or wavering effect.
Scales have been designed where rates of beating are designed to be consistent,
for example.

In a pure Just Intonation scale of all overtone relationships, the "beating"
rate will equal the fundamental. (This is a little hard to explain in one
sentence, and so I apologize in advance.)

That is, the beating rate in something tuned to 15:16:17:18, or something with
whole integers, is exactly 1.

How "in tune" a scale sounds is at least partly dependent on the timbres it is
played with.

The default assumption is timbres that are harmonic, but this doesn't have to be
the case.

Caleb (well, I may have made enough mistakes to bring down the hammer of the
gods, we'll see.)

________________________________
From: zacks <forgiste@...>
To: tuning@yahoogroups.com
Sent: Wed, December 22, 2010 9:36:00 PM
Subject: [tuning] A more perfect 12 tone octave?

First off, I'm curious to know what software programs can be used to play non
12tet tunings and temperaments, and whether or not there are plugins for Logic
Pro or Pro Tools that I could use.

Now, on to the more experimental stuff...

Recently, I examined the absolute frequencies of all the pitches in a 12 tet
octave, and came to the realization that the Perfect 4th and the Perfect 5th are
located at ~33% (1/3rd) and ~50% (1/2) of the total octave, respectively. The
Flat 7th scale degree is at ~75% (3/4ths) of the total octave. If you were to
tune these pitches to the exact fraction, 1/3rd, 1/2, 3/4ths and so on, would
that be a better approximation of how these pitches relate with the root note of
whatever scale you play? For instance, would a Perfect 5th, tuned to the mid
point of the octave (exactly 50%), blend more consonantly to the root?? How
could I experiment with this today, needing no extra hardware besides a computer
with a keyboard?

Hi "Zacks",

Simply put,
1) Yes those ratios (your definitions of the perfect 3rd, 5th, 7th, etc.) ARE
the perfect ratios from a root
2) This perfection becomes scrambled when stacking the tones to form chords. IE
1/1 10/7 3/2 may be a relatively "perfect" suspended major chord from the
root...but between 3/2 and 10/7 the dyad of 21/20 is formed...which is far from
a simple/"perfect" interval.
3) Furthermore, perfection comes scrambled when stacking chord. 4:5:6:7 may be
a perfect major 7th chord...but now make 5 the root...which makes a triad of
5:6:7 from it...which is not as perfect as 4:5:6 and not "even" a perfect minor
chord...and actually sounds more like a diminished triad.

4) Say you stack notes like 4;5:6:7:8:9:10:11:12:13....to achieve "absolute
tonal perfection" relative to the root tone. This is simply a straight harmonic
series. Of course, when you make a spread-out chord starting from a high
harmonic IE 7:9:11...the chord sounds not very tonal at all and the entire scale
begins to sound like a giant chord (not exactly compositionally/emotionally
flexible).

Not to mention you get situations like the dyad 13/12 (or closer)...where
the notes are so close they vibrate fairly violently (critical band dissonance)
causing a feeling of emotional "un-settledness"/tension.

In short...there is no magic bullet to "tonality".
You are always trading one type of perfection for another...and the question
becomes which types of perfection, as a musician, do you value most.
Hence why 12TET is designed to balance between perfection from all possible
roots (though, as many here can tell you, tunings such as 31TET, 53TET, and
quarter comma meantone can do a far better job at that). So when making new
scales, you end up trying to sacrifice some perfection relative to the root in
order to keep perfection relative to other notes in the chord. Programs that
try to change notes dynamically when you switch root tones to partly fix this
issue are called Adaptive Just Intonation. Of course, by switching root tones
you must make other notes shift to accommodate...so you end up with notes
somewhat audibly sliding to form/re-form perfect chords...this is called "comma
shift".

----------------------

Now what can you use (for free) to make some nice/crazy microtonal music? :-)

My favorite program for really quick micro-tonal experimentation is modplug
tracker...which you can download here:
http://sourceforge.net/projects/modplug/files/OpenMPT%20Release%20Candidates/OpenMPT%20v1.17RC2%20%281.17.02.28%29/

You simply make a new MPTM file, create a new instrument (drag a
wave/instrument sample into the instrument box). Next click the "configure
tunings" option at the bottom of the "tunings" drop down box. Then you right
click to create a new tuning and select a type IE geometric for a tuning that
repeats on the octave/tritave/etc.. Then you select a repeating ratio/"period"
IE 2 for the octave and a number of steps IE 12 for 12 tones to the octave. Then
plug in decimal point values IE 1.2 = minor third and 1.5 = perfect fifth. Now
go back to you instrument and click the combo box to select the tuning to apply
it to the instrument. Then you are done...simply use the instrument to make
music.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> I don't know about Logic Pro (I am sure others will chime in) but I use Scala software:
> http://www.huygens-fokker.org/scala/
>
> With this I can create text files to specify experimental intervals and sequences and play them back through MIDI. I know that much more is possible but have not done it myself.

Rather than merely play them back through midi, a tried and true method is to render them. TiMidity++ will do this:

http://sourceforge.net/projects/timidity/

TiMidity can handle MTS midi files, which have some advantages.

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:

> Better still to use ratio notation and express the above as 3:4, 2:3, 4:7, 1:2 and 4:5. Then you can extend to triads, as in the other thread going on at the moment, about 5:6:7, 7:9:11, 10:13:15, 14:18:21 etc.

I know this seems to be the preferred method around here, but I don't like it as the colon is a standard math symbol which people use incorrectly if we assume standard notation. We keep being told that 3:4 and 4:3 are the same thing, denoting the undirected interval between 3 and 4. This is not standard math usage. Standard usage is that 3:4 is the same as 3/4, and 4:3 as 4/3. A continued proportion like 5:6:7 goes down, not up, and so should be written 7:6:5.

The system chosen here works very well for our purposes in music, but is problematic as it does not correspond to notation elsewhere.

I have looked into Just Intonation and the difference between that and equal temperament, although I was never 100% on what the ratios meant, such as 2:3.

I determined the above percentages by taking the frequency of the tone, dividing that by the total frequency range of the octave, and then multiplying that by 100 to get a percentage. I then rounded them to the nearest whole fraction (33% = 1/3). I realized these ratios are completely independent of the ratios other people use, since I'm not multiplying the root by the ratio.

Thanks for clearing that up for me. I actually was in the middle of installing Scala when I posted this.

On 12/22/2010 9:36 PM, zacks wrote:
> First off, I'm curious to know what software programs can be used to
> play non 12tet tunings and temperaments, and whether or not there are
> plugins for Logic Pro or Pro Tools that I could use.
>
> Now, on to the more experimental stuff...
>
> Recently, I examined the absolute frequencies of all the pitches in a
> 12 tet octave, and came to the realization that the Perfect 4th and
> the Perfect 5th are located at ~33% (1/3rd) and ~50% (1/2) of the
> total octave, respectively. The Flat 7th scale degree is at ~75%
> (3/4ths) of the total octave. If you were to tune these pitches to
> the exact fraction, 1/3rd, 1/2, 3/4ths and so on, would that be a
> better approximation of how these pitches relate with the root note
> of whatever scale you play? For instance, would a Perfect 5th, tuned
> to the mid point of the octave (exactly 50%), blend more consonantly
> to the root?? How could I experiment with this today, needing no
> extra hardware besides a computer with a keyboard?

Scala has been mentioned, and that's a great place to start. If you start looking into software synths or samplers, there's already some that can use Scala scales. Pianoteq is a great physically modeled piano synth, with optional plugins for other kinds of instruments like bells, clavinet, or marimba sounds. Garritan has instrument packs like the Garritan Personal Orchestra, World Instruments, and Jazz & Big Band. These are commercial products, but ZynAddSubFX is a free open source software synth that supports retuning.

One thing you'll run into if you try tuning notes to exact ratios (aka just intonation) is that some of the intervals you get will end up very close to a just interval, but far enough away that they sound out of tune. A simple example is that if you try tuning all the fifths just in a chain of fifths from Eb to G#, you'll end up with an out of tune "wolf" fifth G#-Eb, which is off by a comma. There are various ways to deal with this issue, but different solutions work with different kinds of music.

Hi Zack, and welcome to the list!

On Wed, Dec 22, 2010 at 9:36 PM, zacks <forgiste@...> wrote:
>
> Recently, I examined the absolute frequencies of all the pitches in a 12 tet octave, and came to the realization that the Perfect 4th and the Perfect 5th are located at ~33% (1/3rd) and ~50% (1/2) of the total octave, respectively. The Flat 7th scale degree is at ~75% (3/4ths) of the total octave. If you were to tune these pitches to the exact fraction, 1/3rd, 1/2, 3/4ths and so on, would that be a better approximation of how these pitches relate with the root note of whatever scale you play? For instance, would a Perfect 5th, tuned to the mid point of the octave (exactly 50%), blend more consonantly to the root??

That's an excellent realization to have made, and it forms the basis
for what they've been calling the "regular mapping paradigm" around
here.

One useful way to compare 12-tet to the absolute frequency ratios
you're talking about is to put the ratios into cents:

6/5 - minor third - 316 cents
5/4 - major third - 386 cents
4/3 - perfect fourth - 498 cents
3/2 - perfect fifth - 702 cents

where 100 cents is one 12-tet semitone. So a 12-tet major third is 400
cents, a fifth is 700 cents, etc.

So compared with an absolute (or "just") 3/2 and 4/3, 12-tet's
estimations are pretty good. So good, in fact, that you wouldn't be
able to tell the difference if you tried, unless you listened really
hard for evidence of an almost perceptible "beating" around the
harmonics of these notes.

The major and minor thirds aren't as good. 12-tet's major third is 14
cents sharp, and if you play a just major triad in which the
frequencies are in a pure 4:5:6 ratio, you can definitely hear the
difference.

One exciting frontier that microtonal music makes possible is to get
away from temperament entirely and explore writing only music in which
all of the ratios are just ("Just Intonation"). This means that every
chord that you play will be really resonant sounding, and you'll hear
phantom low notes popping out as your brain literally tries to "fuse"
the chords into a single note. Lots of people are fans of this
approach. This also makes a lot of interesting musical sonorities
possible, as you can explore writing music with frequency ratios that
don't pop up in 12-tet at all (like 11/8, for example. Try a chord
like 4:5:6:7:9:11:13).

Another exciting frontier, however, is to go in the opposite
direction: to understand that 400 cents in 12-equal is still
recognizable as a 5/4, and to exploit that property to make different
interesting temperaments with different flavors. This is useful
because tempering makes chord progressions possible that aren't
possible in just intonation.

What do I mean by that? For example, let's explore the "circle of
fifths" of 12-tet and see where it leads. Let's start off by saying
that C is 1/1.

