One answer to Kalle's chord scale question (re-posted)

>> I've found several scales that more or less work as "chord scales"
>> that work well across multiple octaves and can change feel by
>> changing root tone very well.
>Example?
Try this "mangled to fit my tastes" scale (note the odd * notes):
A)
1/1
*10/9*
*17/14*
4/3
3/2
*27/16*
*13/7*
2/1

Now compare it to
B) 18:20:22:24:27:30:33:36

...and even it's simplest 6-tone form (about the ratios in the above scale divided by the denominator of 3 IE 18/3 = 6, 20/3 approximately = 7, 24/3 = 8, etc.
C) 6:7:8:9:10:11:12

There seems to be some odd stuff going on...mathematically you would think C would have a huge advantage over B and B would have a huge advantage over A...but (at least to my ears) they are more different than better or worse and I even consider A notably more consonant than B.
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Another especially odd example because it's not at all based on rational numbers is the simplified silver sections scale IE
1
1.071068
1.17157
1.414214 (silver-period/substitute octave)
....which isn't even based on rational number generation but instead y = 1 + (sqrt(2)-1)^x. The truly bizarre thing IMVHO is how much more consonant this sounds than you'd think simply by mathematically analyzing it so far as JI compliance.
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I have a hunch in both of the above cases, psychoacoustic phenomenon beyond what's explained by JI and/or "harmonic entropy" is occurring...

-Michael

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> >> I've found several scales that more or less work as "chord scales"
> >> that work well across multiple octaves and can change feel by
> >> changing root tone very well.
> >Example?
> Try this "mangled to fit my tastes" scale (note the odd * notes):
> A)
> 1/1
> *10/9*
> *17/14*
> 4/3
> 3/2
> *27/16*
> *13/7*
> 2/1
>
> Now compare it to
> B) 18:20:22:24:27:30:33:36
>
> ...and even it's simplest 6-tone form (about the ratios in the above scale divided by the denominator of 3 IE 18/3 = 6, 20/3 approximately = 7, 24/3 = 8, etc.
> C) 6:7:8:9:10:11:12
>
> There seems to be some odd stuff going on...mathematically you would think C would have a huge advantage over B and B would have a huge advantage over A...but (at least to my ears) they are more different than better or worse and I even consider A notably more consonant than B.

OK, let's listen to the A scale. Melodically it sounds very diatonic,
perhaps like ascending melodic minor. And there is a range of thirds
that all sound like some kind of major or minor third to me. Trying
out different bass tones and playing the scale against it sounds OK,
not particularly remarkable. But that was not what I asked for.
Remember that I was asking if there is a number of chords that all
sound like relatively stable resting points and have different roots
so that there could be satisfying bass lines and chord progressions.
This is what I considered the advantage of for example diatonic and
decatonic scales, scales that can be considered as being made of
relatively low-limit harmonic/subharmonic series segments. Now, can
you show me the stable chords and bass tones in the A scale?

> ****************************************
> Another especially odd example because it's not at all based on rational numbers is the simplified silver sections scale IE
> 1
> 1.071068
> 1.17157
> 1.414214 (silver-period/substitute octave)
> ....which isn't even based on rational number generation but instead y = 1 + (sqrt(2)-1)^x. The truly bizarre thing IMVHO is how much more consonant this sounds than you'd think simply by mathematically analyzing it so far as JI compliance.
> *************************************************
> I have a hunch in both of the above cases, psychoacoustic phenomenon beyond what's explained by JI and/or "harmonic entropy" is occurring...

Actually there are many JI intervals it approximates with varying
success, for example 6:11, 6:7, 5:6, 12:17, 3:5...

Kalle Aho

Kalle>"Remember that I was asking if there is a number of chords that all sound like relatively stable resting points and have different roots so that there could be satisfying bass lines and chord progressions."
What's your definition of a root in this case?

At least to me when I think "root", I think of the LCD of a chord that forms a harmonic series. And a few of the possible exact "roots" by that definition include 2,3,9,14,16, and 18. Then again...I'm guessing you may well mean something different by "root".

>"scales that can be considered as being made of relatively low-limit harmonic/subharmoni c series segments. Now, can you show me the stable chords and bass tones in the A scale? "
Depends how low-limit you think is necessary. I'd have to think how it translates to multiple octaves but, at least within the octave, some of the simple chords include 1/1 10/9 4/3 and 17/14, 3/2, 13/7.
Note how 17/14 nears 11/9 and 27/16 nears 5/3 so you can get 10/9, 17/14(acting as 11/9),4/3, 27/16 (acting as 5/3).

The odd thing is >why< scale A seems to match anywhere from as you said "OK" to fairly well.

If you are looking for a lot of straight harmonic series that are well approximated by scale A in a more obvious form, look at scale B
18:20:22:24: 27:30:33: 36 (40)(44)(48)(54)(60)(66)(72)

Here you can get 18:24:30 = 3:4:5, 20:24:36 = 5:6:9, 24:30:36 = 4:5:6, 27:36:54 = 3:4:6....and tons more if you allow up to 11-limit.
>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

However....I think I do see your point in that "chords" like 27:36:44 don't quite match up across octaves (in that case 27:36:45 would be significantly clearer).
Any ideas how to remedy that?

>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>>

>"Actually there are many JI intervals it approximates with varying success, for example 6:11, 6:7, 5:6, 12:17, 3:5..."
Right, but those all have very different denominators and least common multiples...at least by my research, there's no distinct JI pattern I can see in them. Almost any ratio can be approximated by even a fairly strict JI fraction within 15 cents or so. Actually, once I tried to "round" the Silver Scale to the x/12 harmonic series and found it simply sounded different rather than more consonant or better. It would be amusing to see, for example, if you could manage to convert the reduced "Silver" scale into a JI version and see how it turns out.

-Michael

>"And there is a range of thirds that all sound like some kind of major or minor third to me. "
Right, which is a side effect of the scale's merging toward 7TET. It revolves near the same 4 partials of the same parts of the harmonic series repeated around the 3/2 interval.

>"Now, can you show me the stable chords and bass tones in the A scale? "

A side note...I'm releasing a song based on the scale in A) for a competition. The song, I believe, should break some ground in proving the polyphonic power of the scale. And yes, the song uses "bass lines"...in fact all of the parts to it span at least 3 octaves and are chalk full of droning and highly consonant 6 and 7 note chords. It was done by ear and I haven't had time to find a "proper" music theory for it (since the way chords work in it obviously differ from standard diatonic)...but be on the lookout for the song soon. You might want to try composing with the scale a bit rather than just comparing intervals...I've actually found writing harmony in it easier than in 12TET now that I've become used to it.

-Michael