the evil 27 equal temp scale from outer space

hi, i'm new on this list. i've been composing music (mainly electronic)
for about 20 years but i never really dug into microtonality as i became
increasingly more interested in sound then in notes (except when making
cheesy electrohousetunes or thrashy rocksongs to forget about "serious"
music for a moment). however, my interest has arisen after all because
of a project i'm currently working on.

i've been asked to develop instruments and write music for a klingon
opera. none of the makers involved in this project are trekkies but we
do try to take it as serious as possible. we also want to be clear about
the fact that we're humans interpreting klingon opera, so no ridges on
our foreheads... now, certain aspect of klingon culture have been
covered quite extensively but klingon opera doesn't seem to get further
then a running gag in the deep space 9 series. so canonically speaking
it exists, but nobody has ever seen or heard one. personally, i find the
idea to re-create something that doesn't actually exist yet quite
fascinating.

so my first question was: if there's klingon opera, would they have some
kind of music theory as well? there tradition-minded enough but if i
were a klingon i couldn't care less if i was in or out of tune. then
again, you can't really ignore something properly if it's not there. so
i figured i had to come up with at least some klingon music theory. for
numero-symbolic reasons i wanted to use a tuning system based on the
number 3. it also had to have a lot of dissonants, but i wouldn't want
to miss the 4th and the 5th as even klingons must have some sense of
harmony. this led me to a tritave with 27 equal tempered intervals (so 3
to the power of 1/27th, about 1,0415285).

when i came up with this a few days ago, i didn't know about
bohlen-pierce yet. reading up on that was a bit like realising i'd been
reinventing the wheel. i must admit it was of some comfort to learn that
bohlen, pierce and van prooijen didn't know about eachother at first
either. anyway, so far i haven't found anything on a 27 note based
system, which leads me to believe it's pretty safe to claim the 27 equal
temp scale as a "klingon" scale ;-)

below is a list of the interval ratios. personally i like 1,225619855 a
lot. considering the fact that humans commonly associate minor with
sadness and major with cheerfulness but klingons consider sad things
like death something to look forward to, it seems quite appropriate.
same goes for 1,44224957, which imho would be a very important note for
klingons as it's exactly 1/3 of the tritave. it also pretty closely
resembles our tritone in 12tet. apart from the abudance of numerological
parallels, i like to think that klingons would really like this note if
not despite the fact that humans have called it "the devil in music",
then exactly because of that...
i also like the fact that 1,997145375 (2/3 of the tritave) is almost
(but just not) an octave as from a klingon point of view, a perfect
octave would probably be the most boring note ever.

i'd really like to hear any comments, remarks, tips, ideas or
suggestions.

grtz,
xaf

1
1,041528498
1,084781613
1,129830964
1,176751147
1,225619855
1,276518007
1,329529883
1,384743262
1,44224957
1,502144029
1,564525815
1,629498222
1,697168836
1,767649709
1,841057547
1,917513902
1,997145375
2,080083823
2,16646658
2,256436684
2,350143111
2,447741025
2,549392034
2,655264456
2,765533601
2,880382059
3

Try this
1/1
10/9
16/13
4/3
3/2
5/3
13/7
2/1 (period)

I optimized this scale to improve purity for chords that extend between two octaves trying to approach the strong bass-line possibilities that exist in scales like deca-tonic scales and those formed from "circles of intervals". The more I played around with that idea the more I realized you were right and my "tempered" Ptolemy Homalon scale needed some tweaking.
The scale did gain a tad of critical band roughness due to slightly decreased ratios between dyads, but gained a lot of periodicity due to ratios like 16/13 and 13/7 tempering closely between just pure intervals and doing so better across octaves (IMVHO) than the original 11/6.and 11/9 they replace.

Kalle and others, what do you think of this scale and what do you see as its still-standing weaknesses so far as compositional flexibility?

-Michael

There wasn't any "compositional weakness" in your original "A" scale. Disjunct tetrachords divided for a resulting diatonic octave scale? That's a "classic" scale structure, in a real way, like, "a very large part of the world's music for the last several thousand years has used this same basic structure".

And it sounded really nice harmonically and melodically, not to mention it had stuff like some downright traditional alternate cadences, vii°-i, IV-i on the original tonic for example. IMO you shouldn't dump it yet, though I'd recommend dropping the third a couple of cents, to 63/52 (mirror image of the 13/7, 3/2 the mirror).

