How to Describe Tempered Intervals in Terms of JI

Okay, folks. I need an education here. I've hit a snag in writing my "primer" on alternative EDOs, in the section where I describe the harmonic properties of each EDO. I'd been using the "standard" approach of describing intervals as approximations to JI, but I've realized that I don't quite understand if that's valid to do for some intervals...or rather, if it's valid to use just one JI interval as a descriptor, and if so, how to determine which one.

I'm going crazy because I don't know if odd-limit or prime-limit is a better indicator of how an interval can be "perceived", and also I don't know how much error is acceptable. Really, I guess what I want to know is 1) how do you really judge the "complexity" of a JI ratio, and 2) if an interval is close to a high-complexity ratio but still within 10 to 16 cents of a low-complexity ratio, which one is the better descriptor for that interval?

The best example I've got is the 466.667-cent diminished fourth in 18-EDO. It's close to both 21/16 and 17/13 (in fact, those two intervals are only about 6 cents apart); 21/16 is lower in prime-limit but higher in odd-limit. The tempered interval is about 2 cents closer to 17/13, but 21/16 would be more useful for describing how the interval works in the temperament, since 18-EDO has a very good 9/8 and 7/6 and thus can make a slightly-tempered 16:18:21 triad. Of course, that same triad could also be described as a heavily-tempered 13:15:17 triad, which is higher in prime-limit but closer to the fundamental in the harmonic series.

Another example I have is the 218.182-cent major second common in EDOs divisible by 11. It's midway between 8/7 and 9/8, so it's right on top of 17/15. 17/15 seems to be a very complex interval, and this tempered interval is certainly not within 6 or 7 cents of the near-by simple ratios; yet it's ever-so-slightly closer to 8/7 than 9/8, and in 22-EDO it functions as BOTH a 9/8 and 8/7, depending on the chord. I could see good reasons for describing it as all three ratios, but I've no idea how to narrow it down to one.

Lastly, I'm plagued by trying to figure out 12-tET. In theory, it's supposed to approximate 5-limit ratios, but its thirds are SOOOO close to both 3-prime-limit and 19-odd-limit intervals, it's got me wondering what we're really hearing when we hear 12-tET.

Tell me, folks, is there a way to find the most perceptually-valid JI-related description of tempered intervals like these, or should I just give the two or three possible ratios and not worry about the absolute cents-value deviation? Do I need to understand harmonic entropy to figure this out?

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Okay, folks. I need an education here.
//
> I'm going crazy because I don't know if odd-limit or prime-limit
> is a better indicator of how an interval can be "perceived",

Didn't I just explain this? Odd limit. Or do you not read
my replies?

> Really, I guess what I want to know is 1) how do you really
> judge the "complexity" of a JI ratio,

Square root of product of numerator and denominator when ratio
is in lowest terms. IF said product is less than 70.

> and 2) if an interval is close to a high-complexity ratio but
> still within 10 to 16 cents of a low-complexity ratio, which
> one is the better descriptor for that interval?

JI dyads beyond the 9-limit are generally not perceivable
as JI dyads. They require triads at least. For triads,
the product rule can also be used, but with cuberoot in place
of square root.

> The best example I've got is the 466.667-cent diminished
> fourth in 18-EDO. It's close to both 21/16 and 17/13

Such intervals are not tunable by ear and are not, therefore,
"just". They work only in triads and larger chords. 467 cents
is between the 9-limit intervals 9/7 and 4/3. It is one cent
closer to 4/3, but because, as Partch discovered, the size of
an interval's field of attraction is proportional to its
complexity, even 460 cents is within 4/3's field and therefore
will tend to sound like a wolf 4/3.

Harmonic entropy reproduces the product rule:

/tuning/files/PaulErlich/stearns.jpg

and also incorporates Partch's observation automatically.
Here's a typical harmonic entropy curve for dyads < 1 octave
with minima and maxima labeled:

http://lumma.org/temp/entropy-labels.gif

It says that anything between 448 and 554 cents will tend to
sound like a 4/3, when heard in isolation.

>in 22-EDO it functions as BOTH a 9/8 and 8/7, depending on the
>chord. I could see good reasons for describing it as all three
>ratios, but I've no idea how to narrow it down to one.

Why does it have to be narrowed down to one?

-Carl

Carl>"Didn't I just explain this? Odd limit. Or do you not read
my replies?"
Or my replies which ALSO pointed to odd-limit as the more accurate of the
two.

>"Square root of product of numerator and denominator when ratio
is in lowest terms. IF said product is less than 70."
Isn't that simply the a form of Tenney Height? One thing I hate about that
though...the product must be less than 70 for it to worl...that should mean 9/8
is out as 9*8 = 72...not to mention just about anything 11-limit (even 11/7 is
too high).

>"JI dyads beyond the 9-limit are generally not perceivable as JI dyads."
Ok...I can believe it's an established rule, but don't agree with it. So even
something like 11/6 or 18/11 or 10/9 is not perceivable as a
dyad?!.....ouch!...that's a bit anal-retentive IMVHO. To me (by ear) anything
up to say with a product of up to about 200 (or within 7 cents of such a ratio)
should be OK.

>"For triads, the product rule can also be used, but with cuberoot in place
of square root."
Interesting and makes sense...if you have two good dyads in a triad you have
more slack for the third dyad to be "less JI". So many songs by Igs and other
seems to point toward this as well.

Igs> "The best example I've got is the 466.667-cent diminished
fourth in 18-EDO. It's close to both 21/16 and 17/13"
Carl>"Such intervals are not tunable by ear and are not, therefore,
"just". They work only in triads and larger chords."
Hmm...21/16 and 17/13 1.3125 and 1.30769. Yeah that's within about 7 cents
of each other...thus I'd each could substitute in for the other. For the sake
of speaking "in JI", though, I'd say 17/13 is the more obvious of the two to
round to. Certainly I'd say neither are close enough to 13/10 or 4/3 to round
to those...

Far as Even 1.681 from 12TET is closest to 27/16 and nowhere near 5/3 (not
even with 13 cents of it).
Far as "Such intervals are not tunable by ear"...wow what a pessimistic
assumption. The 6th in 12TET in about 1.681, over 13 cents away from 5/3 and
has it's only ratio within 7 cents as being 27/16 (product of 432!) and yet
people tune it by ear all the time. Not to say 432 is a good product, but I
don't buy for a second that having a high product alone means not tunable by
ear.

