Simple Beauties of JI: The Harmonic Mirror

Back in the 1980s I noticed a simple but elegant structure, which I
have came to call the Harmonic Mirror.

The scales I derive from the Harmonic Mirror are intended to be used
melodically over a harmonic drone. They are highly compatible in this
regard, because as we shall see, they are "genetically" related to
the Harmonic Series.

If we place the ratios with odd numerator on the left, and the
inversions on the right (even numerator), letting each odd limit
terminate before exceeding the 2/1, we obtain the below simple ratio
sequence:

Ratio Cents Ratio Inv. Cents Value
1/1 0.000 2/1 1200.000

3/2 701.955 4/3 498.045

5/4 386.314 8/5 813.686
5/3 884.359 6/5 315.641

7/6 266.871 12/7 933.129
7/5 582.512 10/7 617.488
7/4 968.826 8/7 231.174

9/8 203.910 16/9 996.090
9/7 435.084 14/9 764.916
3/2 701.955 4/3 498.045
9/5 1017.596 10/9 182.404

11/10 165.004 20/11 1034.996
11/9 347.408 18/11 852.592
11/8 551.318 16/11 648.682
11/7 782.492 14/11 417.508
11/6 1049.363 12/11 150.637

13/12 138.573 24/13 1061.427
13/11 289.210 22/13 910.790
13/10 454.214 20/13 745.786
13/9 636.618 18/13 563.382
13/8 840.528 16/13 359.472
13/7 1071.702 14/13 128.298

15/14 119.443 28/15 1080.557
15/13 247.741 26/15 952.259
5/4 386.314 8/5 813.686
15/11 536.951 22/15 663.049
3/2 701.955 4/3 498.045
5/3 884.359 6/5 315.641
15/8 1088.269 16/15 111.731

17/16 104.955 32/17 1095.045
17/15 216.687 30/17 983.313
17/14 336.130 28/17 863.870
17/13 464.428 26/17 735.572
17/12 603.000 24/17 597.000
17/11 753.637 22/17 446.363
17/10 918.642 20/17 281.358
17/9 1101.045 18/17 98.955
etc...

Of course this can be extended as far as one could possibly need to
go - infinitely. A noteworthy quality about this sequence, as one can
easily observe, is that each Odd Limit begins with an odd numerator
Super-Particular Ratio, and with the final inversion of each Odd
Limit, the sequence "closes" with an Even Numerator Super-Particular
Ratio.

So in short, the Harmonic Series is a connecting thread, running
through this sequence. Another interesting fact, is that all the
ratios within each super-particular "framed set", are also easily
found by addition of successive members of the Harmonic Series.

It is a veritable paradise of JI flavors, for both melody and
harmony. There are myriad of ways that one could subset from this,
and I'll provide a couple of methods I've found valuable from a
melodic point of view and stylistic approach.

So let's look a four 13 tone scales derived from this sequence.
Because the Harmonic Mirror scales are so relative to, and derivative
of the harmonic series, it can be an interesting approach to just
merely extract a series of scales in ascending order by odd limit,
taking each functional interval from where it falls in the sequence.
As one can see below, each of these scales has inversional symmetry
at the tritone.

#1
Ratio Cents Value
1/1 0
13/12 138.573
8/7 231.174
6/5 315.641
5/4 386.314
4/3 498.045
7/5 582.512
10/7 617.488
3/2 701.955
8/5 813.686
5/3 884.359
7/4 968.826
24/13 1061.427
2/1 1200.000

#2
Ratio Cents Value
1/1 0
14/13 128.298
9/8 203.910
7/6 266.871
9/7 435.084
13/10 454.214
11/8 551.318
16/11 648.682
20/13 745.786
14/9 764.916
12/7 933.129
16/9 996.090
13/7 1071.702
2/1 1200.000

#3
Ratio Cents Value
1/1 0
15/14 119.443
10/9 182.404
11/9 347.408
14/11 417.508
15/11 536.951
18/13 563.382
13/9 636.618
22/15 663.049
11/7 782.492
18/11 852.592
9/5 1017.596
28/15 1080.557
2/1 1200.000

#4
Ratio Cents Value
1/1 0
16/15 111.731
11/10 165.004
13/11 289.210
16/13 359.472
17/13 464.428
24/17 597.000
17/12 603.000
26/17 735.572
13/8 840.528
22/13 910.790
20/11 1034.996
15/8 1088.269
2/1 1200.000

Number 4, is the scale for the flute I used on my Tuning Punks
piece "The Fourth Order". And that's a very simple demonstration of
what I'm speaking about, as far as my melodic approach to these
tunings.