I think it be a shame to simply abandon a scale with a unique flavor for no real reason.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Try this
> 1/1
> 10/9
> 16/13
> 4/3
> 3/2
> 5/3
> 13/7
> 2/1 (period)
>
> I optimized this scale to improve purity for chords that extend between two octaves trying to approach the strong bass-line possibilities that exist in scales like deca-tonic scales and those formed from "circles of intervals". The more I played around with that idea the more I realized you were right and my "tempered" Ptolemy Homalon scale needed some tweaking.
> The scale did gain a tad of critical band roughness due to slightly decreased ratios between dyads, but gained a lot of periodicity due to ratios like 16/13 and 13/7 tempering closely between just pure intervals and doing so better across octaves (IMVHO) than the original 11/6.and 11/9 they replace.
>
> Kalle and others, what do you think of this scale and what do you see as its still-standing weaknesses so far as compositional flexibility?
>
> -Michael
>

> Try this
> 1/1
> 10/9
> 16/13
> 4/3
> 3/2
> 5/3
> 13/7
> 2/1 (period)

Cameron>"Disjunct tetrachords divided for a resulting diatonic octave scale?
That's a "classic" scale structure"

Kind of seems to go back to Ptolemy's idea that chords should be built from tetrachords vs. the Pythagorean idea of building scales from circles of near perfect 5ths. While I am always looking for flaws in scale I work with to patch, I'm pretty confident tetra-chords general give more harmonic possibilities...especially when working with 1-2 octaves...though circle-of-xth's type scales seem to have an advantage when working over many octaves.

>"I think it be a shame to simply abandon a scale with a unique flavor for no real reason."
Glad you enjoyed the scale. And of course, I'll keep touching it up.

>"IMO you shouldn't dump it yet, though I'd recommend dropping the third a couple of cents, to 63/52 (mirror image of the 13/7, 3/2 the mirror). "
Interesting, I was working on a new version of the scale and all my answers possible hovered around 1.222222. So if I have it right the 63/52 is the same distance from 1.5 that the 13/7 is from 2/1?
Also, perhaps you have a similar technique which explains why my ears gravitate toward the 13/7 instead of more "straight harmonic series-like" 33/18 AKA 11/6?

As a side note, I found the 10/9 in that scale better replaced with a 13/12. Unfortunately I can't find a "mirror around the 5th" trick that works with that one, but that helps resolve the 40/27 (note on the second octave's ratio over a note on the first octave's) to a 39/27 AKA 13/9.

When I get home I'm going to try and compose in this scale with both of our modifications and see what happens....while keeping a copy of the original around, of course.

-Michael

Tetrachords of all varied kinds are generally considered melodic rather than harmonic, but I find the approach ideal for harmonic material as well- you just have to not care whether there is a triadic V-I or not. :-)

24/13 is a great "seventh" (mirror 13/12 around the octave).

Back atcha tomorrow, it is late!

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> > Try this
> > 1/1
> > 10/9
> > 16/13
> > 4/3
> > 3/2
> > 5/3
> > 13/7
> > 2/1 (period)
>
> Cameron>"Disjunct tetrachords divided for a resulting diatonic octave scale?
> That's a "classic" scale structure"
>
> Kind of seems to go back to Ptolemy's idea that chords should be built from tetrachords vs. the Pythagorean idea of building scales from circles of near perfect 5ths. While I am always looking for flaws in scale I work with to patch, I'm pretty confident tetra-chords general give more harmonic possibilities...especially when working with 1-2 octaves...though circle-of-xth's type scales seem to have an advantage when working over many octaves.
>
> >"I think it be a shame to simply abandon a scale with a unique flavor for no real reason."
> Glad you enjoyed the scale. And of course, I'll keep touching it up.
>
> >"IMO you shouldn't dump it yet, though I'd recommend dropping the third a couple of cents, to 63/52 (mirror image of the 13/7, 3/2 the mirror). "
> Interesting, I was working on a new version of the scale and all my answers possible hovered around 1.222222. So if I have it right the 63/52 is the same distance from 1.5 that the 13/7 is from 2/1?
> Also, perhaps you have a similar technique which explains why my ears gravitate toward the 13/7 instead of more "straight harmonic series-like" 33/18 AKA 11/6?
>
>
>
> As a side note, I found the 10/9 in that scale better replaced with a 13/12. Unfortunately I can't find a "mirror around the 5th" trick that works with that one, but that helps resolve the 40/27 (note on the second octave's ratio over a note on the first octave's) to a 39/27 AKA 13/9.
>
> When I get home I'm going to try and compose in this scale with both of our modifications and see what happens....while keeping a copy of the original around, of course.
>
> -Michael
>