>"It says that anything between 448 and 554 cents will tend to
sound like a 4/3, when heard in isolation."
I can believe that, to an extent. 4/3 does have an uncommonly large area
between the 15/11 above it and 9/7 below it (the two nearest intervals near it I
believe sound substantially different)...but an entire semitone difference?!
Looks to me like whoever wrote that must have been smoking a bit too much
12TET...

>"It is one cent closer to 4/3, but because, as Partch discovered, the size of
an interval's field of attraction is proportional to its
complexity"
Makes sense in many cases. Definitely with 3/2, 4/3, and 5/3. Now for 7/4
and above, I'm not so sure: 16/9 (which is very nearby) sounds very much like
it's own in my book and not a "weaker 7/4" by any means. So I'd say...in
general but not always.

Carl>"Why does it (9/8 vs. 8/7) have to be narrowed down to one?
Oddly, that statement seems to go against the grain of much of what you said
before. For one thing, 9/8 is over 70 as a product (as 72). Though I
agree...they are both individual ratios and far enough away from each other to
stand out (certainly, at well over 15 cents apart they are unlikely to be stuck
in each other's "field of attraction".

Hi Carl,
Sorry, I've tried to reply to this twice today but it doesn't seem to be going through. Hope this one fares better! Apologies in advance if all three attempts go through at once.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
> >
> > Okay, folks. I need an education here.
> //
> > I'm going crazy because I don't know if odd-limit or prime-limit
> > is a better indicator of how an interval can be "perceived",
>
> Didn't I just explain this? Odd limit. Or do you not read
> my replies?

Read, yes. Fully understood? Apparently, not quite. But this is no fault of your explanation, I assure you.

> > Really, I guess what I want to know is 1) how do you really
> > judge the "complexity" of a JI ratio,
>
> Square root of product of numerator and denominator when ratio
> is in lowest terms. IF said product is less than 70.

Why 70? What's special about this number?

> JI dyads beyond the 9-limit are generally not perceivable
> as JI dyads. They require triads at least. For triads,
> the product rule can also be used, but with cuberoot in place
> of square root.
>
> Such intervals are not tunable by ear and are not, therefore,
> "just". They work only in triads and larger chords. 467 cents
> is between the 9-limit intervals 9/7 and 4/3. It is one cent
> closer to 4/3, but because, as Partch discovered, the size of
> an interval's field of attraction is proportional to its
> complexity, even 460 cents is within 4/3's field and therefore
> will tend to sound like a wolf 4/3.

What's the limit for triads, then? The context for my questions is the primer I'm attempting to write on alternative EDOs, and I'm assuming most people reading the primer will be interested in triadic harmony. So even if the 9-limit is max for dyads, I'd think that the max limit for triads would be more useful, as far as ratios used to describe tempered intervals goes.

> Harmonic entropy reproduces the product rule:
>
> /tuning/files/PaulErlich/stearns.jpg
>
> and also incorporates Partch's observation automatically.
> Here's a typical harmonic entropy curve for dyads < 1 octave
> with minima and maxima labeled:
>
> http://lumma.org/temp/entropy-labels.gif
>
> It says that anything between 448 and 554 cents will tend to
> sound like a 4/3, when heard in isolation.

Well, I'll be darned! So the field of attraction for a fourth is about +/- 50 cents from 4/3...wasn't I just speculating that that could be the case? Of course, I was imagining it would hold for all interval classes, but harmonic entropy seems to suggest it only holds for a few.

> >in 22-EDO it functions as BOTH a 9/8 and 8/7, depending on the
> >chord. I could see good reasons for describing it as all three
> >ratios, but I've no idea how to narrow it down to one.
>
> Why does it have to be narrowed down to one?

it doesn't, I just assumed it would. But from what I've been learning about harmonic entropy, it seems like for a lot of intervals found in various EDOs, there are multiple valid JI ratios that can be used as descriptors. In fact, it looks like there are only a few intervals that represent significant minima of harmonic entropy. So I guess I should be less concerned with which ratio is the closest to a given tempered interval, and more focused on how many ratios are reasonably close to it. Especially when said tempered interval is close to a maximum of harmonic entropy.

Thank you Carl, this is most helpful.

-Igs

I wrote:

> Harmonic entropy reproduces the product rule:
> /tuning/files/PaulErlich/stearns.jpg

Note it the relationship starts to break down when the
product is about 70.

> Here's a typical harmonic entropy curve for dyads < 1 octave
> with minima and maxima labeled:
> http://lumma.org/temp/entropy-labels.gif

Sorry, that obviously shows dyads up to 2 octaves in size.

-Carl

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> Hi Carl,
> Sorry, I've tried to reply to this twice today but it doesn't
> seem to be going through. Hope this one fares better!
> Apologies in advance if all three attempts go through at once.

No worries.

> > > Really, I guess what I want to know is 1) how do you really
> > > judge the "complexity" of a JI ratio,
> >
> > Square root of product of numerator and denominator when ratio
> > is in lowest terms. IF said product is less than 70.
>
> Why 70? What's special about this number?

As the product gets bigger the ratios it includes start
getting closer together, and the brain apparently hasn't seen
evolutionary pressure to distinguish beyond a certain point.
70 just happens to be that point.

> > Such intervals are not tunable by ear and are not, therefore,
> > "just". They work only in triads and larger chords. 467 cents
> > is between the 9-limit intervals 9/7 and 4/3. It is one cent
> > closer to 4/3, but because, as Partch discovered, the size of
> > an interval's field of attraction is proportional to its
> > complexity, even 460 cents is within 4/3's field and therefore
> > will tend to sound like a wolf 4/3.
>
> What's the limit for triads, then?

sqrt(70) is about 8. So maybe the limit for triads is about 500,
since cubert(500) is about 8? Really, I have no idea, and I
doubt anyone else does either. It would take experiments.

> The context for my questions is the primer I'm attempting to
> write on alternative EDOs, and I'm assuming most people reading
> the primer will be interested in triadic harmony. So even if
> the 9-limit is max for dyads, I'd think that the max limit for
> triads would be more useful, as far as ratios used to describe
> tempered intervals goes.

Certainly, a laser-like focus on dyads will not give a good
picture of how an EDO will work musically. Lots of people have
written about EDOs from a dyads perspective, fewer from the
perspective of larger chords, so it may be a more fertile area.
(Paul was among the first to point out a basic flaw in most
people's dyadic approach -- the best dyads they were measuring
couldn't always fit together into larger chords. The solution
was to test an EDO for something called consistency, and throw
out EDOs that didn't pass. Gene later pointed out that every
EDO tuning can in fact represent several rank 1 temperaments,
all of which are consistent a.k.a. regular by definition, and
we can just compare them to find something satisfactory instead
of throwing any EDOs out...)