C - 1/1
G - 3/2
D - 9/4
A - 27/16
E - 81/32

So four fifths on top of each other is an 81/32 ratio. What happens if
we take the "just" major third, though, like you said - a 5/4 ratio -
and move it up by two octaves? An octave is 2/1.

E = 5/4 * 2/1 * 2/1 = 5/1

So there are actually "two E's," so to speak. You have the E that you
arrive at by stacking a bunch of fifths, and the E that you arrive at
by stacking a 5/4 major third and two octaves. Although in 12-tet
these are "rounded off" (or "tempered") to the same note, in just
intonation they are not. The fifths-based one will be 21.5 cents
sharper than the other one, and the difference between them is an
ultra-small interval (or "comma") with a frequency ratio of 81/80.
12-tet eliminates this comma and rounds the two off to the same note,
and is hence a "meantone" temperament (any temperament eliminating
81/80 is called a meantone. 19-tet is another one).

So if you work with a tuning in which 81/80 doesn't vanish, then you
can distinguish between these two major thirds, and write music that's
more "pure," right? Well, maybe, but something weird ends up happening
with this chord progression:

||: Cmaj -> Am -> Dm -> Gmaj :||

If you do this chord progression in 12-equal (or any tuning in which
81/80 is rounded off to 0 cents), you end up back at the same Cmaj
every time. If you do this chord progression in just intonation, you
will end up at a Cmaj that is 21.5 cents lower than you started at
last time, and the pitch will drift lower and lower every time. So
although this chord progression in 12-tet sounds perfectly "natural,"
it's not really possible in strict JI while keeping one of the notes
common between each adjacent chord. This kind of progression is called
a "comma pump," and the fact that in 12-tet all of those C major
chords end up being the same pitch is called a "pun."

Lots of people are exploring tunings in which the "pure" JI ratios you
spoke of are intentionally detuned in different ways to create new
puns, and this is what I personally find the most interesting.

And then you have the people in between who explore
"microtemperaments," e.g. temperaments that are indistinguishably
close to JI that temper out almost infinitesimally small commas of a
cent or two. A tuning like 53-tet falls into this category, since it
can replicate 5-limit JI almost perfectly, and has only 53 notes
instead of an infinite amount of them. (There's a lot more to this
idea, but this will get you started).

> How could I experiment with this today, needing no extra hardware besides a computer with a keyboard?

Logic + adaptive hermode tuning for the win. If you use the 7-limit
version, then you can play dominant 7 chords and it'll automatically
retune them to 4:5:6:7. But you're really going to want to download
Scala and dive right in: http://www.huygens-fokker.org/scala/

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Hi Zack, and welcome to the list!
>
> On Wed, Dec 22, 2010 at 9:36 PM, zacks <forgiste@...> wrote:
> >
> > Recently, I examined the absolute frequencies of all the pitches in a 12 tet octave, and came to the realization that the Perfect 4th and the Perfect 5th are located at ~33% (1/3rd) and ~50% (1/2) of the total octave, respectively. The Flat 7th scale degree is at ~75% (3/4ths) of the total octave. If you were to tune these pitches to the exact fraction, 1/3rd, 1/2, 3/4ths and so on, would that be a better approximation of how these pitches relate with the root note of whatever scale you play? For instance, would a Perfect 5th, tuned to the mid point of the octave (exactly 50%), blend more consonantly to the root??

> So good, in fact, that you wouldn't be
> able to tell the difference if you tried, unless you listened really
> hard for evidence of an almost perceptible "beating" around the
> harmonics of these notes.

You can hear more difference than that. 702 cents is a touch brighter in sound than 700.

> And then you have the people in between who explore
> "microtemperaments," e.g. temperaments that are indistinguishably
> close to JI that temper out almost infinitesimally small commas of a
> cent or two. A tuning like 53-tet falls into this category, since it
> can replicate 5-limit JI almost perfectly, and has only 53 notes
> instead of an infinite amount of them. (There's a lot more to this
> idea, but this will get you started).

It's just barely possible to tell the difference between 53 and 5-limit JI. But using 118 or 171 on the other hand...

> > How could I experiment with this today, needing no extra hardware besides a computer with a keyboard?
>
> Logic + adaptive hermode tuning for the win.

I produce a lot of very complex microtonal music using nothing but a keyboard and a computer. But it does take time. Scala and TiMidity are basic tools.

On Thu, Dec 23, 2010 at 3:08 PM, genewardsmith
<genewardsmith@...> wrote:
>
> You can hear more difference than that. 702 cents is a touch brighter in sound than 700.

You can, but it's so subtle I doubt I'd do too well in a blind
listening test, especially in timbres where 3 and 6 are removed.

> > And then you have the people in between who explore
> > "microtemperaments," e.g. temperaments that are indistinguishably
> > close to JI that temper out almost infinitesimally small commas of a
> > cent or two. A tuning like 53-tet falls into this category, since it
> > can replicate 5-limit JI almost perfectly, and has only 53 notes
> > instead of an infinite amount of them. (There's a lot more to this
> > idea, but this will get you started).
>
> It's just barely possible to tell the difference between 53 and 5-limit JI. But using 118 or 171 on the other hand...

Ha, I knew you'd chime here. I was just trying to keep it simple :)

-Mike

> > You can hear more difference than that. 702 cents is a
> > touch brighter in sound than 700.
>
> You can, but it's so subtle

700 cents produces audible beating in every circumstance I've
tried. It's pretty hard to miss if you're listening for it.
In brisk music you won't hear it. It's a lot easier to hear
than the difference between 53 and 5-limit JI. For 7-limit
JI, interestingly, ETs up to and including 72 aren't so great.
I should try 99 again. -Carl

> 6/5 - minor third - 316 cents
> 5/4 - major third - 386 cents
> 4/3 - perfect fourth - 498 cents
> 3/2 - perfect fifth - 702 cents
>

Honestly I wasn't aware that the ratios I described were JI, I thought it was something different altogether. Is a Perfect 5th in JI located at the very center of the octave? Because in 12 equal it's not quite, although it close.

Hi Zacks,

> > 6/5 - minor third - 316 cents
> > 5/4 - major third - 386 cents
> > 4/3 - perfect fourth - 498 cents
> > 3/2 - perfect fifth - 702 cents
>
> Honestly I wasn't aware that the ratios I described were JI,
> I thought it was something different altogether. Is a
> Perfect 5th in JI located at the very center of the octave?
> Because in 12 equal it's not quite, although it close.

The ratios you gave were, I believe, in log frequency terms.
That means you're dividing the octave into perceptually equal
bits. The ratios above apply to frequency space (Hz).
Generally we use cents for log frequency (also above).
The perfect fifth in JI is 3/2, whereas it is about 351/600ths
of an octave in log terms.

If you have a tone at 440 Hz, multiplying by 1/2 gives 220 Hz,
and the interval between them will be an octave (2/1).
Halfing an octave in log terms gives 600 cents, a ratio of
sqrt(2) or 600 cents.

Does that make sense?

-Carl

Thank you so much! My questions have all been answered, I hadn't read all the responses before I posted that last one, but it's all very clear now.

I will start using cents as well, for sake of convention and ease of use.

----

On another topic, how many of you have tried applying the techniques of serialism and set theory to microtonality? There's something to be said about foreshadowing and subliminal suggestion in music. You could have an exposition at the beginning of the piece to introduce the listener to a set of tones in a particular order, then break that set down into smaller sub sets and use those to ornament to composition along the way, regardless of the tonality itself. This is atonal composition, and i think it would work well with odd tunings, especially scales with a prime number of tones.

On Thu, Dec 23, 2010 at 3:38 PM, Carl Lumma <carl@...> wrote:
>
> > > You can hear more difference than that. 702 cents is a
> > > touch brighter in sound than 700.
> >
> > You can, but it's so subtle
>
> 700 cents produces audible beating in every circumstance I've
> tried. It's pretty hard to miss if you're listening for it.
> In brisk music you won't hear it.

I can hear it, but it's subtle. I don't think I'd be able to hear it
at all on an acoustic instrument. Maybe.

> It's a lot easier to hear than the difference between 53 and 5-limit JI. For 7-limit
> JI, interestingly, ETs up to and including 72 aren't so great.
> I should try 99 again. -Carl

What about 7-limit harmonies in 72-tet do you not like? The worst
offender for beating in 72-tet, I think, is the 5/4, but I only notice
it if I'm playing in Scala with an unforgiving timbre.

Paul Erlich's 22-tet guitar sounded like it might as well have been
just when I heard it, but 22-tet in Scala doesn't. Even the fifths
weren't noticeably off - sort of. They were sharp, if I thought about
it, but my brain quickly remapped itself to hear them as just since
his timbre hid the beating. Something about the timbre of his guitar
hid everything. The sharpness was no longer unpleasant, it just made
for a really bright fifth.

I think half of the perception of a JI interval is periodicity buzz,
and the other half is VF placement. The former is destroyed pretty
easily, the latter is not.

-Mike

On Thu, Dec 23, 2010 at 3:40 PM, zacks <forgiste@...> wrote:
>
> > 6/5 - minor third - 316 cents
> > 5/4 - major third - 386 cents
> > 4/3 - perfect fourth - 498 cents
> > 3/2 - perfect fifth - 702 cents
> >
>
> Honestly I wasn't aware that the ratios I described were JI, I thought it was something different altogether. Is a Perfect 5th in JI located at the very center of the octave? Because in 12 equal it's not quite, although it close.

Yes. Let's say that the root is a 1/1 ratio, and the octave is 2/1.
The perfect fifth would be 3/2. If we put this in decimal numbers:

Root: 1.0
Perfect fifth: 1.5
Octave: 2.0

The thing is that perceptual pitch scales logarithmically with respect
to the incoming frequency, not linearly, so we hear the tritone as
actually bisecting the octave. The perfect bisection of the octave is
sqrt(2), and the fifth seems sharp of this.

Here's some other ones:

Root: 1/1 - 1.0
Major third: 5/4 - 1.25
Perfect fourth: 4/3 - 1.333333
Perfect fifth: 3/2 - 1.5
Minor sixth: 8/5 - 1.6
Subminor seventh: 7/4 - 1.75
Octave: 2/1 - 2.0

And so there you go.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> I can hear it, but it's subtle. I don't think I'd be able to
> hear it at all on an acoustic instrument. Maybe.

Most of my experience with it comes from hearing it on acoustic
keyboards.

> > It's a lot easier to hear than the difference between 53 and
> > 5-limit JI. For 7-limit JI, interestingly, ETs up to and
> > including 72 aren't so great. I should try 99 again. -Carl
>
> What about 7-limit harmonies in 72-tet do you not like? The
> worst offender for beating in 72-tet, I think, is the 5/4, but
> I only notice it if I'm playing in Scala with an
> unforgiving timbre.

