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Old 21st December 2012, 03:41 AM   #41
calebprime
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Originally Posted by aggle-rithm View Post
Long ago I had the delusion that I could learn to be a piano tuner. The real trick, I soon found out, is to make every interval other than the octave just the right amount OUT of tune.
Yeah, I tried it myself briefly and realized it was something better left to a pro.

There's also the issue of how much you stretch the octaves in the bass.
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Old 21st December 2012, 03:52 AM   #42
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I decided, in the last few days, that I needed to make some small changes to my tuning/scale. It's still 36 pitches in 13.793-cent increments (87EDO) which approximate 11-limit JI, with a 13/12 thrown in there.

To change a Logic Pro "orchestra" with around 50 instruments in it to this tuning took around 5 hours of work -- tedious, but not prohibitive.

This is the way the application Lil' Miss Scale Oven displays the scale, with some editing. It has way too much precision, I don't need anything more accurate than a tenth of a cent.

// Scale Name: "rbhji87"
// Repeat Ratio: 2:1
// Scale Pattern: 0., 110.345, 137.931, 151.724, 165.517, 179.31, 206.897, 234.483, 262.069, 303.448, 317.241, 344.828, 386.207, 413.793, 441.379, 496.552, 537.931, 551.724, 579.31, 620.69, 648.276, 662.069, 703.448, 758.621, 786.207, 813.793, 855.172, 882.759, 910.345, 937.931, 965.517, 993.103, 1020.69, 1034.483, 1048.276, 1089.655
// Scale Pattern Type: Cents Absolute
// as cents difference: 110.345000, 27.586000, 13.793000, 13.793000, 13.793000, 27.587000, 27.586000, 27.586000, 41.379000, 13.793000, 27.587000, 41.379000, 27.586000, 27.586000, 55.173000, 41.379000, 13.793000, 27.586000, 41.380000, 27.586000, 13.793000, 41.379000, 55.173000, 27.586000, 27.586000, 41.379000, 27.587000, 27.586000, 27.586000, 27.586000, 27.586000, 27.587000, 13.793000, 13.793000, 41.379000, 110.345000

// Scale Pattern: 8, 2, 1, 1, 1, 2, 2, 2, 3, 1, 2, 3, 2, 2, 4, 3, 1, 2, 3, 2, 1, 3, 4, 2, 2, 3, 2, 2, 2, 2, 2, 2, 1, 1, 3, 8


// Scale Pattern: 0, 8, 10, 11, 12, 13, 15, 17, 19, 22, 23, 25, 28, 30, 32, 36, 39, 40, 42, 45, 47, 48, 51, 55, 57, 59, 62, 64, 66, 68, 70, 72, 74, 75, 76, 79, 87


Here are my premises:

1) use off-the-shelf gear, not specially-designed gear.

2) no activism -- this is not a social cause.

3) approximate JI, but a little beating, a little waver, is good.

4) number of notes is some multiple of 12 -- 36 is not to much, too few. Reason is that it will be easier to memorize a multiple of 12 pattern on a conventional keyboard layout. (This relates to 1) ).

5) 11-limit and as much 13-limit as you can get with 36 pitches.

6) moveable tuning-base -- (like the slide on a slide trombone).

7) equal step-sizes are not important at all, which is closely related to 8).

8) the 36-note set has many patterns, many subsets embedded in it, but it is not really a "thing" or a gestalt, in itself -- it's just the pitches you most often need to grab at once in one place for convenience.

9) I'm a scale-dweller rather than a scale-nomad: I practice a given scale/tuning for years and years. Others change scales more frequently. Because I practice something for a long time, complexity/difficulty is no objection. So a large number of pitches and unequal step-sizes are ok.

10) scale/tuning should be able to do everything that 12EDO can do and more.

IMO, it turns out that Harry Partch was that rare critter: A better theorist than he was a composer. (Opposite of Hindemith or Schoenberg)
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Old 7th January 2013, 02:08 PM   #43
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Hi Caleb, I just discovered this thread! Been listening to the first three pieces you posted in the OP.... enjoying the tonalities, if that's the word.

I think this third piece is the most pleasing to me so far... the second one was a mite too eery for my comfort... I'm just pottering about getting ready to fly to India day after tomorrow, so it's not full concentration... hence my simple response referencing "comfort", which I realise is not the main consideration in these enterprises...

I'll listen again in February after we get back!
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Old 2nd February 2013, 10:53 AM   #44
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Originally Posted by asydhouse View Post
Hi Caleb, I just discovered this thread! Been listening to the first three pieces you posted in the OP.... enjoying the tonalities, if that's the word.

I think this third piece is the most pleasing to me so far... the second one was a mite too eery for my comfort... I'm just pottering about getting ready to fly to India day after tomorrow, so it's not full concentration... hence my simple response referencing "comfort", which I realise is not the main consideration in these enterprises...

I'll listen again in February after we get back!
Thanks for the listen and comment!

Yeah, that piece is derivative of Bartok's Music for Strings, Percussion and Celeste (though he's far better) which sounds eerie to most normal people.

But not too eerie for...children of the night.
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Old 2nd February 2013, 11:14 AM   #45
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Thought I'd post this here, rather than my old thread called 'another of cp's boring music theory threads' iirc.

I still hope to write many microtonal pieces, but right now I want to write in good old 12-tone tuning, partly because it really is one of the "best" tunings if you like 5ths and not too many pitches!

I thought I'd do another MOF-row piece.

Today I found such a series, after quite a lot of hunting around, so I'm happy. (A rare thing)

(transposition not important) C#,F,G#,D,Bb,F#,E,A,Eb,B,C,G

This is a MOF series (Multiple Order Function). That's the term -- more or less -- for a series which has a version of itself embedded in itself.

There are hundreds of possible complete MOF series, but relatively few that sound good to me. And there are many in which I wouldn't know how to exploit the MOF property in a way that sounds good. But this one works for me.

It's derived by first defining a pattern of self-similarity (see user row), then seeing whether that pattern will support a complete permutation orbit with a chromatic scale (see grid row and orbit). This is an n-1 permutation, meaning that one note doesn't orbit, but stays fixed -- the A. Fortuitously, that pitch happens to land in a place that doesn't mess up the MOF property. Most n-1 perms have messed-up self-similarity (or MOF function).


user row: C..D..Eb.F..G..Ab.Bb.B..C#.E..F#.A
grid row:.Bb.B..C..C#.D..Eb.E..F..F#.G..Ab.A
orbit:....Bb.C..Eb.Ab.F#.C#.F..B..D..G..E..A
series:...C#.F..Ab.D..Bb.F#.E..A..Eb.B..C..G (sounds good!)
intrvals: 4..3..6..8..8..10.5..6..8..1..7..6

...................d.....f#....a..eb.b.....g
.............f........bb....e...........c...
..........c#....g#


1) has tight embedding
2) good hexachords (014579 and 013589)
3) enough intervallic variety
4) good in relation to potential pedal-tones c#,e,g# (C# minor or 037)
5) tonal but not too much, no dead spots, nothing silly
6) Multiple Order Function is usable without it sounding like ca-ca.

The B-C move in relation to C# is actually a sound I like a lot: b7 to 7 -- puts some tension on the tonic (C#) but doesn't wipe it out.
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Old 5th February 2013, 06:00 AM   #46
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Originally Posted by calebprime View Post
...

...another MOF-row piece.

...
(transposition not important) C#,F,G#,D,Bb,F#,E,A,Eb,B,C,G

This is a MOF series (Multiple Order Function). That's the term -- more or less -- for a series which has a version of itself embedded in itself.

...

It's derived by first defining a pattern of self-similarity (see user row), then seeing whether that pattern will support a complete permutation orbit with a chromatic scale (see grid row and orbit). This is an n-1 permutation, meaning that one note doesn't orbit, but stays fixed -- the A. Fortuitously, that pitch happens to land in a place that doesn't mess up the MOF property. Most n-1 perms have messed-up self-similarity (or MOF function).


user row: C..D..Eb.F..G..Ab.Bb.B..C#.E..F#.A
grid row:.Bb.B..C..C#.D..Eb.E..F..F#.G..Ab.A
orbit:....Bb.C..Eb.Ab.F#.C#.F..B..D..G..E..A
series:...C#.F..Ab.D..Bb.F#.E..A..Eb.B..C..G (sounds good!)
intrvals: 4..3..6..8..8..10.5..6..8..1..7..6

...................d.....f#....a..eb.b.....g
.............f........bb....e...........c...
..........c#....g#


1) has tight embedding
2) good hexachords (014579 and 013589)
3) enough intervallic variety
4) good in relation to potential pedal-tones c#,e,g# (C# minor or 037)
5) tonal but not too much, no dead spots, nothing silly
6) Multiple Order Function is usable without it sounding like ca-ca.

The B-C move in relation to C# is actually a sound I like a lot: b7 to 7 -- puts some tension on the tonic (C#) but doesn't wipe it out.
attempt at being more clear:

Purpose of this kind of series: It's one way that the harmonies and lines can be controlled and derived from the same material. Unity, in short. Vaguely fractal idea: The same patterns occur at different speeds.

Problem with this method: It only determines a pattern of self-similarity, it doesn't determine the other qualities of the series it makes -- whether it ends up sounding good is chance*. So you have to make hundreds of these before you find one you like, if you're choosy.

