Yahoo Tuning Groups Ultimate Backup tuning cross-exponent direct lattice connections in blues

ingredients:
- theory complexity (or is it complexity theory?)
- lattices (of course)
- blues power, or, how I got bent last night
- a heady brew of ET and JI

[view this post in a fixed-width font]

--------------------------------------------
preface (yep, it's a long one)
--------------------------------------------

Whoever first used the term 'imperfect consonances'
for the '3rds' and '6ths' certainly had something up his sleeve!

(Was it Jacques de Liege? Theinred of Dover? Walter Odington?
some bloke in a pub?)

Drew Skyfyre wrote:
[way back in TD 3, at the end of last year: 29 Dec 1998]

> In case anyone's interested I found some sites with
> some interesting MIDI files :
>
> Arabic MIDI files
> <http://members.tripod.com/~dalleh/midi.html>
>
> Frank Zappa zipped MIDI files are on this page
> <http://www.pacifier.com/~kurf/>
>
> And here's a nice Etta James Blues MIDI
> (Just be sure to change the main melody instrument
> to a Jazz guitar patch to get it to smoke a bit more)
> <http://www.geocities.com/BourbonStreet/2228/midi/EJames-AtLast.mid>

The Zappa files have been zapped.
Wish I could have heard them.

The Arabic files are *killer* -
lots of microtones in the solo instruments,
really good use of MIDI. I recommend listening.
Have some time on your hands - there are a lot of them.

The great Etta James file is the subject of this posting.
Or the vehicle, anyway.

Go download it now, before it goes the way of Zappa,
listen to it,
and then come back to me.

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

This is the one I promised earlier, where I explore
triangular and other, cross-exponent, prime-lattice connections.

I have been interested for some time in determining
some 'typical' rational and >12-ET representations
of the 'bent' notes one hears in good blues vocal
and guitar work.

This has been an interest ever since first learning
about JI thru reading Partch's _Genesis_,
because I've always loved the blues and used to have
a hard time capturing the 'blues feel' with my
trained-musician chops.

I found out later,
it was because I never learned about microtones in school. :(

Then a few years ago
I happened upon some articles Ezra Sims wrote
analyzing the microtones in an old Louis Armstrong recording
of 'St James Infirmary'.

I analyzed it myself, and found a few notes
that I thought were a tuned a bit differently
than what Sims had said.
This got me even more interested.

Most of you on this list know about the analysis I did
last year of a Robert Johnson vocal, at:
http://www.ixpres.com/interval/monzo/drunken.htm

Recently, particularly in discussion with Paul Erlich
and others regarding the 'musical complexity' and
'lattice metric' measurements, I've been talking about
notes which exhibit quasi-harmonic relationships to a local 1/1
but which are not actual harmonics.

I had also mentioned Partch's idea of a monophonic system,
where these kinds of 'cross-exponent' lattice metrics hold
(at least within the Partch 'diamond').
These are pitch relationships I have discovered to be in use
in some, perhaps most, pop and blues styles.

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

Whether or not this MIDI file is an accurate reflection
of Ms. James's recorded performances, the microtonal
inflections in the 'vocal' part sound very convincing
to me as being typical of this blues idiom.

I was intrigued to see what ratios were being implied
by the '3rd's and '6th's which were the notes having
some microtonal pitch-bend on them.

Apparently the creators of this file
(Bobby Keyes and Gale Bass of Rimfire)
recorded the parts live by playing on synths,
- my guess is that they were wind controllers.
At least that's what it sounds and looks like to me
from the non-quantized start-times
and the curve of the pitch-bend graphs.

So here at last, I had some mathematical underpinning
to go on, when trying to ascertain rational implications
(as opposed to having only my ears to analyze CD tracks).

The theoretical resolution of sequencer pitch-bend
is far finer than any pitch discrimination
discernible by human hearing,
or for that matter, possible on most MIDI equipment.
It leaves open a bandwith of interpretation,
rational or otherwise, for the naming or notation of the note.

I wanted to have the exact pitches someone else recorded,
with an accuracy that would allow me to find
several different allowable numerical interpretations
of characteristic blues vocal gestures.

I've been waiting for a chance to do an analysis like this
with a really good actual musical example,
as opposed to an experiment-type situation.

I found a few surprising things.

I examined all the ratios having numbers under 100
within a cents range of about +/- 10 to 20 cents of
each pitch-bent note. In some cases I noted ratios
with much higher numbers.