They're very good but not as good as 53 in the 5-limit.

> Paul Erlich's 22-tet guitar sounded like it might as well have
> been just when I heard it,

Compare to a real just guitar and I think you'd say different.
Distortion on a JI guitar can knock down buildings.

> I think half of the perception of a JI interval is periodicity
> buzz, and the other half is VF placement. The former is
> destroyed pretty easily, the latter is not.

Agree.

-Carl

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> What about 7-limit harmonies in 72-tet do you not like? The worst
> offender for beating in 72-tet, I think, is the 5/4, but I only notice
> it if I'm playing in Scala with an unforgiving timbre.

For most music, I prefer the very slight sharpness of the odd 7-limit intervals in 99et than the slight flatness of 72et. Of course, 72 works a bit better if you toss in 11, but 99 is still workable and gives you two mappings for the price of one. It's actually an excellent tuning for some significant 11-limit temperaments: zeus, 11-limit hemififths and hitchcock.

> Paul Erlich's 22-tet guitar sounded like it might as well have been
> just when I heard it, but 22-tet in Scala doesn't.

Eh, give me a break. Next you'll tell me 27et sounds just. If you play the same JI piece in 22 and 19, and then in JI, for sure you'll notice the difference. Is the warped canon page still around now that Herman seems to have moved?

Even the fifths
> weren't noticeably off - sort of. They were sharp, if I thought about
> it, but my brain quickly remapped itself to hear them as just since
> his timbre hid the beating. Something about the timbre of his guitar
> hid everything. The sharpness was no longer unpleasant, it just made
> for a really bright fifth.

For sharpness which is quite bright but not unpleasant it's hard to beat the diaschismic temperament family, to which pajara and 22 belong as a class but which really struts its stuff in 58, 80, or echidna.

On Thu, Dec 23, 2010 at 7:22 PM, genewardsmith
<genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> > What about 7-limit harmonies in 72-tet do you not like? The worst
> > offender for beating in 72-tet, I think, is the 5/4, but I only notice
> > it if I'm playing in Scala with an unforgiving timbre.
>
> For most music, I prefer the very slight sharpness of the odd 7-limit intervals in 99et than the slight flatness of 72et. Of course, 72 works a bit better if you toss in 11, but 99 is still workable and gives you two mappings for the price of one. It's actually an excellent tuning for some significant 11-limit temperaments: zeus, 11-limit hemififths and hitchcock.

What do you think about 68-equal for 7-limit stuff? It's quickly
becoming one of my favorite tunings ever. I think if we had all gone
down the 17-equal path instead of the 12-equal path, as per George
Secor's paper, we'd all be talking about 68-tet instead of 72-tet.
It's inconsistent in the 11-limit, though, which is kind of a bummer.

> > Paul Erlich's 22-tet guitar sounded like it might as well have been
> > just when I heard it, but 22-tet in Scala doesn't.
>
> Eh, give me a break. Next you'll tell me 27et sounds just. If you play the same JI piece in 22 and 19, and then in JI, for sure you'll notice the difference. Is the warped canon page still around now that Herman seems to have moved?

Yes, I can obviously hear the differences between the tunings. It's
not like they're indistinguishable. He also had a 31-tet guitar, and
it obviously had a different feel to it than the 22-tet one.

What I'm saying is that before that experience, my only way of playing
around with different tunings was with Timidity and Scala. This means
my palette has been limited to GM sounds. With the usual GM sounds, I
think 22-tet's major triad sounds awful - the fifth is uncomfortably
sharp, and the slightly flat major third always bothered me for some
reason. After hearing the sounds played on an actual guitar, I
realized that part of what I hated most about these tunings is the
beating that pops up with GM's unforgiving timbres.

On the guitar, I could still tell that 22-tet's fifth was a little
wide, but it didn't sound bad anymore, or like I just had to get used
to it or something. It just sounded a bit wider and brighter, which
was fine. He had an Ensoniq keyboard with some magical timbres on it
that even made Mavila fifths sound pretty natural. They were just
really flat fifths, not uncomfortably dirty sonorities that need to be
avoided.

> Even the fifths
> > weren't noticeably off - sort of. They were sharp, if I thought about
> > it, but my brain quickly remapped itself to hear them as just since
> > his timbre hid the beating. Something about the timbre of his guitar
> > hid everything. The sharpness was no longer unpleasant, it just made
> > for a really bright fifth.
>
> For sharpness which is quite bright but not unpleasant it's hard to beat the diaschismic temperament family, to which pajara and 22 belong as a class but which really struts its stuff in 58, 80, or echidna.

Wow, 58 sounds awesome. I might like this better than 68 even, since
it seems to work better in the 11-limit.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> What do you think about 68-equal for 7-limit stuff? It's quickly
> becoming one of my favorite tunings ever.

I like it too.

> It's inconsistent in the 11-limit, though, which is kind of a bummer.

It's actually a good tuning for 11-limit shrutar, so I wouldn't dismiss it as an 11-limit system. If you like Paul's 22et guitar it ought to do for you.

> > For sharpness which is quite bright but not unpleasant it's hard to beat the diaschismic temperament family, to which pajara and 22 belong as a class but which really struts its stuff in 58, 80, or echidna.
>
> Wow, 58 sounds awesome. I might like this better than 68 even, since
> it seems to work better in the 11-limit.

58 actually works well up to the 17 limit, and I should add 13 or 17 limit diaschismic to my mention of echidna--the tunings are pretty close, in fact, even thought one has 3/2 as a generator and the other 9/7. It's something more I've been thinking about lately.

(Caleb waves his hand furiously)

Ooh, Ooh, me, me!

Be forewarned that some here who oppose the academic status quo dislike "set theory" because it is, perhaps, pseudo-scientific or pseudo-mathematical, and not so hearable. Same with serialism.

But no need to argue.

As a composer, one can always say: "I'm doing *this*" -- as long as you're willing to live with the consequences.

To me, extended serial techniques --above and beyond inversion, rotation, transposition, retrograde, and 5m -- are wonderful ways of generating material. But there is no guarantee that what's generated will sound good, of course. And, to be audible, these connections have to be emphasized in the music. There needs to be some simplicity.

However brilliant Boulez is, or Stockhausen was, I don't think anyone really hears the row forms in pieces like Boulez's Piano Sonatas. So whatever charm they have (for me, little) lies in something else.

I'm working on a partly serially-generated piece for which I have yet to decide on the exact tuning system. I'm thinking something close to a 12-tone EDO embedded in 46 tones per octave -- mostly because I've decided on 46EDO as my usual scale, and I have a hard time switching between basic tuning frameworks very fast. It takes me a long time to internalize something.

Somewhat idiosyncratically, I think of a series as being either a contour series or a series of pitch-classes, or a series of broadly-defined intervals -- it can be all those things or none. The series is often thought of as a cycle. The "fields" created by the adjacent notes are as important as anything else.

You generate something with these techniques, and then try to bend things as much as possible, to get a musical result.

I'm very interested in techniques of self-similarity, of which the most famous, perfect example is the Mallalieu series, or the index series of 2 ^ N mod 13. It's an all-interval series.

0 1 4 2 9 5 11 3 8 10 7 6

Here's the entry from my list of all-interval series. It's number 44. In the top line, B = 11, and A = 10.

044 014295B38A76014295B38A76 5*/0 Mallalieu series
C,Db,E,D,A,F,B,Eb,Ab,Bb,G,F#,C,Db,E,D,A,F,B,Eb,Ab,Bb,G,F#
1,3,10,7,8,6,4,5,2,9,11,6,1,3,10,7,8,6,4,5,2,9,11,6 Slonimsky's #1302
11,9,2,5,4,6,8,7,10,3,1,6,11,9,2,5,4,6,8,7,10,3,1,6
7,9,10,1,8,6,4,11,2,3,5,6,7,9,10,1,8,6,4,11,2,3,5,6
5,3,2,11,4,6,8,1,10,9,7,6,5,3,2,11,4,6,8,1,10,9,7,6, Slonimsky's #1308

The entry just says that the 5m is also listed in Slonimsky's Thesaurus of Scales as #1308.

This series has the property that if you start on the second note and take every other note, you have the same series as the original, only transposed up a semitone.

Same with the third note starting on the third note.

Same with the fourth note starting on the fourth.

This means that the series has versions of itself embedded in itself, which means that you can have polyphonic structures where the harmonies and the lines add up to the same thing.

Serialism isn't a hot topic -- like cubism, or Freudianism, or Communism, it might have peaked a few decades ago.

But I'm still addicted.

Happy to discuss, off-list, if anyone wants.

Caleb

On Dec 23, 2010, at 4:09 PM, zacks wrote:

> On another topic, how many of you have tried applying the techniques of serialism and set theory to microtonality? There's something to be said about foreshadowing and subliminal suggestion in music. You could have an exposition at the beginning of the piece to introduce the listener to a set of tones in a particular order, then break that set down into smaller sub sets and use those to ornament to composition along the way, regardless of the tonality itself. This is atonal composition, and i think it would work well with odd tunings, especially scales with a prime number of tones.

--- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:

> I'm very interested in techniques of self-similarity, of which the most famous, perfect example is the Mallalieu series, or the index series of 2 ^ N mod 13. It's an all-interval series.
>
> 0 1 4 2 9 5 11 3 8 10 7 6

It's also A008831 in Sloan's Online Encyclopedia of Integer Sequences, under the heading log2(n) mod 13. I'm not sure why you call "Mallalieu series" famous, as I can find no evidence it is. If you could cite some things here, I could update Sloan's list to include the information.

Heh, Google it, and you'll find mostly posts by me on one forum or another.

I mean famous by the standards of 12-tone series.

There was a discussion of it among composers who wrote serial music in the 70's.

Robert Morris, Milton Babbitt, some others.

http://www.google.com/search?q=%22more+on+0%2C1%2C4%22&hl=en&num=10&lr=&ft=i&cr=&safe=images&tbs=

I don't mean famous like Keith Richards.

But it has unique properties among the 12-tone series.

-c

________________________________
From: genewardsmith <genewardsmith@...>
To: tuning@yahoogroups.com
Sent: Fri, December 24, 2010 12:59:26 PM
Subject: [tuning] Re: A more perfect 12 tone octave?

--- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:

> I'm very interested in techniques of self-similarity, of which the most
>famous, perfect example is the Mallalieu series, or the index series of 2 ^ N
>mod 13. It's an all-interval series.
>
> 0 1 4 2 9 5 11 3 8 10 7 6

It's also A008831 in Sloan's Online Encyclopedia of Integer Sequences, under the
heading log2(n) mod 13. I'm not sure why you call "Mallalieu series" famous, as
I can find no evidence it is. If you could cite some things here, I could update
Sloan's list to include the information.

