Yahoo Tuning Groups Ultimate Backup tuning Progress on Alternative EDO Primer

Hi everybody! I'm making good headway on the primer finally, but I wanted to check in with y'all about the way I'm describing the harmonic properties of each EDO.
Now, I'm dealing with EDOs 5 through 37 (was going to be 36, but Andrew Heathwaite convinced me to go one further), so for the most part, I'm trying to stay away from comparisons to JI and instead am using a re-scaled version of Harmonic Entropy that gives 1:1 a score of 100 and 51 cents a score of 0 (and I'm only dealing with the 1st octave, even though H.E. is not octave-equivalent, because I don't want to overwhelm myself or others). I call these re-scaled H.E. values "Concordance Scores", and for each EDO I give the "Average Concordance Excluding Octave & Unison", and the "Average of the Four Highest Concords" as bases for comparing EDOs according to their relative concordance. Then I go on to describe different harmonic approaches I've found useful in each EDO, different chord-shapes, strengths, weaknesses, synesthetic impressions, etc.--all my subjective data on them, if you will.

My question is: do you all think that this is a useful way to compare EDOs? Is there something I should be including, but am not? I originally thought to show each EDO's approximations to JI, but after consideration I realized that I'd have to give multiple ratios for every interval, I'd have to look at consistency, and I'd probably even have to take potential mappings into account as well. Also, for the majority of EDOs that I'm covering, they just don't get close enough to many "tunable-by-ear" ratios to make it worthwhile. However, when it's relevant in particular cases, I do mention the harmonic series: i.e. 16-EDO gets low concordance scores because of it's very poor fifths, but it has very good 4:5:7 chords--the best of any EDO below 25, I think--so I mention this. But yeah, if I'm leaving out something you think is important, please let me know. Note that this is just the section on harmony, there's a whole other section of the primer that deals with scales.

-Igs

🔗Michael <djtrancendance@...>

IGS>"My question is: do you all think that this is a useful way to compare
EDOs?"

As I've said before, I think using harmonic entropy alone to judge
concordance (no matter what the scaling) is a mistake. At the very least I'd
factor the critical band dissonance curve into the rating for any dyadic
interval of about 6/5 or under.
Harmonic entropy seems to say the quarter tone has the highest entropy and
between there and about 7/6 slopes down evenly so far as discordance. But try
it yourself: I'm betting you'll find the same thing I have...which is that
dissonance shoots up exponentially between about 12/11 and the semi-tone but is
fairly level between 9/8 and 12/11 and decreases linearly between about 12/11
and 6/5 (see http://eceserv0.ece.wisc.edu/~sethares/images/image1.gif).

>"Then I go on to describe different harmonic approaches I've found useful in
>each EDO, different chord-shapes, strengths, weaknesses, synesthetic
>impressions, etc.--all my subjective data on them, if you will."
Agreed...this "human factor" is very important. Perhaps one huge issue (at
least to me) is the function/compositional practices of neutral intervals and
chords using them...you can perhaps compare that to blues' use of quarter-tones
to interpolate between major and minor voicings (and then explain why what you
are doing is so much better than quarter tones, lol).

>"I originally thought to show each EDO's approximations to JI, but after
>consideration I realized that I'd have to give multiple ratios for every
>interval,"

Right. The approach I use is to break each EDO scale into a few sub-scales,
each using one of a set of possible ratios for each notes. The idea is to have
each scale have 2-3 sub-scales and the chords possible in each sub-scale
ultimately summarize a huge percentage of all possible chords. IE I have a
scale with many neutral, major, and minor intervals and break it up into
A) The diatonic scale
B) A single half-step scale with one neutral tone used (IE 15/11 instead of 4/3)
C) A zero half-step scale with no neutral tones used (ALA 7TET, using both the
15/11 and 11/6 "neutrals")
Then I concentrate on the possible chords relative to A,B, or C (one at a
time).
This way....I don't think the use of JI is confusing. Especially if you use
JI up to starting at the 11th harmonic IE x/11 format fractions)...and can
summarize most important ratios within 8 cents accuracy).

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

On Sun, Sep 5, 2010 at 1:19 PM, cityoftheasleep <igliashon@...> wrote:
>
> Hi everybody! I'm making good headway on the primer finally, but I wanted to check in with y'all about the way I'm describing the harmonic properties of each EDO.
> Now, I'm dealing with EDOs 5 through 37 (was going to be 36, but Andrew Heathwaite convinced me to go one further), so for the most part, I'm trying to stay away from comparisons to JI and instead am using a re-scaled version of Harmonic Entropy that gives 1:1 a score of 100 and 51 cents a score of 0 (and I'm only dealing with the 1st octave, even though H.E. is not octave-equivalent, because I don't want to overwhelm myself or others). I call these re-scaled H.E. values "Concordance Scores", and for each EDO I give the "Average Concordance Excluding Octave & Unison", and the "Average of the Four Highest Concords" as bases for comparing EDOs according to their relative concordance. Then I go on to describe different harmonic approaches I've found useful in each EDO, different chord-shapes, strengths, weaknesses, synesthetic impressions, etc.--all my subjective data on them, if you will.

There is an octave-equivalent HE formulation that you might find
interesting. By the way, did you generate the HE curve from scratch to
make these calculations? Or are you just referencing the chart?

> My question is: do you all think that this is a useful way to compare EDOs? Is there something I should be including, but am not? I originally thought to show each EDO's approximations to JI, but after consideration I realized that I'd have to give multiple ratios for every interval, I'd have to look at consistency, and I'd probably even have to take potential mappings into account as well. Also, for the majority of EDOs that I'm covering, they just don't get close enough to many "tunable-by-ear" ratios to make it worthwhile. However, when it's relevant in particular cases, I do mention the harmonic series: i.e. 16-EDO gets low concordance scores because of it's very poor fifths, but it has very good 4:5:7 chords--the best of any EDO below 25, I think--so I mention this. But yeah, if I'm leaving out something you think is important, please let me know. Note that this is just the section on harmony, there's a whole other section of the primer that deals with scales.

I was thinking about this just the other day - basically what you
want, or what I wanted at least, was to come up with some type of
"badness" analogue for EDO's. Something that takes into account both
error and complexity, and maybe other things too. The complexity of an
EDO has to be related to its size. In the 5-limit, 1200tet outshines
almost everything in terms of error, but it's much less "bad" than
53-tet, which is "almost as good" and more practically usable. The
distinction really comes in handy when comparing something like
171-tet and 53-tet, the former being a near-optimal schismatic tuning
with a 386 cent 5/4, and the latter being a near-optimal schismatic
tuning with a 384 cent 5/4, and less than 1/3 the notes.

One idea I had was to make the complexity measure not linearly
proportional to size, but logarithmically proportional. Let's say base
2 log to keep the math simple. So 12-tet would have a normalized
complexity of "1," let's say. 24-tet would have a complexity of 2,
36-tet would have a complexity of 1.584 (log_2(3)), 48-tet would have
a complexity of 3, etc. This is because if you're at 41-tet, the jump
up to 53-tet is going to be less of a pain than the jump from 12 to
19, or so my thinking went, since you're already used to there being a
lot of notes.

I was thinking that that might "underestimate" the complexity though,
so perhaps something halfway in between log and linear would be best.
An average of the two. Not sure which is the best average to take.

-Mike

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:

> There is an octave-equivalent HE formulation that you might find
> interesting. By the way, did you generate the HE curve from scratch to
> make these calculations? Or are you just referencing the chart?

Hmm...yes, that would have been good, but then I'd have to go back and re-enter all the data! I may consider it, though. I got the values from a spreadsheet that Carl sent me, and I wrote a quick and dirty function where I can paste in the cents values of all intervals in a scale into one column and then get their corresponding concordance scores in the column next to it (I don't know jack about spreadsheets so it took me quite a while to figure out how to do that!). I'm not sure what formulation of HE was used to generate the values in the spread-sheet, but I do know it covers the first two octaves and came directly from Paul.

> I was thinking about this just the other day - basically what you
> want, or what I wanted at least, was to come up with some type of
> "badness" analogue for EDO's. Something that takes into account both
> error and complexity, and maybe other things too. The complexity of an
> EDO has to be related to its size. In the 5-limit, 1200tet outshines
> almost everything in terms of error, but it's much less "bad" than
> 53-tet, which is "almost as good" and more practically usable. The
> distinction really comes in handy when comparing something like
> 171-tet and 53-tet, the former being a near-optimal schismatic tuning
> with a 386 cent 5/4, and the latter being a near-optimal schismatic
> tuning with a 384 cent 5/4, and less than 1/3 the notes.

Well, you could do it a bit more straight-forwardly if you used H.E. (or the normalized version of it that I use to give "concordance scores") instead of error. Like I said, looking at EDOs in terms of error is tough to do unless you have a mapping. I mean, in 12-tET, we could look at 400 cents as being an approximation to 5/4, to 81/64, to 24/19, to 14/11, etc. etc. and those are all valid ways of looking at it, so it will have a different error depending on which interval we say it's "supposed to be".

That's what I love about H.E.: it doesn't care about the exact ratio, it's a way of scoring according simultaneously to "proximity to a simple ratio" AND "simplicity of nearest ratios". Of course, what I DON'T like about H.E. is the fact that an out-of-tune fifth still scores higher than a perfectly-in-tune major 3rd or harmonic 7th...but maybe, you know, that is actually okay?

One issue with looking at "average concordance" I've noticed is that discords like small minor 2nds or wide major 7ths or extra-sharp or extra-flat 4ths and 5ths drag down the overall average. For instance, 19-EDO is on average more discordant than 12-tET because its tritones and its subminor 2nd and supermajor 7th are so discordant. This is why I opted to include the average of the four strongest concords--it is there that 19-EDO clearly beats out 12-tET.

> One idea I had was to make the complexity measure not linearly
> proportional to size, but logarithmically proportional. Let's say base
> 2 log to keep the math simple. So 12-tet would have a normalized
> complexity of "1," let's say. 24-tet would have a complexity of 2,
> 36-tet would have a complexity of 1.584 (log_2(3)), 48-tet would have
> a complexity of 3, etc. This is because if you're at 41-tet, the jump
> up to 53-tet is going to be less of a pain than the jump from 12 to
> 19, or so my thinking went, since you're already used to there being a
> lot of notes.