>"Tetrachords of all varied kinds are generally considered melodic rather
than harmonic, but I find the approach ideal for harmonic material as
well- you just have to not care whether there is a triadic V-I or not.
:-)"

Dare I ask what is the "triadic V-I" and why is it so important? Or (just a hunch)...is it a fancy way of saying the first and last notes of each triad supposedly must form a perfect 5th (more or less)?

>"Back atcha tomorrow, it is late!"
Well see you on here tomorrow (hopefully).
Both of your "mirrors" work fantastically...and thank you for the brilliant tips. The combination swapping the 10/9 with the 13/12 and applying your two new "mirror notes" (at least to my ears) make the scale virtually never sour out regardless of the dyad or chord combination, even when the chords reach across different octave.

Just curious: any clue how/why do these mirrors work? I've done similar types of mirroring in my Silver ratio scale, doing reverse sections around the square root of 2 between 1/1 and 2/1, but had no clue any sort of mirroring could be used in JI so effectively... Perhaps it would be fair to say the most important lesson learned is that the brain like to see tones mirrored/made-symmetrical around what it considers relatively simple intervals (almost forming a "magnetic field effect" around those tones)?

-Michael

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >"Tetrachords of all varied kinds are generally considered melodic >rather
> than harmonic, but I find the approach ideal for harmonic material >as
> well- you just have to not care whether there is a triadic V-I or >not.
> :-)"
> Dare I ask what is the "triadic V-I" and why is it so >important? Or (just a hunch)...is it a fancy way of saying the >first and last notes of each triad supposedly must form a perfect >5th (more or less)?

If you play g-b-d, then c-e-g, that's an example of V-I, you'll recognize it immediately as the foundation of a huge amount of Western music. Dominant-tonic. Somewhere in storage I have some xeroxed pages of a text from the late 19th century explaining how
V-I comes from G-d and is how Natural music is constructed, including corrections of Bach's mistakes when he failed to follow the Natural Order, in which Music is Major with V-I cadences. (the book is in the Musikhochschule library in Graz, but I can't remember the title).
>

>
> >"Back atcha tomorrow, it is late!"
> Well see you on here tomorrow (hopefully).
> Both of your "mirrors" work fantastically...and thank you for >the brilliant tips. The combination swapping the 10/9 with the >13/12 and applying your two new "mirror notes" (at least to my ears) >make the scale virtually never sour out regardless of the dyad or >chord combination, even when the chords reach across different >octave.
>
> Just curious: any clue how/why do these mirrors work? I've >done similar types of mirroring in my Silver ratio scale, doing >reverse sections around the square root of 2 between 1/1 and 2/1, >but had no clue any sort of mirroring could be used in JI so >effectively... Perhaps it would be fair to say the most important >lesson learned is that the brain like to see tones mirrored/made->symmetrical around what it considers relatively simple intervals >> (almost forming a "magnetic field effect" around those tones)?

Well in mirroring with rational intervals you maintain the coincidence of the same partials and simplify the overall harmonic structure. When working with higher limits it seems unlikely that
there is any conscious perception of this, but in my experience "everyone" (except among "tuning theorists :-) ) percieves the "yeah, that goes together" feeling of it.

Since I have to test some sounds anyway, I'll make something in your original "A" scale.

Cameron>"V-I comes from G-d"
G-d as in the Lord? Bizarre (not that you omitted the o, it's actually against my religion).
To me g-b-d -> c-e-g is simply the "scale" 1/1 5/4 3/2 15/8 18/8 or 8:10:12:15:18 breaking down into the same chord 4:5:6 starting at C and G (a pretty much perfectly periodic major triad). I guess the two things I see off the bat about it is that
A) It's a chain of two fifths (c to g, and g to d)
B) That form this chord progression and that both chords are major triads in the same key of G major.
I also realize f a c -> c e g has those same properties. So I wonder, why the preference (or is that simply considered the same thing in a different mode)?