Here's my best result so far for a 46-note scale with something closely approximating a 12-tone EDO scale embedded in it.

This was made by taking a JI scale and quantizing it half-way (50%) to 108 EDO (which has a 5th @ 700 cents).

The step sizes are whacked. Formally ugly! But it hits many of the right targets.

I also have about a dozen other candidates, based on a variety of techniques. Amity3dh, diaschismic, 233 cents as generator, 46 notes of 53 tones per octave, and 46 notes of 58 tones per octave are some other possible scales.

If anyone wishes to suggest other possibilties or post other scales, I'd be as grateful as I ever am. Which is more than you probably think.

! Cal's xenophobia
46-note JI/108EDO hybrid
46
!
47.20100
89.40700
102.45600
113.22500
142.38100
165.98100
179.03100
202.63200
231.78800
265.95900
300.12900
313.17900
334.28100
357.88200
387.03800
402.58500
434.24000
465.89400
497.54800
544.74900
586.95500
600.00400
610.77300
634.37400
657.97400
676.57900
702.67800
755.43500
797.64100
810.69000
821.45800
845.05900
868.65900
887.26500
910.86600
934.46600
968.63700
1008.36300
1021.41300
1042.51400
1066.11500
1089.71600
1110.82000
1142.47300
1168.57200
1200.00900

On Dec 24, 2010, at 2:01 PM, Caleb Morgan wrote:

>
> Heh, Google it, and you'll find mostly posts by me on one forum or another.
>
> I mean famous by the standards of 12-tone series.
>
> There was a discussion of it among composers who wrote serial music in the 70's.
>
> Robert Morris, Milton Babbitt, some others.
>
> http://www.google.com/search?q=%22more+on+0%2C1%2C4%22&hl=en&num=10&lr=&ft=i&cr=&safe=images&tbs=
>
> I don't mean famous like Keith Richards.
>
> But it has unique properties among the 12-tone series.
>
> -c
>
> From: genewardsmith <genewardsmith@...>
> To: tuning@yahoogroups.com
> Sent: Fri, December 24, 2010 12:59:26 PM
> Subject: [tuning] Re: A more perfect 12 tone octave?
>
>
>
>
> --- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:
>
> > I'm very interested in techniques of self-similarity, of which the most famous, perfect example is the Mallalieu series, or the index series of 2 ^ N mod 13. It's an all-interval series.
> >
> > 0 1 4 2 9 5 11 3 8 10 7 6
>
> It's also A008831 in Sloan's Online Encyclopedia of Integer Sequences, under the heading log2(n) mod 13. I'm not sure why you call "Mallalieu series" famous, as I can find no evidence it is. If you could cite some things here, I could update Sloan's list to include the information.
>
>
>

Caleb wrote :

> I'm very interested in techniques of self-similarity, of which the > most famous, perfect example is the Mallalieu series, or the index > series of 2 ^ N mod 13. It's an all-interval series.
>
> 0 1 4 2 9 5 11 3 8 10 7 6

Hi Caleb !

Could you explain what means "the index series of 2 ^ N mod 13", in common english please ?
>

> This series has the property that if you start on the second note > and take every other note, you have the same series as the > original, only transposed up a semitone.
> Same with the third note starting on the third note.
> Same with the fourth note starting on the fourth.

That's very interesting. But it seems you have to think of 0 1 4 2 9 5 11 3 8 10 7 6 (-) as a cycle of 13 degrees, right ?

Is there other sequences like this in different numbers of degrees ?
Have you some with 7 ?

> This means that the series has versions of itself embedded in > itself, which means that you can have polyphonic structures where > the harmonies and the lines add up to the same thing.

And polyrhythmic structures as well.
I suppose it could be applied with pertinence to many periodic systems, for example linear temperaments ?

Have you done some music with these sequences ?

Thanks,
- - - - - - -
Jacques

Hi Jacques,

Look for my answer tomorrow. I have two techniques, one which only works to make "perfect" series of length N-1, where N is prime. (So 6 notes, but not 7, or 10 notes.) The second will create 7-note sequences, but with irregular patterns of self-similarity.

I am not very well educated in math, sadly. So it will take me a little time to explain and to try some different things.

This is probably child's play for Gene.

Base: 3 Mod: 7 Period: 6

sequence: 6 2 1 4 5 3

Base: 5 Mod: 7 Period: 6

sequence: 5 3 4 1 0 2

technique #2: irregular 7-note

grid row: 0,1,2,3,4,5,6
user row: 0,2,4,6,1,3,5

user row: 0,3,4,6,1,2,5

user row: 0 3 4 6 1 2 5
grid row: 3 2 1 0 6 5 4
orbit: 3 0 6 1 4 5 2
X-swapped: 1 3 6 0 4 5 2 (this is the irregular sequence, but the self-sim is only backwards)
intervals: 2 3 1 4 1 4 6

user row: 0 3 4 6 1 2 5
grid row: 1 0 6 5 4 3 2
orbit: 1 0 3 2 5 6 4
X-swapped: 1 0 3 2 6 4 5 (same thing)
intervals: 6 3 6 4 5 1 3

Some 10-note "perfect" series, based on mod 11.

Base: 2 Mod: 11 Period: 10
---------------------------------------------------
Power: 1 2 3 4 5 6 7 8 9 10
---------------------------------------------------
Raw: 10 1 8 2 4 9 7 3 6 5
Rank: 9 0 7 1 3 8 6 2 5 4
---------------------------------------------------
Mod 11: 10 1 8 2 4 9 7 3 6 5
Mod 12: 10 1 8 2 4 9 7 3 6 5
Mod 13: 10 1 8 2 4 9 7 3 6 5
===================================================
Base: 6 Mod: 11 Period: 10
---------------------------------------------------
Power: 1 2 3 4 5 6 7 8 9 10
---------------------------------------------------
Raw: 10 9 2 8 6 1 3 7 4 5
Rank: 9 8 1 7 5 0 2 6 3 4
---------------------------------------------------
Mod 11: 10 9 2 8 6 1 3 7 4 5
Mod 12: 10 9 2 8 6 1 3 7 4 5
Mod 13: 10 9 2 8 6 1 3 7 4 5
===================================================
Base: 7 Mod: 11 Period: 10
---------------------------------------------------
Power: 1 2 3 4 5 6 7 8 9 10
---------------------------------------------------
Raw: 10 3 4 6 2 7 1 9 8 5
Rank: 9 2 3 5 1 6 0 8 7 4
---------------------------------------------------
Mod 11: 10 3 4 6 2 7 1 9 8 5
Mod 12: 10 3 4 6 2 7 1 9 8 5
Mod 13: 10 3 4 6 2 7 1 9 8 5
===================================================
Base: 8 Mod: 11 Period: 10
---------------------------------------------------
Power: 1 2 3 4 5 6 7 8 9 10
---------------------------------------------------
Raw: 10 7 6 4 8 3 9 1 2 5
Rank: 9 6 5 3 7 2 8 0 1 4
---------------------------------------------------
Mod 11: 10 7 6 4 8 3 9 1 2 5
Mod 12: 10 7 6 4 8 3 9 1 2 5
Mod 13: 10 7 6 4 8 3 9 1 2 5
===================================================

It's not too hard to understand, once I explain it.

More tomorrow.

caleb

On Dec 25, 2010, at 10:55 AM, Jacques Dudon wrote:

> Caleb wrote :
>
>
>> I'm very interested in techniques of self-similarity, of which the most famous, perfect example is the Mallalieu series, or the index series of 2 ^ N mod 13. It's an all-interval series.
>>
>> 0 1 4 2 9 5 11 3 8 10 7 6
>
> Hi Caleb !
>
> Could you explain what means "the index series of 2 ^ N mod 13", in common english please ?
>>
>
>
>> This series has the property that if you start on the second note and take every other note, you have the same series as the original, only transposed up a semitone.
>> Same with the third note starting on the third note.
>> Same with the fourth note starting on the fourth.
>
> That's very interesting. But it seems you have to think of 0 1 4 2 9 5 11 3 8 10 7 6 (-) as a cycle of 13 degrees, right ?
>
> Is there other sequences like this in different numbers of degrees ?
> Have you some with 7 ?
>
>> This means that the series has versions of itself embedded in itself, which means that you can have polyphonic structures where the harmonies and the lines add up to the same thing.
>
> And polyrhythmic structures as well.
> I suppose it could be applied with pertinence to many periodic systems, for example linear temperaments ?
>
> Have you done some music with these sequences ?
>
> Thanks,
> - - - - - - -
> Jacques
>
>
>
>

--- In tuning@yahoogroups.com, caleb morgan <calebmrgn@...> wrote:

> This is probably child's play for Gene.

Not really. For starters, I don't know what condition you need to satisfy.

Mike wrote :

> I think half of the perception of a JI interval is periodicity buzz,
> and the other half is VF placement. The former is destroyed pretty
> easily, the latter is not.

-Mike

Hi Mike, VF = virtual fundamental ? Just want to be sure I understand your idea here : Do you mean the position of the virtual fundamental is what remains in most cases, regardless of the periodicity buzz, or the audibility of it ?

- - - - - - -
Jacques

On Sat, Dec 25, 2010 at 4:24 PM, Jacques Dudon <fotosonix@...> wrote:
>
> Mike wrote :
>
> I think half of the perception of a JI interval is periodicity buzz,
> and the other half is VF placement. The former is destroyed pretty
> easily, the latter is not.
>
> -Mike
> Hi Mike, VF = virtual fundamental ? Just want to be sure I understand your idea here : Do you mean the position of the virtual fundamental is what remains in most cases, regardless of the periodicity buzz, or the audibility of it ?

Hi Jacques,

Yes, VF = virtual fundamental. As for your latter point, I'm not sure
I understand what you mean. My idea is that the perception of
"periodicity buzz" and the perception of the VF for a just sonority
are caused by two different psychoacoustic mechanisms. I think that
both of these percepts play a role when one hears a JI interval or
triad and "identifies it," and I think the two are subject sometimes
to introspective confusion.

I think this leads to a lot of confusion around here, especially when
folks start talking about "fields of attraction" and the like: 9/7,
for instance, is in 5/4's field of attraction in one sense, but not at
all in the other.