Better explanation of this method of making MOF series:

I'll use numbers instead of pitch-classes.

grid row:.10.11.0..1..2..3..4..5..6..7..8.(9)
user row: 0..2..3..5..7..8..10.11.1..4..6..9

(10 goes to 0, 0 goes to 3, 3 goes to 8, 8 goes to 6, etc. 9 stays fixed here)


orbit:....10.0..3..8..6..1..5..11.2..7..4.(9) one-line permutation

series:...1..5..8..2..10.6..4..9..3..11.0..7 swap values, positions of one-line perm
intrvals: 4..3..6..8..8..10.5..6..8..1..7..6 differences

...................2.....6.....9..3..11....7 (series + 1, in two cycles of series)
.............5........10....4...........0...
..........1.....8.


This is Mod 12 arithmetic -- everything is thought of like a clock...

C=0, Db and C#=1, D=2, D# and Eb=3, E=4, etc.

Start with the pattern of embedding. Is there some alignment of that series of numbers with some version of 0,1,2,3,4, etc. that produces a complete orbit? Or second-best, one where 11 elements orbit, and one stays fixed?

Having come up with a one-line permutation, think of that series as a set of paired order-numbers (0 to 11) and values, like so:

(10.0..3..8..6..1..5..11.2..7..4.)(9) one-line permutation

is 0,10 1,0 2,3 3,8 4,6 5,1 pairs of order-numbers and values

Then, to get the final result, you swap these pairs.

Instead of 10 in position 0, you have 0 in position 10
Instead of 0 in position 1, you have 1 in position 0
Instead of 3 in position 2, you have 2 in position 3
etc.

to get 1..5..8..2..10.6..4..9..3..11.0..7

converting these numbers to pitch-classes, you get your series.

Now, does it sound good? Could it be the source of a piece? Probably not.

Rinse and repeat.

*Well, not chance really, but not foreseeable or "forehearable" if you will
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Old 20th February 2013, 07:07 PM   #47
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Originally Posted by calebprime View Post
...

I thought I'd do another MOF-row piece.

...
(transposition not important) C#,F,G#,D,Bb,F#,E,A,Eb,B,C,G

...
https://www.box.com/s/litnhwpcyk8swpk46vo7

What's on paper seems good to me, but this sketch is ponderous. Only took about three hours from paper to this 9+ minute realization.

So I'm calling it Sea Bass with Frickin' Lasers.

It will take a lot more work to make something with more subtlety than this.

One problem is that I suck on piano. Another is that this was just slapped together. In that light, it came out pretty good.

Trying to exactly make the Multiple-Order embedding sound good turned out to be harder than it looked on paper.

This is only an .aif file, so it's 95+ megs, so it may not play directly, you may need to download it.
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Old 22nd February 2013, 05:11 AM   #48
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A few years ago, I was still thinking about finding myself a teacher, late in life, late in career. The teacher was to be a paid substitute for a composition buddy. In your twenties, it's easy to find and keep alliances with some other composers. These are your composition buddies. You play your work for each other, and comment.

Later on, it becomes harder and harder to find such people. You find this trusting relationship only in your marriage. The comparison with marriage is apt, because a trusting friendship with another composer is as risky and intimate as marriage. I'm thinking of the friendship between Edward Thomas and Robert Frost. (Poets, in this case, but the dynamic is the same.)

I gave up looking for a teacher, for several reasons. It was really swimming against the current. It didn't work. Music is really a big pyramid with the successful older musicians feeding off the younger ones, who are feeding off their parents and student loans. Any other relationship has to be improvised, and it just doesn't make sense.

Another reason -- and here's the point of this post -- is that being a successful teacher in music, from what I could tell, requires some combination of insensitivity, pedantry, lack of imagination, authoritarianism, or just plain unscrupulousness. Or it causes these traits to become more pronounced over time. There is a vast oversupply, and little demand, for composers, so they go into teaching whether they really like it or not. The ones that like it have to teach the same things over and over to undergrads, or they become administrators -- "chairs". This seems to deaden their spirits, or something.

My sample size here is maybe twenty people. Not a large group.

The thing I'm really trying to get at here is an observation about so-and-so who I used to know as an energetic young composer, full of curiosity. Fast forward thirty years, and he's changed. He's internalized the effect of teaching solfege to sophomores. He's accustomed to being the leader in the room, and knowing the most. He has his routines. What's good isn't new, and what's new isn't good. So-and-so is a composite of a handful of people.

This was true of the composers I checked out as potential teachers. What struck me was that they'd long since lost their curiosity. Or some others were basically stealing money from their students.

I'm not at all sure this is universally true, or why there are certain age-limits and age-ratios required for a composer/mentor relationship, but it seems to be true.

I don't get out much, but when I did -- to look for a teacher/witness/mentor/sponsor/enabler -- whatever -- I was surprised at the price some of these composers seem to have paid for their success in academia.

Looking over what I've just written, it seems to be only a truism, but what struck me was how much real life was really like that.
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Old 22nd February 2013, 06:05 AM   #49
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Looking for another kernel to base a piece on.

The disadvantage of the kind of series I just used is that the self-similarity is irregular, unlike another kind of series, the indices of power-residue series.

The indices of power residues have perfect, regular self-similarity. There are only two basic sources for such series in the land of conventional 12-tone tuning, because there are only two primes in the vicinity of 12: 11 and 13. The series derived from these primes will have one less element, so 10 and 12 numbers, respectively.

mod 13 generates the Mallalieu series, so let's try mod 11.

2^n mod 11: 1,2,4,8,5,10,9,7,3,6
indices: 0,1,8,2,4,9,7,3,6,5

eta: If we Google this series, we see that it's well-known in number-theory, etc:
http://www.google.com/search?q=%220%....1.L_kl1vTeSrE

eta2: also, in the OEIS: http://oeis.org/search?q=0%2C1%2C8%2...lish&go=Search

6^n mod 11 indices (0 to 9): 9,8,1,7,5,0,2,6,3,4 is simply inversion of above

7^n mod 11 indices: 9,2,3,5,1,6,0,8,7,4 [eta4: removed error]

8^n mod 11 is only the inversion of above.

However, the resulting tables of number series can be mapped to any set of pitches. It can be thought of as a source of contours rather than a more literal 0=C, 1=C#, 2=D, mapping.

Here's such a table:

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


Here's the 7^n mod 11 table

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




When the series are mapped to more-or-less familiar scales, you can easily generate polyrhythmic textures moving at 1:2:3:4:5 speeds -- or however complicated you want -- with the overall sound being a pleasing "chord-scale" rather than the more gnarly 12-notes-at-once sound.

And these scales can be tuned to more-or-less conventional tunings. 10 pitches per octave (10EDO) would be one obvious choice. (And I like the sound of 10EDO, somewhat.)

Or the "scale" can be any arbitrary set of pitches ranked from low to high.
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Old 23rd February 2013, 12:04 AM   #50
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Originally Posted by calebprime View Post
https://www.box.com/s/litnhwpcyk8swpk46vo7

What's on paper seems good to me, but this sketch is ponderous. Only took about three hours from paper to this 9+ minute realization.

So I'm calling it Sea Bass with Frickin' Lasers.

It will take a lot more work to make something with more subtlety than this.

One problem is that I suck on piano. Another is that this was just slapped together. In that light, it came out pretty good.

Trying to exactly make the Multiple-Order embedding sound good turned out to be harder than it looked on paper.

This is only an .aif file, so it's 95+ megs, so it may not play directly, you may need to download it.
That was rather lovely, caleb. Subtle, but not too subtle. Love the title!
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Old 24th February 2013, 08:44 AM   #51
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Thanks, Nay. Yeah, I couldn't get the sharks I wanted, so I had to settle for sea bass.

Here's a very good explanation of self-similarity for musicians:

David Feldman reviews composer Tom Johnson's interesting little book.

Robert Morris is a modernist, Tom Johnson is a minimalist, and David Feldman turns out to be a former student of Robert Morris.

A review of Self-Similar Melodies.

http://kalvos.org/johness2.html

Musicians without a lot of math training need things to be spelled out very concretely. Musical mathematics is very concerned with particular results -- the answers are as important as the process. In applied mathematics of this kind, formality and proofs don't help at all, so the way math is taught in some of the traditional texts is not helpful. Musicians have an intense, narrow interest in certain outcomes, whereas traditional math takes a much broader, more rigorous view of things. So I found that trying to learn about permutation, number theory, and the like from texts by mathematicians was simply the wrong way to go. You couldn't tell in advance whether some avenue would be a dead-end with no relevance.

That's why articles like this, and books like Johnson's are so helpful.

from the review:
Quote:
"...The theorem, the commodity central to mathematical life, barely plays a role here. Theorems delineate the interrelations between the properties of mathematical objects, but for Johnson the interest lies directly with the mathematical objects themselves, rendered palpable and lambent by translation into music."


In learning about this stuff, it turns out to be more useful to work backwards from certain good results -- such as certain specific number series -- to the method that generated them.

Most of the number series in the OEIS are of no musical interest. http://oeis.org/

Also, there's a subtle but key difference between similarity and invariance. In musical art, things only need to resemble each other or be similar enough to make a mental connection: This is like this. "Invariance" is often both more restrictive and less interesting, and may not even be better at creating the necessary psychological connections than mere similarity.
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Old 24th February 2013, 10:41 AM   #52
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Still looking for the next little kernel.

