Yahoo Tuning Groups Ultimate Backup tuning-math MOSes that minimize average pairwise harmonic entropy

I would be surprised if someone hadn't done exactly this before, because it seems like such an obvious thing to do, but I've made some plots of MOSes versus their average octave-equivalent HE.

Without further ado, here are the plots:

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGNTAyNDBmMzItODI1Mi00YzFjLTk4MDgtMzlmZGNhNGNmZDQ2&hl=en

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGNTE1MzRjYTItYTdhYS00NmZjLWFiYzktOGQxZjQwMWIzOWQw&hl=en

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGYmIwZDJlYjAtOGM4Yy00MzhjLWI3NTktODM2YmNjOTBjMTgw&hl=en

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGMmIzZDY4NTQtMzViYy00MWYyLTgyZDYtNTlhZGY1ODU5YWY2&hl=en

https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGNWJkYzE5NzYtMGM0Zi00NjY4LTg4NGQtY2QxNDRhMDVlZWYz&hl=en

"Coarse", "fine", and "extra fine" refer to harmonic entropy curves with sigma values of 20 cents, 10 cents, and 7 cents respectively. I meant "extra fine" to be the finest anyone could actually hear, even with ideal conditions (high register, rich and clear timbre, and the right amount of intermodulation distortion). But I might still go to an even smaller sigma and call it "ridiculously fine".

Here are the actual HE curves if you're interested. The second column is the "raw" HE; the third column is the "corrected" HE with the overall density trend subtracted out, which is what I actually use. There's also one called "medium" with sigma = 14 cents.
https://docs.google.com/document/d/1_i1Xh6UTsOem1wmlF1DHjuAEUKfRpEGJ1d-uS0EzkDI/edit?hl=en
https://docs.google.com/document/d/1NVHPu7k3q3eP-vWsbqVp0GyWXhVCYqmxdql-3U3E6GE/edit?hl=en
https://docs.google.com/document/d/1l0HseL6EzOdvIkwmyMyNOvC-CD_E-ooezr9UGoQuV-Y/edit?hl=en
https://docs.google.com/document/d/1ShItleHPXCuqnIXZCW0dAdkEvpJYf1dgxclRury0XW0/edit?hl=en

For each possible MOS, I calculate the average HE for all intervals in the scale, with multiplicity. For example, a certain meantone pentatonic scale has 4 intervals of 503 cents, 3 of 194, 2 of 309, and 1 of 388. So I put these 10 total intervals into some averaging function.

At first I used the simple arithmetic mean of the HE values, but then I thought "Why penalize scales with some really high-HE dissonances if the rest of the intervals all have low HE?". I tried using the median, but eventually settled on throwing away the 30% of intervals with the highest HE values and taking the mean of the other 70%. The overall shapes of the curves are quite insensitive to the specific numbers here. For the pentatonic example, this means I throw away all three of the 194-cent intervals, and average the HE values of the 7 remaining ones (which all happen to be great consonances).

So the curves you see are just this function, plotted for all the MOSes that actually exist (had to play around a lot with Farey series and the extended Euclidean algorithm to get that to happen automatically...), without the spurious low values at the ends of the MOS range where some pairs of pitches are collapsing into one. What's left is all the "real" MOSes where all the pitches are effectively distinguishable according to the chosen HE function.

Let me know if any of the labels are incorrect or misleading. Also, there is one MOS for which I couldn't identify any temperament: its generator is a quarter of a 3/1 (~475 cents) and it's marked "???" on the plots.

Note that these are just MOSes, not temperaments with mappings! That's why some of them are labeled with multiple temperaments. An example of this is "Hanson/Keemun": since this analysis is based on limit-agnostic harmonic entropy, it doesn't know the difference between 5-limit hanson and 7-limit keemun (and similarly it doesn't care about the specific mapping of 7). Similarly with "Liese/Triton": these are totally different temperaments, but their generators are so similar that their small MOSes are indistinguishable, and they show up as a single valley in the plot.