-Mike

I only get half of your idea, but never mind.
When you say "9/7 would be in 5/4's field of attraction in one sense",
I guess it can't be for its VF placement, so it must be for what you call "periodicity buzz" -
then it's clear that I don't know what is meant by "periodicity buzz".
Is there any page somewhere that refers to it ?
Then may be I will be able to understand why it would be "destroyed easily", and the other not.
- - - - - - -
Jacques

>> > Mike wrote :
>> >
>> > I think half of the perception of a JI interval is periodicity >> buzz,
>> > and the other half is VF placement. The former is destroyed pretty
>> > easily, the latter is not.
>> >
>> > -Mike
>
>
> > Hi Mike, VF = virtual fundamental ? Just want to be sure I > understand your
> idea here : Do you mean the position of the virtual fundamental is > what remains
> in most cases, regardless of the periodicity buzz, or the > audibility of it ?
>
> Hi Jacques,
>
> Yes, VF = virtual fundamental. As for your latter point, I'm not sure
> I understand what you mean. My idea is that the perception of
> "periodicity buzz" and the perception of the VF for a just sonority
> are caused by two different psychoacoustic mechanisms. I think that
> both of these percepts play a role when one hears a JI interval or
> triad and "identifies it," and I think the two are subject sometimes
> to introspective confusion.
>
> I think this leads to a lot of confusion around here, especially when
> folks start talking about "fields of attraction" and the like: 9/7,
> for instance, is in 5/4's field of attraction in one sense, but not at
> all in the other.
>
> -Mike

Hi Jacques,

I'm posting this on the tuning in list in case anyone else is interested. My
grasp of terminology is pretty shaky, so I'll try to keep things very basic.

I'll describe three methods. Each has limitations. They are all related, but
have important differences.

I was able to find some irregular 7-note self-similar "rows" which I'll post,
below. There are certainly other ones, but these will give you some ideas.

Before I begin, there's a useful article by Feldman called A Review of
Self-Similar Melodies (by composer Tom Johnson) -- a minimalist composer.

http://www.kalvos.org/johness2.html

Also, if you have access to a Mac and a little patience, perhaps you can get
some of these applications that I use to work. I can't get to them from this
computer at the moment, but I'll post links tomorrow.

I'm typing this in fixed-width courier font -- it might make a difference below
if anything needs to align.

Method 1 -- This is straight out of an old book called _A Table of Indices and
Power Residues_. (Norton) The book only uses one "base" -- the thing you
multiply -- the lowest base, so it is not a complete list.

(These are also called, I think, "discrete logarithms".

Let's use 2 ^ N mod 11 as an example.

In this technique, the modulus must be a prime, and the base must be (relatively
prime?) such that it makes a complete cycle of length modulus-1.

You can start with 1, or 2 -- since it's a cycle, it makes no difference to the
final result.

> multiply 2 by itself mod 11. (When the result exceeds 11, you subtract 11 from
>the result and multiply that number by the base, that is, 2.) This will give you
>the "power residue" sequence.

1,2,4,8,5,10,9,7,3,6

> Pair each of these numbers with the "ordinal number" -- just a number showing
>its place in the sequence, 0,1,2,3,4,etc. Actually it doesn't matter if you
>start with 1, the result will just be a "transposition" in the musical sense.

0,1 1,2 2,4 3,8, 4,5, 5,10, 6,9, 7,7, 8,3, 9,6

> Swap the pairs -- and reorder, such that what *was* the power residue sequence
>is now in order -- to get the "indices" or index sequence.

1,0 2,1, 3,8 4,2 5,4 6,9 7,7 8,3 9,6 10,5

And, taking the second number of each pair, we have our sequence:

0,1,8,2,4,9,7,3,6,5,(x) (x is a placeholder to make cycles more regular)

Taking every other number from the second number:

1,2,9,3,5,0,8,4,7,6 (the sequence plus 1, mod 11)

Taking every third number from the third number:

8,9,6,0,2,7,5,1,4,3 (the sequence plus 8 or minus 3, mod 11.)

So these are "perfect" sequences -- they are all-interval within whatever
modulus, and they are perfectly self-similar at each starting-point.

Method 2 -- This is something I either stole from my sketchy reading of Robert
Morris or perhaps I added a slight variation. I'm honestly not sure. This works
for any length sequence where you can find another sequence that makes a
complete orbit with it. It can make irregularly or regularly self-similar
sequences, so it is more inclusive. You can make the sequence of 2 ^ N mod 11,
above with it, and many more.

The user specifies the intervals of self-similarity only, then looks -- by
trial-and-error -- for whether there is a complete orbit that will make a
sequence.

I'll use length 7 for my example.

For the best sequence, we want some number sequence that will make a complete
orbit with 0,1,2,3,4,5,6. With no number the same, and the sequence
0,1,2,3,4,5,6 not backwards, but forwards.

There are no working combinations with 0,2,4,6,1,3,5 that produce a complete
orbit. There are many combinations where the orbit is 6 plus 1 -- where 1 number
matches, but all of these have the "grid row" going backwards. So the self-sim
will be backwards, and the self-sim will also be flawed.

Would you like to find (1) orbits of size n, (2) orbits of size n - 1, or (3)
both? 3

(n.b.--because I chose "both", it finds the less-optimal, less-audible
combinations listed below.)

grid row: 0,1,2,3,4,5,6
user row: 0,2,4,6,1,3,5

user row: 0 2 4 6 1 3 5
grid row: 6 5 4 3 2 1 0
orbit: 6 0 5 2 1 3 4
X-swapped: 1 4 3 5 6 2 0
intervals: 3 6 2 1 3 5 1

user row: 0 2 4 6 1 3 5
grid row: 5 4 3 2 1 0 6
orbit: 5 0 3 4 2 6 1
X-swapped: 1 6 4 2 3 0 5
intervals: 5 5 5 1 4 5 3

user row: 0 2 4 6 1 3 5
grid row: 4 3 2 1 0 6 5
orbit: 4 0 1 6 3 2 5
X-swapped: 1 2 5 4 0 6 3
intervals: 1 3 6 3 6 4 5

user row: 0 2 4 6 1 3 5
grid row: 3 2 1 0 6 5 4
orbit: 3 0 6 1 4 5 2
X-swapped: 1 3 6 0 4 5 2
intervals: 2 3 1 4 1 4 6

user row: 0 2 4 6 1 3 5
grid row: 2 1 0 6 5 4 3
orbit: 2 0 4 3 5 1 6
X-swapped: 1 5 0 3 2 4 6
intervals: 4 2 3 6 2 2 2

user row: 0 2 4 6 1 3 5
grid row: 1 0 6 5 4 3 2
orbit: 1 0 2 5 6 4 3
X-swapped: 1 0 2 6 5 3 4
intervals: 6 2 4 6 5 1 4

user row: 0 2 4 6 1 3 5
grid row: 0 6 5 4 3 2 1
orbit: 6 2 3 1 5 4 0
X-swapped: 6 3 1 2 5 4 0
intervals: 4 5 1 3 6 3 6

But we could try a less-regular self-similarity pattern such as 0,3,5,1,2,4,6.
And allow an orbit of 6 notes with 1 note left out, or just going to itself:

The computer finds these three possibilities. (Or you could do it without a
computer with trial-and-error, just as the computer does.)

user row: 0 3 5 1 2 4 6
grid row: 2 3 4 5 6 0 1
orbit: 2 0 4 5 1 6 3
X-swapped: 1 4 0 6 2 3 5
intervals: 3 3 6 3 1 2 3

user row: 0 3 5 1 2 4 6
grid row: 3 4 5 6 0 1 2
orbit: 3 0 2 6 1 4 5
X-swapped: 1 4 2 0 5 6 3
intervals: 3 5 5 5 1 4 5

user row: 0 3 5 1 2 4 6
grid row: 6 0 1 2 3 4 5
orbit: 6 0 3 2 1 5 4
X-swapped: 1 4 3 2 6 5 0
intervals: 3 6 6 4 6 2 1

This is not all that promising, but we can get an ok result, if not the best
result. Let's take the first one as an example.

We have to map these numbers to some 7-tone scale.

Let's make the self-similarity be at the fourth.

To do this, we can match numbers and pitches like so:

0123456
beadgcf

Back to the example. The computer found a combination that had a 6-number orbit
with the 3 going to itself.

user row: 0 3 5 1 2 4 6
grid row: 2 3 4 5 6 0 1
orbit: 2 0 4 5 1 6 3

We just stick the 3 on the end.

This gives us the one-line permutation of
2,0,4,5,1,6,3

Mapping these numbers to pitches-names we have:

e....g....b....f....a....d....c....

....................a---------c-----
e---------b--------------d---------
-----g---------f

-------------------------d----------
---------------f....a..............
e....g.........................c....
..........b..............

These should align on your screen such that you can see that the pitch-sequence
a,c,e,b,d,g,f is embedded in e,g,b,f,a,d,c at intervals of every 2 notes except
for b to d, where the interval is 3 notes, and f back to a, where the interval
is only 1 note.

As far as I have found, there are no "perfect" 7-note self-similar sequences.
They are all based on orbits 6 plus 1, and/or they all have irregular intervals
of self-similarity. (Not to mention that in a diatonic scale, the
self-similarity will always be slightly inexact, because the scale is 6 fourths
plus one tritone.) In 7EDO, the self-similarity would be more exact.

Method 3 -- probably of even less interest. You can take any length of any
power-residue sequence and it will be inexactly, or loosely, or fuzzily
self-similar.

Let's take length 12, and pick a modulus arbitrarily.

Say, mod 53.

The first 12 numbers of the indices of the power-residue series is:

Raw: 52 49 1 46 15 50 10 43 2 12 34 47

If we rank them from highest to lowest:

Rank: 11 9 0 7 4 10 2 6 1 3 5 8

and assign that to 12EDO pitches:

Rank: 11 9 0 7 4 10 2 6 1 3 5 8

B..A..C..G..E..Bb.D..F#.C#.D#.F.G#.

now look for partial, inexact self-similarity by aligning with every-other note
from the second note:

B..A..C..G..E..Bb.D..F#.C#.D#.F..G#.

A..G..Bb.F#.D#.G#.B..C..E..D..C#.F..

intervals in 12EDO:

10.10.10.11.11.10.9..6..3..11.8..9..

The first half of the row is self-similar +-1 (9,10,11)

Different mod-base combinations will be a little better or a little worse by
this method. This one was pretty typical.

Fuzzy self-similarity might be useful, because imitation in tonal music is often
inexact -- a major third instead of a minor third, that kind of thing.

Jacques, I'll attempt to send you the Macintosh apps that I'm using, in a
separate post.

Phew. I hope some of that made sense.

Now I'll read your post, and think about what *you've* said.

Caleb

________________________________
From: Jacques Dudon <fotosonix@...>
To: tuning@yahoogroups.com
Sent: Sun, December 26, 2010 4:11:28 AM
Subject: [tuning] Re: A more perfect 12 tone octave?

I only get half of your idea, but never mind.
When you say "9/7 would be in 5/4's field of attraction in one sense",
I guess it can't be for its VF placement, so it must be for what you call
"periodicity buzz" -
then it's clear that I don't know what is meant by "periodicity buzz".
Is there any page somewhere that refers to it ?
Then may be I will be able to understand why it would be "destroyed easily", and
the other not.