I wonder if that's true. I'll say that on guitar, the number of notes seems to matter less than the open-string tuning. For instance, 19-EDO is actually easier to deal with (for diatonic purposed) than 17-EDO, and 31-EDO is a bit easier than 22-EDO in a lot of ways. All the 5n-EDOs, like 15, 20, and 25, are easier than any near-by EDOs because you can tune all the strings in 4ths and span 2 octaves exactly. So on guitar, scoring complexity just by number of notes isn't necessarily helpful.

-Igs

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> As I've said before, I think using harmonic entropy alone to judge
> concordance (no matter what the scaling) is a mistake. At the very least I'd
> factor the critical band dissonance curve into the rating for any dyadic
> interval of about 6/5 or under.
> Harmonic entropy seems to say the quarter tone has the highest entropy and
> between there and about 7/6 slopes down evenly so far as discordance. But try
> it yourself: I'm betting you'll find the same thing I have...which is that
> dissonance shoots up exponentially between about 12/11 and the semi-tone but is
> fairly level between 9/8 and 12/11 and decreases linearly between about 12/11
> and 6/5 (see http://eceserv0.ece.wisc.edu/~sethares/images/image1.gif).

That curve doesn't look all that different than the H.E. curve to me, and at any rate, I really don't think I can tell which curve my listening experience seems to describe. It's really hard for me to say how much more concordant I find 12/11 to be than 18/17. I really don't hear 12/11 as being equally concordant to 11/10, 10/9, and 9/8, but I do find 9/8 and 8/7 and 7/6 to all be fairly close. Looking at the "normalized" H.E. values that I'm using also gives a different curve, since it's based on the exponential version of H.E. (i.e. "e^(H.E.)" and is reversed so that low H.E. = high concordance score. I should plot these scores on a graph to show what the curve really looks like.

> Agreed...this "human factor" is very important. Perhaps one huge issue (at
> least to me) is the function/compositional practices of neutral intervals and
> chords using them...you can perhaps compare that to blues' use of quarter-tones
> to interpolate between major and minor voicings (and then explain why what you
> are doing is so much better than quarter tones, lol).

Yeah, I think looking only at concordance misses the fact that intervals can have "moods" independent of their concordance. I wax fairly poetic about the quality of neutral intervals at a few points, too. Intervals that straddle interval classes, like the 450-cent sub-4th/supermajor-3rd--it's a neat interval, really, despite being discordant as all hell.

> Right. The approach I use is to break each EDO scale into a few sub-scales,
> each using one of a set of possible ratios for each notes. The idea is to have
> each scale have 2-3 sub-scales and the chords possible in each sub-scale
> ultimately summarize a huge percentage of all possible chords.

Yeah, see, I kind of want to keep the "harmonic properties" of each EDO *separate* from their scalar properties. When I discuss scales, I'm going to do it by MOS families, and then break down the harmonic idiosyncrasies of each EDO's instantiation of the pattern. I really don't think looking to JI to define the "chords" possible in each EDO is very helpful, because there can be inconsistency and it can be confusing to think of one interval having multiple identities.

This primer is meant to be the absolute BASICS, and if people want to learn more about JI and how various EDOs relate to it, that's a more advanced subject. All I want to convey to them is "what sounds good, what doesn't sound good, what emotional effects are supported, and what patterns relate different EDOs". Also, I really hate the "JI bias" that is so common in tuning literature. JI may be a sort of "ideal" as far as psychoacoustics is concerned, but it doesn't explain everything about non-JI scales...in fact, it may not actually explain ANYTHING about them. What, in the end, matters most? The "numerological properties" a tempered interval has, or how concordant it sounds?

-Igs

🔗Carl Lumma <carl@...>

Hi Igs,

> Now, I'm dealing with EDOs 5 through 37 (was going to be 36, but
> Andrew Heathwaite convinced me to go one further), so for the
> most part, I'm trying to stay away from comparisons to JI and
> instead am using a re-scaled version of Harmonic Entropy that
> gives 1:1 a score of 100 and 51 cents a score of 0

Which rescaling is that? Did you see this message:

/tuning/topicId_91940.html#91961

> I call these re-scaled H.E. values "Concordance Scores", and
> for each EDO I give the "Average Concordance Excluding Octave
> & Unison", and the "Average of the Four Highest Concords" as
> bases for comparing EDOs according to their relative
> concordance.

"Average of the four highest" sounds a lot more sensible
than an overall average. Also, though it may not be an
issue only going up to 37, but make sure each of the four
highest are approximating different concordances (have
different local minima). You only want to score the best
approximation of each concordance.

> 16-EDO gets low concordance scores because of it's very poor
> fifths, but it has very good 4:5:7 chords--the best of any
> EDO below 25, I think--so I mention this.

A list of all 7-limit triads and the best ET < 37 for each
might be interesting...

> But yeah, if I'm leaving out something you think is important,

Sounds like a good approach, but hard to say without seeing it!

-Carl

🔗Michael <djtrancendance@...>

IGS>"That curve doesn't look all that different than the H.E. curve to me"
Agreed for the most part...except for the lower range (IE under 6/5....the
part I was describing in which to focus on critical band instead of Harmonic
Entropy (HE), in other words).

>"but I do find 9/8 and 8/7 and 7/6 to all be fairly close"
Well...at least we can agree on that for the most part apparently (I think
7/6 has a not-so-significant advantage over 9/8 and there's a significant
concordance gap between 7/6 and 6/5). IE that somewhere a bit below 9/8
dissonance starts shooting up exponentially. It's funny because before I recall
you described 12/11 as an "excellent minor second" and 13/12 a not-so-hot one.

>"I should plot these scores on a graph to show what the curve really looks
>like."
Agreed. Just thinking of it in my head seems to give a version with less
drastic peaks and troughs, rather than a completely different curve....perhaps
I'm mistaken.

>"Yeah, see, I kind of want to keep the "harmonic properties" of each EDO
>*separate* from their scalar properties. When I discuss scales, I'm going to do
>it by MOS families, and then break down the harmonic idiosyncrasies of each
>EDO's instantiation of the pattern."
Sounds good...so long as the person "only" wants to concentrate on scales
under EDOs that are also MOS.

>"I really don't think looking to JI to define the "chords" possible in each EDO
>is very helpful, because there can be inconsistency and it can be confusing to
>think of one interval having multiple identities. "
But isn't it that inconsistency that gives microtonal scales so much of their
flavor? Then again, you do say you go on a bit about the possible uses of
things like neutral intervals (correct?)...maybe that will help introduce some
of the possibilities and "values of inconsistency".

>"All I want to convey to them is "what sounds good, what doesn't sound good,
>what emotional effects are supported, and what patterns relate different EDOs"."
Agreed...I guess you could just say I'm a bit wary of the idea of an MOS
bias...the same way I am of a "strict-JI" bias...as overly simplifying things to
the point it seems to almost say "such is the only way to do things" and
restrict growth beyond that.

>Also, I really hate the "JI bias" that is so common in tuning literature. JI may
>be a sort of "ideal" as far as psychoacoustics is concerned"
Agreed. To me JI (IE the type focused on creating/reproducing in scales ONLY
very low numbered harmonic series segments as 3+ note chords) simply explains
the ultimate arrangement so far as periodicity is concerned. Even taking a c
fairly settled hord like C E F A seems to violate what so many academic types
think of JI because it's "not periodic enough"...even if in reality it sounds
quite stable. Meanwhile the 5:7:9 chord sounds quite unsettled despite being
highly periodic.

JI (alone) ignores critical band dissonance, emotional effects of possible
chords, how many sets of tones the mind can handle efficiently, how far away
from periodicity you can go without the mind getting confused, dyadic dissonance
(IE chords not based on perfect harmonic series segments ALA 7:8:10), root
tonality (IE the 5:7:9 example), and a ton of other concepts.

>"but it doesn't explain everything about non-JI scales...in fact, it may not
>actually explain ANYTHING about them. "
Strict JI, dare I say it, seems to even throw out most things like intervals
different than 12TET (IE neutral, supermajor, etc. intervals). 7-limit, to me,
is often barely different than 12TET and where things start really getting is 9+
limit...which is exactly what academic JI seems to try so hard to avoid.

------------------------------------
You made some fascinating points on things like how a nasty 16/11 dyad can be
part of a quite "pure" sounding triad...and I'd hate to think your tutorial
would avoid such things because they are "too close to JI". I'm not saying you
need to write a "JI essay"...but at least a quick summary of how
non-strict/"open" dyadic JI may in many cases help people determine the overall
consonance (not just concordance!) of a chord would IMVHO be much encouraged.
-------------------------------------

Now taking dyadic JI analysis of triads and such, as Harmonic Entropy (based
on JI, but not on enforcing periodicity ALA 3+ notes with "the same
denominator") I think we can both agree matters...otherwise you wouldn't be
using HE (right)?

>"What, in the end, matters most? The "numerological properties" a tempered
>interval has, or how concordant it sounds?"
How concordant it sounds which, to a decent though not entire extent, can
faithfully be determined (IE believe) by things like summarizing chords in terms
of JI dyads ALA Paul Erlich's tetrachord dissonance rating charts Carl posted a
while back. And sure, there's always that other says 30% of what makes a chord
work in a context that can never be summarized in math....but that means
different things to different people IE depends much on the mood of the musician
(though most, say 70%, of musicians may agree for the most part on many types of
chords as being most concordant that can't be "numerically proven").

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> Well...at least we can agree on that for the most part apparently (I think
> 7/6 has a not-so-significant advantage over 9/8 and there's a significant
> concordance gap between 7/6 and 6/5). IE that somewhere a bit below 9/8
> dissonance starts shooting up exponentially. It's funny because before I recall
> you described 12/11 as an "excellent minor second" and 13/12 a not-so-hot one.

I didn't mean to suggest I don't hear a difference along the Minor 2nd-->Neutral 2nd continuum, just that the difference in discordance between 16/15 and 12/11 isn't really all that sharp. To be sure, 12/11 is notably more concordant, but it's not like the difference between e.g. 9/7 and 4/3. IOW, I wouldn't call 12/11 "concordant". But it's certainly less-discordant than its lower-pitched neighbors.