I've found that one can extend this sequence very far, and still
produce scales that sound compatible with a harmonic drone - I have
explored ratios up to 37. As above, there are many ways to subset
from the sequence, and I have worked with large scale structures too,
even have for years explored tunings where I have a different tuning
in each octave, mapped across the midi range, so I have this morphing
tuning happening as I play.

The detailing of all the subset methods, and variations on the theme
would be extremely bulky to present here, but I hope this little
window into this harmonic tuning approach may inspire independent
investigations. As always, feel free to "Steal this Scale" (remember
Hoffman's funny "Steal this Book"?), and please let me know what you
think. These may fail the "lattice test" but perhaps the success that
one will find in using them melodically over a harmonic drone, will
show the truth that "Man cannot live on Lattice Alone". And we still
are without solid theory about melodic concerns, which can be applied
to any musical scenario. Still the final "proofs" for melodic theory
are those that live manifest in actual music.

One microtonalist I asked about the Harmonic Mirror sequence, told me
that there are many ways to create similar "framing set" sequences
like this without using the harmonic series at all. I hope to be
taken under his guidance to learn what this might be, but I have also
worked with a Phi weighted version of this sequence as well, which
likely qualifies.

Thanks,

Jacky Ligon

--- In tuning@y..., ligonj@n... wrote:

Hi Jacky, you wrote:

"The scales I derive from the Harmonic Mirror are intended to be used
melodically over a harmonic drone."

I got interested because of the words 'melodically' and 'drone'.
I am trying to see if it has any relevance to Indian music. Can you
elaborate [so that an Indian musician with no background in Tuning
and temperament, can follow]? What is a harmonic drone? If this is
too irritatingly elementary for the group, you may reply through an
email (I am not familiar with any of such conventions).

I know I have yet to reply to some of your queries. Later,
definitely.

Thanks. Regards,
Haresh.

Jacky wrote:

> Back in the 1980s I noticed a simple but elegant structure,
> which I have came to call the Harmonic Mirror.

> <<4 examples>>

Hi Jacky!

OK I charged up these beauties and activated them to see
what they were made of. I should say that going in to this
I am a little bit skeptical of this whole symmetry thing
as a theory.

First I viewed the scales graphically and found them quite
beautiful and intriguing to look it. I found my eyes
moving from line to line, examining the spacing and the
symmetry. Interesting indeed. Intriguing. But how would
they sound?

Tuning #1: Ho hum. Yeah OK. It's probably just me but I
just do not get excited over overly consonant scales.

Tuning #4: Holy Cow!!!!!!!!!!!!!

Yes tuning #4 is really something. It is totally
astonishing. And I do not get too excited about most
new tuning ideas.

I didn't play it over the drone. I'm bad. I just
tuned a piano with it and stamped it up and down
in octave repetition.

There's something about these degrees:

17/13,24/17,17/12,26/17

First that 24/17,17/12 in the middle there is
just plain silly but I gotta admit it works.

And I greatly enjoyed the shimmering effects I got from
those close scale degrees & their interactions with other
notes in tunings #2 and #3.

Now that 26/17 interval is pretty darn neat.
It does something to me.
And I started moving around, playing.
And I was getting all overcome with these weird feelings.
And I actually started crying.
And I thought, now this is really something what is going
on here? These beat patterns have got ahold of me and are
wrenching me around this way and that and I feel like I am
floating or something or just what is going on anyway.

I stopped and I felt totally overwhelmed -- like I had
been through some incredibly intense emotional experience
that changes your life but I can't quite say what it was.

Kind of like a weird dimesnion of euphoria or something?
Maybe if I was a drug fiend I could compare it to some
particular kind of trip but I just don't ever touch any of
those things 'cos I don't want to screw up my creativity
by randomly rewiring my brain which I have spent a great
deal of time and effort to wire properly. (Besides, my
friends have described their drug trips to me and I am
always unimpressed because I have to tell them "So you are
saying that for you it is not normal to see and feel those
things and you have to you drugs to get there? Dude, like,
that's what I go through when I write music!")

So that's it for tuning #4 so far.

I'll take a look again at tunings #2 and #3 again later
but it seems on first listening to me that the general
trend is that the tunings progress upwards in terms of
profundity and depth and dimensionality.

All tripped out and stuff, man. Freaking way to go, Jacky!
You the man!