- - - - - - -
Jacques

> Mike wrote :
>>>
>>> I think half of the perception of a JI interval is periodicity buzz,
>>> and the other half is VF placement. The former is destroyed pretty
>>> easily, the latter is not.
>>>
>>> -Mike
>
>> Hi Mike, VF = virtual fundamental ? Just want to be sure I understand your
>idea here : Do you mean the position of the virtual fundamental is what remains
>in most cases, regardless of the periodicity buzz, or the audibility of it ?
>
>Hi Jacques,
>
>Yes, VF = virtual fundamental. As for your latter point, I'm not sure
>I understand what you mean. My idea is that the perception of
>"periodicity buzz" and the perception of the VF for a just sonority
>are caused by two different psychoacoustic mechanisms. I think that
>both of these percepts play a role when one hears a JI interval or
>triad and "identifies it," and I think the two are subject sometimes
>to introspective confusion.
>
>I think this leads to a lot of confusion around here, especially when
>folks start talking about "fields of attraction" and the like: 9/7,
>for instance, is in 5/4's field of attraction in one sense, but not at
>all in the other.
>
>-Mike

Oops, in my Method Two example below, I forgot one step.

After getting the 1-line permutation, you have to do the same kind of index
swap, or x-swap, that you did in the first method.

So, value and ordinal number pairs are swapped to give:

X-swapped: 1 4 0 6 2 3 5

Then, these are mapped to some pitch-classes such as 0=b, 1=e, etc.

Sorry for the confusion.

Caleb

________________________________
From: Caleb Morgan <calebmrgn@yahoo.com>
To: tuning@yahoogroups.com
Sent: Sun, December 26, 2010 8:36:08 AM
Subject: Re: [tuning] Re: techniques for generating self-similar number/tonal
series

Hi Jacques,

I'm posting this on the tuning in list in case anyone else is interested. My
grasp of terminology is pretty shaky, so I'll try to keep things very basic.

I'll describe three methods. Each has limitations. They are all related, but
have important differences.

I was able to find some irregular 7-note self-similar "rows" which I'll post,
below. There are certainly other ones, but these will give you some ideas.

Before I begin, there's a useful article by Feldman called A Review of
Self-Similar Melodies (by composer Tom Johnson) -- a minimalist composer.

http://www.kalvos.org/johness2.html

Also, if you have access to a Mac and a little patience, perhaps you can get
some of these applications that I use to work. I can't get to them from this
computer at the moment, but I'll post links tomorrow.

I'm typing this in fixed-width courier font -- it might make a difference below
if anything needs to align.

Method 1 -- This is straight out of an old book called _A Table of Indices and
Power Residues_. (Norton) The book only uses one "base" -- the thing you
multiply -- the lowest base, so it is not a complete list.

(These are also called, I think, "discrete logarithms".

Let's use 2 ^ N mod 11 as an example.

In this technique, the modulus must be a prime, and the base must be (relatively
prime?) such that it makes a complete cycle of length modulus-1.

You can start with 1, or 2 -- since it's a cycle, it makes no difference to the
final result.

> multiply 2 by itself mod 11. (When the result exceeds 11, you subtract 11 from
>the result and multiply that number by the base, that is, 2.) This will give you
>the "power residue" sequence.

1,2,4,8,5,10,9,7,3,6

> Pair each of these numbers with the "ordinal number" -- just a number showing
>its place in the sequence, 0,1,2,3,4,etc. Actually it doesn't matter if you
>start with 1, the result will just be a "transposition" in the musical sense.

0,1 1,2 2,4 3,8, 4,5, 5,10, 6,9, 7,7, 8,3, 9,6

> Swap the pairs -- and reorder, such that what *was* the power residue sequence
>is now in order -- to get the "indices" or index sequence.

1,0 2,1, 3,8 4,2 5,4 6,9 7,7 8,3 9,6 10,5

And, taking the second number of each pair, we have our sequence:

0,1,8,2,4,9,7,3,6,5,(x) (x is a placeholder to make cycles more regular)

Taking every other number from the second number:

1,2,9,3,5,0,8,4,7,6 (the sequence plus 1, mod 11)

Taking every third number from the third number:

8,9,6,0,2,7,5,1,4,3 (the sequence plus 8 or minus 3, mod 11.)

So these are "perfect" sequences -- they are all-interval within whatever
modulus, and they are perfectly self-similar at each starting-point.

Method 2 -- This is something I either stole from my sketchy reading of Robert
Morris or perhaps I added a slight variation. I'm honestly not sure. This works
for any length sequence where you can find another sequence that makes a
complete orbit with it. It can make irregularly or regularly self-similar
sequences, so it is more inclusive. You can make the sequence of 2 ^ N mod 11,
above with it, and many more.

The user specifies the intervals of self-similarity only, then looks -- by
trial-and-error -- for whether there is a complete orbit that will make a
sequence.

I'll use length 7 for my example.

For the best sequence, we want some number sequence that will make a complete
orbit with 0,1,2,3,4,5,6. With no number the same, and the sequence
0,1,2,3,4,5,6 not backwards, but forwards.

There are no working combinations with 0,2,4,6,1,3,5 that produce a complete
orbit. There are many combinations where the orbit is 6 plus 1 -- where 1 number
matches, but all of these have the "grid row" going backwards. So the self-sim
will be backwards, and the self-sim will also be flawed.

Would you like to find (1) orbits of size n, (2) orbits of size n - 1, or (3)
both? 3

(n.b.--because I chose "both", it finds the less-optimal, less-audible
combinations listed below.)

grid row: 0,1,2,3,4,5,6
user row: 0,2,4,6,1,3,5

user row: 0 2 4 6 1 3 5
grid row: 6 5 4 3 2 1 0
orbit: 6 0 5 2 1 3 4
X-swapped: 1 4 3 5 6 2 0
intervals: 3 6 2 1 3 5 1

user row: 0 2 4 6 1 3 5
grid row: 5 4 3 2 1 0 6
orbit: 5 0 3 4 2 6 1
X-swapped: 1 6 4 2 3 0 5
intervals: 5 5 5 1 4 5 3

user row: 0 2 4 6 1 3 5
grid row: 4 3 2 1 0 6 5
orbit: 4 0 1 6 3 2 5
X-swapped: 1 2 5 4 0 6 3
intervals: 1 3 6 3 6 4 5

user row: 0 2 4 6 1 3 5
grid row: 3 2 1 0 6 5 4
orbit: 3 0 6 1 4 5 2
X-swapped: 1 3 6 0 4 5 2
intervals: 2 3 1 4 1 4 6

user row: 0 2 4 6 1 3 5
grid row: 2 1 0 6 5 4 3
orbit: 2 0 4 3 5 1 6
X-swapped: 1 5 0 3 2 4 6
intervals: 4 2 3 6 2 2 2

user row: 0 2 4 6 1 3 5
grid row: 1 0 6 5 4 3 2
orbit: 1 0 2 5 6 4 3
X-swapped: 1 0 2 6 5 3 4
intervals: 6 2 4 6 5 1 4

user row: 0 2 4 6 1 3 5
grid row: 0 6 5 4 3 2 1
orbit: 6 2 3 1 5 4 0
X-swapped: 6 3 1 2 5 4 0
intervals: 4 5 1 3 6 3 6

But we could try a less-regular self-similarity pattern such as 0,3,5,1,2,4,6.
And allow an orbit of 6 notes with 1 note left out, or just going to itself:

The computer finds these three possibilities. (Or you could do it without a
computer with trial-and-error, just as the computer does.)

user row: 0 3 5 1 2 4 6
grid row: 2 3 4 5 6 0 1
orbit: 2 0 4 5 1 6 3
X-swapped: 1 4 0 6 2 3 5
intervals: 3 3 6 3 1 2 3

user row: 0 3 5 1 2 4 6
grid row: 3 4 5 6 0 1 2
orbit: 3 0 2 6 1 4 5
X-swapped: 1 4 2 0 5 6 3
intervals: 3 5 5 5 1 4 5

user row: 0 3 5 1 2 4 6
grid row: 6 0 1 2 3 4 5
orbit: 6 0 3 2 1 5 4
X-swapped: 1 4 3 2 6 5 0
intervals: 3 6 6 4 6 2 1

This is not all that promising, but we can get an ok result, if not the best
result. Let's take the first one as an example.

We have to map these numbers to some 7-tone scale.

Let's make the self-similarity be at the fourth.

To do this, we can match numbers and pitches like so:

0123456
beadgcf

Back to the example. The computer found a combination that had a 6-number orbit
with the 3 going to itself.

user row: 0 3 5 1 2 4 6
grid row: 2 3 4 5 6 0 1
orbit: 2 0 4 5 1 6 3

We just stick the 3 on the end.

This gives us the one-line permutation of
2,0,4,5,1,6,3

Mapping these numbers to pitches-names we have:

e....g....b....f....a....d....c....

....................a---------c-----
e---------b--------------d---------
-----g---------f

-------------------------d----------
---------------f....a..............
e....g.........................c....
..........b..............

These should align on your screen such that you can see that the pitch-sequence
a,c,e,b,d,g,f is embedded in e,g,b,f,a,d,c at intervals of every 2 notes except
for b to d, where the interval is 3 notes, and f back to a, where the interval
is only 1 note.

As far as I have found, there are no "perfect" 7-note self-similar sequences.
They are all based on orbits 6 plus 1, and/or they all have irregular intervals
of self-similarity. (Not to mention that in a diatonic scale, the
self-similarity will always be slightly inexact, because the scale is 6 fourths
plus one tritone.) In 7EDO, the self-similarity would be more exact.

Method 3 -- probably of even less interest. You can take any length of any
power-residue sequence and it will be inexactly, or loosely, or fuzzily
self-similar.

Let's take length 12, and pick a modulus arbitrarily.

Say, mod 53.

The first 12 numbers of the indices of the power-residue series is:

Raw: 52 49 1 46 15 50 10 43 2 12 34 47

If we rank them from highest to lowest:

Rank: 11 9 0 7 4 10 2 6 1 3 5 8

and assign that to 12EDO pitches:

Rank: 11 9 0 7 4 10 2 6 1 3 5 8

B..A..C..G..E..Bb.D..F#.C#.D#.F.G#.

now look for partial, inexact self-similarity by aligning with every-other note
from the second note:

B..A..C..G..E..Bb.D..F#.C#.D#.F..G#.

A..G..Bb.F#.D#.G#.B..C..E..D..C#.F..

intervals in 12EDO:

10.10.10.11.11.10.9..6..3..11.8..9..

The first half of the row is self-similar +-1 (9,10,11)

Different mod-base combinations will be a little better or a little worse by
this method. This one was pretty typical.