> Agreed. Just thinking of it in my head seems to give a version with less
> drastic peaks and troughs, rather than a completely different curve....perhaps
> I'm mistaken.

It will be flipped, actually, so peaks will become valleys, and it will span 0 to 100, rather than ~2.5 to about 4- or 5-point-something.

> Sounds good...so long as the person "only" wants to concentrate on scales
> under EDOs that are also MOS.

Well, I could go into other possibilities, but that multiplies the amount of information I'd have to convey exponentially. MOS scales have the benefit of being the simplest scalar patterns, and satisfy a lot of Rothenbergian ideals (like being strictly proper, etc.). They are also obviously analogous to the diatonic scale, in that they are "generated" by a "circle" of some interval. This makes them a little more intelligible, I think, from a beginner's standpoint. Non-MOS scales are certainly important, but come on--I have something like 40 or 50 scales to cover, just restricting it to MOS! I've identified at least 29 just with an octave period, and I want to cover scales with 1/2-octave, 1/3-octave, 1/4-octave, and 1/5-octave periods as well. Though I certainly intend to explain that I *have* omitted these other scales, and that they do exist and are worth exploring. Just like I explain that JI exists and is an option but I'm not covering it for various reasons.

> But isn't it that inconsistency that gives microtonal scales so much of their
> flavor?

By "inconsistency", I mean in terms of prime-limit JI, i.e. how something like 13-EDO might approximate some 7-limit intervals, but not approximate the full 7-limit spectrum consistently--so you can't effectively use it to approximate 7-limit JI.

> Agreed...I guess you could just say I'm a bit wary of the idea of an MOS
> bias...the same way I am of a "strict-JI" bias...as overly simplifying things to
> the point it seems to almost say "such is the only way to do things" and
> restrict growth beyond that.

It's not a bias if I touch on the existence of other possibilities. A lot of the literature on tuning doesn't even suggest that there are alternatives to JI to look at.

> You made some fascinating points on things like how a nasty 16/11 dyad can be
> part of a quite "pure" sounding triad...

I think you're getting confused. I've never used 16/11 in a triad. 22/15, sure, or even 40/27 (though actually I've never used those strict intervals but their tempered approximations in EDOs like 9-EDO, 16-EDO, etc).

And at any rate, I'm rather convinced that the concordance of those dyads makes more sense from a H.E. standpoint than it does from a JI standpoint. One thing H.E. suggests is that "bad fifths" can be stronger concords than "good thirds", or so it seems anyway (maybe my interpretation is unfounded). For instance, a fifth of 675 cents still has lower H.E. than a pure 7/6 subminor third.

-Igs

🔗Mike Battaglia <battaglia01@...>

On Sun, Sep 5, 2010 at 5:34 PM, cityoftheasleep <igliashon@...> wrote:
>
> Well, you could do it a bit more straight-forwardly if you used H.E. (or the normalized version of it that I use to give "concordance scores") instead of error. Like I said, looking at EDOs in terms of error is tough to do unless you have a mapping. I mean, in 12-tET, we could look at 400 cents as being an approximation to 5/4, to 81/64, to 24/19, to 14/11, etc. etc. and those are all valid ways of looking at it, so it will have a different error depending on which interval we say it's "supposed to be".
>
> That's what I love about H.E.: it doesn't care about the exact ratio, it's a way of scoring according simultaneously to "proximity to a simple ratio" AND "simplicity of nearest ratios". Of course, what I DON'T like about H.E. is the fact that an out-of-tune fifth still scores higher than a perfectly-in-tune major 3rd or harmonic 7th...but maybe, you know, that is actually okay?
>
> One issue with looking at "average concordance" I've noticed is that discords like small minor 2nds or wide major 7ths or extra-sharp or extra-flat 4ths and 5ths drag down the overall average. For instance, 19-EDO is on average more discordant than 12-tET because its tritones and its subminor 2nd and supermajor 7th are so discordant. This is why I opted to include the average of the four strongest concords--it is there that 19-EDO clearly beats out 12-tET.

I suppose that the complexity measure will be built into it if you
have more notes, since most of them will be hitting discordant
intervals, thus bringing it down.

HE is something that I think is primarily good for a very specific
psychoacoustic phenomenon: predicting how intervals, specifically pure
tones, will sound when tempered. The fact that a tempered fifth is
higher than a perfectly in tune major 3rd is because a tempered fifth
simply presents less "options" for misperception than a perfectly in
tune major 3rd. There are a few ways to interpret the HE curve:

1) Is it measuring, in a musical sense, how "consonant" the interval is? (No.)
2) Is it measuring, in a timbral sense, how strong or loud the
fundamental being produced is? (Possibly indirectly, but not
directly.)
3) Is it measuring, in a timbral sense, how many fundamentals can
match a certain dyad, if that dyad is allowed to be mistuned? (Yes.)

The second one, IMO, is what a lot of people think HE is measuring,
but that it isn't really. The fact that 5/4 produces a weaker
fundamental than 3/2 isn't for the same reason that HE has it listed
as more discordant, for example. It's listed as more discordant
because there's a chance it'll be perceived as a mistuned 6/5, and a
mistuned 7/6, and a mistuned 9/7, and a mistuned 4/3. 3/2 doesn't have
nearly as many strong intervals in the way, so it's perceived as being
less confusing.

If the reason that the 5/4 fundamental is sonically lower in volume
than the 3/2 one happens to be because the 5/4 one confuses your brain
more or something - then HE would be measuring that too. If. But, I
don't think that's something we know for sure. It could also be that
there is a neural filterbank capturing in the incoming data, and that
each filter simply has a rolloff so that higher overtones resonate
less strongly than lower ones. In fact, there could be any explanation
for how it works, really, so it's best not to make assumptions about
that just yet.

The point of this long tirade? HE is a great concept, and it's the
best thing we have, but there are a few practical considerations that
have to be made before you really use it as an HE <-> consonance
measure. "Concordance" is a different psychoacoustic percept than most
people think it is, and your concerns about what the curve shows are
very valid :) Entropy models in the future might take additional
factors into consideration, and this might alter the shape of the
curve slightly to match the phenomena you are describing.

An alternate, but shorter, explanation, is that a tempered fifth will
probably "fuse" more easily than a pure major third if they're both
played with sine waves. The fact that the pure major third sounds
better with harmonic timbres has primarily to do with beating
partials, I think.

-Mike

🔗Mike Battaglia <battaglia01@...>

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi Igs,
>
> Which rescaling is that? Did you see this message:
>
> /tuning/topicId_91940.html#91961

Yes, and I opted to use the one suggest by S.J. Martin in this one:

/tuning/topicId_91940.html#91949

Which is basically the same principle (exponentials) but gives a range of 0-100, which I like.

> "Average of the four highest" sounds a lot more sensible
> than an overall average. Also, though it may not be an
> issue only going up to 37, but make sure each of the four
> highest are approximating different concordances (have
> different local minima). You only want to score the best
> approximation of each concordance.

Yeah, I just checked, and sure enough that is a problem with 35 & 37-EDO. How do you think I should deal with that? Just use the highest for each interval class?

> A list of all 7-limit triads and the best ET < 37 for each
> might be interesting...

7-odd-limit, I presume? 7-prime-limit would be an awfully long list... Though I'm betting for 7-odd-limit, it'll be 31 for all chords where 5 is included, and 36 for all chords where 5 is omitted.

> Sounds like a good approach, but hard to say without seeing it!

I'll post a draft as soon as I finish the "subjective" descriptions.

BTW, did you have a good Burn? I take it you didn't stay for the Temple burn?

-Igs

🔗Carl Lumma <carl@...>

Hi Mike,

> I was thinking about this just the other day - basically what
> you want, or what I wanted at least, was to come up with some
> type of "badness" analogue for EDO's.

No need for an analog... badness is defined for EDOs perfectly
well.

> One idea I had was to make the complexity measure not linearly
> proportional to size, but logarithmically proportional. Let's
> say base 2 log to keep the math simple. So 12-tet would have a
> normalized complexity of "1," let's say. 24-tet would have a
> complexity of 2, 36-tet would have a complexity of 1.584
> (log_2(3)), 48-tet would have a complexity of 3, etc. This is
> because if you're at 41-tet, the jump up to 53-tet is going to
> be less of a pain than the jump from 12 to 19, or so my thinking
> went, since you're already used to there being a lot of notes.

That's one way to look at it. Another is that the number of
possible chords goes up quadratically (Metcalfe's law).

Gene's "logflat badness" is

error * complexity^(pi(p)/(pi(p)-r)

where pi(p) is the number of primes, r is the rank (1 for
EDOs) and ^ is exponentiation. It makes sense in that we
expect higher-rank systems to be more accurate for a given
number of notes. For instance, rank 1 meantone requires
31 notes to provide the same accuracy that rank 2 meantone
provides in 5. And it happens that 5^3 ~~ 31^(3/2).

Gene found that if you plot a histogram showing the number
of temperaments below a given badness as complexity goes to
infinity, the histogram is roughly flat if the complexity
axis is logarithmic and you used the above badness formula
(hence its name). In other words, the best ET < 100 notes
should often be the 2nd best ET < 1000 notes.

-Carl

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

🔗Mike Battaglia <battaglia01@...>

🔗Michael <djtrancendance@...>

Igs>"I didn't mean to suggest I don't hear a difference along the Minor
2nd-->Neutral 2nd continuum, just that the difference in discordance between
16/15 and 12/11 isn't really all that sharp."

Hmm... To me if I play, say, a triad from a 12/11 and 11/10 to make a 6/5
interval, for example, it seems to pass to me as a "very jazzy ultra diminished
kind of chord". I'm even using it in my Un-twelve competition entry. Trying
the same thing by stacking 16/15 and 15/14 to make 7/6 just doesn't seem to cut
it; it sounds like neighboring tones jumbled together not a "real" chord. It
may not be a "huge" difference...but it seems large enough to change its
possible usage to me. Or do you think that...the gap isn't enough to change the
usage?