> > Harmonic entropy reproduces the product rule:
> >
> > /tuning/files/PaulErlich/stearns.jpg
> >
> > and also incorporates Partch's observation automatically.
> > Here's a typical harmonic entropy curve for dyads < 1 octave
> > with minima and maxima labeled:
> >
> > http://lumma.org/temp/entropy-labels.gif
> >
> > It says that anything between 448 and 554 cents will tend to
> > sound like a 4/3, when heard in isolation.
>
> Well, I'll be darned! So the field of attraction for a fourth
> is about +/- 50 cents from 4/3...wasn't I just speculating that
> that could be the case?

Maybe. I sometimes don't read your messages carefully. :P

> > >in 22-EDO it functions as BOTH a 9/8 and 8/7, depending on the
> > >chord. I could see good reasons for describing it as all three
> > >ratios, but I've no idea how to narrow it down to one.
> >
> > Why does it have to be narrowed down to one?
>
> it doesn't, I just assumed it would. But from what I've been
> learning about harmonic entropy, it seems like for a lot of
> intervals found in various EDOs, there are multiple valid
> JI ratios that can be used as descriptors. In fact, it looks
> like there are only a few intervals that represent significant
> minima of harmonic entropy.

Yes, it's very depressing to numerologists. But nobody's ever
been able to tune ratios like 17/13 by ear in an experimental
setting. Most or all of the harmonic entropy minima have been
tuned by ear in experiments. However, the ratios of 17 are still
significant in chords! And tetradic harmonic entropy would
surely show minima for chords like 10:12:15:17.

-Carl

On Sun, Jul 25, 2010 at 2:47 PM, Carl Lumma <carl@...> wrote:
>
> As the product gets bigger the ratios it includes start
> getting closer together, and the brain apparently hasn't seen
> evolutionary pressure to distinguish beyond a certain point.
> 70 just happens to be that point.
>
> Yes, it's very depressing to numerologists. But nobody's ever
> been able to tune ratios like 17/13 by ear in an experimental
> setting. Most or all of the harmonic entropy minima have been
> tuned by ear in experiments. However, the ratios of 17 are still
> significant in chords! And tetradic harmonic entropy would
> surely show minima for chords like 10:12:15:17.
>
> -Carl

What is your take on a compound interval like 15/8? That one's
generally easy to tune, despite having a product of 120.

Would you say that it's because people generally hear that interval on
some level as a third on top of a fifth (or vice-versa), thus
imagining it as part of a faux-triad or something similar?

-Mike

(caleb plays 10:12:15:17 with 10=ca. f4)

Oh yeah, a 'minor 6th' chord, or nearly indistinguishable from one!

To my surprise, it sounds so close to 1/1, 6/5, 3/2, 27/16 that I couldn't tell
them apart. They're the same, except for the 17.

17/15= 1.1333=216.6 cents

27/16 to 3/2 = 9/8 = 1.125=203.9 cents

In this context, I don't really hear the 12.7 cents' difference.

Or, 1/1, 6/5, 3/2, 12/7, where the interval of 12/7 to 3/2 is around 231 cents.

Again, that's still in the zone of 'minor added 6th'.

Some chords seem to have wide tolerance; some seem to require more accuracy. A
chord like 6:7:9:11 has to be within a few cents--it seems--to sound like
itself.

6:7:9:10 is again a 'minor 6th', but the 6:7:9 makes it a harmonic series chord,
and therefore, perhaps, more exacting?

Chords other than harmonic-series chords don't need as much accuracy?

I'm contradicting myself, because 10:12:15:17 would be a 'harmonic series' chord
by some definitions. But not really, to my ear, because you don't hear its
relation to '1'.

Carry on. It's just interesting to me.

________________________________
From: Carl Lumma <carl@...>
To: tuning@yahoogroups.com
Sent: Sun, July 25, 2010 2:47:50 PM
Subject: [tuning] Re: How to Describe Tempered Intervals in Terms of JI

... And tetradic harmonic entropy would
surely show minima for chords like 10:12:15:17.

-Carl

Carl>"As the product gets bigger the ratios it includes start
getting closer together, and the brain apparently hasn't seen
evolutionary pressure to distinguish beyond a certain point.
70 just happens to be that point."
Wait a minute, I've heard from Gene and countless others that the ratio of
indistinguish-ability is around 7 cents and even the comma (where notes no
longer blend) as typically at more like 20 cents. How can 70 (something around
27 cents) be the limit at the same time? Also to note...again even 9/8 (the
usual JI whole tone) is greater than 70 (although barely). Seems to be rather
sharply biased (and highly pessimistic) aesthetics.

>"Certainly, a laser-like focus on dyads will not give a good picture of how an
>EDO will work musically. "
Agreed, though the flip problem is how do you make scales based on combinations
of desirable triads without severely limiting the number of chords available?
That a whole lot of combinatorics...though Lord bless whoever can figure out an
efficient way to compute that sort of thing.

>"And tetradic harmonic entropy would surely show minima for chords like
>10:12:15:17."
Interesting...now how would you calculate tetradic harmonic entropy?

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> Wait a minute, I've heard from Gene and countless others that the ratio of
> indistinguish-ability is around 7 cents

I never said that. I agreed that when you get within seven cents, the approximation begans to sound pretty decent. But the fifth of 31et, not to mention that of Lucy Tuning, can easily be told from JI. Plus it all depends on the type of interval and the harmonic context anyway.

Gene>"I never said that. I agreed that when you get within seven cents, the
approximation begans to sound pretty decent. But the fifth of 31et, not to
mention that of Lucy Tuning, can easily be told from JI"
So you are saying even 7 cents (what I though you had said before was close
enough) is still too much...in which case you seem even further in disagreement
with Carl's apparent idea that around 27 cents and under (according to harmonic
entropy) is when two ratios sound indistinguishable...correct?

Ahoy, Carl,

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@> wrote:
>
> > Why 70? What's special about this number?
>
> As the product gets bigger the ratios it includes start
> getting closer together, and the brain apparently hasn't seen
> evolutionary pressure to distinguish beyond a certain point.
> 70 just happens to be that point.