It occurs to me that if I'm going to be mapping number series to scales with 7 elements -- such as most of the familiar scales -- I should be looking for self-similar sequences of either 7 elements or 14 elements.

If my goal is the standard "every-other" permutation pattern, there turn out to be no perfect patterns of 7 elements. But there are a lot of near-perfect patterns of 14 elements, so I'll be looking at these over the next few days.

The actual series -- the results -- are here called "x-swapped". The series, from 0 to 13, will be mapped onto 2 octaves of a 7-note scale.

Here are what seem to be the only four candidates, each with its quirks.

user row: 0 2 4 6 8 10 12 1 3 5 7 9 11 13
grid row: 1 2 3 4 5 6 7 8 9 10 11 12 13 0
orbit: 1 0 13 11 7 12 9 3 4 6 10 5 8 2
X-swapped: 1 0 13 7 8 11 9 4 12 6 10 3 5 2
intervals: 13 13 8 1 3 12 9 8 8 4 7 2 11 13

user row: 0 2 4 6 8 10 12 1 3 5 7 9 11 13
grid row: 3 4 5 6 7 8 9 10 11 12 13 0 1 2
orbit: 3 0 9 12 5 4 2 13 7 8 10 1 11 6
X-swapped: 1 11 6 0 5 4 13 8 9 2 10 12 3 7
intervals: 10 9 8 5 13 9 9 1 7 8 2 5 4 8

user row: 0 2 4 6 8 10 12 1 3 5 7 9 11 13
grid row: 11 12 13 0 1 2 3 4 5 6 7 8 9 10
orbit: 11 0 6 5 3 12 2 10 13 4 1 8 9 7
X-swapped: 1 10 6 4 9 3 2 13 11 12 7 0 5 8
intervals: 9 10 12 5 8 13 11 12 1 9 7 5 3 7

user row: 0 2 4 6 8 10 12 1 3 5 7 9 11 13
grid row: 13 0 1 2 3 4 5 6 7 8 9 10 11 12
orbit: 13 0 2 6 1 4 10 9 7 3 8 5 12 11
X-swapped: 1 4 2 9 5 11 3 8 10 7 6 13 12 0
intervals: 3 12 7 10 6 6 5 2 11 13 7 13 2 1


Suppose we take the last one, and map this to A major for the moment, just to see how it works.

0=A1, 1=B1, 2=C#2, 3=D2...etc. to 6=G#2, 7=A2, 8=B2, 9=C#3...13=G#3 || 14 would equal A3

We get the sequence of pitches B1,E2,C#2,C#3,F#2,E3,D2,B2,D3,A2,G#2,G#3,F#3,A1 (repeat)

and embedded in this, starting on the third note, is the sequence at "every other note" C#2,F#2,D2,D3,G#2,F#3,E2,C#3,E3,B2,A2,(G#3,A1(or adjusted)

Which is identical to the first sequence, up a diatonic step -- except for the last two notes.

This process of embedding can be carried out as far as one would like, with any scales of 7 notes, not just major scales. I like music where the scales change, because of my jazzoid background.

I'm going to try this for my next piece, unless I see some fatal flaw or get a better idea in the meantime.
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Old 25th February 2013, 03:36 AM   #53
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Originally Posted by calebprime View Post
Still looking for the next little kernel.

It occurs to me that if I'm going to be mapping number series to scales with 7 elements -- such as most of the familiar scales -- I should be looking for self-similar sequences of either 7 elements or 14 elements.

If my goal is the standard "every-other" permutation pattern, there turn out to be no perfect patterns of 7 elements. But there are a lot of near-perfect patterns of 14 elements, so I'll be looking at these over the next few days.

The actual series -- the results -- are here called "x-swapped". The series, from 0 to 13, will be mapped onto 2 octaves of a 7-note scale.

Here are what seem to be the only four candidates, each with its quirks.

user row: 0 2 4 6 8 10 12 1 3 5 7 9 11 13
grid row: 1 2 3 4 5 6 7 8 9 10 11 12 13 0
orbit: 1 0 13 11 7 12 9 3 4 6 10 5 8 2
X-swapped: 1 0 13 7 8 11 9 4 12 6 10 3 5 2
intervals: 13 13 8 1 3 12 9 8 8 4 7 2 11 13

user row: 0 2 4 6 8 10 12 1 3 5 7 9 11 13
grid row: 3 4 5 6 7 8 9 10 11 12 13 0 1 2
orbit: 3 0 9 12 5 4 2 13 7 8 10 1 11 6
X-swapped: 1 11 6 0 5 4 13 8 9 2 10 12 3 7
intervals: 10 9 8 5 13 9 9 1 7 8 2 5 4 8

user row: 0 2 4 6 8 10 12 1 3 5 7 9 11 13
grid row: 11 12 13 0 1 2 3 4 5 6 7 8 9 10
orbit: 11 0 6 5 3 12 2 10 13 4 1 8 9 7
X-swapped: 1 10 6 4 9 3 2 13 11 12 7 0 5 8
intervals: 9 10 12 5 8 13 11 12 1 9 7 5 3 7

user row: 0 2 4 6 8 10 12 1 3 5 7 9 11 13
grid row: 13 0 1 2 3 4 5 6 7 8 9 10 11 12
orbit: 13 0 2 6 1 4 10 9 7 3 8 5 12 11
X-swapped: 1 4 2 9 5 11 3 8 10 7 6 13 12 0
intervals: 3 12 7 10 6 6 5 2 11 13 7 13 2 1


Suppose we take the last one, and map this to A major for the moment, just to see how it works.

0=A1, 1=B1, 2=C#2, 3=D2...etc. to 6=G#2, 7=A2, 8=B2, 9=C#3...13=G#3 || 14 would equal A3

We get the sequence of pitches B1,E2,C#2,C#3,F#2,E3,D2,B2,D3,A2,G#2,G#3,F#3,A1 (repeat)

and embedded in this, starting on the third note, is the sequence at "every other note" C#2,F#2,D2,D3,G#2,F#3,E2,C#3,E3,B2,A2,(G#3,A1(or adjusted)

Which is identical to the first sequence, up a diatonic step -- except for the last two notes.

This process of embedding can be carried out as far as one would like, with any scales of 7 notes, not just major scales. I like music where the scales change, because of my jazzoid background.

I'm going to try this for my next piece, unless I see some fatal flaw or get a better idea in the meantime.
Heh. Not a fatal flaw exactly.

When I took out two pitches and mapped this number series to a good-old chromatic scale, I came up with:

C,Eb,Db,Ab,E,Bb,D,G,A,Gb,F,B

First thing I noticed was that this is a rather good 12-tone MOF. Maybe too good.

Next thing I did was write down the intervals: 3,10,7,8,6,4,5,2,9,11,6,1.

Uh oh. Good MOF, and is an all-interval series.

Next thing I did was look it up in my data-base of all-interval series.

Yep. It's the Mallalieu series. There's nothing wrong with it, it's the only perfect series there is. But I've spent years looking for alternatives to it, for series that have many of its properties but lack perfection in some way.

Here's the entry in the my list. Mallalieu is #44.

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, 19^n mod37 also Slonimsky's #1308


Maybe I should just write a piece based on Mallalieu series, or maybe I should keep looking. Thing is, I like the way it sounds starting on the second note.
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Old 25th February 2013, 06:32 AM   #54
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Originally Posted by calebprime View Post
Heh. Not a fatal flaw exactly.

When I took out two pitches and mapped this number series to a good-old chromatic scale, I came up with:

C,Eb,Db,Ab,E,Bb,D,G,A,Gb,F,B

First thing I noticed was that this is a rather good 12-tone MOF. Maybe too good.

Next thing I did was write down the intervals: 3,10,7,8,6,4,5,2,9,11,6,1.

Uh oh. Good MOF, and is an all-interval series.

Next thing I did was look it up in my data-base of all-interval series.

Yep. It's the Mallalieu series. There's nothing wrong with it, it's the only perfect series there is. But I've spent years looking for alternatives to it, for series that have many of its properties but lack perfection in some way.

Here's the entry in the my list. Mallalieu is #44.

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, 19^n mod37 also Slonimsky's #1308


Maybe I should just write a piece based on Mallalieu series, or maybe I should keep looking. Thing is, I like the way it sounds starting on the second note.
How about applying your 7+7 mapping idea to pentatonic scales, and using the mod11 generator to make a 10-note pattern, apply that to a 5+5 mapping? There's a lot of wiggle-room with pentatonic scales. You've got yer Balinese, Javanese, yer hemitonic & anhemitonic, yer shredder-pentatonic, and yer bro-step pentatonic. Everyone likes pentatonics!

Here's the 7^n mod 11 table. map any row to a 2-octave pentatonic, or 10EDO tuning, back to that idea?

9, 2, 3, 5, 1, 6, 0, 8, 7, 4
2, 5, 6, 8, 4, 9, 3, 1, 0, 7
5, 8, 9, 1, 7, 2, 6, 4, 3, 0
8, 1, 2, 4, 0, 5, 9, 7, 6, 3
1, 4, 5, 7, 3, 8, 2, 0, 9, 6
4, 7, 8, 0, 6, 1, 5, 3, 2, 9
7, 0, 1, 3, 9, 4, 8, 6, 5, 2
0, 3, 4, 6, 2, 7, 1, 9, 8, 5
3, 6, 7, 9, 5, 0, 4, 2, 1, 8
6, 9, 0, 2, 8, 3, 7, 5, 4, 1
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Old 25th February 2013, 12:49 PM   #55
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Here's a perfect 10-note self-similar pattern, all-interval mod10.