I can make plots for the higher period MOSes (3,4,5..) if you want, but they're not nearly as interesting as periods 1 and 2. There is basically only a single valley for each period: 3 is augmented, 4 is diminished, 5 is blackwood, and 6 is not even hexe - it's 12-EDO. For period 3 you can see little bumps for "semiaug" and "triforce" but they're dwarfed by augmented. It can't tell the difference between august, augene, and all those. Similarly, "extra-fine" period-2 looks pretty much the same as "fine" period-2.

For the "fine" HE curve and 30%-trimmed-mean averaging, here are the "winners":

Meantone pentatonic 0.801 proper
Pythagorean pentatonic 0.802 proper

Meantone diatonic 0.815 proper
(Neutral thirds [4]) 0.816 proper
(Semaphore[4]) 0.817 proper

Pythagorean diatonic 0.825 slightly improper
Srutal/Pajara[6] 0.826 ssLssL is proper (not sssLsL)
Semaphore[5] 0.828 proper

Srutal/Pajara[8] 0.833 improper
Slendric[6] 0.835 proper
Neutral third scale 0.838 proper
Decatonic 0.841 proper
Augmented[6] 0.842 LsLsLs is proper (not LLsLss)
Meantone chromatic 0.843 proper
Porcupine[6] 0.847 improper
Magic[7] 0.847 improper
Srutal/Pajara[12] 0.848 proper
Porcupine[7] 0.849 proper
Magic[10] 0.849 improper
Pythagorean/Helmholtz chromatic 0.849 proper
Liese/Triton[7] 0.850 improper
Semaphore[9] 0.851 improper
Augmented[9] 0.851 slightly improper
Neutral thirds [10] 0.852 slightly improper
12-EDO 0.852 proper
Hanson/Keemun[11] 0.853 improper
Magic[13] 0.853 improper
Tetracot[7] 0.854 proper
Slendric[11] 0.854 improper

One nice thing is that the neutral thirds scale ("dicot", "mohajira", whatever), pops out very clearly. I think people here are often talking about whether that scale is "really" a 2.3.11 scale or a 2.3.(non-JI neutral third) scale, and what makes it appealing or popular, and I think here we have an answer: it simply has a low average HE for a MOS.

Keenan

🔗Mike Battaglia <battaglia01@gmail.com>

🔗Carl Lumma <carl@lumma.org>

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Keenan Pepper <keenanpepper@gmail.com>

🔗Mike Battaglia <battaglia01@gmail.com>

On Fri, Apr 22, 2011 at 5:19 PM, Keenan Pepper <keenanpepper@gmail.com> wrote:
>
> I would be surprised if someone hadn't done exactly this before, because it seems like such an obvious thing to do, but I've made some plots of MOSes versus their average octave-equivalent HE.
>
> Without further ado, here are the plots:
>
> https://docs.google.com/viewer?a=v&pid=explorer&chrome=true&srcid=0B9CMyeCjAMQGNTAyNDBmMzItODI1Mi00YzFjLTk4MDgtMzlmZGNhNGNmZDQ2&hl=en

Semaphore is particularly magical, eh? I noticed that the other day.
How does the following omnitetrachordal variant of semaphore rank up?
It's not an MOS, and further it has 10 notes instead of 9, but it
sounds amazing to my ears at least. The scale is (in 24-tet notation)

C C^ D D^ F F^ G G^ A A^ C

So basically just two pentatonic scales that are offset by 1\24.

> One nice thing is that the neutral thirds scale ("dicot", "mohajira", whatever), pops out very clearly. I think people here are often talking about whether that scale is "really" a 2.3.11 scale or a 2.3.(non-JI neutral third) scale, and what makes it appealing or popular, and I think here we have an answer: it simply has a low average HE for a MOS.

I've been trying to develop for a long time a regular mapping variant
based around this property - as I mentioned before, if you just plot
Thomae's function and convolve the whole thing with a Gaussian, the
resulting curve behaves very similarly to HE - minima ordered in the
same order, maxima in about the same places, smooth slope between
them, etc. You could use this principle to realize a JI lattice in
which each point is surrounded by a multivariate Gaussian that
represents the probability that some point in the area will get sucked
into the perception of that interval.