>"but it's not like the difference between e.g. 9/7 and 4/3. "
Nothing is much like 3/2 and 4/3...they are so periodic they almost sound too
consonant, giving the impression of the exact same root tone-feel. I swear this
explains the musical usage of why 1/1 3/2 2/1 6/2 tone-using arpeggios are used
so often in things like trance music...to give a psychedelic feeling of
"shifting notes without shifting root tones".

>"It will be flipped, actually, so peaks will become valleys, and it will span 0
>to 100, rather than ~2.5 to about 4- or 5-point-something."
Ok, but still it's really just looking at the same curve with a different unit
skew (IE rates "height of consonance" instead of "lowness of discordance"),
correct?

>"They are also obviously analogous to the diatonic scale, in that they are
>"generated" by a "circle" of some interval. "
Right. Actually I think it's an issue I have with micro-tonality: while MOS
is all good stuff, I see very few people messing around with the idea of
alternating generators to get rid of would-be commatic build-up.

>"I have something like 40 or 50 scales to cover, just restricting it to MOS!
>I've identified at least 29 just with an octave period, and I want to cover
>scales with 1/2-octave, 1/3-octave, 1/4-octave, and 1/5-octave periods as well."
Agreed...it's a tough nut. I'd suggest trying to
A) Single out scales you think have many things in common and group them
B) Take one or two examples from each group to explain in detail, giving care to
cover general chords/moods/instrument-mappings etc. for such scales.
C) Give good "if interested" references to existing documents that explain the
rest...or perhaps link to an "in constant development" web page that lists
detail about increasingly more scales over time and labels them under each
"group type".

>"By "inconsistency", I mean in terms of prime-limit JI, i.e. how something like
>13-EDO might approximate some 7-limit intervals, but not approximate the full
>7-limit spectrum consistently--so you can't effectively use it to approximate
>7-limit JI."
Right, but then again by having to say "7=limit (and nothing but!)", that
seems to be overly simplifying it. Would you see any reason it would be a crime
to say, for example "it gives many good 7-limit intervals (list), plus a handful
of 11-limit and a couple of 13-limit ones". I agree though...you certainly
can't say many TET scales are "just built for one type of limit".

>"It's not a bias if I touch on the existence of other possibilities. A lot of
>the literature on tuning doesn't even suggest that there are alternatives to JI
>to look at."
It just sounded to me before like you were going to completely ignore JI on
purpose...but now I agree with your point.

>"I think you're getting confused. I've never used 16/11 in a triad. "
In the first song on "Map of An Internal Landscape" I swear you told me you
did and also swear it was some sort of alternative fifth. Maybe it was 22/15?
I'd be interested to know which one...because (again) it completely fooled my
ear "in context" into thinking it was a "pure" 5th and the triads sounded
surprisingly stable.

>"One thing H.E. suggests is that "bad fifths" can be stronger concords than
>"good thirds", or so it seems anyway (maybe my interpretation is unfounded)."
Hmm...I'd say yes and no. Anything around 13-20 cents of 3/2 (IE around
1.48) seems to really really bite my ears....but things like 22/15 and even 14/9
often seem passable to the same degree, say, 5/4 is...and in the case of 22/15
perhaps a bit better. So I agree there is a lean toward
near-simpler-intervals...but I don't think it's nearly so strong as HE presents
it at (also HE puts a small dip in dissonance near 22/15, while I hear a huge
sudden dip/bump at 22/15 relative to both its near neighbors 16/11 and 40/27).
40/27, to me, actually sounds as or more dissonant than something like 12/11,
despite being "near the 5th" on the HE curve!

It (HE) often sound to me a case of right idea, wrong degrees of exaggeration,
giving too much credits to exact versions of a tone like 3/2 and not enough
credit to its alternative versions.
Try it yourself (on your own ears) and let me know what you think.....

🔗Michael <djtrancendance@...>

🔗genewardsmith <genewardsmith@...>

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> Hmm... To me if I play, say, a triad from a 12/11 and 11/10 to make a 6/5
> interval, for example, it seems to pass to me as a "very jazzy ultra diminished
> kind of chord". I'm even using it in my Un-twelve competition entry. Trying
> the same thing by stacking 16/15 and 15/14 to make 7/6 just doesn't seem to cut
> it; it sounds like neighboring tones jumbled together not a "real" chord. It
> may not be a "huge" difference...but it seems large enough to change its
> possible usage to me. Or do you think that...the gap isn't enough to change the
> usage?

Well, the bare dyad is one thing, a stack of two dyads is quite another. At any rate, 16/15 plus 15/14 doesn't make 7/6, it makes 8/7. 14/13 and 13/12 make 7/6, and I'm not entirely sure that that's a whole lot less concordant than your 12/11-11/10 example. I will say though that a stack of 12/11's sounds better to me than a stack of 13/12's or 14/13's. But that's a stack, not the bare dyad. Here's an experiment for you: try taking two triangle waves at the same pitch, and slowly increase the pitch of one without being able to see what pitch it's at (a MIDI pitch-bend wheel will probably work for this). Stop increasing the pitch when it gets to a point that you like. Repeat this several times, and then measure the pitches you stopped at and compute the ratio, then post the results!

> Nothing is much like 3/2 and 4/3...they are so periodic they almost sound too
> consonant, giving the impression of the exact same root tone-feel. I swear this
> explains the musical usage of why 1/1 3/2 2/1 6/2 tone-using arpeggios are used
> so often in things like trance music...to give a psychedelic feeling of
> "shifting notes without shifting root tones".

Yes, it's quite true. I was bothered at first by how Tenney Height suggested that a pure 3/2 is more than 3 times as concordant as a pure 5/4, until I started thinking of concordance as a measure of "how obvious it is that two tones are sounding, rather than one". For this reason, I think B-P "tritave equivalence" may very well be a possibility.

> Ok, but still it's really just looking at the same curve with a different unit
> skew (IE rates "height of consonance" instead of "lowness of discordance"),
> correct?

Yes. Precisely. It's just a transformation of the H.E. data, so it's fundamentally going to be saying the same thing, just in (what I consider to be) a more readable way.

> Right. Actually I think it's an issue I have with micro-tonality: while MOS
> is all good stuff, I see very few people messing around with the idea of
> alternating generators to get rid of would-be commatic build-up.

Huh? What commatic build-up? And how do alternating generators (i.e. rank-3 temperaments, IIRC) eliminate commas any better than rank-1 or rank-2 temperaments?

> >"I have something like 40 or 50 scales to cover, just restricting it to MOS!
> >I've identified at least 29 just with an octave period, and I want to cover
> >scales with 1/2-octave, 1/3-octave, 1/4-octave, and 1/5-octave periods as well."
> Agreed...it's a tough nut. I'd suggest trying to
> A) Single out scales you think have many things in common and group them
> B) Take one or two examples from each group to explain in detail, giving care to
> cover general chords/moods/instrument-mappings etc. for such scales.
> C) Give good "if interested" references to existing documents that explain the
> rest...or perhaps link to an "in constant development" web page that lists
> detail about increasingly more scales over time and labels them under each
> "group type".

Believe me, I've considered my options. Choosing to stick with MOS scales is an informed choice.

> Right, but then again by having to say "7=limit (and nothing but!)", that
> seems to be overly simplifying it. Would you see any reason it would be a crime
> to say, for example "it gives many good 7-limit intervals (list), plus a handful
> of 11-limit and a couple of 13-limit ones". I agree though...you certainly
> can't say many TET scales are "just built for one type of limit".

That's the thing, though: unless the H.E. of a tempered interval is very low, you can't say for sure what limit it is. I mean, *is* the 240-cent interval of 5-EDO a 7-limit interval? Yeah, you could say that it's an approximation of 8/7, but it's also a 23-limit interval of 23/20, or a 13-limit interval of 15/13. You could argue that 8/7 is simpler, so is a more "valid" interpretation, but I don't think that's a valid argument. In 13-limit JI, 15/13 is NOT 8/7, they're completely distinct intervals. Or take the 300-cent minor 3rd found in many EDOs: is it a 5-limit interval? A 13-limit interval? A 19-limit interval? A 25-limit interval? It's pretty darn close to 19/16 and 25/21, and it's closer to 13/11 than it is to 6/5, so how do we even say what its odd- or prime-limit is?

The point is, if you're looking at "n-limit" intervals because you want to approximate JI, then you will want to have consistency. If you're looking at n-limit intervals because you're using n-limit as a gauge of concordance, you're better off using H.E. anyway. I mean, look at the EDO charts here:

http://www.microtonal-synthesis.com/scales.html

They give one ratio that each interval is supposed to approximate, and they give error in approximation. What sort of useful stuff does this really tell you? To figure out concordance, you have to know how concordant the JI ratios are, and then how much mistuning effects their concordance. H.E. tells you this directly. But then look at it from a JI standpoint: look at their 22-TET page. They give the 327.27-cent interval as a "40/33", but they give the 381.82-cent interval as 5/4 and the 709.09-cent interval as a 3/2. Yet, you can't have a 5/4 and a 3/2 *without* having a 6/5. So either than 40/33 is actually a 6/5, that 3/2 is actually a 200/132, or that 5/4 is actually a 99/80. But since it's NOT JI we're dealing with, ALL OF THE ABOVE is correct. There is an infinite number of ways to interpret any EDO in terms of JI, and even narrowing it down to three or four is hopelessly arbitrary.
> In the first song on "Map of An Internal Landscape" I swear you told me you
> did and also swear it was some sort of alternative fifth. Maybe it was 22/15?

It was 40/27. That song is in 16-EDO, so the fifth is 675 cents, which is actually about 5 cents off, but I think 40/27 is a good-enough approximation of it.

> >"One thing H.E. suggests is that "bad fifths" can be stronger concords than
> >"good thirds", or so it seems anyway (maybe my interpretation is unfounded)."
> Hmm...I'd say yes and no. Anything around 13-20 cents of 3/2 (IE around
> 1.48) seems to really really bite my ears....but things like 22/15 and even 14/9
> often seem passable to the same degree, say, 5/4 is...and in the case of 22/15
> perhaps a bit better.

As Mike said, what you are hearing is the partials clashing. Try it with sine waves or maybe triangle waves.

> It (HE) often sound to me a case of right idea, wrong degrees of exaggeration,
> giving too much credits to exact versions of a tone like 3/2 and not enough
> credit to its alternative versions.
> Try it yourself (on your own ears) and let me know what you think.....