I came up with it in a weird way, as I'll show.

It doesn't seem to be Mallalieu minus a couple pitches, nor does it seem to be the same as the series you get from power-residue index series.

That I've discovered a genuinely new thing seems very unlikely, but I don't recognize this.

As Robert DiDomenica used to say, "You'll be disappointed."

Seems too good to be true:

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

and its inversion

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

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

I got this result by entering these series into my permutation algorithm-thingy, and looking for complete orbits only:

grid row: 0,7,3,6,2,9,5,1,8,4,
user row: 8,6,4,2,0,9,7,5,3,1

and it found:

user row: 8 6 4 2 0 9 7 5 3 1
grid row: 6 2 9 5 1 8 4 0 7 3
orbit: 6 8 9 4 7 3 1 0 5 2
X-swapped: 7 6 9 5 3 8 0 4 1 2

intervals: 9 3 6 8 5 2 4 7 1 5

You're too good to be true
Can't take my eyes off you.
I love you baby, and if it's quite alright...


Can I generalize this at all -- to at least a rule of thumb, and find more?

It's probably merely some transformation of a power-residue series in mod10 arithmetic.
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Old 25th February 2013, 12:59 PM   #56
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Originally Posted by calebprime View Post
...
Seems too good to be true:

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

...

It's probably merely some transformation of a power-residue series in mod10 arithmetic.
yep, a quick Google shows it's just a log series, or power-res index series.

http://www.galaxyng.com/adrian_atana.../cript/c12.pdf

De aici rezult˘a imediat tabelul logaritmilor ˆın baza 6:
β 1 2 3 4 5 6 7 8 9 10
log6β 0 9 2 8 6 1 3 7 4 5
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Old 25th February 2013, 02:41 PM   #57
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Here's another perfect one that I don't recognize yet:


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

eta: oh, that's just 8 ^n mod 11 indices , but no one seems to have mentioned it so that it shows up in Google.
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Old 26th February 2013, 03:23 AM   #58
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I have another app written by 69Dodge that finds series with self-similarity within some error tolerance. That's a way of getting around the fact that there are no new perfect series under the sun.

Maybe if one of the series below is mapped to a 6+6 arrangement -- 2 octaves of a 6-note scale -- there will be self-similarity that is close enough for Cal, and with the necessary euphony.

The numbers after the dash on the right mean that each of these has 10 positions where the self-sim is exactly 2, the maximum error of any position is +-1, and the total sum of errors is 2.


0 1 7 3 6 9 10 5 4 8 2 11 - 10 1 2
0 2 7 3 6 9 10 5 4 8 1 11 - 10 1 2 most of these are basically mallalieu intervals x2, +-1
0 2 7 4 5 9 10 6 3 8 1 11 - 10 1 2
0 2 7 4 6 9 10 5 3 8 1 11 - 10 1 2
0 2 8 4 5 9 10 6 3 7 1 11 - 10 1 2
0 2 8 4 5 10 9 6 3 7 1 11 - 10 1 2
0 2 8 4 7 10 11 6 5 9 3 1 - 10 1 2
0 2 8 4 7 11 10 6 5 9 3 1 - 10 1 2
0 2 9 4 6 11 10 7 5 8 3 1 - 10 1 2
0 2 9 4 7 11 10 6 5 8 3 1 - 10 1 2
0 2 9 5 6 11 10 7 4 8 3 1 - 10 1 2
0 3 9 5 6 11 10 7 4 8 2 1 - 10 1 2

These can be considered either as standard 12-tone rows or as the result of their mapping to a 6-note, 2-octave scale, or both.

The series below, with the search parameters above them, are interesting because the perms are as free of unpleasant or dorky contiguous harmonic relations as the listed series. So the permutatated series sounds as good as the listed one. These are self-similar at 4 semitones, or a major third.


M) modulus: 12
P) permutation: 1,3,5,7,9,11,0,2,4,6,8,10
C) center interval of self-similarity: 4
O) minimum number of occurrences of center: 5
D) maximum deviation from center: 1
S) maximum sum of all deviations: 7
F) fixed-position notes: 0:0
E) excluded cells within range: 0,1,2:3 0,3,6,9:4
R) excluded intervals within range:
A) filter original solution: on
B) filter permuted solution: on
X) permutation depth: 3
N) number of solutions to print: 1000
W) wrap: on

0 3 4 7 1 9 8 11 10 5 6 2 - 6 1 6
0 3 4 7 2 9 8 11 10 5 6 1 - 6 1 6
0 4 2 7 10 6 8 11 5 3 1 9 - 6 1 6
0 4 2 8 10 6 7 11 5 3 1 9 - 6 1 6
0 4 3 8 9 6 7 11 5 2 1 10 - 6 1 6
0 4 3 8 10 6 7 11 5 2 1 9 - 6 1 6
0 4 5 8 2 10 9 1 11 6 7 3 - 6 1 6
0 4 5 8 3 10 9 1 11 6 7 2 - 6 1 6
0 4 6 8 2 10 9 1 11 5 7 3 - 6 1 6
0 4 6 8 2 11 9 1 10 5 7 3 - 6 1 6
0 5 4 9 10 7 8 1 6 3 2 11 - 6 1 6
0 5 4 9 11 7 8 1 6 3 2 10 - 6 1 6
found 12 solutions.
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Old 28th February 2013, 06:54 AM   #59
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A new tuning for the next piece. 31 Pitches per 2/1 (octave). Basically a modified Just Intonation (JI) approach with a complete circle of near-5ths. The tuning base can be moved so that 1/1 can be any frequency. Modulation -- or moving the tonal center -- is achieved by common-tones between the old tonal center and the new one that one gets to by pitch-bending the whole structure.

There are so many criteria for this tuning that I won't even attempt to list them all -- unless someone wants to know. This took a lot of thought, the help of the Lil' Miss Scale Oven (now sadly no more) and some things I learned from the microtonal folks. However, it's neither a Partch tuning nor one of the new tunings or temperaments that people such as Gene Ward Smith and Graham Breed are developing.

This will be for sampled instruments in Logic retuned via Lil Miss Scale Oven.

It's in Scala format, so anyone who uses that can just plug it in. (Technology is sometimes a wonderful thing!) The numbers are in cents (2/1 or octave divided into 1200 parts)

There might be a flaw, but I haven't found any so far. It might be possible to smooth some rough edges by slightly adjusting the whole thing*, but I don't know of any easy way to do that with my software, and I can't think of any pitches I don't need.

*eta: such as quantizing the pitches to 140EDO, or something???

!Caleb 31comp#1.scl
!
First try at <36 with complete circle of 5ths, 4ths, lots of 13, 701.62 or 111.86 generator

31
! ! 0.00
91.9 ! m2 small pyth
111.86 ! 16/15
138.573 ! 13/12
182.404 ! 10/9
203.24 ! 9/8
231.174 ! 8/7
266.871 ! 7/6
!
305.12 ! pyth min third
315.1 ! 6/5 t
386.52 ! 5/4 t
406.48 ! 81/64 t
416.98 ! 14/11
434.414 ! 9/7
498.38 ! 4/3
!
551.32 ! 11/8, no sym
590.28 ! pyth 7/5
!
609.72 ! pyth 10/7
!
701.62 ! 3/2
765.586 ! 14/9
783.02 ! 11/7 and small pyth min 6th
813.48 ! 8/5 t
840.528 ! 13/8
885.42 ! 5/3
904.86 ! pyth maj 6th
!
932.794 ! 12/7 t
968.826 ! 7/4
996.09 ! 16/9
1049.36 ! 11/6
1088.14 ! 15/8 t
1108.1 ! maj 7th, large pyth
!
1200.00


**eta2: Here are the above pitches quantized to 140EDO, so they make more consistent step-sizes. Not sure if there are any audible advantages, have to try it:

0., 94.286, 111.429, 137.143, 180., 205.714, 231.429, 265.714, 300., 317.143, 385.714, 402.857, 420., 437.143, 497.143, 548.571, 591.429, 608.571, 702.857, 762.857, 780., 814.286, 840., 882.857, 900., 934.286, 968.571, 994.286, 1045.714, 1088.571, 1105.714
and in "relative srutis": 11, 2, 3, 5, 3, 3, 4, 4, 2, 8, 2, 2, 2, 7, 6, 5, 2, 11, 7, 2, 4, 3, 5, 2, 4, 4, 3, 6, 5, 2, 11


nah, 140edo version is too screwed up to fix easily.
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Old 28th February 2013, 01:37 PM   #60
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Added on more pitch to tuning: 9/5 -- which I couldn't seem to live without -- bringing the total to 32.

Quantizing to 140EDO seems, actually, to be ok, but I'm not sure if it sounds better yet. (I arrived at it by adding 53EDO and 87EDO, two fairly well-known tunings.)

32 pitches per 2/1 is different enough from my previous scale of 36 pitches per 2/1 that it was worth the effort, and different enough that when I practice it, the patterns won't get too confused with the old scale patterns -- always a problem if you try to practice with more than one tuning system. It's like learning a language. At some point it becomes second nature.
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Old 1st March 2013, 04:26 AM   #61
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Nah, not happy with it. Too little /11 and /13 and 13/ available.