So each point on this lattice would represent multiple intervals, each
with a weighting specifying how strong the interval is at that point
on the lattice. So you'd end up with a "fuzzy abelian group"
structure, where three "fuzzy" 10/9's gets you to a "fuzzy" 4/3, and
so on. Whether you want to give each interval the same size detuning
Gaussian is a point best left up in the air for now - if you want to
model things like HE, you want the Gaussians to be the same size, but
other algorithms that may be more accurate (SWIPE) would likely model
things such that more complex intervals get smaller Gaussians (e.g.
are more sensitive to mistuning).

-Mike

🔗Mike Battaglia <battaglia01@gmail.com>

A last note, Keenan - you might also want to consider looking at the
FFT of the octave-equivalent HE curve. This will tell you, in a very
broad sense, what the strongest equal divisions within the curve are.
You'll see things like 3 and 4 popping up, representing that 128/125
and 648/625 are really good commas to temper out. Integrating provides
a good list of the "best" ETs within the HE function.

Here's a thread I made some time ago about it, which unfortunately
didn't garner much interest:

/tuning-math/message/18585

I was trying to extend this at the time to work with linear
temperaments, but had a bit of trouble in formalizing the approach. As
the octave-equivalent HE curve is a periodic function, you will never
get generators that don't perfectly subdivide the octave. One way
around this would be to use damped sinusoids as basis vectors instead
of normal sinusoids (aka use the Laplace transform instead of the
Fourier transform), which can be easily done by taking several periods
of the octave-equivalent HE curve, strung together, and multiplying
the whole thing pointwise by e^-(at), where a is some kind of damping
factor. At least that's what I was thinking at the time, but hit some
snags early on and never got started trying it again.

-Mike

🔗Herman Miller <hmiller@IO.COM>

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗Herman Miller <hmiller@IO.COM>

🔗genewardsmith <genewardsmith@sbcglobal.net>

🔗cityoftheasleep <igliashon@sbcglobal.net>

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> I would be surprised if someone hadn't done exactly this before, because it seems like
> such an obvious thing to do, but I've made some plots of MOSes versus their average
> octave-equivalent HE.

This is awesome. Stuff like this is why I check in here occasionally. Question: can you figure out the maxima? I.e. the "worst" MOS scales of 5-10 notes?

> Liese/Triton[7] 0.850 improper

I've never seen this one mentioned before...looks like the generator is around an 11/8? If it's competing with Semaphore [9], I should check it out.

> One nice thing is that the neutral thirds scale ("dicot", "mohajira", whatever), pops out
> very clearly. I think people here are often talking about whether that scale is "really" a
> 2.3.11 scale or a 2.3.(non-JI neutral third) scale, and what makes it appealing or
> popular, and I think here we have an answer: it simply has a low average HE for a MOS.

Makes sense. You can really "fake" something Pythagorean-sounding with the 7-note MOS if you leave out the neutral 3rds and 2nds.

-Igs

🔗Herman Miller <hmiller@IO.COM>

🔗Keenan Pepper <keenanpepper@gmail.com>

--- In tuning-math@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
> This is awesome. Stuff like this is why I check in here occasionally. Question: can you figure out the maxima? I.e. the "worst" MOS scales of 5-10 notes?

Thanks! The *absolute* maxima all have really small generators and are really improper, for example the maximum-entropy 5-note MOS is something like [0,30,60,90,120,1200] cents, with four tiny steps and one huge step. But the non-silly maxima might be interesting to look at. For example you could look for the *proper* scales that maximize HE.

But even that might not be so interesting because these plots tend to have one or two minima that really stand out (e.g. meantone), and a large number of maxima that are all about equal.

> > Liese/Triton[7] 0.850 improper
>
> I've never seen this one mentioned before...looks like the generator is around an 11/8? If it's competing with Semaphore [9], I should check it out.