The more I think about it, the more sense this makes. Harmonic entropy isn't a measure of the stability of an interval (i.e. how periodic it is/how well the partials line up with the harmonic series) but rather how strongly the brain can identify it. Yes, it is theoretically possible to tune 10/9 and 9/8 by ear, since they're both stable/beatless, but you'd be hard-pressed to be able to identify which one you'd tuned to! Same thing with 11/8 and 18/13 and 15/11. What really fascinates me is how uneven the graph of harmonic entropy is, suggesting that stable ratios are not evenly spread out across the pitch-spectrum but instead cluster in certain areas.

> > What's the limit for triads, then?
>
> sqrt(70) is about 8. So maybe the limit for triads is about 500,
> since cubert(500) is about 8? Really, I have no idea, and I
> doubt anyone else does either. It would take experiments.

I'm frankly amazed that so many people have drawn conclusions about harmony without conducting experiments on triads.

> Certainly, a laser-like focus on dyads will not give a good
> picture of how an EDO will work musically. Lots of people have
> written about EDOs from a dyads perspective, fewer from the
> perspective of larger chords, so it may be a more fertile area.
> (Paul was among the first to point out a basic flaw in most
> people's dyadic approach -- the best dyads they were measuring
> couldn't always fit together into larger chords. The solution
> was to test an EDO for something called consistency, and throw
> out EDOs that didn't pass. Gene later pointed out that every
> EDO tuning can in fact represent several rank 1 temperaments,
> all of which are consistent a.k.a. regular by definition, and
> we can just compare them to find something satisfactory instead
> of throwing any EDOs out...)

Yeah, I think this is the way I need to go. Of course, I'm not always very good about figuring out the harmonic series representation of certain triads. But I'll get better, I'm sure. I remember Cameron Bobro pointed out to me one day that 23-EDO could be represented very well almost entirely with 21-odd-limit ratios. Perhaps other EDOs will prove similar.

> > > Why does it have to be narrowed down to one?
> >
> > it doesn't, I just assumed it would. But from what I've been
> > learning about harmonic entropy, it seems like for a lot of
> > intervals found in various EDOs, there are multiple valid
> > JI ratios that can be used as descriptors. In fact, it looks
> > like there are only a few intervals that represent significant
> > minima of harmonic entropy.
>
> Yes, it's very depressing to numerologists. But nobody's ever
> been able to tune ratios like 17/13 by ear in an experimental
> setting. Most or all of the harmonic entropy minima have been
> tuned by ear in experiments. However, the ratios of 17 are still
> significant in chords! And tetradic harmonic entropy would
> surely show minima for chords like 10:12:15:17.

I guess my biggest question about these higher-limit ratios is, if you played them on an electric guitar with heavy distortion/fuzz, would they fall apart the way 12-tET's thirds do? I suppose the best measure for that is periodicity. Is there a standard way to calculate periodicity? Is it just the product of the numerator and denominator? I know that to find the common meter for a polyrhythm, you multiply the numerators of the meters of both rhythms (so 5/4 over 3/4 fits into 1 bar of 15/4), so I'd suspect frequency ratios would be the same way. I'd be really excited if I could figure out the highest-limit ratios that can survive a slight tempering and still sound "stable" with the timbre of a distorted electric guitar. I know 13-odd-limit ratios work well enough, and my guess is that the limit is something like 15/11.

Thanks for helping me with this.

-Igs

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Gene>"I never said that. I agreed that when you get within seven cents, the
> approximation begans to sound pretty decent. But the fifth of 31et, not to
> mention that of Lucy Tuning, can easily be told from JI"
> So you are saying even 7 cents (what I though you had said before was close
> enough) is still too much...in which case you seem even further in disagreement
> with Carl's apparent idea that around 27 cents and under (according to harmonic
> entropy) is when two ratios sound indistinguishable...correct?
>
You're misinterpreting what's being said here, Michael. Gene said that 7 cents is about the limit for a tempered interval to sound "close enough" to a Just interval. Of course, he also stipulated that this really depends on the ratio and the musical context, and is not a universal constant. On the other hand, that "27 cents" value has more to do with two somewhat-simple ratios that are close together, like 10/9 and 9/8. While both are stable enough to be tuned by ear, they are not "unique enough" from each other to be easily told apart. If you look at that graph of harmonic entropy that Carl posted, you'll see that it's very uneven across the pitch spectrum--what this means is that some areas of the pitch spectrum are denser with intervals of similar complexity, while others are not. In the range of a major second, you have 10/9, 9/8, 11/10, 8/7, and such...yes, they are all of different complexity, but the range is small compared to the range of a perfect fifth. There, you have 3/2, 22/15, 32/21, 50/33, 40/27, etc. You can clearly see that 3/2 is of a drastically lower complexity than anything near-by, so there is lower harmonic entropy at that point.

So basically, the point of harmonic entropy is that if you have two ratios of similar complexity, and they're within about 27 cents of each other, it will be hard to tell which ratio you're hearing (assuming you hear the dyad in isolation; I'm sure if you alternated between 10/9 and 9/8, you could tell the difference). In other words, for some interval classes, it is much harder to define which ratio defines the interval class, because there are too many ratios of similar complexity close together in that one range to just "pick one".

You should really study this! I think this theory explains a lot of things very well. Like Mike B.'s and my observation that the size of the third in a minor triad seems to be able to vary quite a bit without losing the "feel" of a minor triad...the minor third range on the harmonic entropy graph is a pretty flat area! So it stands to reason any interval on that range will work as well as any other.

-Igs

On 26 July 2010 17:31, cityoftheasleep <igliashon@...> wrote:

> I guess my biggest question about these
> higher-limit ratios is, if you played them on
> an electric guitar with heavy distortion/fuzz,
> would they fall apart the way 12-tET's thirds do?
>  I suppose the best measure for that is periodicity.
>  Is there a standard way to calculate periodicity?
>  Is it just the product of the numerator and
> denominator?  I know that to find the common
> meter for a polyrhythm, you multiply the
> numerators of the meters of both rhythms
> (so 5/4 over 3/4 fits into 1 bar of 15/4), so
> I'd suspect frequency ratios would be the same way.
>  I'd be really excited if I could figure out the highest-
> limit ratios that can survive a slight tempering and
> still sound "stable" with the timbre of a distorted
> electric guitar.  I know 13-odd-limit ratios work
> well enough, and my guess is that the limit is
> something like 15/11.