Well, H.E. does have that variable "s" value for a listener's ability of discrimination, so maybe giving a different s-value would give results you'd find more agreeable?

Either way, my ears would suggest that H.E. gives *too much* credit to "alternative 5ths", but that's because I'm using harmonic timbres for everything. With sine or even triangle waves, I find the H.E. curve fits pretty much dead-on.

-Igs

🔗cityoftheasleep <igliashon@...>

🔗Carl Lumma <carl@...>

🔗cityoftheasleep <igliashon@...>

🔗Carl Lumma <carl@...>

🔗genewardsmith <genewardsmith@...>

🔗cityoftheasleep <igliashon@...>

🔗Michael <djtrancendance@...>

>"At any rate, 16/15 plus 15/14 doesn't make 7/6, it makes 8/7. 14/13 and 13/12
>make 7/6,"
Doh, yes you're right 16/14 = 8/7, etc.

>"Here's an experiment for you: try taking two triangle waves at the same pitch,
>and slowly increase the pitch of one without being able to see what pitch it's
>at (a MIDI pitch-bend wheel will probably work for this). Stop increasing the
>pitch when it gets to a point that you like. Repeat this several times, and then
>measure the pitches you stopped at and compute the ratio, then post the
>results!"
Will try....now where can I get a good triangle wave? Or will it kill me to
use a sawtooth wave instead?

>> Right. Actually I think it's an issue I have with micro-tonality: while MOS
>> is all good stuff, I see very few people messing around with the idea of
>> alternating generators to get rid of would-be commatic build-up.
>Huh? What commatic build-up?
Well say if you take 3/2^12 (ALA Pythagorean Diatonic). You get about
129.7...and the nearest octave/"power of 2" is 128. Thus each multiplication of
3/2 builds up to the "commatic" error...which you have to "suddenly" kill at the
very end of the scale to make it match the octave.

Now say you make the scale out of a "fifths" of slightly under 22/15 and one
of about 50/33. Multiply all those and you can get about 128 even. So this way
it forms a "moment of symmetry" at the octave without having to meet the "MOS
criterion" of having a single generator, and you can kill the comma in many
different ways and use those ways to enable mini-max balance of error between
intervals (IE nothing perfectly pure but nothing so sour it can readily be heard
as impure...such as something to the effect of 1/4 comma mean-tone but without
the wolf fifth).

Me>> Agreed...it's a tough nut. I'd suggest trying to
>> A) Single out scales you think have many things in common and group them
>> B) Take one or two examples from each group to explain in detail, giving care
>>to
>>
>> cover general chords/moods/instrument-mappings etc. for such scales.
>> C) Give good "if interested" references to existing documents that explain the
>
>> rest...or perhaps link to an "in constant development" web page that lists
>> detail about increasingly more scales over time and labels them under each
>> "group type".
Igs>Believe me, I've considered my options. Choosing to stick with MOS scales
is an informed choice.

Ugh...I was just saying that I can't see why "having too many scales" would
be an impenetrable/"non-hackable" road block to influence you to base your
tutorial around only MOS scales. But now you seem to just be telling me "this
topic is not up for discussion"...so I kind of wonder why you seemed to be
asking for input....but fair enough; you've decided.

>"And how do alternating generators (i.e. rank-3 temperaments, IIRC) eliminate
>commas any better than rank-1 or rank-2 temperaments?"
If that really is what "alternating generator" means perhaps I have my
definition wrong.
I always thought higher rank meant the temperament can be generated by
taking ONE OF A SET of multiple generators and creating a scale using only that
generator (x^y letting you change x to be a constant value for each generation).
For example, Miracle has a secor, which multiplied by itself gives about
8/7, and 8/7^3 gives a fifth. So any one of these three can be taken to a power
and create and octave...but in NO case do I have to use one and not the other.
Taking about 8/7^5 gives the octave. A little under 3/2^12 gives the octave.
But it all falls along the line of using one of a set of generators, not
switching/alternating between them in the middle of the scale creation process.

Try reaching the octave using either a pure 3/2 fifth OR near-22/15
"alternative fifth" within 12 tones (IE either 3/2^x or about 22/15^x). Can't
do it! Now try alternating between using 3/2 and near 22/15. IE 3/2 * 3/2 *
about 22/15 * 3/2 etc. ...and you can. That's what I'm talking about.

>"Yeah, you could say that it's an approximation of 8/7, but it's also a 23-limit
>interval of 23/20, or a 13-limit interval of 15/13. You could argue that 8/7 is
>simpler, so is a more "valid" interpretation"
I simply argue round it to the simplest interval within 7 cents of the exact
value. Usually, at worst, you can round it to the nearest interval up to x/11
and the two will sound and feel fairly alike.

>"25-limit interval? It's pretty darn close to 19/16 and 25/21, and it's closer
>to 13/11 than it is to 6/5, so how do we even say what its odd- or prime-limit
>is?"

Now that is one HUGE gray area between 13/11 and 6/5. it's about 13 cents
from each of those "nearest points". My impulse would be to say "that interval
is just too far gone...give it up!" But if you had to give it a class...I'd
call it in a "noise" class and forget trying to round it to JI. Personally,
those two are also so close to each other my ear can't tell the difference
between them. Haha...you win on this example...it's one of those cases where I
don't think even fairly extended JI can classify it well at all.

>"It was 40/27. That song is in 16-EDO, so the fifth is 675 cents, which is
>actually about 5 cents off, but I think 40/27 is a good-enough approximation of
>it."
Right, 40/27 is close enough. And it's awesome how it actually works
consider it must be just about my least favorite "5th" other than the
misery-mooded 16/11.

>"Yet, you can't have a 5/4 and a 3/2 *without* having a 6/5."
Sure you can. My scales routinely have neutral stacked thirds that act as
both 3/2 and 6/5 and still count as "just" within 7 cents or so.

>"There is an infinite number of ways to interpret any EDO in terms of JI, and
>even narrowing it down to three or four is hopelessly arbitrary."
>"they give the 381.82-cent interval as 5/4 and the 709.09-cent interval as a
>3/2"
Hmm....the difference between the two is about 328 cents, about 12 cents off
3/2. But there is absolutely NO x/11, x/9...fraction nearby for it to compete
with...the closest one is 14/11 and that's about 18 cents off! Thus I'd call
it a bit of a nowhere interval, but round it to 6/5. My rules of thumb: if the
nearest x/11 does not work and it's no more than 15 cents from the nearest
simple JI interval, round it to the nearest simple fraction (assuming harmonic
entropy takes over and pushes it a bit). If x/11 and another interval of x/11
or lower format is almost equally near by (ALA 13 cents or so away)...the
interval is generally registered as noise and not able to be explained in JI.

>Me> Hmm...I'd say yes and no. Anything around 13-20 cents of 3/2 (IE around
>
>> 1.48) seems to really really bite my ears....but things like 22/15 and even
>>14/9
>>
>> often seem passable to the same degree, say, 5/4 is...and in the case of 22/15
>
>> perhaps a bit better.
Igs>As Mike said, what you are hearing is the partials clashing. Try it with
sine waves or maybe triangle waves.

It sounds much better with sine waves indeed...but for almost any timbre
(even in-harmonic ones) the 22/15 and 14/9 seem to have an obvious advantage.
What it (and Mike B?!) seem to say is that in some cases critical band
dissonance factors can over-ride harmonic entropy factors in terms of their
contribution to overall sense of consonance.

>"Well, H.E. does have that variable "s" value for a listener's ability of
>discrimination, so maybe giving a different s-value would give results you'd
>find more agreeable? "
Right, but that doesn't give me much of a tool to "optimize the
'imaginary/varying/user-generated' curve of s". :-D

>"Either way, my ears would suggest that H.E. gives *too much* credit to
>"alternative 5ths", but that's because I'm using harmonic timbres for
>everything. With sine or even triangle waves, I find the H.E. curve fits pretty
>much dead-on."
My ears agree for sine waves but not triangular ones. When I think of it the
reason it works for sine waves is obvious: a sine wave only recognize the
root-tone...and HE is based largely on root-tone periodicity. Now if you could
take fully fledged instruments with many harmonics (yes, including the option of
weird FM synthesized sounds) and make them follow the HE curve at "near 100%"
accuracy, that would certainly make me think twice if there's a way to make HE
summarize nearly all psycho-acoustics behind music. But as for now I'm still
split something like 60% HE, 40% critical band so far as how much they explain
relative to each other.

🔗Mike Battaglia <battaglia01@...>

On Mon, Sep 6, 2010 at 2:49 PM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> > > > Paul's response was that it wasn't right because his curve
> > > > slopes "down" and mine sloped "up." That is, 7/1 is supposed
> > > > to have a lower entropy than 2/1, for example.
> > >
> > > It is ??
> >
> > Apparently so.
>
> Did you mean higher entropy?

No, I meant lower entropy. My curve, which mirrors the curve you
posted, has 7/1 having higher entropy than 2/1. His curve, which
slopes down and to the right, has 7/1 having lower entropy than 2/1.
He said that this was the "correct" curve and the other one resulted
from an improper generation of the Farey series, if I remember
correctly. I am just as displeased with this as you seem to be. I
think it would result from the Farey series generation algorithm
producing less frequencies in the 7/1 range, than it does in the 2/1
range, hence there are less ratios up there to confuse your brain
with, and hence the equation produces a lower entropy.

This is one of the reason that the whole concept of using a Farey
series for this bums me out so much.

> > > I don't see any evidence of slope in either of the plots I
> > > linked to
> > > /tuning/topicId_91940.html#91961
> > > Then again I'm about to pass out so I might be missing it.
> >
> > He was referring to the slope of the minima. Look at 1/1, 2/1,
> > 3/1, 4/1, 5/1, and note that the line connecting those points
> > has a positive slope. Mine also has a positive slope.
> > Paul said that those points -should- make a line with a
> > negative slope
>
> All the entropy plots I have show entropy increasing as we
> go up the harmonic series like this. So I'm baffled that he
> said it should be going down.

So am I. I'll dig up the picture he sent me.