The generator is actually 7/5, but if you want you can certainly temper out 56/55 and make it serve as 11/8 also, in which case you get this guy (unnamed): http://x31eq.com/cgi-bin/rt.cgi?ets=15_2&limit=11&key=3_-5_-6_-1_1_0_5_6_4

I'm currently preparing plots for a slight improvement of this search intended to weed out scales that appear as small minima just because they have one or two 4/3's, but nothing else of interest.

🔗cityoftheasleep <igliashon@sbcglobal.net>

--- In tuning-math@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> Thanks! The *absolute* maxima all have really small generators and are really improper, >for example the maximum-entropy 5-note MOS is something like [0,30,60,90,120,1200] > cents,

Ha, I should have guessed that!

> But the non-silly maxima might be interesting to look at. For example you could look
> for the *proper* scales that maximize HE.

> But even that might not be so interesting because these plots tend to have one or two
> minima that really stand out (e.g. meantone), and a large number of maxima that are all > about equal.

So the non-silly maxima are all about equal, eh? That's really interesting.

> The generator is actually 7/5,

Ah yes. I looked these up on the xenwiki, they seem to be compatible with 19-EDO and give off scales about as improper as magic. In other words, they produce dyads that are all really consonant with the root-tone but not necessarily with each other.

>
> I'm currently preparing plots for a slight improvement of this search intended to weed
> out scales that appear as small minima just because they have one or two 4/3's, but
> nothing else of interest.

Good idea. Do you think you could up the resolution of the x-axis, too? Maybe to 50 cent increments instead of 100 cent?

-Igs

🔗Keenan Pepper <keenanpepper@gmail.com>

Better plots:

The main difference here is that in addition to throwing away 30% of the largest HE values (to allow scales to have some dissonant intervals without penalizing them), I also throw away 10% of the smallest HE values. This avoids having little dips appear wherever one of the intervals in the scale happens to pass by 4/3. Now even small dips are guaranteed to be somewhat meaningful.

Notice that two of the labeled scales do not correspond to regular temperaments at all. I've temporarily called them "pseudo-slendric[11]" and "pseudo-semaphore[9]" for lack of better names. Let's take a look at pseudo-semaphore[9] to understand what's going on here.

This is a 9-note MOS with a generator of 244 cents, just slightly over 1\5 of an octave. The intervals are therefore
1 step: 20 or 224
2 steps: 244 or 448
3 steps: 264 or 468
4 steps: 488 or 692
and 5-8 step intervals are of course the octave complements of these.

The reason this is not a regular temperament is that the 4-step interval of 692 cents represents 3/2, but the 5-step interval of 712 cents also represents 3/2!

Since both 4-step intervals represent 3/2 or its octave equivalent, that means this 9-note scale has 9 separate occurrences of (some version of) 3/2 in it. The only way this is possible in a regular temperament is if it is closed, that is, an EDO. But this 9-note scale has 3/2's that are only detuned by 10 cents (in both directions), which is more accurate than both 5-EDO and 7-EDO, and much better than 9-EDO.

I thought about how to force this into the regular mapping framework, but it's pretty gnarly. The simplest way I can describe it is as a 2.3.3'.7 temperament, where 3'>3. Then pseudo-slendric is the unique temperament that tempers out the commas 49/48' (that is, 7^2/(16*(3')^2)) and 2^8*3^2/(3')^7. It should be obvious that if we also temper out 3'/3, then we get 5-EDO, but pseudo-semaphore is not 5-EDO.

Similarly, pseudo-slendric is the 2.3.3'.7 temperament that tempers out 1029'/1024 and 2^8*3^3/(3')^8.

There are also some more "???"s on the plots. Each ??? you identify will win you one internet.