- Jeff

--- In tuning@y..., "Haresh BAKSHI" <hareshbakshi@h...> wrote:
> --- In tuning@y..., ligonj@n... wrote:
>
> Hi Jacky, you wrote:
>
> "The scales I derive from the Harmonic Mirror are intended to be
used
> melodically over a harmonic drone."
>
> I got interested because of the words 'melodically' and 'drone'.
> I am trying to see if it has any relevance to Indian music. Can
you
> elaborate [so that an Indian musician with no background in Tuning
> and temperament, can follow]? What is a harmonic drone?

Haresh,

Hello! Actually "harmonic drones", and the use of them in my music,
has an important parallel to the use of the Tanpura in Indian music.
The tanpura, is probably one of the most important models for my
purposes, and when I speak of "harmonic drones", it is really a
personal metaphor for the function that the tanpura serves in Indian
Classical music.

If you have the ability to play MP3 files, you may be interested to
hear an example of a harmonic drone that I spoke of during our thread
about Meditation and Music. Here's the URL to our files section where
I have a 35 second long harmonic drone (552 KB) in the key of
Concert "E":

/tuning/files/Ligon%20FFT/

This is an attempt to create a sound that will serve a function
similar to that of the tanpura. The challenge with such electronic
sound design, is to create a drone with a pleasant sound, that one
can easily enjoy hearing for long periods of melodic improvisation.
As Alison pointed out, it is movement in the harmonics that makes it
work, and this is something that the tanpura does in a wonderfully
natural way. If I had never heard the tanpura in Indian music, I feel
certain I would not have been able to conceive of this. Thanks India!

Something I failed to mention last evening, is that if anyone who
wishes to try out the scales, and is in need of a drone, you may play
these scales with the drone here, to see the phenomenon
of 'compatibility with the harmonic series' I mentioned. Of course,
it may become necessary to transpose either the drone to another
desired key center, or else make your tuning in concert "E".

I believe that you will have superior results of beauty, if you do in
fact use a real tanpura with the scales I posted. As I mentioned in
the post, I've found that even with the extended scales, which at the
point where I stopped at ratios of 37, still sound correct over the
drone, and inspire melodic possibilities.

> If this is
> too irritatingly elementary for the group, you may reply through an
> email (I am not familiar with any of such conventions).

No question is too elementary, and I have many to ask about the
practices of Indian music. As above, the convention of the tanpura
stands as a great model to me, for using drones and melodies.

> I know I have yet to reply to some of your queries. Later,
> definitely.

Looking forward to this, and take you time.

>
> Thanks. Regards,
> Haresh.

Kindest thanks to you,

Jacky Ligon

--- In tuning@y..., "J Scott" <xjscott@e...> wrote:
> Jacky wrote:
>
> > Back in the 1980s I noticed a simple but elegant structure,
> > which I have came to call the Harmonic Mirror.
>
> > <<4 examples>>
>
> Hi Jacky!
>
> OK I charged up these beauties and activated them to see
> what they were made of. I should say that going in to this
> I am a little bit skeptical of this whole symmetry thing
> as a theory.

Jeff,

Good Morning! Yes, and from a theoretical point of view, which may
seek consonance in complex chords, this approach would invite healthy
skepticism, but I've found that the inversional symmetry at the 4th
(modal), or the tritone, has incredible melodic power and
possibilities. It's a virtue that the intervals are mirrored above,
and lends a unique "evenness" to a melodic line over a drone. Usually
I do not play chords at all with these scales, but harmony results
when many different timbres are playing melodically with them
(sometimes several scales at once are used!). One can make very
complex tunings work in a musical texture with clever timbral and
intervallic partitioning techniques.

>
> First I viewed the scales graphically and found them quite
> beautiful and intriguing to look it. I found my eyes
> moving from line to line, examining the spacing and the
> symmetry. Interesting indeed. Intriguing. But how would
> they sound?

That you ask this question, inspires me that I've found a kindred
spirit!

>
> Tuning #1: Ho hum. Yeah OK. It's probably just me but I
> just do not get excited over overly consonant scales.

I must admit that they become more interesting for me too, when
derived from the more complex members of the higher primes. That
first one is a little "conservative".

>
> Tuning #4: Holy Cow!!!!!!!!!!!!!
>
> Yes tuning #4 is really something. It is totally
> astonishing. And I do not get too excited about most
> new tuning ideas.

Yes, things start getting sweet around that area of the harmonic
mirror sequence. There is some great melodic tension that inspires me
with the inclusion of more 'exotic' thirds, and the alternate fourths
and fifths that become available.