> Paul reported an overall slope to the entropy curve when
> using mediant-to-mediant widths to compute the probabilities,
> which disappeared when he moved to 1/sqrt(n*d) widths.
> Maybe that's what you're thinking of.

Ah. He sent me a Sethares paper which described the algorithm as using
mediant-to-mediant widths. So that was supplanted later on by the
1/sqrt(n*d) approach? Where did the notion of using 1/sqrt(n*d) come
from, specifically - is that proportional to the mediant-to-mediant
width that each ratio takes as the number of Farey terms approaches
infinity?

-Mike

🔗Mike Battaglia <battaglia01@...>

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

On Mon, Sep 6, 2010 at 5:50 PM, Michael <djtrancendance@...> wrote:
>
> >"At any rate, 16/15 plus 15/14 doesn't make 7/6, it makes 8/7. 14/13 and 13/12 make 7/6,"
> Doh, yes you're right 16/14 = 8/7, etc.
>
> >"Here's an experiment for you: try taking two triangle waves at the same pitch, and slowly increase the pitch of one without being able to see what pitch it's at (a MIDI pitch-bend wheel will probably work for this). Stop increasing the pitch when it gets to a point that you like. Repeat this several times, and then measure the pitches you stopped at and compute the ratio, then post the results!"
> Will try....now where can I get a good triangle wave? Or will it kill me to use a sawtooth wave instead?

Run a square wave through a lowpass filter to get sort of a triangle
wave, if you can't get anything else. If you really want to be
precise, make the filter cutoff point be relative to the key you're
playing, so that it actually changes the timbre of the note instead of
imposing a strict rolloff across the entire spectrum. Sawtooth will
probably defeat the purpose of this experiment.

> Well say if you take 3/2^12 (ALA Pythagorean Diatonic). You get about 129.7...and the nearest octave/"power of 2" is 128. Thus each multiplication of 3/2 builds up to the "commatic" error...which you have to "suddenly" kill at the very end of the scale to make it match the octave.
> Now say you make the scale out of a "fifths" of slightly under 22/15 and one of about 50/33. Multiply all those and you can get about 128 even. So this way it forms a "moment of symmetry" at the octave without having to meet the "MOS criterion" of having a single generator, and you can kill the comma in many different ways and use those ways to enable mini-max balance of error between intervals (IE nothing perfectly pure but nothing so sour it can readily be heard as impure...such as something to the effect of 1/4 comma mean-tone but without the wolf fifth).

That would still be an MOS with a single generator. The generator
would be slightly under 22/15, and the period would be an octave. The
50/33 would end up being equal to an octave minus 11 generators or
something like that.

> >"And how do alternating generators (i.e. rank-3 temperaments, IIRC) eliminate commas any better than rank-1 or rank-2 temperaments?"
> If that really is what "alternating generator" means perhaps I have my definition wrong.
> I always thought higher rank meant the temperament can be generated by taking ONE OF A SET of multiple generators and creating a scale using only that generator (x^y letting you change x to be a constant value for each generation).

> For example, Miracle has a secor, which multiplied by itself gives about 8/7, and 8/7^3 gives a > Igs>As Mike said, what you are hearing is the partials clashing. Try it with sine waves or maybe triangle waves.
> It sounds much better with sine waves indeed...but for almost any timbre (even in-harmonic ones) the 22/15 and 14/9 seem to have an obvious advantage. What it (and Mike B?!) seem to say is that in some cases critical band dissonance factors can over-ride harmonic entropy factors in terms of their contribution to overall sense of consonance.

And I think that their fit to some predetermined mode of listening, or
internal "map," is more important than both of them. Hence for someone
used to diatonic hearing, and used to linked 5-limit chords - a major
7th is is high up on both the entropy and the critical band dissonance
charts, but sounds great if it reminds you of the pleasant sound of a
major 7 chord. And you can get used to the 14-cent sharp major thirds
of 12-tet too. And you can also get used to the ~675-cent fifths of
mavila if you want as well.

-Mike

🔗Mike Battaglia <battaglia01@...>

On Mon, Sep 6, 2010 at 7:49 PM, Carl Lumma <carl@...> wrote:
>
> Mike wrote:
>
> > > Did you mean higher entropy?
> >
> > No, I meant lower entropy. My curve, which mirrors the curve
> > you posted, has 7/1 having higher entropy than 2/1. His curve,
> > which slopes down and to the right, has 7/1 having lower entropy
> > than 2/1.
>
> Which curve is that? Like I said, all the curves I have from
> him (posted here, to the harmonic_entropy list, etc) show an
> upward-right slope.

I can't find the conversation now, just this quote:

"Also, I looked at your curves. How exactly are you
defining/constructing your Farey series and mapping them into interval
space? If you do it like I do, I think your curves should have an
overall downward slope, not upward. Are you sure you're using an
appropriate metric on the x-axis (a log-ratio measure such as cents)?"

The answer to the last question is "yes" (not sure why he thought my
curves were in linear frequency space). I think we followed up in a
facebook chat or something, which is when he said that the tuning list
curves were the older one. But now I'm confused, because I just found
this picture from the tuning list in Paul's files:

http://f1.grp.yahoofs.com/v1/gICFTJ3KbWCmRKLC-Kjz8imNf8kstKUY7miJSQ6CpgAVCFl5bmzS05kDHOsCd8e_MHt-iJrwA-bX_CY2Wmo3EHZA-1nmDH0/PaulErlich/ent_015.jpg
http://f1.grp.yahoofs.com/v1/gICFTLC9oeymRKLCtugQwzP_KgUcECe9rohQzrZZMkjTd-ejmEVcZnere6VGOsMLq8m77RS3B-4ShTlP9uoBbOX-aZ3O32k/PaulErlich/harment.gif

Note how the curve slopes down. But that's from 2000, and the 2004
ones produce an upward slope (or an overall neutral slope, with the
minima going upward and to the right). So perhaps that's when he
switched away from the Farey series to the Mann/Tenney series
approach. I don't know enough about the difference to really comment -
he didn't mention any of it to me.

I also just noticed this, where the Farey series slope down and the
Mann series slope up:
http://sonic-arts.org/td/erlich/entropy-erlich.htm

-Mike

🔗Carl Lumma <carl@...>

🔗Mike Battaglia <battaglia01@...>

🔗Graham Breed <gbreed@...>

On 6 September 2010 01:19, cityoftheasleep <igliashon@...> wrote:

You can't get away from consistency. Consistency is what tells you
that you can use certain intervals together in a chord. Whether you
define those intervals as approximating ratios or not, you need to say
how they add up.

Avoiding JI comparisons is a bit bizzarre. Harmonic entropy is one
way of scoring closeness to JI. Every equal temperament will have
some approximations to tunable-by-ear ratios. If you think you can
make the explanations simpler by ignoring this, go for it. I've tried
to avoid numbers in the past but it really isn't easy when it comes to
harmony.

What you don't need to do is score each equal temperament by a
consecutive prime limit.

Graham

🔗Mike Battaglia <battaglia01@...>

On Mon, Sep 6, 2010 at 10:06 PM, Graham Breed <gbreed@...> wrote:
>
> On 7 September 2010 07:30, Mike Battaglia <battaglia01@...> wrote:
>
> > Yeah, I think it's that propriety requires L:s to be 2:1, and strict
> > propriety requires it to be less than 2:1, and impropriety requires it
> > to be more than 2:1. Another way of putting this is that given a 5+2
> > scale, if it's strictly proper, the next MOS produced will be 7+5. If
> > it's improper, the next MOS produced will be 5+7. If it's proper but
> > not strictly proper, the next MOS produced will be 12-tet (if you want
> > to call it an MOS).
>
> There might be corner cases where you have a L L ... L s scale. Let's
> try 2 2 1.
>
> 2 2 1
> 4 3 3
> 5 5 5
>
> That's strictly proper, but it has a 2:1 ratio. I worked out some rules here:
>
> http://x31eq.com/proof.html

This is immensely useful, thanks for this. I was also hoping you'd
weigh in on the last conversation that Igs and I had, the one about
relating different MOS's to different temperaments as well, especially
since your name keeps popping up with regard to it :)

If you weren't following, the idea is - although there's no real limit
to how detuned a generator for linear temperament can get, I was
throwing around the idea of defining the boundaries by when the MOS's
produced flip around. This could perhaps, I figure, be used to fit
temperaments into related groups as well.

So for example, anything between the fifth of 7-tet and 5-tet makes
sense to be viewed as a type of meantone. Right when the fifth gets
flatter than 7-tet is when it starts to sound like mavila instead. But
when we say "sounds like meantone" and "sounds like mavila," that
really correlates to "produces 5+2" and "produces 2+5" scales,
respectively. Meantone[5] and Mavila[5] can be made to sound pretty
similar, so they're related in that they both produce 2+3 scales.

Either way, from a 5+2 standpoint, even 22-tet style superpyth
diatonic scales can sound like meantone - both perceptually, and by
this definition. So the temperaments whose optimal generators produce
5+2 scales could perhaps be said to belong the "diatonic" temperament
group, although Margo has proposed the name "isotonic" for all of
those temperaments.

The 5+2 MOS can be further subdivided into 7+5, which is even closer
to what we call "meantone." Anything producing 5+7 scales generally
fits into more of a superpyth type category (although around 12-equal
the two converge, and could perhaps be said to approximate dominant
temperament). So by this definition, anything with a fifth sharper
than that of 7-tet, and up to that of 12-tet, would be in the
"meantone" realm.

And within the "meantone" realm you can subdivide further by what
produces 12+7 scales and 7+12 scales. Margo has proposed the name
"inframeantone" for the latter, which deals with anything with a fifth
less than 19-tet. And 19-tet functions as a very clear perceptual
boundary for me as far as meantones go.

Anyway, we were having a good old time trying to figure out how to do
this most rationally, and organize the space, and your name kept
coming up. Have you worked this out before? I know you've done a lot
of work on this sort of thing.

> p.s. I insist that the Pythagorean diatonic is proper, for the average
> listener, if you follow Rothenberg's cognitive definition of the
> ordering.

I would say that the 22-tet superpyth diatonic is also proper, by the
same definition. At least, I generally perceive the aug4 and the dim5
to be "roughly equivalent tritones." Lyd #2 doesn't fare quite as
well, since C-D# is 5/4.