There's one radical solution that I've never tried. Could be awesome.

A tuning that's designed purely for keyboard and computer realization.

It would have far fewer pitches in the lowest register, then more in higher registers per 2/1. (Like the overtone series, but not the same as the overtone series.)

The problem is that a full 13-limit system has too many pitches to fit into an 88-note keyboard -- you end up with less than two full octaves.

The opportunity, however, arises from the fact that you might not need 11/8 or 13/8 in the lowest registers -- for me, they are pitches that only sound good in a higher register. So there's waste in most tunings. That's an opportunity.

The other opportunity is provided by the fact of the moveable tuning-base. So all pitches are available in theory. What's in the scale on the keyboard is what you need to grab quickly and at the same time.

This would be the most efficient use of keyboard space, but also the ugliest and hardest-to-memorize pattern -- a full 88-note scale with no repeating patterns at all. (The practical solution is simply to label every key by writing on a strip of tape right above the keys.)

A true monstrosity, and well outside the kind of temperaments that the highly competent folks in the microtonal community are designing.

Scale to follow. To be called Pyramid Scheme, or something.
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Old 1st March 2013, 07:55 AM   #62
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Here's the basic idea. Three tiers.


I present: The Güber Mensch, or Cal's Inverted Pyramid

bass -- 16-tone subset
mid -- 29-tone subset
treble -- 42-tone set plus last 2/1 = 43
----------------------------
=88 keys

Tones are derived from 13-limit JI, with a 19/16 stuck in there, and then tempered to something like 111edo, which gives a very good approximation of everything.

Sadly, there are trade-offs even with an eff'd up scheme like this.


Bass
1/1, 16/15, 12/11, 9/8, 8/7, 6/5, 16/13, 5/4, 4/3, 10/7, 16/11, 3/2, 8/5, 5/3, 16/9, 15/8

Mid
1/1, 16/15, 12/11, 10/9, 9/8, 8/7, 7/6, 6/5, 16/13, 5/4, 9/7, 4/3, 18/13, 7/5, 10/7, 16/11, 3/2, 20/13, 14/9, 8/5, 18/11, 5/3, 12/7, 7/4, 16/9, 9/5, 20/11, 24/13, 15/8

Treble
1/1, 16/15, 14/13, 13/12, 12/11, 11/10, 10/9, 9/8, 8/7, 7/6, 19/16, 6/5, 11/9, 16/13, 5/4, 81/64, 14/11, 4/3, 11/8, 18/13, 7/5, 45/32, 10/7, 16/11, 3/2, 20/13, 14/9, 11/7, 8/5, 13/8, 18/11, 5/3, 27/16, 12/7, 7/4, 16/9, 9/5, 20/11, 11/6, 24/13, 13/7, 15/8, 2/1
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Last edited by calebprime; 1st March 2013 at 07:58 AM.
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Old 1st March 2013, 08:45 AM   #63
calebprime
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GuberMensch Tuning in Scala Format -- 88keys

!GuberMensch Tuning
!
3-tier tuning from 13L-JI with 16tones in bass, 29 in mid, and 42 tones in treble
! Look Upon my Peas You Mighty, and Eat! Goodness, How Delicious!
88
! ! 1/1
16/15
12/11
9/8
8/7
6/5
16/13
5/4
4/3
10/7
16/11
3/2
8/5
5/3
16/9
15/8
!
2/1
32/15
24/11
20/9
9/4
16/7
7/3
12/5
32/13
5/2
18/7
8/3
36/13
14/5
20/7
32/11
6/2
40/13
28/9
16/5
36/11
10/3
24/7
7/2
32/9
18/5
40/11
48/13
15/4
!
4/1
64/15
56/13
52/12
48/11
22/5
40/9
9/2
32/7
14/3
19/4
24/5
44/9
64/13
5/1
81/16
56/11
16/3
11/2
72/13
28/5
45/8
40/7
64/11
6/1
80/13
56/9
44/7
32/5
52/8
72/11
20/3
27/4
48/7
7/1
64/9
36/5
80/11
22/3
96/13
52/7
15/2
8/1
3600.00
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Last edited by calebprime; 1st March 2013 at 08:57 AM. Reason: 11/2 !???
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Old 1st March 2013, 09:02 AM   #64
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GuberMensch Tuning quantized to 111EDO, Scala Format

!
GuberMensch 16+29+42 111EDO
88
!
108.10800
151.35100
205.40500
227.02700
313.51400
356.75700
389.18900
497.29700
616.21600
648.64900
702.70300
810.81100
886.48600
994.59500
1091.89200
1200.00000
1308.10800
1351.35100
1383.78400
1405.40500
1427.02700
1470.27000
1513.51400
1556.75700
1589.18900
1632.43200
1697.29700
1762.16200
1783.78400
1816.21600
1848.64900
1902.70300
1945.94600
1967.56800
2010.81100
2054.05400
2086.48600
2129.73000
2172.97300
2194.59500
2216.21600
2237.83800
2259.45900
2291.89200
2400.00000
2508.10800
2529.73000
2540.54100
2551.35100
2562.16200
2583.78400
2605.40500
2627.02700
2670.27000
2702.70300
2713.51400
2745.94600
2756.75700
2789.18900
2810.81100
2821.62200
2897.29700
2951.35100
2962.16200
2983.78400
2994.59500
3016.21600
3048.64900
3102.70300
3145.94600
3167.56800
3178.37800
3210.81100
3243.24300
3254.05400
3286.48600
3308.10800
3329.73000
3372.97300
3394.59500
3416.21600
3437.83800
3448.64900
3459.45900
3470.27000
3491.89200
3600.00000
3600.00000
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Old 1st March 2013, 12:52 PM   #65
calebprime
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I like this idea, and it's very workable. I'm testing it with a Pianoteq piano sound. Mostly, I'm very happy with what I'm hearing.

It turns out to be not hard to play if the ratios are labelled on the keyboard (assuming you already know JI theory, which I do.)

Unfortunately, I need a little more in the bass and mid, in order to get "5ths", (or 3/2 ratios) down there -- which means robbing a few pitches off the top.

If only there were mass-produced midi controllers with velocity, etc. with more than 88 keys! None of this would be necessary!

While it's tempting to just declare the process done, that would be premature closure. I'm going to be married to this scale for years if the concept works out, so it makes sense to put up with the slight aggravation of making a few adjustments, based on listening to it.

I need a 12/7 in the bass, and maybe another pitch or two, or at most 3 more.

I can get rid of 45/32 in the treble, for starters, maybe a few others.

It's a tedious process from here on.

No tuning is perfect. There are always trade-offs. Partch stopped at the 11-limit. It's not that the 13th partial doesn't sound good, or /13 ratios -- they sound great on piano if voiced correctly. It's that there's a proliferation of pitches with a 13-limit, and the whole system takes a while to tweak.

Just labelling the keys takes an hour or so.

Gubermensch 2.0 soon. This time, it's personal.
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Old 2nd March 2013, 12:40 AM   #66
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Here's the version I'm going with for now.

Had to rob some /13 and /11 ratios in the top, add a bunch of 3/2 ratios to the bass.

So the difference between the number of pitches in the bass and the number in the treble is now less pronounced.

To my eye, this is one ugly scale.

To my ear, it's all the good harmonic relationships that will fit in 88 keys.

If certain pitches are needed but unavailable, the whole tuning bass can be moved using a table converting cents to pitch-bend info in Logic. I've tested this with simple sine waves in the high register (to be able to hear even very slow beating) and the accuracy is perfect if you use the right number. In this system, the pitch-bend range of the instruments is set to +-12 semitones.

It wouldn't work for improv or real-world instruments, but it's the simplest, most accurate system for composition using Logic. (Notation is simply conventional notation with ratios written above the pitches, and the issue of pretty notation doesn't interest me in any case.)

A few days of testing, and I marry this scale, probably in some cheap chapel in Las Vegas, with a preacher dressed like Elvis.

It's strange how the desire for elegance on paper, and the desire to get with the program (i.e., use one of the tunings people in the tuning community have named) has prevented me from arriving at this solution.

She ain't a looker -- this tuning -- but she do the job.

bass: 1/1, 16/15, 12/11, 9/8, 8/7, 32/27, 6/5, 16/13, 5/4, 4/3, 10/7, 16/11, 3/2, 8/5, 5/3, 27/16, 12/7, 16/9, 24/13, 15/8,

mid: 2/1, 32/15, 24/11, 20/9, 9/4, 16/7, 7/3, 64/27, 12/5, 32/13, 5/2, 18/7, 8/3, 36/13, 14/5, 20/7, 32/11, 3/1, 40/13, 28/9, 16/5, 36/11, 10/3, 27/8, 24/7, 7/2, 32/9, 18/5, 40/11, 48/13, 15/4,

treble: 4/1, 64/15, 56/13, 13/3, 48/11, 22/5, 40/9, 9/2, 32/7, 14/3, 128/27, 24/5, 44/9, 5/1, 81/16, 56/11, 16/3, 11/2, 28/5, 40/7, 6/1, 80/13, 44/7, 32/5, 13/2, 72/11, 20/3, 27/4, 48/7, 7/1, 64/9, 36/5, 80/11, 22/3, 96/13, 15/2, 8/1
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Last edited by calebprime; 2nd March 2013 at 12:51 AM.
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Old 3rd March 2013, 03:02 AM   #67
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This might be final version. I'm cautiously optimistic.