In all cases, my goal was to find
the least complex rationalization possible
within a few cents (i.e., an 'inaudible difference')
of the given pitch-bend.

By 'least complex rationalization',
I mean reduction into factors, according to my theories:
the best accomodation of both
the lowest prime-base and the lowest exponent
for a given musical aesthetic effect.
I also gave consideration to odd- or integer- factors
where it seemed reasonable to include.

The rationalization is based on the idea that the
ratios are being perceived as harmonics above the bass note.
(This may not necessarily be 100% valid,
- especially considering that the notes under consideration
are appearing in mainly a melodic,
and only secondarily a harmonic, context -
but regardless, it is the method I used here.)

So the 1:1 of each lattice diagram is the bass note of the chord.

I also give the nearest 72-Eq degree,
as I think this is a decent practical notation for microtones.

Also, because of the numerous even subdivisions of this scale,
quite a few notes crop up which are found in these smaller subsets
(18-, 24-, and 36-Eq).

There are several cases in this track
(and the entire 'sax' solo)
where pitch-bend is used only
to raise the pitch by a 12-Eq semitone or tone,
but there are 3 instances (in the 'flute' part)
which have microtonal 'targets'.

I will examine here in excruciating detail
these 3 particular uses of pitch-bend,
one from measure 7, one from measure 16, and one from
measure 18 in the 'flute' lead vocal track.

------------------------------------------------
Introductory: assumptions, structure, notations
------------------------------------------------

All ratios here are notated assuming 'octave'-equivalence,
and ignoring the 'octave' placement of the notes.

The key of the piece is F# major,
so the overall 'root' is F# n^0 [= 1/1].

The chord progression of the basic part of the tune
is I - VI - II - V, the roots and bass line of which are:

F# n^0 maj - D# 2^(9/12) min - G# 2^(2/12) min - C# 2^(7/12) dom7

After an intro, the vocal line starts at measure 5.
I am examining here only the 'flute melody' track
and only the first part ('A section') of the tune.

The basic structure of the tune is thus,
(measure numbers in Arabic numerals and bass line in Roman):

5 6 7 8 9 10 11 12
I VI II V I VI II V I VI II V I VI II V

13 14 15 16 17 18 19 20
I VI II V I VI II V I VI II V I IV I I >> to the 'bridge' ...

The precise timing is expressed as Measure.Beat.Tick,
with 480 Ticks or pulses per quarter-note (ppq).

A pitch-bend amount of 4096 units per 12-eq semitone
is assumed.

I use an adaptation of Ezra Sims's 72-eq notation,
to illustrate the pitches in my explanations with
letter-names which come within ~8.3 cents of any ratio.

This notation uses, in addition to letter-names,
the following accidentals:

# and b for semitones (as in usual 12-Eq),
^ and v for quarter-tones,
> and < for sixth-tones,
+ and - for twelfth-tones, as follows:

cents sharp flat

1/2-tone 100 # b

1/4-tone 50 ^ v

1/6-tone 33.3 > <

1/12-tone 16.7 + -

Using the division of the first 12-eq 'whole tone' to
illustrate these accidentals gives the following table,
in 1/4-, then 1/6-, then 1/12-tones,
with cents given at the bottom:

0 1/4 2/4 3/4 4/4 = 1
C C^ C# (C#^) D
(Dbv) Db Dv

0 1/6 2/6 3/6 4/6 5/6 6/6 = 1
C C> C#< C# C#> D< D
Db< Db Db>

0 1/12 2/12 3/12 4/12 5/12 6/12 7/12 8/12 9/12 10/12 11/12 12/12 = 1
C C+ C> C^ C#< C#- C# C#+ C#> D< D- D
Db< Db- Db Db+ Db> Dv

0 17 33 50 67 83 100 117 133 150 167 183 200

The fractions actually also indicate the exponents of 2.

It would be possible to combine many of these symbols into
further 'enharmonic equivalents' beyond those presented here,
(as, for example, the two I indicated in parentheses)
but I think that is a needless complication
and I prefer to simplify as much as possible,
using as few total symbols as possible, and
giving equivalents only where both members of a pair
present equally complicated notations.

In addition, the prime-factor or ratio
always gives the exact JI or ET intonation.

In addition to giving cents values for all pitches,
I also provide the pb [= amount of pitch-bend]
to facilitate comparison of all notes on the pitch-bend graphs
and to make it easy to look into the MIDI file itself
and examine what I'm discussing.