-Mike

🔗Carl Lumma <carl@...>

🔗Graham Breed <gbreed@...>

On 7 September 2010 10:32, Mike Battaglia <battaglia01@...> wrote:

> This is immensely useful, thanks for this. I was also hoping you'd
> weigh in on the last conversation that Igs and I had, the one about
> relating different MOS's to different temperaments as well, especially
> since your name keeps popping up with regard to it :)

I don't think I did have anything to add. There aren't clear
boundaries and I said that.

> If you weren't following, the idea is - although there's no real limit
> to how detuned a generator for linear temperament can get, I was
> throwing around the idea of defining the boundaries by when the MOS's
> produced flip around. This could perhaps, I figure, be used to fit
> temperaments into related groups as well.

Yes, you can do that, and there are problems that we already dealt with.

> So for example, anything between the fifth of 7-tet and 5-tet makes
> sense to be viewed as a type of meantone. Right when the fifth gets
> flatter than 7-tet is when it starts to sound like mavila instead. But
> when we say "sounds like meantone" and "sounds like mavila," that
> really correlates to "produces 5+2" and "produces 2+5" scales,
> respectively. Meantone[5] and Mavila[5] can be made to sound pretty
> similar, so they're related in that they both produce 2+3 scales.

Yes, that's a clear boundary. But it assumes you knew you wanted to
look at those two temperament classes. You can also call 12 the
boundary at the other end, because it divides meantone from
schismatic. But it gets more complicated in higher limits and when
you look at more marginal temperaments.

I encourage you to write out the scale tree, on multiple sheets of
paper, and try and fit temperament classes to it. It works reasonably
well and you will learn something from it.

Graham

🔗Michael <djtrancendance@...>

MikeB>"That would still be an MOS with a single generator. The generator would
be slightly under 22/15, and the period would be an octave. The 50/33 would end
up being equal to an octave minus 11 generators or something like that."
In that case...wouldn't it still be counted as needing the 50/33 instead of
the 22/15 for that final "gap" to the octave?

>"Hence for someone used to diatonic hearing, and used to linked 5-limit chords -
>a major 7th is is high up on both the entropy and the critical band dissonance
>charts"
It isn't in the dyadic chart (AKA Plomp and Llevelt's curve) as a single dyad
relative to nothing else: the root tones are obviously very well separated. But
agreed...it is high on the HE chart...not to mention being highly odd-limit.

>"but sounds great if it reminds you of the pleasant sound of a major 7 chord. "
Hmm...I asked that same question to Kraig Grady a month or so ago: why it
sounded so good even with full instruments and not just sine waves.
And his answer (as I recall) was that 15/8 has the property of being straight
along the harmonic series with 5/4 and 3/2 (all x/8). So in a way that sounds
like a twist on what you are saying. I wonder if that also at least partly
solves the puzzling founding my ears seem to have that 15/8 sounds more relaxed
than the much lower limit 9/5.

If you look at the partials between a 9/5 and 15/8 dyad you'll see for
A) Between the overtones of 15/8 and 1/1 the 7th chord dyads of both 5/4 and
15/8 are formed from the third overtone of the root:
1
15/8 (root)
2
3 (8/8)
15/4 (15/4 over 3 = 15/12 = !!!!5/4!!!! FIRST OVERTONE)
4
45/8 (45/8 over 3 = 45/24 =!!!!!15/8!!!!.......forms an 8:10:15 chord
with 3 SECOND OVERTONE)

And for 9/5 (has "only" a 10:11:18 chord or 9/5 dyad linking the first and
second tones)
9/5
2
3
18/5 (18/5 over 3 = 11/10)
4
5
27/5 (27/5 over 3 = !!!!!!!!9/5!!!!!!....linked to 11/10 as 18/10
forming 10:11:18)

My guess is the brain sees the 5/4 and 15/8 in the overtones of the 15/8
dyad and thinks "aha...this must be an x/8 harmonic series"...whereas it has
more trouble doing so for 9/5 given the (10 'missing')11:18 with the 10 linking
to 5 as the root...instead of the 8:10:15 chord formed within the 15/8 overtones
which directly implies a tonal base of 8).

Has anyone else ran into this "original root tone also implied within the
mesh of overtones" phenomenon and questioned it?

🔗Mike Battaglia <battaglia01@...>

On Mon, Sep 6, 2010 at 11:06 PM, Graham Breed <gbreed@...> wrote:
> > If you weren't following, the idea is - although there's no real limit
> > to how detuned a generator for linear temperament can get, I was
> > throwing around the idea of defining the boundaries by when the MOS's
> > produced flip around. This could perhaps, I figure, be used to fit
> > temperaments into related groups as well.
>
> Yes, you can do that, and there are problems that we already dealt with.

What problems were there besides just that temperament error is
unbounded, and so any linear temperament of sufficient error could
produce any MOS?

> > So for example, anything between the fifth of 7-tet and 5-tet makes
> > sense to be viewed as a type of meantone. Right when the fifth gets
> > flatter than 7-tet is when it starts to sound like mavila instead. But
> > when we say "sounds like meantone" and "sounds like mavila," that
> > really correlates to "produces 5+2" and "produces 2+5" scales,
> > respectively. Meantone[5] and Mavila[5] can be made to sound pretty
> > similar, so they're related in that they both produce 2+3 scales.
>
> Yes, that's a clear boundary. But it assumes you knew you wanted to
> look at those two temperament classes. You can also call 12 the
> boundary at the other end, because it divides meantone from
> schismatic. But it gets more complicated in higher limits and when
> you look at more marginal temperaments.

Right, my idea was more to find all of the temperaments that fit a
certain MOS, and keep subdividing into further MOS's and finding out
when the temperaments diverge. That is, mavila, meantone, superpyth,
and schismatic all match 2+3, but mavila diverges for 2+5. Meantone,
superpyth, and schismatic all match 5+2, but meantone diverges for
7+5. Superpyth and schismatic all match 5+7, but probably diverge for
something down the line. The reasoning goes that you'd end up then
with different "shades" of meantone and I hypothesize the shades will
switch when some MOS down the line switches (and the further down the
line, the less perceptually relevant).

Really what I was hoping to do was come up with an algorithm that
automatically fits n-limit rank 1 temperaments to certain MOS
families, so that it can be seen that meantone is, on the whole,
optimum for 5+2 scales, with superpyth in second and schismatic in
third (or something like that). Complexity could be taken into account
as well, so perhaps superpyth would rank higher than schismatic, or
something like that. Do you know if something like that has ever been
sorted out?

> I encourage you to write out the scale tree, on multiple sheets of
> paper, and try and fit temperament classes to it. It works reasonably
> well and you will learn something from it.

I will have to do that.

-Mike

🔗Mike Battaglia <battaglia01@...>

On Mon, Sep 6, 2010 at 11:15 PM, Michael <djtrancendance@...> wrote:
>
> MikeB>"That would still be an MOS with a single generator. The generator would be slightly under 22/15, and the period would be an octave. The 50/33 would end up being equal to an octave minus 11 generators or something like that."
> In that case...wouldn't it still be counted as needing the 50/33 instead of the 22/15 for that final "gap" to the octave?

Yes. If it didn't, it wouldn't be rank 2, but rank 1. If it ended up
closing at the octave, you'd have an equal temperament. Since it
doesn't, you basically have a generator of ~50/33, and a period of an
octave. The ~22/15 can be represented as a combination of the two of
those.

> >"but sounds great if it reminds you of the pleasant sound of a major 7 chord. "
> Hmm...I asked that same question to Kraig Grady a month or so ago: why it sounded so good even with full instruments and not just sine waves.
> And his answer (as I recall) was that 15/8 has the property of being straight along the harmonic series with 5/4 and 3/2 (all x/8). So in a way that sounds like a twist on what you are saying. I wonder if that also at least partly solves the puzzling founding my ears seem to have that 15/8 sounds more relaxed than the much lower limit 9/5.

I'm not entirely sure that's what it is though. I like 8:10:12:15 more
than 8:10:12:13, for example. Even if you voice these two chords in
inversion such that there's no roughness (since I know you hate that
:)) I still prefer 8:10:12:15 to 8:10:12:13.

I think it has to do with the musical meaning that emerges when you
play a major 7 chord, which is that it's really happy, even more so
than a regular major chord. And at this point I think that musical
meaning emerges from the mental map you have of what's going on,
rather than the ideal JI intonation of the chord. Note that the C maj
will sound different in Bbm/Db->Cmaj than Gmaj7->Cmaj - the first one
is much darker and sounds like it's going to resolve to Fm, the second
one sounds resolved and happy. It all has to do, I think, with the
"background notes" in your mind that are produced from what you
perceive as the current tonal set and where the current chord fits
into that. This, in turn, has to do with your mental map of what's
going on (are you thinking diatonically, or in porcupine[7], or what?)

-Mike

🔗Michael <djtrancendance@...>

>"I'm not entirely sure that's what it is though. I like 8:10:12:15 more
than 8:10:12:13, for example. Even if you voice these two chords in
inversion such that there's no roughness (since I know you hate that
:)) I still prefer 8:10:12:15 to 8:10:12:13."
You know me too well...yep, making the 8:10:12:13 to 13:16:20:24 definitely
kills that nasty root-tone beating in 13/12. :-D Then again with 13:16:20:24
you get all those sour x/13 dyads...which seems to point to a suspicion I have
in general that too many sour dyads within a chord can make a sour chord. My
guess is that may well have something to do with it...giving up and saying it's
a cultural thing we think 7ths are "happy" is just way too easy an excuse and
far too hard to use for optimizations IMVHO. :-D

>"Note that the C maj will sound different in Bbm/Db->Cmaj than Gmaj7->Cmaj - the
>first one
is much darker and sounds like it's going to resolve to Fm, the second one
sounds resolved and happy."

I can definitely buy that....otherwise why would music theory even care
about things like relative minor keys that contain the same notes? Then again,
that seems to obviously apply to consecutive chords rather than lone-standing
ones.

Assuming Bbm = A# C# F....I see that C major does not seem to fit in well
harmonically IE C E G has almost nothing in common with A# C# F.