The period's the virtual 1/1. So a 4:5:6:7 chord has the same period
as the 4:5:6 chord with the same tuning and should deal as well with
distortion. 10:12:15:17 will give a period related to the major third
below the root. The smallest number in the ratio's the thing to look
at. I'm sure 15-limit chords will work, but it helps to reinforce the
root and fifth, and you start to run out of strings.

With distortion, you get sum and difference tones produced. So you
can also try things like 7:9:11 or phi tunings. But the latter work
better with a phi timbre and guitars aren't likely to produce one of
them.

Graham

On Mon, Jul 26, 2010 at 1:17 PM, Graham Breed <gbreed@...> wrote:
>
> With distortion, you get sum and difference tones produced. So you
> can also try things like 7:9:11 or phi tunings. But the latter work
> better with a phi timbre and guitars aren't likely to produce one of
> them.
>
> Graham

I wouldn't expect phi tunings to sound so hot under distortion even if
a phi timbre is used. Combination tones aren't just produced by mixing
different component frequencies in a sound together, but also by
mixing each component frequency with itself. So you get the sum tone
of some frequency f and itself, which is 2f. Then you might get 2f +
f, which is 3f, and 3f + f, and 3f + 2f, and 5f - 3f, etc. So
distortion tends to generate a complete harmonic series all by itself.

Given that, I'd expect phi tunings to sound absolutely terrible, but
maybe the resultant amplitude of the "harmonic" distortion would be
low enough to not produce too much of a clash. Would be an interesting
experiment.

-Mike

>"Harmonic entropy isn't a measure of the stability of an interval (i.e. how
>periodic it is/how well the partials line up with the harmonic series) but
>rather how strongly the brain can identify it."
I'll agree with that, even by experience generated by experiments I performed
on my own hearing before I knew the term. My question becomes, why does "how
strongly the brain can identify it" matter so much?

>"I guess my biggest question about these higher-limit ratios is, if you played
>them on an electric guitar with heavy distortion/fuzz, would they fall apart
>the way 12-tET's thirds do? I suppose the best measure for that is
>periodicity. Is there a standard way to calculate periodicity? Is it just the
>product of the numerator and denominator?"

I believe it is the product. For example

For 3/2...
100hz * 3 = 300
100hz * 2 = 200
exactly periodic at 600 or 6 cycles of 100hz

100hz * 11 = 1100
100hz * 6 = 600
exactly periodic at 6600 or 66 cycles of 100hz

However from person to person I've found periodicity alone often fails (IE
some people like 15/8 better than 11/6 or vice-versa). In the past there has
been much discussion about splitting fractions up IE the brain's saying that
15/8 = 3/2 * 5/4. Come to think of it 22/15 = 4/3 * 11/10 and 11/6 = 5/3 *
11/10 or approximately 8/5 * 8/7.
So add to the equation the idea of using such "split fractions" to define
larger ratios within 7 cents (and not just exactly) and you get a whole lot of
possibilities.

Bottom line...there seem to be a lot more ways to achieve stability than
limiting yourself to low(odd)-limit JI. If we were truly limited to low-limit
JI why would we even bother making anything other than low-limit adaptive JI
(how boring would that be)?

_

Igs>"While both (10/9 and 9/8) are stable enough to be tuned by ear, they are
not "unique enough" from each other to be easily told apart."
Ok, so it's a question of "can they be used without changing the emotional
impact".

>"In the range of a major second, you have 10/9, 9/8, 11/10, 8/7, and
>such...yes, they are all of different complexity, but the range is small
>compared to the range of a perfect fifth. There, you have 3/2, 22/15, 32/21,
>50/33, 40/27, etc. You can clearly see that 3/2 is of a drastically lower
>complexity than anything near-by, so there is lower harmonic entropy at that
>point. "
I have seen that curve countless times, however I still question not the
on-the-average validity of but the usefulness of its results.

What information you gave that IS new to me is the idea that harmonic entropy
isn't about how "clean" a ratio sounds but rather which ratios define the
emotional sound of ratios around them (IE the idea of complex ratios in an area
being approximates of simpler ones).

However, I simply don't agree with many stipulations that potentially come
from that line of thinking, such as the idea that everything around 3/2 must be
a "weak version of a pure fifth", though I do understand in many, if not most
cases, that does hold. It's kind of like saying all metals are conductive when
you can still take a nickel coin in front of a magnet and have it do nothing.

>"I think this theory explains a lot of things very well...Like Mike B.'s and my
>observation that the size of the third in a minor triad seems to be able to
>vary quite a bit without losing the "feel" of a minor triad"

Well I think the idea can be useful if used to promote the idea of tempering
between two "versions" of a note in a chord without losing much (if any) of the
sense of composure. The problem is, according to the curve, that "trick" can
only be applied to certain areas, such as seconds and thirds.

>"the minor third range on the harmonic entropy graph is a pretty flat area!"

Another example. Under the theory, as I understand it, ideas like
substituting 4ths, 5ths, and 6ths pretty much go down the toilet because those
areas have such extreme low points. In reality I've found several uses for
ratios like 22/15 and 11/8 which, in theory, end up smack at the top of Paul's
curves (nowhere near those so-called flat center), as unique centers of
sound/emotion. So it seems to work a good of the time, but also outright
disagree with what my ear tells me enough of the time to make me think it's
making a few too many limiting assumptions.

_,_._,___

On 26 July 2010 18:48, Mike Battaglia <battaglia01@...> wrote:

> Given that, I'd expect phi tunings to sound absolutely terrible, but
> maybe the resultant amplitude of the "harmonic" distortion would be
> low enough to not produce too much of a clash. Would be an interesting
> experiment.

I've done it, and they sounded okay, but obviously not so great that I
wanted to record them.

Graham

Mike wrote:

> What is your take on a compound interval like 15/8? That one's
> generally easy to tune, despite having a product of 120.

It's not so easy to tune -- have you tried it? Nevertheless,
the product rule is only a rule of thumb. If we consult Paul's
plot again

/tuning/files/PaulErlich/stearns.jpg

you'll see that 11/9 has lower entropy than 11/7.

> Would you say that it's because people generally hear that
> interval on some level as a third on top of a fifth
> (or vice-versa),

Nope.

-Carl

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> What information you gave that IS new to me is the idea that harmonic entropy
> isn't about how "clean" a ratio sounds but rather which ratios define the
> emotional sound of ratios around them (IE the idea of complex ratios in an area
> being approximates of simpler ones).
>
> However, I simply don't agree with many stipulations that potentially come
> from that line of thinking, such as the idea that everything around 3/2 must be
> a "weak version of a pure fifth", though I do understand in many, if not most
> cases, that does hold. It's kind of like saying all metals are conductive when
> you can still take a nickel coin in front of a magnet and have it do nothing.