(and most importantly - to listen to it!)

-----------------------------------------
example 1: measure 7 (7.1.97 -- 7.1.204)
-----------------------------------------

1st measure of 2nd part of 1st phrase

Bass: F# n^0 [= I]

Melody: C# 2^(7/12) [= '5th'],
bending upwards to a 'neighboring-tone' near D# [= '6th'],
then back down to C#.

The melody cycles back to the beginning of the next phrase here
landing on C# 2^(7/12), the '5th' degree of the scale.

The 'note-on' event happens 16 pulses,
or just 1 pulse more than a 32nd-note,
before the start of measure 7.

(This is just a 'human-feel'
anticipation for what would be notated as beginning right
on the first beat of measure 7.)

The MIDI-file uses pitch-bend to raise the pitch of this note
according to the following time-vs-frequency graph:

6.4.464 NOTE ON: C# 2^(7/12) ...>> 6.4.480 = 7.1.1 ...>>
______________________________________________________|
|
[timing pulses]
7.1.96 100 104 108 112 116 120 124 164 188 192 196 200 204 >>

[pitch-bend amounts]
6288
6080
5872
5664
5456
5040 5040
4608
4400
3984
3568
3136
2720
2304
1680
1456
1040
416 416
0 0

(the note sustains on C# 2^(7/12) well beyond this point)

The total time involved in the pitch-bend,
from pulse 96 to pulse 204, is exactly 9/40 of a quarter-note,
or just over 2/9, i.e., 2/3 of a triplet divsion of the triplet
eighth-note rhythm played on the piano:

Measure: 7
Beat: 1 2 etc.
Triplet 8ths: 1 2 3 1 2 3
Triplet of triplet: 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3
\pb/

\pb/ signifies approximately where the pitch-bend occurs
in this measure, and thus the timing of the above graph.

The pitch-bend values of 6080 and 6288 occupy 37% and 22% respectively
of the total time of the pitch-bend, thus are by far the most
important indicators of the perceived pitch of this short note.

A pitch-bend value of 6080 is more than a semitone [= 4096] above
the original starting pitch, to be more precise, ~148.4 cents.
Likewise, 6288 = ~153.5 cents.

Thus the two most significant pitches in this series of pitch-bends
lie 1.6 cents below and 3.5 cents above an exact quarter-tone
of D^ 2^(51/72) [= 2^(17/24) = 850 cents = 7 +6144 pb].

(24-Eq, 72-Eq, and cents)

_______
16/24 | 17/24 | 18/24
D | D^ | D#
| (Ebv) | Eb
| |
| |
48/72 49/72 50/72 | 51/72 | 52/72 53/72 54/72
D D+ D> | D^ | D#< D#- D#
| | Eb< Eb- Eb
| |
800 817 833 | 850 | 867 883 900
-------

This is an excellent ET representation
of this microtonally-inflected 'neghboring tone'.

The only ratios
with a combination of low primes and low exponents
which lie near this range
are 13/8 [= 13^1 = ~840.5 cents = 7 +5756 pb]
and 18/11 [= 3^2*11^-1 = (3^2)/11 = ~852.6 cents = 7 +6250 pb].

18/11 is so close to the quarter-tone that D^ may
represent it well, only 2.6 cents low.

Representing 13/8 is the thorn in the side of 72-Eq.
It is the worst of the representations in 72-Eq as far as
the most basic rational intervals are concerned.
D^ is ~9.4 cents too high, and D> is ~7.2 cents too low.
Of course, this is still not *that* bad compared with 12-Eq.

In any case, 13/8 seems to me a little too low to serve
as the ratio in this particular case, especially considering
that the pitch of ~848.4 cents has the longest duration.
I would say that if a rational 'target' had to be pinpointed
for this little 'vocalism', it would be 18/11.

On the odd-factor triangular lattice, it would already
have a 1-step connection.

( )
/|\
/ | \
/ | \
1:1---------3:2 \
'-. /. ' ' .\
18:11--------( )

(I'm treading on thin ice with this 9-limit stuff
- perhaps one of you 'odd' fellows can help out
with a nice 9-odd-limit triangular lattice)

On the prime-factor rectangular lattice (ahh...that's better),
it's two rungs up the 3-axis and one rung down the 11-axis.