Fuzzy self-similarity might be useful, because imitation in tonal music is often
inexact -- a major third instead of a minor third, that kind of thing.

Jacques, I'll attempt to send you the Macintosh apps that I'm using, in a
separate post.

Phew. I hope some of that made sense.

Now I'll read your post, and think about what *you've* said.

Caleb

________________________________
From: Jacques Dudon <fotosonix@...>
To: tuning@yahoogroups.com
Sent: Sun, December 26, 2010 4:11:28 AM
Subject: [tuning] Re: A more perfect 12 tone octave?

I only get half of your idea, but never mind.
When you say "9/7 would be in 5/4's field of attraction in one sense",
I guess it can't be for its VF placement, so it must be for what you call
"periodicity buzz" -
then it's clear that I don't know what is meant by "periodicity buzz".
Is there any page somewhere that refers to it ?
Then may be I will be able to understand why it would be "destroyed easily", and
the other not.

- - - - - - -
Jacques

> Mike wrote :
>>>
>>> I think half of the perception of a JI interval is periodicity buzz,
>>> and the other half is VF placement. The former is destroyed pretty
>>> easily, the latter is not.
>>>
>>> -Mike
>
>> Hi Mike, VF = virtual fundamental ? Just want to be sure I understand your
>idea here : Do you mean the position of the virtual fundamental is what remains
>in most cases, regardless of the periodicity buzz, or the audibility of it ?
>
>Hi Jacques,
>
>Yes, VF = virtual fundamental. As for your latter point, I'm not sure
>I understand what you mean. My idea is that the perception of
>"periodicity buzz" and the perception of the VF for a just sonority
>are caused by two different psychoacoustic mechanisms. I think that
>both of these percepts play a role when one hears a JI interval or
>triad and "identifies it," and I think the two are subject sometimes
>to introspective confusion.
>
>I think this leads to a lot of confusion around here, especially when
>folks start talking about "fields of attraction" and the like: 9/7,
>for instance, is in 5/4's field of attraction in one sense, but not at
>all in the other.
>
>-Mike

Here's a link to all the applications I'm using. All custom.

http://www.box.net/shared/m37jhti1og#/shared/m37jhti1og/1/61968314

There's some directions to go with each one.

Handalg2 and DiscreteLogCalc are the two for generating series.

permRot is for looking at tables of permutations in which each line is a permutation of the one above it. Can use any symbols.

DiscreteLogCalc is a Java program -- should run on any platform with Java.

The others run on Mac from Terminal.

You may need to make the files executable:

For example, you can make lincho2.1 executable
(if that's the name of the file) by typing

chmod +x lincho2.1

I've been able to get copies of these programs working on both older and newer Macs, both Intel and PowerPC.

If anyone is interested, I'd be happy to provide more info.

(These are the apps I always wanted -- but you may have no use for them.)

Lincho is a custom 12-tone series manipulator -- it finds combinations of versions of a given row. But using it requires learning a vocabulary of special commands, listed in the directions. You can do things like making "chorales" or arrays where every row is some version of the series.

Caleb

On Dec 26, 2010, at 8:44 AM, Caleb Morgan wrote:

>
> Oops, in my Method Two example below, I forgot one step.
>
> After getting the 1-line permutation, you have to do the same kind of index swap, or x-swap, that you did in the first method.
>
> So, value and ordinal number pairs are swapped to give:
>
> X-swapped: 1 4 0 6 2 3 5
>
>
>
>
>
> Then, these are mapped to some pitch-classes such as 0=b, 1=e, etc.
>
>
>
> Sorry for the confusion.
>
>
>
> Caleb
>
>
> From: Caleb Morgan <calebmrgn@yahoo.com>
> To: tuning@yahoogroups.com
> Sent: Sun, December 26, 2010 8:36:08 AM
> Subject: Re: [tuning] Re: techniques for generating self-similar number/tonal series
>
>
>
> Hi Jacques,
>
> I'm posting this on the tuning in list in case anyone else is interested. My grasp of terminology is pretty shaky, so I'll try to keep things very basic.
>
> I'll describe three methods. Each has limitations. They are all related, but have important differences.
>
> I was able to find some irregular 7-note self-similar "rows" which I'll post, below. There are certainly other ones, but these will give you some ideas.
>
> Before I begin, there's a useful article by Feldman called A Review of Self-Similar Melodies (by composer Tom Johnson) -- a minimalist composer.
>
> http://www.kalvos.org/johness2.html
>
> Also, if you have access to a Mac and a little patience, perhaps you can get some of these applications that I use to work. I can't get to them from this computer at the moment, but I'll post links tomorrow.
>
> I'm typing this in fixed-width courier font -- it might make a difference below if anything needs to align.
>
> Method 1 -- This is straight out of an old book called _A Table of Indices and Power Residues_. (Norton) The book only uses one "base" -- the thing you multiply -- the lowest base, so it is not a complete list.
>
> (These are also called, I think, "discrete logarithms".
>
> Let's use 2 ^ N mod 11 as an example.
>
> In this technique, the modulus must be a prime, and the base must be (relatively prime?) such that it makes a complete cycle of length modulus-1.
>
> You can start with 1, or 2 -- since it's a cycle, it makes no difference to the final result.
>
> > multiply 2 by itself mod 11. (When the result exceeds 11, you subtract 11 from the result and multiply that number by the base, that is, 2.) This will give you the "power residue" sequence.
>
> 1,2,4,8,5,10,9,7,3,6
>
> > Pair each of these numbers with the "ordinal number" -- just a number showing its place in the sequence, 0,1,2,3,4,etc. Actually it doesn't matter if you start with 1, the result will just be a "transposition" in the musical sense.
>
> 0,1 1,2 2,4 3,8, 4,5, 5,10, 6,9, 7,7, 8,3, 9,6
>
> > Swap the pairs -- and reorder, such that what *was* the power residue sequence is now in order -- to get the "indices" or index sequence.
>
> 1,0 2,1, 3,8 4,2 5,4 6,9 7,7 8,3 9,6 10,5
>
> And, taking the second number of each pair, we have our sequence:
>
> 0,1,8,2,4,9,7,3,6,5,(x) (x is a placeholder to make cycles more regular)
>
> Taking every other number from the second number:
>
> 1,2,9,3,5,0,8,4,7,6 (the sequence plus 1, mod 11)
>
> Taking every third number from the third number:
>
> 8,9,6,0,2,7,5,1,4,3 (the sequence plus 8 or minus 3, mod 11.)
>
> So these are "perfect" sequences -- they are all-interval within whatever modulus, and they are perfectly self-similar at each starting-point.
>
> Method 2 -- This is something I either stole from my sketchy reading of Robert Morris or perhaps I added a slight variation. I'm honestly not sure. This works for any length sequence where you can find another sequence that makes a complete orbit with it. It can make irregularly or regularly self-similar sequences, so it is more inclusive. You can make the sequence of 2 ^ N mod 11, above with it, and many more.
>
> The user specifies the intervals of self-similarity only, then looks -- by trial-and-error -- for whether there is a complete orbit that will make a sequence.
>
> I'll use length 7 for my example.
>
> For the best sequence, we want some number sequence that will make a complete orbit with 0,1,2,3,4,5,6. With no number the same, and the sequence 0,1,2,3,4,5,6 not backwards, but forwards.
>
> There are no working combinations with 0,2,4,6,1,3,5 that produce a complete orbit. There are many combinations where the orbit is 6 plus 1 -- where 1 number matches, but all of these have the "grid row" going backwards. So the self-sim will be backwards, and the self-sim will also be flawed.
>
> Would you like to find (1) orbits of size n, (2) orbits of size n - 1, or (3) both? 3
>
> (n.b.--because I chose "both", it finds the less-optimal, less-audible combinations listed below.)
>
> grid row: 0,1,2,3,4,5,6
> user row: 0,2,4,6,1,3,5
>
> user row: 0 2 4 6 1 3 5
> grid row: 6 5 4 3 2 1 0
> orbit: 6 0 5 2 1 3 4
> X-swapped: 1 4 3 5 6 2 0
> intervals: 3 6 2 1 3 5 1
>
> user row: 0 2 4 6 1 3 5
> grid row: 5 4 3 2 1 0 6
> orbit: 5 0 3 4 2 6 1
> X-swapped: 1 6 4 2 3 0 5
> intervals: 5 5 5 1 4 5 3
>
> user row: 0 2 4 6 1 3 5
> grid row: 4 3 2 1 0 6 5
> orbit: 4 0 1 6 3 2 5
> X-swapped: 1 2 5 4 0 6 3
> intervals: 1 3 6 3 6 4 5
>
> user row: 0 2 4 6 1 3 5
> grid row: 3 2 1 0 6 5 4
> orbit: 3 0 6 1 4 5 2
> X-swapped: 1 3 6 0 4 5 2
> intervals: 2 3 1 4 1 4 6
>
> user row: 0 2 4 6 1 3 5
> grid row: 2 1 0 6 5 4 3
> orbit: 2 0 4 3 5 1 6
> X-swapped: 1 5 0 3 2 4 6
> intervals: 4 2 3 6 2 2 2
>
> user row: 0 2 4 6 1 3 5
> grid row: 1 0 6 5 4 3 2
> orbit: 1 0 2 5 6 4 3
> X-swapped: 1 0 2 6 5 3 4
> intervals: 6 2 4 6 5 1 4
>
> user row: 0 2 4 6 1 3 5
> grid row: 0 6 5 4 3 2 1
> orbit: 6 2 3 1 5 4 0
> X-swapped: 6 3 1 2 5 4 0
> intervals: 4 5 1 3 6 3 6
>
>
> But we could try a less-regular self-similarity pattern such as 0,3,5,1,2,4,6. And allow an orbit of 6 notes with 1 note left out, or just going to itself:
>
> The computer finds these three possibilities. (Or you could do it without a computer with trial-and-error, just as the computer does.)
>
> user row: 0 3 5 1 2 4 6
> grid row: 2 3 4 5 6 0 1
> orbit: 2 0 4 5 1 6 3
> X-swapped: 1 4 0 6 2 3 5
> intervals: 3 3 6 3 1 2 3
>
> user row: 0 3 5 1 2 4 6
> grid row: 3 4 5 6 0 1 2
> orbit: 3 0 2 6 1 4 5
> X-swapped: 1 4 2 0 5 6 3
> intervals: 3 5 5 5 1 4 5
>
> user row: 0 3 5 1 2 4 6
> grid row: 6 0 1 2 3 4 5
> orbit: 6 0 3 2 1 5 4
> X-swapped: 1 4 3 2 6 5 0
> intervals: 3 6 6 4 6 2 1
>
>
> This is not all that promising, but we can get an ok result, if not the best result. Let's take the first one as an example.
>
> We have to map these numbers to some 7-tone scale.
>
> Let's make the self-similarity be at the fourth.
>
> To do this, we can match numbers and pitches like so:
>
> 0 1 2 3
> 4 5 6
>
> b e a d g c f
>
>
>
> Back to the example. The computer found a combination that had a 6-number orbit with the 3 going to itself.
>
> user row: 0 3 5 1 2 4 6
>
> grid row: 2 3 4 5 6 0 1
>
> orbit: 2 0 4 5 1 6 3
>
>
>
>
>
> We just stick the 3 on the end.
>
>
>
> This gives us the one-line permutation of
>
> 2,0,4,5,1,6,3
>
>
>
> Mapping these numbers to pitches-names we have:
>
>
>
>
>
>
> e....g....b....f....a....d....c....
>
>
>
> ....................a---------c-----
>
> e---------b--------------d---------
>
> -----g---------f
>
>
>
>
>
> -------------------------d----------
>
> ---------------f....a..............
>
> e....g.........................c....
>
> ..........b..............
>
>
>
>
>
> These should align on your screen such that you can see that the pitch-sequence a,c,e,b,d,g,f is embedded in e,g,b,f,a,d,c at intervals of every 2 notes except for b to d, where the interval is 3 notes, and f back to a, where the interval is only 1 note.
>
>
>
> As far as I have found, there are no "perfect" 7-note self-similar sequences. They are all based on orbits 6 plus 1, and/or they all have irregular intervals of self-similarity. (Not to mention that in a diatonic scale, the self-similarity will always be slightly inexact, because the scale is 6 fourths plus one tritone.) In 7EDO, the self-similarity would be more exact.
>
>
>
>
>
> Method 3 -- probably of even less interest. You can take any length of any power-residue sequence and it will be inexactly, or loosely, or fuzzily self-similar.
>
>
>
> Let's take length 12, and pick a modulus arbitrarily.
>
>
>
> Say, mod 53.
>
>
>
> The first 12 numbers of the indices of the power-residue series is:
>
>
>
> Raw: 52 49 1 46 15 50 10 43 2 12 34 47
>
>
>
> If we rank them from highest to lowest:
>
>
>
> Rank: 11 9 0 7 4 10 2 6 1 3 5 8
>
>
>
>
>
> and assign that to 12EDO pitches:
>
>
>
> Rank: 11 9 0 7 4 10 2 6 1 3 5 8
>
>
>
> B..A..C..G..E..Bb.D..F#.C#.D#.F.G#.
>
>
>
> now look for partial, inexact self-similarity by aligning with every-other note from the second note:
>
>
>
> B..A..C..G..E..Bb.D..F#.C#.D#.F..G#.
>
>
>
> A..G..Bb.F#.D#.G#.B..C..E..D..C#.F..
>
>
>
> intervals in 12EDO:
>
>
>
> 10.10.10.11.11.10.9..6..3..11.8..9..
>
>
>
> The first half of the row is self-similar +-1 (9,10,11)
>
>
>
> Different mod-base combinations will be a little better or a little worse by this method. This one was pretty typical.
>
>
>
> Fuzzy self-similarity might be useful, because imitation in tonal music is often inexact -- a major third instead of a minor third, that kind of thing.
>
>
>
>
>
> Jacques, I'll attempt to send you the Macintosh apps that I'm using, in a separate post.
>
>
>
> Phew. I hope some of that made sense.
>
>
>
> Now I'll read your post, and think about what *you've* said.
>
>
>
> Caleb
>
>
>
>
>
> From: Jacques Dudon <fotosonix@...>
> To: tuning@yahoogroups.com
> Sent: Sun, December 26, 2010 4:11:28 AM
> Subject: [tuning] Re: A more perfect 12 tone octave?
>
>
> I only get half of your idea, but never mind.
>
> When you say "9/7 would be in 5/4's field of attraction in one sense",
> I guess it can't be for its VF placement, so it must be for what you call "periodicity buzz" -
> then it's clear that I don't know what is meant by "periodicity buzz".
> Is there any page somewhere that refers to it ?
> Then may be I will be able to understand why it would be "destroyed easily", and the other not.
> - - - - - - -
> Jacques
>
>
>>> > Mike wrote :
>>> >
>>> > I think half of the perception of a JI interval is periodicity buzz,
>>> > and the other half is VF placement. The former is destroyed pretty
>>> > easily, the latter is not.
>>> >
>>> > -Mike
>>
>>
>> > Hi Mike, VF = virtual fundamental ? Just want to be sure I understand your
>> idea here : Do you mean the position of the virtual fundamental is what remains
>> in most cases, regardless of the periodicity buzz, or the audibility of it ?
>>
>> Hi Jacques,
>>
>> Yes, VF = virtual fundamental. As for your latter point, I'm not sure
>> I understand what you mean. My idea is that the perception of
>> "periodicity buzz" and the perception of the VF for a just sonority
>> are caused by two different psychoacoustic mechanisms. I think that
>> both of these percepts play a role when one hears a JI interval or
>> triad and "identifies it," and I think the two are subject sometimes
>> to introspective confusion.
>>
>> I think this leads to a lot of confusion around here, especially when
>> folks start talking about "fields of attraction" and the like: 9/7,
>> for instance, is in 5/4's field of attraction in one sense, but not at
>> all in the other.
>>
>> -Mike
>
>
>
>