Well, as Carl pointed out, this curve only holds for dyads. So if you're playing triads, there might be much more leeway. Tetrads, perhaps even more so.

> >"I think this theory explains a lot of things very well...Like Mike B.'s and my
> >observation that the size of the third in a minor triad seems to be able to
> >vary quite a bit without losing the "feel" of a minor triad"
>
> Well I think the idea can be useful if used to promote the idea of tempering
> between two "versions" of a note in a chord without losing much (if any) of the
> sense of composure. The problem is, according to the curve, that "trick" can
> only be applied to certain areas, such as seconds and thirds.

Not necessarily. There is somewhat of a paradox here, I believe. Check it: some interval classes have a small difference between local minima and maxima, so you can say they are "vaguely defined" and thus not sensitive to "mistuning". Having a "weak identity", in other words, implies that it's hard to make that interval sound very "out-of-tune". On the other hand, intervals with a large difference between their local minima and maxima have a strong identity, and so getting too far away from the minima leads to a "mistuned" sound. However, because the identity is so strong, the "musical meaning" will be maintained, even under mistuning. And of course, the mistuning may only be evident when played as dyads. I've found time and time again that a wolf dyad can become a consonant triad if the right note(s) is (are) put in the middle.

What harmonic entropy suggests to me is really a model of the character of various interval classes. Some interval classes get their character from having a simple ratio at their center, while others get their character from having a sort of "ambiguity" to them. Maybe the musical meaning of an interval has less to do with ratios and more to do with, oh, I dunno, maybe something like the derivative of the curve of harmonic entropy? (Sorry if that doesn't make sense, my recollection of calculus is a bit dim). What I mean is, maybe the "feel" of an interval comes from the relationship of the harmonic entropy of that interval to the harmonic entropy of its neighbors, i.e. the "rate of change" of harmonic entropy at that interval--that's a derivative, right?

I dunno, maybe I have the whole concept upside-down and backwards. I haven't seen too many people interpreting harmonic entropy in this way, so perhaps that just means I'm way off-base.

-Igs

I wrote:

> > Why 70? What's special about this number?
>
> As the product gets bigger the ratios it includes start
> getting closer together, and the brain apparently hasn't seen
> evolutionary pressure to distinguish beyond a certain point.
> 70 just happens to be that point.

Another way to look at sqrt(70) ~~ 8 is that dyads formed
between harmonics < 8 will tend to sound just, whereas those
formed between higher harmonics won't. This coincidentally
fits with data from Dale Purves' lab
(view with fixed-width font)

Highest amplitude in speech corpus:

n prob.

2 20
3 25
4 25
5 15
6 10
7 5

They analyzed a huge corpus of recorded speech. The above
means, if you pick a random spot in those recordings and ask
what the loudest partial is, there's a 25% chance it'll be
the 3rd partial, etc.

-Carl

Hello, all.

To a very rich thread, please let me add a few ideas and a link.
What I should emphasize is that there's not necessarily any one
best way to describe temperaments, equal or otherwise.

Furthermore, a ratio need not be tuneable by ear, in my
view, to be useful in contrapuntal or harmonic as well as
melodic practice, and possibly also to be a very useful
landmark on the interval continuum in describing temperaments,
equal and other.

Here's a paper I wrote a few years and revised with some helpful
suggestions from David Keenan and Andrew Heathwaite, to whom I
express my gratitude while emphasizing that the views expressed
are, of course, my responsibility and not theirs:

<http://www.bestII.com/~mschulter/IntervalSpectrumRegions.txt>

Rational ratios are in my view _one_ helpful landmark -- not
the only one, or the one always most applicable. However, I
do find complex as well simple ratios as friendly and welcoming
landmarks when exploring a familiar region of the spectrum, or
possibly a new one.

For example, suppose I were writing a brief description of 29-EDO.
Here are some possible ideas:

* * *

In describing or exploring 29-EDO, we can make the fertile territory
of this temperament a bit more orderly by looking at three main
"interval families" available.

The first of these families consists of regular diatonic intervals
built with chains of from one to six of the near-pure 29-EDO fifths
(17/29 octave or about 703.45 cents). Since the fifths are less
than 1.5 cents wide of a pure 3:2 (701.96 cents), these regular
intervals are quite close to those of Pythagorean tuning using
these just fifths -- a feature very conducive to some styles,
and not so conducive to others.

These wide 29-EDO fifths -- extended fifths, as they are sometimes
called -- result in a kind of accentuated Pythagorean-like tuning
which can be beautifully apt for the classic 13th-14th century
styles of Europe, but quite unlikely the meantone or well-tempered
systems that developed in the 15th-19th centuries. Major thirds
at around 414 cents (slightly wide of 33:26), and minor thirds at
290 cents (just a tad wide of 13:11), very nicely fit a medieval
style where these complex intervals, like the Pythagorean 81:64
and 32:27 at 408 and 294 cents, typically resolve to stable
intervals such as fifths and fourths. For those who relish
these styles, the near-pure fourths and fifths of 29-EDO contrast
nicely with the more complex thirds and sixths.

A big asset of 29-EDO for these polyphonic progressions as well
as pure melody is the regular diatonic semitone or limma
(2/29 octave) at 83 cents, or somewhere between 22:21 (81 cents)
and 21:20 (84 cents).

From these regular intervals we move to the next family, augmented
or diminished intervals formed from chains of 7-11 fifths.

Again, people looking for subtly complex thirds and sixths will
be happy, while those seeking out the simplest ratios (e.g. 5:4,
6:5, 7:6) may be disappointed. A diminished fourth (e.g. C#-F)
from 8 fifths down or fourths up is 372 cents, a bit wide of
the large neutral or submajor third at 26:21, while an augmented
second from 9 fourths up at 331 cents (e.g. F-G#) is just narrow
of of a small neutral or supraminor third at 63:52 (332 cents).

Along with these thirds, this family also includes a chromatic
semitone from 7 fifths up (e.g. C-C#), also known as an apotome,
at 124 cents, a step a bit narrow of 14:13 (128 cents). A
diminished third at 166 cents from ten fifths down (e.g. C#-Eb),
precisely twice the size of the usual 83-cent semitone, yields
a virtually just 11:10.