/|
/ |
/ |
C# 3:2 |
/ |
/ |
/ |
F# 1:1 |
|
|
D^ 18:11
850~853c

But if it can be analyzed as funtioning here as
a 'pseudo-harmonic' of the overall fundamental of F#,
as my hearing of it tells me it can,
it would have a direct link to 1/1
even on the prime-factor lattice,
similar to the way 13/8 already does.

So the 3 steps of the original lattice metric become 1 step.

Or perhaps what's happening is that the 12.1-cent xenharmonic bridge
between 13/8 and 18/11 is being crossed in my hearing of this.
So on my lattice, I would draw a 'unison vector' between these
two notes, and by extension, all other points on the lattice
which embody this same metric relationship.

But it still indicates the existence of a direct link
from 18/11 to 1/1.

/|
/ |
/ |
c# 3:2 |
/ |
/ |
/ |
F# 1:1 . |
. |
. |
D^ 18:11
850~853c

This particular example gives added weight to an
odd-factor, rather than prime-factor, orientation.

---------------------------------------------
example 2: measure 16 (16.3.004 -- 16.3.156)
---------------------------------------------

2nd measure of 2nd part of 2nd phrase

Bass: C# 2^(7/12) [= V]

Melody: D# 2^(9/12) [= 'major 2nd' or '9th'],
bending upwards to near E, the 'minor 3rd' or 'sharp 9th'.

I was misled a bit by the graphs I saw in Cakewalk
when I first looked at this. I'll present it here
divided into two halves. Only the first half will
turn out to be relevant.

duration = 152 pulses

1st HALF:

16.3.004 NOTE ON: D# 2^(9/12) ................. NOTE OFF 156

[timing]
4 92 96 100 104 108 112 116 120 124 128 132 140 156

[pitch-bend]
2928
2720
2512
2304
2096
1888
1680
1456
1248
1040
832
624
416
208
0

2nd HALF:

[timing]
156 188 196 224 228 232 236 240 244

[pitch-bend]
3568
3360 3360
3136

2512

1888

1456

1040

416

I could hear that the note duration did not extend to
cover the entire pitch-bend graph, because the note
bent upwards and then cut off without returning back down
as the graph shows. I did not at first realize how short
the note was, however.

So I examined what I thought was the significant range
of pitch-bend values, based on their length of time,
as above in example 1, and got ~71.5 and ~87.1 cents
for pitch-bend values 2720 and 3568 respectively.

Considering the notes as intervals from the bass note,
we add in the original 200 cents of the D# 2^(9/12),
and thus have a range from ~271.5 -- ~287.1 cents.

The 72-eq notes closest to these are, respectively,

E< [= ~266.7 cents = 2^(16/72), '1/6th-low E' = 2 +3413 pb],
~4.8 cents too low, and

E- [= ~283.3 cents = 2^(17/72), '1/12th-low E' = 2 +2731 pb],
~3.8 cents too low.

(18-,36-,72-Eq, and cents)

_____________
3/18 | 4/18 |
D# | E< |
| |
| |
6/36 7/36 | 8/36 | 9/36
D# D#> | E< | E
Eb> | |
| |
| |
12/72 13/72 14/72 15/72 | 16/72 17/72 | 18/72
D# D#+ D#> Ev | E< E- | E
Eb Eb+ Eb> | |
| |
200 217 233 250 | 267 283 | 300
-------------

I think these are both close enough so that
72-Eq notation represents these intervals well.

There are also two very close low-prime-low-exponent
rational interpretations for these notes, respectively:

7/6 [= ~266.9 cents = 2 +2739 pb], ~4.6 cents too low for
pitch-bend value 2928, but very close to 2720, which was
also held for a very short length of time.

13/11 [= ~289.2 cents = 2 +3654 pb], ~2.1 cents too high, and

As far as their usefulness in representing the ratios,
E< is only ~0.2 cents lower than 7/6, and
E- is ~5.9 cents lower than 13/11.

On the odd lattice, again, both ratios are already
one-step connections to 1/1.

I'm sorry but since it would have 4 dimensions,
I refrain from attempting to draw one.
(again, if anyone else wants to do it ...)

I can at least show where 7/6 goes on
one version of the triangular:

7:6---------( )
.-' \'-. .-'/
(4:3)-----\--1:1--/
\ \ /| /
\ \ | /
\ / \|/
\ / ( )
\ /
( )

On the prime lattice, in both cases, we have
a combination of 1 utonal and 1 otonal step
transforming into a direct 1-step connection, this time
to C# 2^(7/12) [i.e., the bass note],
or C# 3/2 if we ignore the ~2-cent difference.

