Comparing Intervals in Unusual Equal Tempered Scales

[Genetic_Junk]

Suppose I wanted to compare (in cents) intervals from equal tempered scales to corresponding just intervals. So in an equal tempered scale with 17 notes the fifth 3/2=1.5 is approximated to 2^(10/17)=1.503. To determine the difference in cents we (ln(3/2)/ln(2)-ln(2^(10/17))/ln(2))*1200 cents in an octave. This comes out to about 4 cents. So we know that the fifth in 17 note equal tempered scale is 4 cents wide of the 3/2 interval.

Does this seem right?
Thanks in advance for any help.

[jancivil]

close enough for jazz.

[Genetic_Junk]

yeah if the math is correct, 4 cents isn't too bad. But what about the major third.

5/4=1.25=386.313cents
2^(5/17)=1.226=352.9411cents
cents difference 33.3725cents too low

or the fifth in a 25 note equal tempered scale

3/2=1.5=701.955cents
2^(15/25)=1.5157=720cents
difference is 18.054cents too high

Just downloaded Scala and using that with Albino 3. Fifth sounds pretty good in the 17-TET (is that the right terminology) scale (with a pure sinusoid). Testing out some others now.

Thanks for the reply jan.

[jancivil]

does these 3rds compare badly with 12-ET?

[Genetic_Junk]

if I'm doing this math right, 12-ET isn't that great

2^(4/12)=1.2599=400cents and 5/4=1.250=386.31 cents
so ln(2^(4/12)/ln(2)-ln(5/4)/ln(2)*1200=13.686 so the major third is 13.686 cents wide in 12-ET

but in 25-ET it is
2^(8/25)=1.2483=383cents
difference it is 3.313 cents too low so the third is closer to the just interval in 25-ET.

I was just messing with 17 and 25. I haven't checked other equal tempered scales.

[jancivil]

yeah, that third in 12-ET is well wide of a just third. The fifths are close enough for jazz as they say, which tempers the fact of those thirds.

It's my thought that this lack of harmonic 'justness' is behind all the decisions to keep gilding the lily in western music, cause it's not much of a lily. A major chord, you'd think it'd be richer.

Like, a single low fundamental with a certain amt of harmonic distortion, sounds richer (to me, I'm sure I'm not lonely in this) than you'd imagine, from experience with a 'tempered' major (etc) 'harmony'.

[Genetic_Junk]

I'm not sure if there is something intrinsic to humans (like frequencies beating in critical bands) that makes low integer ratios between frequencies sound "sweet", or if slight beating sounds better to some individual for reasons other than constant exposure to 12-ET. In any case, it is fun to see how often certain theories (Helmholtz's is easiest) get things right by creating scales in Scala. Although "getting things right" runs up against the problem of theory-laden observation.

And you are not only in liking harmonic distortion. Does it matter whether it is odd, even or both?

[jancivil]

I kind of have leaned towards odd numbered partials in my attempts at drawing waveforms, just trial-and-error, I've seen it mentioned somewhere, but I have no theory.

I like that 7th, and particularly big ol' 11.

[Aroused by JarJar]

It's my thought that this lack of harmonic 'justness' is behind all the decisions to keep gilding the lily in western music, cause it's not much of a lily. A major chord, you'd think it'd be richer.

That's a great comparison, can I quote you? I have to give a little lecture/demonstration on these things next month.

Like, a single low fundamental with a certain amt of harmonic distortion, sounds richer (to me, I'm sure I'm not lonely in this) than you'd imagine, from experience with a 'tempered' major (etc) 'harmony'.

Exactly- distortion usually strongly brings out the 3d and 5th partials, the "natural fifth" and "natural M3" which are in their "first position voicing", 3:1 and 5:1, in the harmonic series. So you're hearing a Just major chord in a single tone.

If you just ride up and down the first seven harmonics you'll hear the structural foundation of probably most of the world's musics. The major exceptions in Africa and S-E Asia are precisely where the leading instruments are percussive and have different spectra, they have tunings which sound good with their drums and bells.

The current European version, contrary to what Hindemith and many theorists keep insisting, doesn't go beyond the third partial, it is audibly based on the third partial. It is nothing but a series of almost, but not quite, pure 3:2. There is no major or minor third coming out of the harmonic series, just irrational constructs allegedly representing them which must be learned by rote.