I made some small tweaks, but decided to preserve the 81/16 to 14/11 difference, also substituted 16/13 for 11/9 in the top register.

Here's a version quantized to 111edo (111=3x37)

This is a working Scala file that I'm trying out with PianoTeq:

!
GuberMensch 2.8 111edo
87
!
108.10800
151.35100
205.40500
227.02700
291.89200
313.51400
356.75700
389.18900
497.29700
616.21600
648.64900
702.70300
810.81100
886.48600
908.10800
929.73000
994.59500
1059.45900
1091.89200
1200.00000
1308.10800
1351.35100
1383.78400
1405.40500
1427.02700
1470.27000
1491.89200
1513.51400
1556.75700
1589.18900
1632.43200
1697.29700
1762.16200
1783.78400
1816.21600
1848.64900
1902.70300
1945.94600
2010.81100
2054.05400
2086.48600
2108.10800
2129.73000
2172.97300
2194.59500
2216.21600
2237.83800
2248.64900
2259.45900
2291.89200
2400.00000
2508.10800
2529.73000
2540.54100
2551.35100
2562.16200
2583.78400
2605.40500
2627.02700
2670.27000
2691.89200
2713.51400
2756.75700
2789.18900
2810.81100
2821.62200
2897.29700
2951.35100
2983.78400
3016.21600
3102.70300
3145.94600
3178.37800
3210.81100
3243.24300
3254.05400
3286.48600
3308.10800
3329.73000
3372.97300
3394.59500
3416.21600
3437.83800
3448.64900
3459.45900
3491.89200
3600.00000
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Old 3rd March 2013, 06:24 AM   #68
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And here, finally, is a version that at least has no more weirdness than the basic idea requires.

A few more fixes, and quantized to 87EDO.

Viable Scala file format:


!
GuberMensch 3.0 87#4 9/7
87
!
110.34500
151.72400
206.89700
234.48300
289.65500
317.24100
358.62100
386.20700
496.55200
620.69000
648.27600
703.44800
813.79300
882.75900
910.34500
937.93100
993.10300
1062.06900
1089.65500
1200.00000
1310.34500
1351.72400
1379.31000
1406.89700
1434.48300
1462.06900
1489.65500
1517.24100
1558.62100
1586.20700
1641.37900
1696.55200
1765.51700
1779.31000
1820.69000
1848.27600
1903.44800
1944.82800
2013.79300
2055.17200
2082.75900
2110.34500
2137.93100
2165.51700
2193.10300
2220.69000
2234.48300
2248.27600
2262.06900
2289.65500
2400.00000
2510.34500
2524.13800
2537.93100
2551.72400
2565.51700
2579.31000
2606.89700
2634.48300
2662.06900
2689.65500
2717.24100
2758.62100
2786.20700
2813.79300
2835.08
2896.55200
2951.72400
2963.38
2979.31000
3020.69000
3103.44800
3144.82800
3213.79300
3241.37900
3255.17200
3282.75900
3310.34500
3337.93100
3365.51700
3393.10300
3420.69000
3434.48300
3448.27600
3462.06900
3489.65500
3600.00000
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Old 4th March 2013, 04:06 AM   #69
calebprime
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Had to tweak one note of the above tuning, to have a 16/11 up in the high register.

Done.

Time to start making some music.

I'm now married to an 87edo approximation of a sort of 13-limit Just Intonation scale, with the world's most horrible keyboard layout. There seems to be no other way.

Time to start making some frickin' music.

Enough frickin' Fokker bloody blocks. Enough commas.

http://en.wikipedia.org/wiki/Fokker_periodicity_blocks
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Old 6th March 2013, 04:22 AM   #70
calebprime
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One more post about the tuning/scale to show how I've nearly come full circle.


I think this is the framework I'm going with.

Two keyboards with 88 keys, 4 octaves to hand, total, with an upper 4/1 on both keyboards.

43 pitches of 87 edo. Same number of pitches as Harry Partch, but I squeeze in the 13-limit by leaving out some pitches he included.

A pair of "minor seconds" or major 7ths -- one at 15/8, the other at 82.75 cents to make a decent 4th (4/3) with 7/5, or 5th (3/2) with 10/7.

Each key labeled with the closest ratio, and I'm going to add labels with number of steps ("srutis") in 87 edo, until the two numbers become synonymous in my mind.

Heavy reliance on a pair of sustain pedals at my feet.

Pitch-bend set to +-12, and reference to a conversion chart (1 full octave of pitch bend = 8192 steps in Logic).

I think I've gotten rid of all the eccentricities except the uneven fingering patterns. There always has to be a compromise, some extra difficulty.

steps of 87edo difference: 6, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 4, 1, 1, 3, 1, 1, 4, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 6

steps of 87edo absolute: 0, 6, 8, 9, 11, 12, 13, 15, 17, 19, 21, 23, 25, 26, 28, 30, 32, 33, 36, 40, 41, 42, 45, 46, 47, 51, 54, 55, 57, 59, 61, 62, 64, 66, 68, 70, 72, 74, 75, 76, 77, 79, 81, 87


Scala file:

!
43 tones of 87 #3
43
!
82.75862
110.34483
124.13793
151.72414
165.51724
179.31034
206.89655
234.48276
262.06897
289.65517
317.24138
344.82759
358.62069
386.20690
413.79310
441.37931
455.17241
496.55172
551.72414
565.51724
579.31034
620.68966
634.48276
648.27586
703.44828
744.82759
758.62069
786.20690
813.79310
841.37931
855.17241
882.75862
910.34483
937.93103
965.51724
993.10345
1020.68966
1034.48276
1048.27586
1062.06897
1089.65517
1117.24138
1200.00000
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Last edited by calebprime; 6th March 2013 at 04:26 AM.
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Old 7th March 2013, 07:31 AM   #71
calebprime
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Originally Posted by calebprime View Post
...

I think this is the framework I'm going with.

Two keyboards with 88 keys, 4 octaves to hand, total, with an upper 4/1 on both keyboards.

43 pitches of 87 edo. Same number of pitches as Harry Partch, but I squeeze in the 13-limit by leaving out some pitches he included.

...
I think I've gotten rid of all the eccentricities except the uneven fingering patterns. There always has to be a compromise, some extra difficulty.

steps of 87edo difference: 6, 2, 1, 2, 1, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 3, 4, 1, 1, 3, 1, 1, 4, 3, 1, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 2, 2, 6

steps of 87edo absolute: 0, 6, 8, 9, 11, 12, 13, 15, 17, 19, 21, 23, 25, 26, 28, 30, 32, 33, 36, 40, 41, 42, 45, 46, 47, 51, 54, 55, 57, 59, 61, 62, 64, 66, 68, 70, 72, 74, 75, 76, 77, 79, 81, 87

...
1) basic intuition: low-number frequency-ratios can sound good. 12-tone tuning approximates only multiples of 3-ratios.

2) second intuition: harmony that moves, that changes, sounds good. Tuvan throat-singing sounds great for about 15 seconds, then you start to smell the Yak-dung, get bored. But it sure is harmonic.

3) Working out: 87 divisions of the 2/1 ratio is the lowest-numbered EDO to approximate the overtone series to prime number 13 .

4) 87 pitches too much for one human playing a keyboard to handle.

5) choose most important of those 87.

6) 43 tones are do-able, barely.

7) Harry Partch already did this, but stopped at prime 11.

8) The 43-tone "Genesis" scale of Partch is designed to have nearly consistent interval sizes.

9) The "GuberMensch Supreem" of Calebprime has inconsistent interval sizes, but gets the 13 ratios.

10) How do you master such a scale? Practice.

Here are the fingering patterns to play, essentially, 8 transpositions of an 8-tone overtone series. The series is: 8,9,10,11,12,13,14,15, (16)

This inconsistency is enough to make the GuberMensch lose any chance of winning any design awards in the microtonal community.

We keep our poor GuberMensch child chained to the radiator in the basement, feeding it only leftover scraps and water from the same radiator.

Eight different fingering patterns to play the same goddam thing!

/8: 7,7,5,6,5,5,6,2 (sum=43)

/9: 7,6,6,6,5,4,5,4

/10: 8,6,6,5,6,4,4,4

/11: 7,7,5,4,6,5,4,5

/12: 7,7,7,4,4,5,5,4

/13: 7,6,7,7,3,3,6,4

/14: 8,6,6,6,6,3,3,5

/15: 9,7,7,4,5,7,2,2


Who would win? GuberMensch or Genesis?
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Last edited by calebprime; 7th March 2013 at 07:36 AM.
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Old 10th March 2013, 09:26 AM   #72
calebprime
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Originally Posted by calebprime View Post
...
Eight different fingering patterns to play the same goddam thing!

/8: 7,7,5,6,5,5,6,2 (sum=43)

/9: 7,6,6,6,5,4,5,4

/10: 8,6,6,5,6,4,4,4

/11: 7,7,5,4,6,5,4,5

/12: 7,7,7,4,4,5,5,4

/13: 7,6,7,7,3,3,6,4

/14: 8,6,6,6,6,3,3,5

/15: 9,7,7,4,5,7,2,2


Who would win? GuberMensch or The 43-tone Harry Partch Genesis scale?