C# 1:1 .
/| . .
/ | . .
/ | . .
(F# 4:3)------------E< 7:6 .
| 267c E- 13:11
| _.-' 283~289c
| _.-'
| _.-'
| _.-'
| -'
( )

These connections are also already represented
on triangular prime lattices.

So these interpretations worked on paper.
However, there are two important things I overlooked
about this example:

The first thing to notice is that the note is turned off
after a duration of 152 pulses, which means at 4 + 152 = 156.
So all the pitch-bend information after this is irrelevant,
because it's not audible.

The other thing is that the first 88 pulses,
or 58% of the total time for the note, are occupied by
the exact (unbent) 12-eq D# 2^(9/12).

So only 7/6, or in 72-Eq, E<, would be implied for this note.
Of course this is no real news - its been long established
that 7/6 is an important rational value for the "minor 3rd".

I had done my analysis of both pitches first,
so I tried extending the note value
to cover the entire graph of the pitch-bend,
to hear what that higher value sounded like.

What I found was that the phrase still sounded very musical,
and entirely in keeping with the idiom,
with a very subtle difference from the lower value.
So E- or 13/11 could also be implied very well in this context.

These notes are functioning, simultaneously,
harmonically, as 'minor 3rd' or 'sharp 9th' of the C# bass
and melodically, as a kind of 'harmonic 7th' of the F# 'tonic'.

7/6, of course, *is* in this case the harmonic 7th of F#.
(again, we're ignoring the ~2 cent difference between JI and 12-Eq)

13/11, in relation to F# n^0, is closer to the E 16/9 of F#
than to its harmonic '7th'.

In neither case with this example is there anything gained
or lost by replacing one type of factorization with the other,
as all ratio terms are already prime.

---------------------------------------------
example 3: measure 18 (18.2.472 -- 18.2.316)
---------------------------------------------

(This was the one I analyzed first, and find it to be
the most interesting case of the three.)

2nd measure of 3rd part of 2nd phrase

Bass: C# 2^(7/12) [= V]

Melody: A 2^(3/12) [= 'flat 6th' or 'flat 13th'],
bending upward to near A# [= '6th' or '13th'],
then back down to C#.

This pitch-bend occurs during the cadence of
the second statement of the tune.

Again, as in measure 7, the note under consideration
has a 'human-feel' anticipation and starts just before
the third beat.

duration = 324 pulses

1st HALF:

18.2.472 NOTE ON: A 2^(3/12)...>> 480 = 19.1.1 ...>>
________________________________________|
|
[timing]
19.1.140 152 156 164 168 176 180 188 192 200 204 212 216

[pitch-bend]
2720
2512
2304
2096
1888
1680
1456
1248
1040
832
624
416
208
...0

2nd HALF:

.................. NOTE OFF 316
[timing]
216 312 316 320 324 328

[pitch-bend]
2720
2512

2096

1680
1456

832

208
0

We see here a situation similar to the one above in
example 2: the note duration does not extend to the
full length of the pitch-bend graph, rather cutting off
before the upward pitch-bend comes back down.

It is different, however, in that because of its length
and the relative shortness of the other values,
the only pitch-bend amount under serious consideration
in the perception of the pitch of this note is 2720.

This pitch-bend value is held for 96 pulses,
or very close to 30% of the duration of the entire note.
It is equivalent to ~66.4 cents.

'A' 2^(3/12) here is functioning as a 'flat 6th' or 'flat 13th'
800 cents above the bass note of C# 2^(7/12),
thus the size of the bent interval is ~866.4 cents.

There is again a very close 72/36/18-Eq note
which can represent this pitch extremely well:

A#< [= '1/6th-low A#', or possibly '1/3-high A' = 2^(52/72)
= 2^(26/36) = 2^(13/18) = ~866.7 cents = 8 +2731 pb],
only ~0.3 cent higher than the pitch in the MIDI file.

(18-,36-,72-Eq, and cents)

_______
12/18 | 13/18 |
A | A#< |
| Bb< |
| |
| |
24/36 25/36 | 26/36 | 27/36
A A> | A#< | A#
| Bb< | Bb
| |
| |
48/72 49/72 50/72 51/72 | 52/72 | 53/72 54/72
A A+ A> A^ | A#< | A#- A#
| Bb< | Bb- Bb
| |
800 817 833 850 | 867 | 883 900
-------

So finding an ET note for this one was easy.