Caleb wrote :

> Hi Jacques,
>
> I'm posting this on the tuning in list in case anyone else is
> interested. My
> grasp of terminology is pretty shaky, so I'll try to keep things
> very basic.

Thanks Caleb for this fully detailed description of the techniques.
I think myself this field of research can have applications in
microtonal music.

> Before I begin, there's a useful article by Feldman called A Review of
> Self-Similar Melodies (by composer Tom Johnson) -- a minimalist
> composer.
>
> http://www.kalvos.org/johness2.html

I am totally fan of Tom Johnson ! A true dadaïst composer and an
immense poet of the melody. I was just listening recently to his
album "Rational melodies", recorded in Paris by the Dedalus Ensemble.
Waow, this article is serious too ! But I don't know this other work
of him, have to get into it.

> Method 1 -- This is straight out of an old book called _A Table of
> Indices and
> Power Residues_. (Norton) The book only uses one "base" -- the
> thing you
> multiply -- the lowest base, so it is not a complete list.
>
> (These are also called, I think, "discrete logarithms".
>
> Let's use 2 ^ N mod 11 as an example.
>
> In this technique, the modulus must be a prime, and the base must
> be (relatively
> prime?) such that it makes a complete cycle of length modulus-1.
>
> You can start with 1, or 2 -- since it's a cycle, it makes no
> difference to the
> final result.
>
> > multiply 2 by itself mod 11. (When the result exceeds 11, you
> subtract 11 from
> >the result and multiply that number by the base, that is, 2.) This
> will give you
> >the "power residue" sequence.
>
> 1,2,4,8,5,10,9,7,3,6
>
> > Pair each of these numbers with the "ordinal number" -- just a
> number showing
> >its place in the sequence, 0,1,2,3,4,etc. Actually it doesn't
> matter if you
> >start with 1, the result will just be a "transposition" in the
> musical sense.
>
> 0,1 1,2 2,4 3,8, 4,5, 5,10, 6,9, 7,7, 8,3, 9,6
>
> > Swap the pairs -- and reorder, such that what *was* the power
> residue sequence
> >is now in order -- to get the "indices" or index sequence.
>
> 1,0 2,1, 3,8 4,2 5,4 6,9 7,7 8,3 9,6 10,5
>
> And, taking the second number of each pair, we have our sequence:
>
> 0,1,8,2,4,9,7,3,6,5,(x) (x is a placeholder to make cycles more
> regular)

Amazing ! I could do it with different lengths, so I think I got it,
thanks !
However I noticed it does not always work in any base :
I could resolve this nice modulo 17 one, in base 2 :
0, 14, 1, 12, 5, 15, 11, 10, 2, 3, 7, 13, 4, 9, 6, 8

and mod 29 as well with base 2, and this mod 7 one, but with base 3 :
0, 2, 1, 4, 5, 3

and mod 23 with base 5, and 41 with base 7, cool !

... Can one same length have several solutions (I mean with different
patterns) ?

> So these are "perfect" sequences -- they are all-interval within
> whatever
> modulus, and they are perfectly self-similar at each starting-point.
>
> Method 2 -- This is something I either stole from my sketchy
> reading of Robert
> Morris or perhaps I added a slight variation. I'm honestly not
> sure. This works
> for any length sequence where you can find another sequence that
> makes a
> complete orbit with it. It can make irregularly or regularly self-
> similar
> sequences, so it is more inclusive. You can make the sequence of 2
> ^ N mod 11,
> above with it, and many more.

That's interesting for sure, but I will stick to the the 1st
technique for now !
Also because I want first to make a few experiments with the
"perfect" sequences.
Actually these researches on discrete logarithms are completely
related with some technique I've been using for my photosonic disks,
that I call "angles cycles", that generates many different waveforms,
also related with FM synthesis (frequency modulation).
So I want to check if these sequences produce more particular sounds !
Will let you know.
I also want to test them as polyrhythmic sequences, because that's
what they are basically I think.
Then of course they can be applied for sequencing scales, and it's
true that edos of the same lengths, having the same periodic number
of steps will provide certain musical illustrations. But not only
edos, not only octave divisions, and in many other possible forms
than in the classical serial music.

Thanks for sharing your enthusiasm, and let us hear what music you
make with that stuff !
- - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> ... Can one same length have several solutions (I mean with different
> patterns) ?

Any primitive root

http://en.wikipedia.org/wiki/Primitive_root_modulo_n

will work.

Sorry Jacques, I missed this in the holiday shuffle. Back to it now:

On Sun, Dec 26, 2010 at 4:11 AM, Jacques Dudon <fotosonix@...> wrote:
>
> I only get half of your idea, but never mind.
>
> When you say "9/7 would be in 5/4's field of attraction in one sense",
> I guess it can't be for its VF placement, so it must be for what you call "periodicity buzz" -

No, I do mean for its VF placement. Although it is possible to hear
9/7 generate a VF that points to its actual 1/1, it's also very common
to hear it produce a VF as though it were a distorted 5/4. The latter
case is more likely in my experience.

> then it's clear that I don't know what is meant by "periodicity buzz".
> Is there any page somewhere that refers to it ?
> Then may be I will be able to understand why it would be "destroyed easily", and the other not.

Periodicity buzz is, I think, related to your concept of equal
beating. When you play a just interval (or triad, or tetrad, etc)
there will be partials that beat against other partials, so the
overtones all undergo AM in some kind of polyrhythm with one another.

I haven't heard a satisfactory explanation of why it occurs, so my
best guess right now is that it occurs because the time-frequency
response of the ear isn't fixed across the board.

-Mike