Of course it is not written that we must follow the harmonic series, contrasting with it is part of the fun. What is written, in the laws of physics and psychoacoustics, is that we cannot pretend that spectra don't exist, except when we are trained, heavily and relentlessly, to do so.

By the way the secret of 17 equal is that it has no Major Third at all. It has a supramajor third right inbetween 14/11 and 9/7 (at 23/18), and it has a great median third less than 2 cents from 27/22, the "Zalzal wosta of
al-Farabi", you will immediately recognize it from Middle Eastern music. The minor third is great, too, a little darker than 13/11 but not nearly as dark as 7/6.

[Genetic_Junk]

Thanks for the contribution jarjar. So the supramajor third of 17 would be the the the sixth note (2^(6 / 17) = 1.27716168 which is (ln(2^(6 / 17)) / ln(2)) * 1 200 = 423.529412cents? So its sharp of of a major third by 37.216412 cents whilst theoretically it should be 48.7703814 cents wide?

[Aroused by JarJar]

Thanks for the contribution jarjar. So the supramajor third of 17 would be the the the sixth note (2^(6 / 17) = 1.27716168 which is (ln(2^(6 / 17)) / ln(2)) * 1 200 = 423.529412cents? So its sharp of of a major third by 37.216412 cents whilst theoretically it should be 48.7703814 cents wide?

Yes that's the one. If you play a 14/11 at about 418 cents, it sounds like M3 sometimes does in an orchestral setting, especially in "expressive intonation" of Romantic music. 9/7 at about 432 cents is really pushing it as far as being a "third", but it works perfectly in tunings based on the seventh partial, it's really smooth in that context. I'd say 7/6 (266 cents)and 9/7 are the limits of minor and major for thirds, beyond them you're very concretely in diminshed and augmented sounding territory.

The supramajor third of 17 is well sharp of both 12-tET and Just M3s as you describe but still sounds like some kind of M3. So it is pretty dangerous in my opinion, as far as being used "instead of" a 12-tET or Just M3, as it is going to sound wildly bright or out of tune, and requires suitable music.

34-equal, or 17 divided by two, on the other hand gives you all the 17-et intervals as well as a nearly perfect 6/5, 5/4 and their inversions, and lots more.

Any of these equal temperaments (including 12-tET) sound much, much better if you make them a little bit lumpy, tweak them by ear a couple of cents here and there. With 34 you can be Just in the near keys and further off in the far keys for example, and it still circulates perfectly, aka a "well-temperament", like in Wohltemperierte Klavier. In my opinion.

[Genetic_Junk]

I'm going to try making a "custom" 12-ET in Scala. Staying in 12 is better for my music because I integrate lots of guitar (and I don't have a guitar built to another scale). Trying out different equal temperament scales is fun if I'm just using synths with .tun loading capability.
Thanks again for the comments.

[jancivil]

It's my thought that this lack of harmonic 'justness' is behind all the decisions to keep gilding the lily in western music, cause it's not much of a lily. A major chord, you'd think it'd be richer.

That's a great comparison, can I quote you? I have to give a little lecture/demonstration on these things next month.

Like, a single low fundamental with a certain amt of harmonic distortion, sounds richer (to me, I'm sure I'm not lonely in this) than you'd imagine, from experience with a 'tempered' major (etc) 'harmony'.

Exactly- distortion usually strongly brings out the 3d and 5th partials, the "natural fifth" and "natural M3" which are in their "first position voicing", 3:1 and 5:1, in the harmonic series. So you're hearing a Just major chord in a single tone.

If you just ride up and down the first seven harmonics you'll hear the structural foundation of probably most of the world's musics. The major exceptions in Africa and S-E Asia are precisely where the leading instruments are percussive and have different spectra, they have tunings which sound good with their drums and bells.

The current European version, contrary to what Hindemith and many theorists keep insisting, doesn't go beyond the third partial, it is audibly based on the third partial. It is nothing but a series of almost, but not quite, pure 3:2. There is no major or minor third coming out of the harmonic series, just irrational constructs allegedly representing them which must be learned by rote.

Of course it is not written that we must follow the harmonic series, contrasting with it is part of the fun. What is written, in the laws of physics and psychoacoustics, is that we cannot pretend that spectra don't exist, except when we are trained, heavily and relentlessly, to do so.