It turns out that the Genesis scale has inconsistent-enough fingering on a standard keyboard that you'd have to learn it the same way I'm having to learn my own 43-note scale.

A scale with perfectly consistent fingering means that you can always find the right pitch (given good relative pitch) by ear and the right pattern of key-relationships.

But Partch's 43-note scale (tuning, whatever) doesn't really do that.

So, with both scales, you have to label and color-code the keys.

I tried to upload a photo of my keyboard, but it will have to wait 'til I can cut the size of the .JPG file down.

With 8 different colors or styles of marking, it's easy to distinguish at least 8 different basic overtone-series patterns by eye. (Memorizing all the other myriad scales will come after that first task.)

Here, for anyone whose software synth can read a Scala file, is the Harry Partch Genesis scale, taken from some website (not confirmed accurate, but it looks good.)



!
Partch Genesis scale
43
!
21.506
53.273
84.47
111.731
150.637
165.004
182.404
203.91
231.174
266.871
294.135
315.641
347.408
386.313
417.51
435.08
470.78
498.045
519.551
551.318
582.512
617.488
648.68
680.449
701.955
729.219
764.916
782.492
813.687
852.592
884.359
905.865
933.129
968.826
996.091
1017.596
1034.996
1049.363
1088.269
1115.533
1146.727
1178.494
1200.00000
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Last edited by calebprime; 10th March 2013 at 09:30 AM.
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Old 11th March 2013, 04:55 PM   #73
CplFerro
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Dear calebprime,

Thanks for posting your recent musical work. I have listened to the fugazzi one twice now, and it occurs to me to ask you about it. I'm not trained in music and have no idea why people composed what they composed. I'm afraid I'm mystified by complex instrumental music. What do Mozart's pieces mean, for instance? So, I'm wondering what your piece means, or is it a Rorschach inkblot in which we see whatever we bring to it?

Cpl Ferro
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Old 12th March 2013, 07:09 AM   #74
calebprime
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Originally Posted by CplFerro View Post
Dear calebprime,

Thanks for posting your recent musical work. I have listened to the fugazzi one twice now, and it occurs to me to ask you about it. I'm not trained in music and have no idea why people composed what they composed. I'm afraid I'm mystified by complex instrumental music. What do Mozart's pieces mean, for instance? So, I'm wondering what your piece means, or is it a Rorschach inkblot in which we see whatever we bring to it?

Cpl Ferro
I know from reading your posts that you are civil and like polite back-and-forth, so I'll try to answer in that spirit.

You raise very fundamental issues: Questions that sound simple, but have complicated answers. Or rather, if the answers I give sound cryptic, it's because the underlying reality is complicated.

Here goes: There is an objective aspect to music. It is very complex, but it can be explained to some extent. It has simple elements. Like anything -- investment, law enforcement, religion, physics -- it may require participation to really understand.

There is also a subjective aspect to music. Every individual will bring a somewhat different mind and musical sensibility to a musical experience. On different days, the same individual may be in different moods and hear the exact same recording in different ways.

However, his or her subjective reactions occur within predictable limits, usually.

Having said that, here's some info about the Fuggazi piece.

Here's a decent Wiki explanation of a fugue:
http://en.wikipedia.org/wiki/Fugue

>It's sort of a fugue, but a 20th century-style fugue.

>A fugue typically has a theme enter alone in one voice. (This is called the subject.) Then the same theme enters in a second voice while the first voice continues. Except that when the second voice enters, the theme is a little different. (In a fugue from the Baroque period, it typically is transposed or shifted up a perfect fifth.)

>My theme is based on a carefully-constructed 12-note series, with certain special properties. Because I'm trying to have the whole design of the piece come from those properties, the musical intervals at which each voice enters are derived from that.

>The piece is neither tonal like Baroque music, nor strictly atonal. It begins and ends on C#, and has many points of repose where the chords have a definite tonal sound. In between, there are chords that are like the weird cousins of jazz or something.

>Unlike some of the stuff I'm currently trying to work out, it's based on a conventional 12-pitch-per-octave tuning.

>Part of the inspiration for it, and the idea for the design, come from a particular Bartok piece.

These are more-or-less objective facts.

Now, when my composer friend Andy listened, he tracked the pitches in his head (could write them down if given a few tries), and anticipated what I was going to do next, and commented very astutely on some of my choices.

He liked the piece, and knew that it wasn't an earth-shattering masterpiece, but a fairly well-crafted piece in many ways.

He could hear that it was based on 12-tone rows, but managed not to suffer from some of the lack of musicality of 12-tone rows, because that's the skill involved.

He could hear that the rhythm was "floaty" and not strict, so the piece sounded to him like clouds forming and reforming.

Now, he wasn't concerned about whether the clouds resembled a camel or a whale. But he has a visceral reaction to harmony and handling of dissonance, and he did with this piece, too.

Because the rhythm is "floaty" and not pulsed, the piece lacks the momentum that Bartok's piece has, and some of tight design and brilliance of Bartok, even some of Bartok's audacity. My ear for harmony is more conservative than Bartok's, and Andy could hear that, right away.

Someone else without Andy's ear and mind might be a more typical listener. They would hear the piece with reference to how the music would be used in a film-score, most probably. In that case, it would sound like something tense or uncanny or creepy or dark or sad.

So, objective craft, subjective reaction. Neither a bat, a camel, nor a whale.

When I've listened recently, I think that I got it 90% right, but there are things about it that don't completely satisfy me. It's not perfect.

The Bartok piece that inspired it is a masterpiece. This isn't. But I'm doing something in some ways slightly harder technically, fwiw.

The Bartok is better.

Another inspiration for the piece is the beginning of Mahler's 10th Symphony. That piece also is in that twilight area between tonal and atonal.

The Mahler is better.

eta: Upon listening to it after a long time away, it sounded good to me. Not botched.

Now Mozart: I'm not worthy to sniff the sweat from his etc. An entirely different order of human being. Tchaikovsky once made the remark that for him, Mozart was the musical Christ, or words to that effect. I could work for ten years on a piece and never hope to have it come close to what he could write in a week or less.
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Old 12th March 2013, 12:35 PM   #75
CplFerro
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Thanks for that explanation, calebprime, and thanks also for publishing your works here; I'm enjoying your Fugazzi piece.

If you would, though, another question, pardon me if I'm being thick: You talked about objective technical aspects of music, and how listeners have a subjective reaction to the products of this, but I'm wondering what your intention was with it? Is your goal to achieve technical excellence, a kind of puzzle-solving, or is there an idea you are trying to express? E.g. Beethoven's pastoral expresses the glory of nature.

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Old 12th March 2013, 12:46 PM   #76
calebprime
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Heh, funny, I imagined you might ask me that. I'll try to answer more tomorrow.

Briefly, though, something on the beautiful/sad side of things.

But it might be that my ear is attracted to some harmonies I find beautiful, which have a certain amount of tension in them. Then, can I make four voices play those lines without it turning to crap? That is, keeping the beauty of those harmonies going, while increasing the complexity, not letting the piece devolve into aimlessness?

That's part of the challenge of fugue. But here the rhythm is like a cloud or like an amoeba, so entropy or aimlessness will occur if the pitches aren't right, maybe even faster than if the piece had a strong rhythm to carry it along.

The process might go: Like harmonies. Fit into mold of fugue. Admire Bartok. Be myself.

Then, listening, like everyone, I imagine scenarios.

I'd say my inspiration is first harmonies, then textures, rhythms, forms, melodies.

Beethoven's Pastoral was perhaps his most programmatic piece. Most are less obviously about something.

eta: Oh, another thing: I'm a lot like Ynot trying to make a self-powered helicopter fly. It's the challenge of making some long-cherished technical idea work. But not to show off. More to make the idea sing to you, give you something back for your pains.
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Old 14th March 2013, 03:47 AM   #77
calebprime
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Originally Posted by calebprime View Post
Still looking for the next little kernel.

It occurs to me that if I'm going to be mapping number series to scales with 7 elements -- such as most of the familiar scales -- I should be looking for self-similar sequences of either 7 elements or 14 elements.

If my goal is the standard "every-other" permutation pattern, there turn out to be no perfect patterns of 7 elements. But there are a lot of near-perfect patterns of 14 elements, so I'll be looking at these over the next few days.

The actual series -- the results -- are here called "x-swapped". The series, from 0 to 13, will be mapped onto 2 octaves of a 7-note scale.

Here are what seem to be the only four candidates, each with its quirks.

user row: 0 2 4 6 8 10 12 1 3 5 7 9 11 13
grid row: 1 2 3 4 5 6 7 8 9 10 11 12 13 0
orbit: 1 0 13 11 7 12 9 3 4 6 10 5 8 2
X-swapped: 1 0 13 7 8 11 9 4 12 6 10 3 5 2
intervals: 13 13 8 1 3 12 9 8 8 4 7 2 11 13

user row: 0 2 4 6 8 10 12 1 3 5 7 9 11 13
grid row: 3 4 5 6 7 8 9 10 11 12 13 0 1 2
orbit: 3 0 9 12 5 4 2 13 7 8 10 1 11 6
X-swapped: 1 11 6 0 5 4 13 8 9 2 10 12 3 7
intervals: 10 9 8 5 13 9 9 1 7 8 2 5 4 8

user row: 0 2 4 6 8 10 12 1 3 5 7 9 11 13
grid row: 11 12 13 0 1 2 3 4 5 6 7 8 9 10
orbit: 11 0 6 5 3 12 2 10 13 4 1 8 9 7
X-swapped: 1 10 6 4 9 3 2 13 11 12 7 0 5 8
intervals: 9 10 12 5 8 13 11 12 1 9 7 5 3 7

user row: 0 2 4 6 8 10 12 1 3 5 7 9 11 13
grid row: 13 0 1 2 3 4 5 6 7 8 9 10 11 12
orbit: 13 0 2 6 1 4 10 9 7 3 8 5 12 11
X-swapped: 1 4 2 9 5 11 3 8 10 7 6 13 12 0
intervals: 3 12 7 10 6 6 5 2 11 13 7 13 2 1


Suppose we take the last one, and map this to A major for the moment, just to see how it works.