>"When working with higher limits it seems unlikely that there is any conscious perception of this, but in my experience
"everyone" (except among "tuning theorists :-) ) percieves the "yeah,
that goes together" feeling of it. "

Exactly...it does feel more together. Another cool side effect is I've found you can obtain that feeling of symmetry while avoid the "grating" sound that comes with periodic distortion and straight harmonic series by using the "mirroring" technique you've described. Playing a 18:20:22:24:27:30:33 chord or any close 5-tone subset sounds downright like an engine knocking to me, but using mirrored and slightly de-tuned version begins to sound smooth, relaxed, and "rainbow-ish".

>"Since I have to test some sounds anyway, I'll make something in your
original "A" scale."
I'll still say/admit the version of "my" scale with your "mirroring" improvements sounds significantly better than the original due to the sense of togetherness/confidence (keeps the togetherness of a straight harmonic series with the relaxed/non-grating nature of tempered scales)...but, good luck! :-)

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Cameron>"V-I comes from G-d"
> G-d as in the Lord? Bizarre (not that you omitted the o, it's >actually against my religion).

It's not stated so bluntly, but the idea is clear. Obviously the guy was very confident in the authority of his belief, citing Bach's "mistakes". These kinds of thinking persist in music theory to this day, just in more subtle guises, and Darwin takes the place of divinity.

I was just reading this fine book:
http://www.amazon.com/Theories-Fugue-Josquin-Eastman-Studies/dp/1580460291

and notice that the author mentions this very problem, even uses the word "Darwinism". In fugal theory, early fugues have long been
judged as if they were all "less evolved" Bach fugues, rather than
be judged on their own terms.

> To me g-b-d -> c-e-g is simply the "scale" 1/1 5/4 3/2 15/8 >18/8 or 8:10:12:15:18 breaking down into the same chord 4:5:6 >starting at C and G (a pretty much perfectly periodic major >triad). I guess the two things I see off the bat about it is that
> A) It's a chain of two fifths (c to g, and g to d)
> B) That form this chord progression and that both chords are major >triads in the same key of G major.
> I also realize f a c -> c e g has those same properties. So >I wonder, why the preference (or is that simply considered the same >thing in a different mode)?

f-a-c to c-e-g is called IV-I, and is considered an alternate cadence (plagal, or "Amen" etc). The dominance of dominant-tonic thinking has
been so strong that to this day some maintain that IV-I isn't a
proper cadence.

A little thought will show that if you were honestly looking for
a "natural" cadence, and a "natural scale", the cadence and scale
presenting themselves more immediately and logically in the harmonic series would be v-I, with v septimal (G,Bb,D, where Bb is 7/4 above C), and a harmonic scale of C,D,E,F#,G,Ab,Bb,C, where F# is 11/8,
Ab is 13/8, and Bb is 7/4. It would still be nutty to claim any
scale or cadence as "divine", but at least you'd have some actual physics to back up your claim (the idea that V-I is "natural" assumes the function of the leading tone, B resolving to C, which is a cultural and stylistic thing).

>
> >"When working with higher limits it seems unlikely that there is >any conscious perception of this, but in my experience
> "everyone" (except among "tuning theorists :-) ) percieves >the "yeah,
> that goes together" feeling of it. "
>
> Exactly...it does feel more together. Another cool side >effect is I've found you can obtain that feeling of symmetry while >avoid the "grating" sound that comes with periodic distortion and >straight harmonic series by using the "mirroring" technique you've >described. Playing a 18:20:22:24:27:30:33 chord or any close 5->tone subset sounds downright like an engine knocking to me, but >using mirrored and slightly de-tuned version begins to sound smooth, >relaxed, and "rainbow-ish".

Rainbowish is a nice description. I like the feeling that there is
great variety, but a continuous oneness at the same time.
>
> >"Since I have to test some sounds anyway, I'll make something in >your
> original "A" scale."
> I'll still say/admit the version of "my" scale with >your "mirroring" improvements sounds significantly better than the >original due to the sense of togetherness/confidence (keeps the >togetherness of a straight harmonic series with the relaxed/non->grating nature of tempered scales)...but, good luck! :-)
>

Well with that <4 cent tweak on the third, your A scale is doing just
fine!