By the way the secret of 17 equal is that it has no Major Third at all. It has a supramajor third right inbetween 14/11 and 9/7 (at 23/18), and it has a great median third less than 2 cents from 27/22, the "Zalzal wosta of al-Farabi", you will immediately recognize it from Middle Eastern music. The minor third is great, too, a little darker than 13/11 but not nearly as dark as 7/6.

Sure, quote away! Great!
This puts me in mind of when I had a minimoog, eons ago. The big WOW when you realize the harmonic series concretely.

I'm really liking those arab thirds more and more; historically I've been partial to the smoothness of Indian intonation, but...

[Aroused by JarJar]

It's my thought that this lack of harmonic 'justness' is behind all the decisions to keep gilding the lily in western music, cause it's not much of a lily. A major chord, you'd think it'd be richer.

That's a great comparison, can I quote you? I have to give a little lecture/demonstration on these things next month.

Like, a single low fundamental with a certain amt of harmonic distortion, sounds richer (to me, I'm sure I'm not lonely in this) than you'd imagine, from experience with a 'tempered' major (etc) 'harmony'.

Exactly- distortion usually strongly brings out the 3d and 5th partials, the "natural fifth" and "natural M3" which are in their "first position voicing", 3:1 and 5:1, in the harmonic series. So you're hearing a Just major chord in a single tone.

If you just ride up and down the first seven harmonics you'll hear the structural foundation of probably most of the world's musics. The major exceptions in Africa and S-E Asia are precisely where the leading instruments are percussive and have different spectra, they have tunings which sound good with their drums and bells.

The current European version, contrary to what Hindemith and many theorists keep insisting, doesn't go beyond the third partial, it is audibly based on the third partial. It is nothing but a series of almost, but not quite, pure 3:2. There is no major or minor third coming out of the harmonic series, just irrational constructs allegedly representing them which must be learned by rote.

Of course it is not written that we must follow the harmonic series, contrasting with it is part of the fun. What is written, in the laws of physics and psychoacoustics, is that we cannot pretend that spectra don't exist, except when we are trained, heavily and relentlessly, to do so.

By the way the secret of 17 equal is that it has no Major Third at all. It has a supramajor third right inbetween 14/11 and 9/7 (at 23/18), and it has a great median third less than 2 cents from 27/22, the "Zalzal wosta of al-Farabi", you will immediately recognize it from Middle Eastern music. The minor third is great, too, a little darker than 13/11 but not nearly as dark as 7/6.

Sure, quote away! Great!
This puts me in mind of when I had a minimoog, eons ago. The big WOW when you realize the harmonic series concretely.

I'm really liking those arab thirds more and more; historically I've been partial to the smoothness of Indian intonation, but...

I can't bring myself to call them "neutral thirds" becuase they're so full of different characters and emotions. 16/13 is a groovy one because it is leaning towards the Just M3 of 5/4 a little but it's still neither major nor minor, and it's very smooth though not as smooth as 11/9. And the inversion at 13/8 is awesome too.

@Genetic_Junk, have you tried tunings that divide up 12-equal? You could use 24 equal, or some tasteful subset of 48 or 72 (there are several different whole "schools" of 72-equal musics, it's very versatile) and for example if you are multitrack recoding do a couple of passes with the whole thing tuned down a quarter or eighth tone etc. If you tune down an eighth tone the strings are slacker and you can bend thirds up to Just. Apparently this is what Hendrix did, and it certainly sounds that way some times. But supposedly his strings were so massive and high that noone ever got an exact hands-on take on exactly how his guitars were set up, don't know if that's just legend.

Edits- missspellingz and I used the adjective "groovy" twice.

[Genetic_Junk]

@Genetic_Junk, have you tried tunings that divide up 12-equal? You could use 24 equal, or some tasteful subset of 48 or 72 (there are several different whole "schools" of 72-equal musics, it's very versatile) and for example if you are multitrack recoding do a couple of passes with the whole thing tuned down a quarter or eighth tone etc. If you tune down an eighth tone the strings are slacker and you can bend thirds up to Just. Apparently this is what Hendrix did, and it certainly sounds that way some times. But supposedly his strings were so massive and high that noone ever got an exact hands-on take on exactly how his guitars were set up, don't know if that's just legend.

Haven't had a chance yet, but I'll definitely try out the subsets of multiplies of twelve.

Thanks again