Meanwhile your suggestion of F A# C as a point of resolution has two notes
in common...and you can even form an F A# C F chord by stacking the two (minus
that odd c#). Plus try to find a note that sounds relaxed under A# C F and F
simply seems to work best. So the transition comes off to me as going from one
part of an F A# C F chord to another for the most part.

Meanwhile Gmaj7 and Cmaj have 5ths between all of their first three notes
linking them. (IE between CEG and GBD is a 5th).

Now what puzzles me...is that the A# C# F seems to resolve better to F A# C
than to F A# C#! And the only thing I can think of off the top of my head is
"well....they are both minor chords, and perhaps that minor-ness can take
precedence over how many notes each chord has that work well with notes in the
other chord?". Any other ideas?

Come to think of it...I'm wondering if it's fair to say the brain in general
likes to resolve from a certain pattern of intervals (IE major chord) to the
same pattern (IE another major chord)...especially considering your Gmaj7 ->
Cmaj example has that same restriction (either maj->maj or min -> min).

>" It all has to do, I think, with the "background notes" in your mind that are
>produced from what you
perceive as the current tonal set and where the current chord fits into that. "
I think we're on the same page...it's almost as if the mind tries to mold the
two into one chord, to an extent.

🔗Mike Battaglia <battaglia01@...>

On Tue, Sep 7, 2010 at 12:30 AM, Michael <djtrancendance@...> wrote:
>
> I can definitely buy that....otherwise why would music theory even care about things like relative minor keys that contain the same notes? Then again, that seems to obviously apply to consecutive chords rather than lone-standing ones.
> Assuming Bbm = A# C# F....I see that C major does not seem to fit in well harmonically IE C E G has almost nothing in common with A# C# F.

C E G has F harmonic minor in common with Bb Db F. Bbm is the iv
chord, and Cmaj is the V chord. iv6 (meaning in first inversion) -> V
is what they call a phrygian half cadence. it sounds like it's going
to go to Fm. Voice it like this:

Db-Bb-F-Bb
C-C-E-G
F-C-F-Ab

> Meanwhile your suggestion of F A# C as a point of resolution has two notes in common...

?? I didn't suggest that... It would be F-Bb-C, btw. A Fsus4 chord. Do
you mean F Ab C for F minor?

Assuming you mean Fm, all 3 of those chords fit into F harmonic minor,
so they have lots of notes in common :) They have, in fact, F harmonic
minor as a full "mental reference frame" in common.

> Now what puzzles me...is that the A# C# F seems to resolve better to F A# C than to F A# C#! And the only thing I can think of off the top of my head is "well....they are both minor chords, and perhaps that minor-ness can take precedence over how many notes each chord has that work well with notes in the other chord?". Any other ideas?

I'm going to assume you mean F Ab C again. I think the logical
culmination of the concept of chords "sharing notes that work with one
another" is the concept of the scale. Paul threw out as a hypothesis
that whatever causes musical meaning to arise has something to do with
a low-level cognitive procedure taking place right on your mental map
of what's going on, which has something to do with a scale.

But if I could figure out why iv in minor resolves so nicely to i, I'd
be a very happy man. I think it has something to do with the structure
of the F minor scale. This is my current obsession to figure out - how
scales work and how mental maps work. I've heard a few answers for why
things "resolve" nicely and I still haven't found anything that
satisfies me 100% yet. They're usually put in tandem with why only
Aeolian and Ionian are usable as tonal systems that emphasize a
certain note as the tonic. Coming from my musical background, I don't
think that only Aeolian and Ionian ARE usable as tonal systems, so
these theories usually leave me wanting. I think that Rothenberg has
some of it figured out with his stuff, and I think that Paul has a lot
figured out as well in his 22-tone paper. Someone also put out the
idea a while ago that it was because only Aeolian and Ionian have
dissonant intervals (the tritone) at an interval adjacent to the root
(I think Carl, citing some other author). Perhaps all of these have
something to do with it, but I still personally haven't seen "the big
picture" yet.

I don't think the minorness has much to do with it, because note how
well Bbm resolves to Fmaj as well. Try Dbaug -> Bbm -> F7. Voice it
like this:

Db-C-F-A
Bb-Db-F-Bb
F-Eb-F-C

It sounds like some kind of mystical cavern or something. Explain that
one to me :) You can make it Fmaj too instead of F7 and the feeling is
pretty much the same.

Another neat technique is to use modes as "chords" and the chromatic
scale as the "scale." That is, chords are mini-reference frames, and
the diatonic scale is the full reference frame in common practice
music. A beautiful modern harmony trick is to use different diatonic
scales as mini-reference frames, and the chromatic scale as the full
frame. So try something like this chord progression:

||: Gm11 -> Dbmaj9 -> Dm7 -> A13 :||

Voice it like this

G-F-G-A-Bb-C
Db-Eb-F-Ab-Bb-C
D-E-F-G-A-C
A-F#-G-A-B-C#

Or pick your own roughness free voicings if you want, but try to do
some nice voice leading if you can :)

You can keep the stream of consciousness type thing going for as long
as you want, and it's possible to get very fluid with this approach,
just as fluid as you can with common practice harmony. Every chord
changes the "background scale" in your mind, and every change has a
certain function that occurs, but you can make it sound as tonal or
non-tonal as you want. You can, if you want, make it sound like a
12-tet version of the eikosany, where every chord is harmonic, but
there's no root, and where the basic tonal set is C-E-G-B-D instead of
4:5:6:7:9:11.

||: Gm9 -> Dbmaj9 -> Dm9 -> A13 -> Ebm7 -> Fm9 -> Cmaj9 -> F#m9 ->
Ebm7 -> Abmaj7/Bb -> Emaj/F# -> Ebm7 :||

I'll upload a listening example in a bit to get an idea of what I'm
talking about :) There are lots of things that resolve in different
ways besides what is usually expressed in common practice harmony. If
I could generalize the above concept to something like porcupine
temperament, I'd be set :)

> Come to think of it...I'm wondering if it's fair to say the brain in general likes to resolve from a certain pattern of intervals (IE major chord) to the same pattern (IE another major chord)...especially considering your Gmaj7 -> Cmaj example has that same restriction (either maj->maj or min -> min).

For a less extreme example, try Dm/F -> Dm -> Amaj.

> >" It all has to do, I think, with the "background notes" in your mind that are produced from what you
> perceive as the current tonal set and where the current chord fits into that. "
> I think we're on the same page...it's almost as if the mind tries to mold the two into one chord, to an extent.

You should read Rothenberg.

-Mike

🔗Graham Breed <gbreed@...>

On 7 September 2010 11:37, Mike Battaglia <battaglia01@...> wrote:

> What problems were there besides just that temperament error is
> unbounded, and so any linear temperament of sufficient error could
> produce any MOS?

Sometimes they overlap. Like the diaschismic territory would look
something like

Pajara |12| 58&46 |34| 56&22 |22| Pajara

Pajara would be 22&12. In the middle of the range, there are two
alternative mappings for the 7 that have reasonable complexity. But
if Pajara is valid at either end, it must be in the middle as well.
And the 56&22 range includes the fashionable tunings of Pajara.

> Really what I was hoping to do was come up with an algorithm that
> automatically fits n-limit rank 1 temperaments to certain MOS
> families, so that it can be seen that meantone is, on the whole,
> optimum for 5+2 scales, with superpyth in second and schismatic in
> third (or something like that). Complexity could be taken into account
> as well, so perhaps superpyth would rank higher than schismatic, or
> something like that. Do you know if something like that has ever been
> sorted out?

You can find MOS families for an equal temperament by choosing
different sizes of the generator. When the number of steps to the
generator and period share a common factor, divide through by it.

What you're trying to do, though, is segment MOS space into different
temperament classes. For that, you need to start with a list of
temperament classes. You can find the intersection of any pair using
linear algebra. You can also find nodes on the scale tree and get
rank 2 temperaments by pairing them off. Any node that works with 2
different rank 2 classes will be the boundary between them. Remember
to look at all mappings when they're inconsistent. You'll need some
rule to decide which class dominates.

Graham

🔗Michael <djtrancendance@...>

🔗Mike Battaglia <battaglia01@...>

On Tue, Sep 7, 2010 at 3:10 AM, Michael <djtrancendance@...> wrote:
>
> I'm going to have to response to this at length later...wow I'm tired. :-D
>
> MikeB>"||: Gm11 -> Dbmaj9 -> Dm7 -> A13 :||
>
> Voice it like this
> G-F-G-A-Bb-C
> Db-Eb-F-Ab-Bb-C
> D-E-F-G-A-C
> A-F#-G-A-B-C#"
> Call my ears picky...but this doesn't quite gel for my ears. It feels to me like sudden transpositions.

Haha, well I like it :) I ended up recording a minute long improv on
the concept, check it out:
http://www.mikebattagliamusic.com/music/ModalResolutions.mp3

It's late, so I ended up meandering a bit at the end, but you get the
idea. What I always wonder is, if I was in 19-tet, would there be even
more colors to play around with? And even more so in 31-tet?

> >"it sounds like it's going
> to go to Fm. Voice it like this:
> Db-Bb-F-Bb
> C-C-E-G
> F-C-F-Ab"
> Now that sounds beautiful. A bit tense, but very cohesive. The F-minor resolve works beautifully. But my mind still hears the first and the third as the "focused" chords, and the C major as a sort of throw in intermediary. And, not too surprisingly, I can play those two chords at the same time and have them mix fairly well. I agree with you though now...I don't think minor chords transferring well to other minor chords explains it.

How about Dm/F -> Cmaj7 -> Dm7? Voice it like this

F-A-D-A
C-B-E-B
D-C-F-C

That's all from D Dorian, and you can play the whole scale at once
without it clashing too bad. Try in a wide-fifth tuning for bonus
points.

> Only you can take, say, C E G and D A B and they are both in the same scale...but squeal a bit when played at the same time (at least to my ears).
>
> >"You should read Rothenberg."
> What are some of the best links, in your opinion, to his ideas on compositional theory relative to scale/tuning theory?

I'm not sure what he's done regarding compositional theory, but his
stuff on pattern perception (which is in Carl's tctmo) I found rather
enlightening, although I don't think it explains entirely everything.

-Mike