I put the rankings in the form of a "top 10" for each category. I arbitrarily decided to include 4-note scales for "coarse", but only 5-note scales and larger for the other categories. The format is

Ranking. Name (generator, average harmonic entropy)

Coarse:

1. Meantone pentatonic (502.7, 0.7994)
2. Meantone diatonic (503.1, 0.8218)
3. Semaphore[4] (249.0, 0.8237)
4. Neutral thirds [4] (351.0, 0.8294)
Tied for 5. 5-EDO (240, 0.8306)
Tied for 5. Semaphore[5] (249.0, 0.8309)
7. Superpyth diatonic (485.2, 0.8380)
Tied for 8. 12-EDO (100, 0.8476)
Tied for 8. Decatonic (104.7, 0.8479)
Tied for 8. Porcupine[7] (163.4, 0.8479)
Honorable mention: 7-EDO, Neutral thirds [7], Porcupine[8], Semaphore[9], Blackwood[10]

Medium:

1. Meantone pentatonic (503.0, 0.7997)
2. Meantone diatonic (503.4, 0.8217)
3. Semaphore[5] (249.0, 0.8347)
Tied for 4. Meantone chromatic (503.5, 0.8507)
Tied for 4. Superpyth diatonic (485.2, 0.8509)
Tied for 4. Decatonic (105.4, 0.8510)
Tied for 7. Porcupine[7] (163.4, 0.8522)
Tied for 7. Slendric[5] (233.8, 0.8527)
Tied for 9. Neutral thirds scale (349.3, 0.8537)
Tied for 9. Slendric[6] (233.8, 0.8543)
Tied for 9. Porcupine[8] (163.1, 0.8544)
Honorable mention: Semaphore[9], Pseudo-semaphore[9], 12-EDO

Fine:

1. Meantone pentatonic (503.0, 0.8079)
2. Pythagorean pentatonic (498.2, 0.8139)
3. Meantone diatonic (503.4, 0.8284)
4. Semaphore[5] (249.1, 0.8435)
5. Pythagorean diatonic (497.5, 0.8450)
Tied for 6. Slendric[5] (233.7, 0.8550)
Tied for 6. Meantone chromatic (503.5, 0.8559)
Tied for 6. Slendric[6] (233.6, 0.8566)
Tied for 9. Decatonic (105.3, 0.8581)
Tied for 9. Neutral thirds scale (350.8, 0.8583)
Honorable mention: Srutal/Pajara[12], Porcupine[7], Porcupine[8], Magic[7], Semaphore[9], Helmholtz/Garibaldi chromatic

Extra fine:

1. Pythagorean pentatonic (498.0, 0.8084)
2. Meantone pentatonic (503.1, 0.8195)
3. Meantone diatonic (503.4, 0.8358)
Tied for 4. Pythagorean diatonic (497.3, 0.8410)
Tied for 4. Semaphore[5] (249.0, 0.8417)
6. Slendric[5] (233.7, 0.8516)
7. Slendric[6] (233.7, 0.8533)
8. Neutral thirds scale (350.9, 0.8556)
Tied for 9. Decatonic (105.1, 0.8584)
Tied for 9. Meantone chromatic (503.5, 0.8590)
Honorable mention: Srutal/Pajara[12], Helmholtz/Garibaldi chromatic, Hanson/Keemun[7]

Keenan

🔗Carl Lumma <carl@lumma.org>

🔗Keenan Pepper <keenanpepper@gmail.com>

--- In tuning-math@yahoogroups.com, Carl Lumma <carl@...> wrote:
>
> Aren't some of the top-10s missing srutal/pajara (.836, .843 etc)?

Whoops, I knew I forgot something.

I got tired of seeing all these scales like porcupine(5), porcupine(6)... that have one large gap per period, which can easily be filled in just by adding one more note. I think of them as "incomplete", because you haven't yet got to the point where the scale runs over a whole period and new notes start appearing between the existing notes (rather than in the large gap).

So, define an "incomplete" MOS as one to which you can add one more note per period to create a larger MOS with the same smallest interval, that is, you don't introduce any new smallest interval by adding one more note per period. Equivalently, if the MOS comes from the Farey pair a/b, c/d, it is incomplete if both a=0 (implying c=1), and the generator is smaller than c/(d+2b).

My top ten lists are only the "complete" MOSes, so srutal/pajara(6) and (8) aren't there. The complete version, decatonic, places quite high, as you can see.

Sorry for the confusion.

Keenan