I searched hard to find the best rational interpretation
with relatively low primes and exponents.
I took note that this pitch lies almost mid-way between
the 26th and 27th harmonics, of ~841 and ~906 cents respectively.

I found this to be very interesting, as I have already noticed
the wide variety of ratios that can be used as '6ths':
not only 26 [= 13] and 27 [= 3^3], but also 5/3 [= ~884 cents].
And of course, the 5-limit JI variant of these 'major 6ths'
is the 8/5 [= ~814 cents] 'minor 6th'.

The harmonic which is the arithmetic mean between 26 and 27
is 53, with an interval size of ~873.5 cents [=
, 7.1 cents too high.
This is pretty close, but 53 is quite a high prime,
so it had to be excluded on those grounds.

Just to see how high one had to go to find a harmonic
that accurately represented this pitch, I went two 'octaves'
further up the harmonic series, and found that 211
[= ~865.3 cents] was only ~1.1 cent too low. But again,
211 is prime, and very high.

The most obvious ratio to consider first would be 5/3
[= ~884.4 cents = 8 +3455 pb], but it's ~18 cents too high
- too much of a difference.

A better candidate was 105/64
[= 3*5*7 = ~857.1 cents = 8 +2339 pb],
~9.3 cents too low, but with only 7-limit complexity.
But 9 cents is still an appreciable difference,
and 105 has quite high odd- and integer-complexity.

The best 'fit' I could find with both low primes and exponents
was 33/20 [= 3^1*5^-1*11^1 = (3*11)/5 = ~867.0 cents = 8 +2743 pb],
only ~0.6 cent higher than the note in the file.

This is found very interesting.

On the odd-factor triangular lattice,
this ratio is 2 steps away from the bass note:
one step along the 6/5 axis, and one rung up the 11-axis.

( )---------( )
\'-. .-'/ \'-.
\ 1:1--/---\--( )
\ |\ / \ /|\
\ | / \ | \
\|/ \ / \| \
( )--------33:20 \
'-.\ /.-' '-.\
6:5---------( )

(again, I'm not sure if I put the 11-ratio in the right place)

On the prime-factor rectangular lattice, it's 3 steps:
one rung up the 3-axis, one rung up the 11-axis,
and one rung down the 5-axis.

( )
|`'-,_
| ` A#< (3*11):5
| ~867c
|
|
|
|
|
|
|
( )
/
/
/
C# 1:1
/
/
/
[F#]

That gives it a moderately high lattice metric complexity.

But in this MIDI sequence, it's clearly being used
with the same effect as a harmonic of C#,
thus implying a direct 1-step connection.

( )
|`'-,_
| ` A#< (3*11):5
| ~867c
| .
|
| .
|
| .
|
.
( )
.
/
/.
C# 1:1
/
/
/
[F#]

In this example, I see no advantage in
odd-limit over prime-limit because
it would be on a 33-axis, which I think is
too high to be a primary axis
*in regular odd-limit theorizing*.

But as listening to this proves, it must be admitted
that the 33/5-axis has a significant musical meaning.

There is no other 'simpler' (less-complex) ratio
anywhere near it in pitch-space,
and it is functioning distinctly as a pseudo-harmonic
of either the V or the I, or both.

Take note of its dual function as botha flattened 'major 6th' over C#
and
a flattened ('blue') 'major 3rd' over the melodic tonic F#.

-----------------------------------------
conclusion
-----------------------------------------

I take this to be indisputible evidence,
even tho I've been arguing virtually the opposite,
that a strictly harmonic-series model of 'harmonic' consonance
- i.e., in tri[+]ads - is not valid,
or at least, that the 'bridges' between harmonics
and these types of 'pseudo-harmonics'
must be taken into account.

I hear an electronic psuedo-Etta James 'singing'
a note that starts a 'half-step' flat then bends upwards,
and I think it's going to bend a 'half-step'
up to its 'regular' pitch,
but it stops short only partway up,
and this tiny subtlety has an enormous impact.

I go back and listen again and again,
and still I can't get enough.

I think there's an awesome amount of information
packed into what's going here.

-monzo
http://www.ixpres.com/interval/monzo/homepage.html

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cross-exponent direct lattice connections in blues

🔗Joe Monzo <joe_monzo@xxxxxxx.xxxx>

🔗Joe Monzo <joe_monzo@xxxxxxx.xxxx>