>
> I didn't play it over the drone. I'm bad. I just
> tuned a piano with it and stamped it up and down
> in octave repetition.

That still qualifies as a harmonic drone! You know, when I improvise
with these scales, without using a drone, I still get some sense of
there being an implied drone by virtue of the harmonic derivation of
the pitches.

>
> There's something about these degrees:
>
> 17/13,24/17,17/12,26/17

I would agree, and I really like the melodic energy of the 17/13, and
26/17 altered fourths and fifths - that 736 cents fifth is wonderful.

Unfortunately, when I woke up this morning, I found a severed
Chocolate Bunny Head in my bed with an accompanying note from Pat and
David, reminding me of their recent copyright of 17. So I must be
cautious about spreading undue "Seventeenness" around. Sorry about
that fellers! : )

>
> First that 24/17,17/12 in the middle there is
> just plain silly but I gotta admit it works.

I remember David Beardsley had something such as this in a recent
piece, but can't recall the intervals. I do enjoy the melodic
possibilities of extremely small commas, and where one may be using a
mode from this scale which omits the fourth, having the two tritones,
becomes a very nice thing.

>
> And I greatly enjoyed the shimmering effects I got from
> those close scale degrees & their interactions with other
> notes in tunings #2 and #3.
>
> Now that 26/17 interval is pretty darn neat.
> It does something to me.
> And I started moving around, playing.
> And I was getting all overcome with these weird feelings.
> And I actually started crying.
> And I thought, now this is really something what is going
> on here? These beat patterns have got ahold of me and are
> wrenching me around this way and that and I feel like I am
> floating or something or just what is going on anyway.

It has a similar emotional effect on me, especially after I go into a
long improv of about a half an hour or so. After I have internalized
the tuning and my hands are warm, and I'm in the "Zone", the essence
of the tuning just channels right through, and my hands gravitate to
its unique internal qualities.

I think it noteworthy to point out, that I've found allot of common
ground with Margo Schulter in regard to her use of such "wide"
intervals in her Neo Gothic music practice. There is something so
energetic, and "Gamelan-Like" about an interval like 26/17.

Isn't it a great wonder and blissful mystery how we may harness these
beautiful musical effects for the emotional power that is latent in
their structures? I just love this! I like to point out that we are
but clever incidental organizers of phenomenon of nature, and the
wonderful emotional power of microtonal tunings is something that we
only assemble into desired structures, but the reasons why it works
and has the emotive power it does, is outside our control. The
tuning, rhythm and melody, all represent the most atomic level
aspects of music.

>
> I stopped and I felt totally overwhelmed -- like I had
> been through some incredibly intense emotional experience
> that changes your life but I can't quite say what it was.

I do know exactly what you mean, and as above, I love to do long
improvisations over drones - a highly meditative practice and
experience. Even though I really enjoy complex tunings, I will always
hold special interest for the other extreme of simplicities. There is
something to be said for intimate musics, where a single instrument
explores the possibilities of a melodic tuning in a very naked way.
It can touch deeply.

>
> Kind of like a weird dimension of euphoria or something?

Definitely euphoric for me too.

>
> So that's it for tuning #4 so far.
>
> I'll take a look again at tunings #2 and #3 again later
> but it seems on first listening to me that the general
> trend is that the tunings progress upwards in terms of
> profundity and depth and dimensionality.

A very interesting statement, and I agree, that as one ascends up
this series, the treasures found become more and more wonderful. If
it pleases folks, and there is further interest, I will be happy to
post some more about this, in small installments. This is something
I've used for a very long time, so there's lots more there.

>
> All tripped out and stuff, man. Freaking way to go, Jacky!
> You the man!
>
> - Jeff

Thanks so much Jeff, and I really wish I could have been there when
you were playing the scales. I love to listen to folks improvise on
them - and a curious note is that there have been times when I have
let non-microtonalists improvise on these scales when they come to
visit, and it was interesting to note how they tended to gravitate
toward similar attributes of the scales that I find in them too. To
watch them discover the melodic value of small intervals was a treat.

Is there perhaps some kind of archetypal interpretation possible with
scales? Makes me wonder if many others would also tend to gravitate
toward these qualities in a scale - either experienced with
microtonality or not. I like to fancy that there are universals at
work, although I'm sure it would be difficult to pin down in the lab.

I have pondered too, that we may enjoy these sounds because they are
so exotically different from the sounds of 12 tET, and perhaps we
gravitate toward the unique melodic qualities which immediately give
us pleasing and different flavors.

In gratitude,

Jacky Ligon