For this family of intervals, at least two stylistic approaches
are possible. One approach, for people looking for a harmonic
approximation of 12-EDO or meantone, is to treat a 372-cent
or 331-cent third as an inaccurate approximation of 5:4 (386
cents) or 6:5 (316 cents). In such a context, both the
notable impurity or complexity of the thirds and the wide
nature of the 124-cent apotome which here serves in effect
as a diatonic semitone, can create what is sometimes described
as a languid or melancholy quality -- different, but not
neessarily unpleasant.

However, one may alternatively look on these steps and
intervals as neutral, or at least semi-neutral, possibly
representing, for example, 26:21, 63:52, and 14:13. In
some Persian and Turkish contexts, and possibly in
certain Kurdish ones also, steps of 124 and 165 cents
may nicely fit the needs of the music -- and likewise
the semi-neutral thirds, in polyphonic styles.

For example, here is a basic tuning for a Turkish
interpretation of Makam Huseyni, using a conventional
note spelling to reflect the chains of fifths:

|-------------| |-------------|
0 4 7 12 17 21 24 29
C# Eb E F# G# Bb B C#
0 166 290 497 703 869 993 1200

Makam Huseyni in its basic "textbook" form consists
of two tetrachords, with a tone between them, each
having a large neutral second (ideally around 11:10),
a small neutral second, and an upper tone. It's
interesting that a small step of 124 cents, found
here at Eb-E and Bb-B, may occur in certain Persian
modes.

The third interval family of 29-EDO consists of rather
complex intervals favored by many of the world's musical
traditions, here termed _interseptimal_ intervals because
they occur in the middle region between two septimal
ratios such as the large 8:7 tone (231 cents) and the
small 7:6 minor third (267 cents). In 29-EDO, we get
an intriguing size from 14 fifths up of 248 cents --
sometimes called a hemifourth, because it's equal
more or less (here precisely) to half of a fourth.

Hemifourths might suggest a simple septimal ratio
like 8:7 or 7:6 -- and can delightfully be used
contrapuntally in some styles as either a very
wide tone or a very narrow minor third -- but are
really a breed unto themselves. As it happens,
the 29-EDO interval of 248 cents is a virtually
just 15:13, while we also find intervals of
455 cents (near 13:10), 745 cents (near 20:13),
and 952 cents (near 26:15).

These intervals are used in world musics such as
Balinese or Javanese gamelan, and additionally may
be introduced into other types of contexts where
they provide distinctly "xenharmonic" material.

Interseptimal intervals are closely related to
another member of this third family: the diesis
of 1/29 octave, or 41 cents, by which 12 fifths
exceed 7 octaves. This family thus begins with
a chain of 12 fifths producing the diesis itself,
and continues as various regular intervals are
altered by this diesis. Thus a regular 414-cent
major third plus the diesis, or 497-cent fourth
minus it, produces a 455-cent interval almost
exactly at 13:10, somewhere between a usual
septimal major third at 9:7 (435 cents) and
a 4:3 fourth. The diesis might, for example,
be interpreted in JI terms as a 40:39 (44 cents),
the difference between 13:10 and 4:3.

For the adventurous, the 41-cent diesis may
sometimes serve as an ultranarrow equivalent
of a semitone, as may the larger 50-cent step
in 24-EDO, for example. However, the diesis
plays a vital role in the structure of the
tuning for styles where it is unlikely to
occur as a direct melodic step: for example,
gamelan modes using the 248 cent hemifourths
and other interseptimal intervals.

In short, 29-EDO is a versatile system supporting
an "accentuated Pythagorean" rendition of 13th-14th
century European music; a family of "semi-neutral"
intervals very useful for some Near Eastern styles,
and sometimes applied to get a "different" sound
for European music that might normally be played
in a system like meantone, a well-temperament,
or 12-EDO; and interseptimal intervals fitting
various musical styles including gamelan as well
as adventurous xenharmonicism.

* * *

This is just a sample about how I might approach
an EDO system -- or a regular or irregular temperament
of some other variety, for that matter. In 31-EDO,
for example, we would have regular meantone, septimal,
and neutral families. Knowing where these intervals
fall on the spectrum, with rational ratios providing
one set of useful landmarks, is only part of the
question; what styles the reader of such a description
is interested in, or might become interested in, is
another.

Best,

Margo Schulter
mschulter@...

Igs>"Well, as Carl pointed out, this curve only holds for dyads. So if you're
playing triads, there might be much more leeway. Tetrads, perhaps even more so."
Agreed but, at the same time, I think a lot of the assumptions are too
extreme, even for dyads. The whole IMVHO rather Puritanist "dyadic 5th must be
a few cents of 3/2 or die an ugly and useless death" and "dyadic 4th must be
4/3" assumptions still hold.

>"On the other hand, intervals with a large difference between their local minima
>and maxima have a strong identity, and so getting too far away from the minima
>leads to a "mistuned" sound."
Right, which seems to say that too far away from a pure 4th, 5th, or 6th must
be "wrong".

>"Some interval classes get their character from having a simple ratio at their
>center, while others get their character from having a sort of "ambiguity" to
>them."
I guess that's where my ear is odd. I don't hear it as "ambiguity" or
"incompleteness"...to me it's like putting strawberries on your chocolate
ice-cream (for the "weird intervals" instead of chocolate chips (the more
"normal" choice b/c the flavor of the "topping" is the same). Or like setting a
different tint setting on a photo of a painting. You can feel the difference
but, at the same time, it's obvious the "root" source is the same (IE ice-cream,
the original painting, or the "pure" interval version).

>"However, because the identity is so strong, the "musical meaning" will be
>maintained, even under mis-tuning."
That makes much more sense to me, though I've never heard it mentioned that
way.
Though that alone hardly explains things like, at least as I hear it, why your
suggestion of 22/15 over 17/11 works so well (IE why 22/15 seems to work as a
fifth while 17/11 doesn't despite both being almost exactly the same distance
from 3/2). Or why the tonal "color" on those two is so much different despite
their admittedly sounding a lot more like fifth than any other nearby interval
class (IE than the 4th or 6th). There seems to be a missing layer of
granularity here that keeps the curve from being truly continuous...I just
wonder what accounts for it.

>"What harmonic entropy suggests to me is really a model of the character of
>various interval classes. "
Exactly...though it doesn't seem to acknowledge much of the "subtle-ties"
within those classes (which often pop up as very flat on the graph or,
especially in the case of 4ths, 5ths, and 6ths, often not at all).