0=A1, 1=B1, 2=C#2, 3=D2...etc. to 6=G#2, 7=A2, 8=B2, 9=C#3...13=G#3 || 14 would equal A3

We get the sequence of pitches B1,E2,C#2,C#3,F#2,E3,D2,B2,D3,A2,G#2,G#3,F#3,A1 (repeat)

and embedded in this, starting on the third note, is the sequence at "every other note" C#2,F#2,D2,D3,G#2,F#3,E2,C#3,E3,B2,A2,(G#3,A1(or adjusted)

Which is identical to the first sequence, up a diatonic step -- except for the last two notes.

This process of embedding can be carried out as far as one would like, with any scales of 7 notes, not just major scales. I like music where the scales change, because of my jazzoid background.

I'm going to try this for my next piece, unless I see some fatal flaw or get a better idea in the meantime.
The 43-note scale is ok to play on two keyboards, but trying to compose a self-similar piece with it is a nightmare, because the range is so narrow. To do the beginnings of a piano sketch, I already had to have 6 tracks of piano, one in each octave! Lines were being divided between tracks. Madness.

So I'm back to a slightly-modified version of my 11-limit JI scale with 36 pitches per octave. The orchestra is all ready to go.

I'm back to the 14-tone 7+7 mapping idea as sketched out above.

But I'm confused.

I entered what I thought were exactly the same search parameters, and came up with a different answer, or so it seems in the bleary light of day.


Would you like to find (1) orbits of size n, (2) orbits of size n - 1, or (3) both? 3
grid row: 0,1,2,3,4,5,6,7,8,9,10,11,12,13
user row: 1,3,5,7,9,11,13,0,2,4,6,8,10,12

user row: 1 3 5 7 9 11 13 0 2 4 6 8 10 12
grid row: 4 5 6 7 8 9 10 11 12 13 0 1 2 3
orbit: 4 1 8 9 11 0 6 5 3 12 2 10 13 7
X-swapped: 5 1 10 8 0 7 6 13 2 3 11 4 9 12
intervals: 10 9 12 6 7 13 7 3 1 8 7 5 3 7

user row: 1 3 5 7 9 11 13 0 2 4 6 8 10 12
grid row: 6 7 8 9 10 11 12 13 0 1 2 3 4 5
orbit: 6 1 4 10 9 7 3 8 5 12 13 0 2 11
X-swapped: 11 1 12 6 2 8 0 5 7 4 3 13 9 10
intervals: 4 11 8 10 6 6 5 2 11 13 10 10 1 1

user row: 1 3 5 7 9 11 13 0 2 4 6 8 10 12
grid row: 8 9 10 11 12 13 0 1 2 3 4 5 6 7
orbit: 8 1 0 13 11 7 12 9 3 4 6 10 5 2
X-swapped: 2 1 13 8 9 12 10 5 0 7 11 4 6 3
intervals: 13 12 9 1 3 12 9 9 7 4 7 2 11 13

user row: 1 3 5 7 9 11 13 0 2 4 6 8 10 12
grid row: 10 11 12 13 0 1 2 3 4 5 6 7 8 9
orbit: 10 1 11 3 0 9 12 5 4 2 13 7 8 6
X-swapped: 4 1 9 3 8 7 13 11 12 5 0 2 6 10
intervals: 11 8 8 5 13 6 12 1 7 9 2 4 4 8


The hi-lighted one seems ok to base a piece on. Is this really different from what I posted above/before, and why? More coffee is needed.
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Old 14th March 2013, 05:32 AM   #78
calebprime
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Originally Posted by calebprime View Post
...
But I'm confused.

I entered what I thought were exactly the same search parameters, and came up with a different answer...

Would you like to find (1) orbits of size n, (2) orbits of size n - 1, or (3) both? 3
grid row: 0,1,2,3,4,5,6,7,8,9,10,11,12,13
user row: 1,3,5,7,9,11,13,0,2,4,6,8,10,12

user row: 1 3 5 7 9 11 13 0 2 4 6 8 10 12
grid row: 4 5 6 7 8 9 10 11 12 13 0 1 2 3
orbit: 4 1 8 9 11 0 6 5 3 12 2 10 13 7
X-swapped: 5 1 10 8 0 7 6 13 2 3 11 4 9 12
intervals: 10 9 12 6 7 13 7 3 1 8 7 5 3 7

user row: 1 3 5 7 9 11 13 0 2 4 6 8 10 12
grid row: 6 7 8 9 10 11 12 13 0 1 2 3 4 5
orbit: 6 1 4 10 9 7 3 8 5 12 13 0 2 11
X-swapped: 11 1 12 6 2 8 0 5 7 4 3 13 9 10
intervals: 4 11 8 10 6 6 5 2 11 13 10 10 1 1

user row: 1 3 5 7 9 11 13 0 2 4 6 8 10 12
grid row: 8 9 10 11 12 13 0 1 2 3 4 5 6 7
orbit: 8 1 0 13 11 7 12 9 3 4 6 10 5 2
X-swapped: 2 1 13 8 9 12 10 5 0 7 11 4 6 3
intervals: 13 12 9 1 3 12 9 9 7 4 7 2 11 13

user row: 1 3 5 7 9 11 13 0 2 4 6 8 10 12
grid row: 10 11 12 13 0 1 2 3 4 5 6 7 8 9
orbit: 10 1 11 3 0 9 12 5 4 2 13 7 8 6
X-swapped: 4 1 9 3 8 7 13 11 12 5 0 2 6 10
intervals: 11 8 8 5 13 6 12 1 7 9 2 4 4 8


The hi-lighted one seems ok to base a piece on. Is this really different from what I posted above/before, and why? More coffee is needed.
The user row you entered was trivially different, and should have produced the same results. Hence* the confusion.

0,2,4,6,8,10,12,1,3,5,7,9,11,13 is only a rotation of 1,3,5,7,9,11,13,0,2,4,6,8,12. So, different paramater entered but it should produce the same results.

I have to understand why this is. Otherwise I'm just chasing my tail.

Let's look at an example: These two should produce the same results.

user row: 1 3 5 7 9 11 13 0 2 4 6 8 10 12
grid row: 8 9 10 11 12 13 0 1 2 3 4 5 6 7
orbit: 8 1 0 13 11 7 12 9 3 4 6 10 5 2
X-swapped: 2 1 13 8 9 12 10 5 0 7 11 4 6 3
intervals: 13 12 9 1 3 12 9 9 7 4 7 2 11 13

and

user row: 0 2 4 6 8 10 12 1 3 5 7 9 11 13
grid row: 1 2 3 4 5 6 7 8 9 10 11 12 13 0
orbit: 1 0 13 11 7 12 9 3 4 6 10 5 8 2
X-swapped: 1 0 13 7 8 11 9 4 12 6 10 3 5 2
intervals: 13 13 8 1 3 12 9 8 8 4 7 2 11 13

with the eat-space forum formatting, these can't be aligned easily, but they should produce the same results, and they're not.

Software bug or interesting real result that is an opportunity?


Answer: When we designed this, we decided -- in cases of n-1 orbits, where one set of numbers matches and the rest orbit -- to stick the matching number at the end, as you would in the standard way of notating a one-line permutation. So depending on where the orbit starts, different results will be produced in n-1 cases, because it's as if one note can move around in the series.

This is an opportunity, but I still don't know whether it's an opportunity because I'm not that hep, or a real mathematical principle. It means, I think, that in n-1 cases with 14 elements, there will be 14 slightly different variants of each basic orbit. (?)

It's an opportunity to move the non-matching part of the self-similarity to the end of the series, where it's least audible. (?)

But, to my surprise, each result is different, not merely a rotation of the other results. (?!)




* A good word.
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Old 14th March 2013, 06:23 AM   #79
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Originally Posted by CplFerro View Post
Dear calebprime,

Thanks for posting your recent musical work. I have listened to the fugazzi one twice now, and it occurs to me to ask you about it. I'm not trained in music and have no idea why people composed what they composed. I'm afraid I'm mystified by complex instrumental music. What do Mozart's pieces mean, for instance?

Cpl Ferro
It just so happens that I've written a blog post on this subject recently:

http://www.johnmreese.net/music/

I mainly use this blog to bounce ideas around in my head. Kind of like caleb does with his threads.
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Old 14th March 2013, 08:22 AM   #80
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Wow, I scanned that quickly, and it's a very serious attempt to make sense of things.

I'll read it again.

Maybe I can ask you some intelligent questions about it.

But when it comes to reasoning on that level of abstraction, I don't do that well.

More on that later, after I've worked out the mechanics for today.
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