>"What I mean is, maybe the "feel" of an interval comes from the relationship of
>the harmonic entropy of that interval to the harmonic entropy of its neighbors,
>i.e. the "rate of change" of harmonic entropy at that interval--that's a
>derivative, right?"
People make calculus seem way too complex :-D: yes, derivative = slope = rate
of change. Problem is rate of change would be VERY high around things like
"impure" fifths, again seeming to allude to that, beyond a certain rather
conservative point, 5ths like 22/15 or 13/9 or like comparing chocolate
ice-cream to vanilla wafers instead of chocolate ice-cream and chocolate chips
to chocolate ice-cream and strawberries.

>"I dunno, maybe I have the whole concept upside-down and backwards. I haven't
>seen too many people interpreting harmonic entropy in this way, so perhaps that
>just means I'm way off-base."
I'll say this much though...while I don't get your derivative concept for this
I fully agree on your concept that "a stronger/steeper interval allows more
intervals around it to substitute for it" (kind of like their all being sucked
into a tornado, but in a good way). :-D

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:

> So you are saying even 7 cents (what I though you had said before was close
> enough) is still too much...in which case you seem even further in disagreement
> with Carl's apparent idea that around 27 cents and under (according to harmonic
> entropy) is when two ratios sound indistinguishable...correct?
>

Correct? I hardly think it's likely Carl said anything so fatuous.

Carl>"It says that anything between 448 and 554 cents will tend to sound like a 4/3, when heard in isolation."

Michael> ... 4/3 does have an uncommonly large area between the 15/11 above it and 9/7 below it ... but an entire semitone difference?!

Well, the graph of Harmonic Entropy that Carl posted confirms this (almost - it says 448-545 cents actually). Note, that is (almost) an entire semitone difference between the lowest and highest in the range, but only a quartertone between either of these and the "target" 4/3 (remember Carl said "heard in isolation").

Steve M.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> > with Carl's apparent idea that around 27 cents and under
> > (according to harmonic entropy) is when two ratios sound
> > indistinguishable...correct?
>
> Correct?

Certainly not.

-Carl

cityoftheasleep wrote:

> Ahoy, Carl,

Splice the mainbrace!

> The more I think about it, the more sense this makes. Harmonic
> entropy isn't a measure of the stability of an interval (i.e. how
> periodic it is/how well the partials line up with the harmonic
> series) but rather how strongly the brain can identify it. Yes,
> it is theoretically possible to tune 10/9 and 9/8 by ear,

It is? Have you tried it? In my experience, 9/8 is an
outside possibility, 10/9 is pretty much impossible.

> but you'd be hard-pressed to be able to identify which one you'd
> tuned to! Same thing with 11/8 and 18/13 and 15/11.

Again, 11/8 maybe in a full moon on a Sunday. No way on
18/13 or 15/11. Not without using specially prepared timbres
or something.

> I'm frankly amazed that so many people have drawn conclusions
> about harmony without conducting experiments on triads.

Paul is the first person, to my knowledge, to point out that
sensory dissonance models like Sethares' fall flat on their
face when triads and larger chords are considered.

The only published work I know of that explicitly considers
triads is that of Cook & Fujisawa, and their approach is pretty
primitive.

One should note in these things that understanding consonance
is somewhere below understanding drain cleaner on society's
list of priorities, so it's probably not fair to blame
researchers.

> I guess my biggest question about these higher-limit ratios is,
> if you played them on an electric guitar with heavy
> distortion/fuzz, would they fall apart the way 12-tET's
> thirds do? I suppose the best measure for that is periodicity.

I think 12-ET 3rds fall apart under distortion because of
their size, but I'm not a guitarist and I don't have direct
experience playing various things under distortion.

> Is there a standard way to calculate periodicity?

Of a specific signal? Yes: autocorrelation. Of a ratio
played however? Yes: the product of the numerator and
denominator. :)

> I know that to find the common meter for a polyrhythm, you
> multiply the numerators of the meters of both rhythms (so
> 5/4 over 3/4 fits into 1 bar of 15/4), so I'd suspect
> frequency ratios would be the same way.

Right-o.

-Carl

Not so.

My band (in the dark ages) used to cover Sunshine of Your Love by
Cream (read Eric Clapton) with parallel major 3rds.

Now minor thirds are a difference story. But major 12 et 3rds are a
part of many power chords....

Chris

>
> > I guess my biggest question about these higher-limit ratios is,
> > if you played them on an electric guitar with heavy
> > distortion/fuzz, would they fall apart the way 12-tET's
> > thirds do? I suppose the best measure for that is periodicity.
>
> I think 12-ET 3rds fall apart under distortion because of
> their size, but I'm not a guitarist and I don't have direct
> experience playing various things under distortion.
>

Gene> > "with Carl's apparent idea that around 27 cents and under
> > (according to harmonic entropy) is when two ratios sound
> > indistinguishable...correct?"
>
> Correct?

Carl>"Certainly not."

"The individual must not merely wait and criticize, he must defend the cause the
best he can." -Albert Einstein

Furthermore, I really think it would help if we all put more of an effort in
respecting the art of teaching as an art of simplifying things rather than
complicating them to the point many people get confused.
So Carl, what is your idea so far as harmonic entropy (you used the terms "27
cents" and "indistinguishable" in the same sentence to describe what you meant
before, but now seem to be denying correlation between those terms and your take
or harmonic entropy)?

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Not so.
>
> My band (in the dark ages) used to cover Sunshine of Your Love by
> Cream (read Eric Clapton) with parallel major 3rds.
>
> Now minor thirds are a difference story. But major 12 et 3rds are a
> part of many power chords....
>
> Chris
>

Well, some distortions hold up better than others. Ever tried an Octavia or a Boss HyperFuzz? Fuzz effects like these, which IIRC excite the odd harmonics more, totally murder anything other than a root-fifth-octave power-chord.

Also, the maj3's in power chords are really more like 10ths, since they're in the 2nd octave up from the root. And they're also covered up pretty well by the doubled fifth and tripled octave (in a full barre chord). But I digress, some distortions do allow the thirds to survive more or less, though there's a bit of clouding no matter what you do.

-Igs

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> cityoftheasleep wrote:
> > The more I think about it, the more sense this makes. Harmonic
> > entropy isn't a measure of the stability of an interval (i.e. how
> > periodic it is/how well the partials line up with the harmonic
> > series) but rather how strongly the brain can identify it. Yes,
> > it is theoretically possible to tune 10/9 and 9/8 by ear,
>
> It is? Have you tried it? In my experience, 9/8 is an
> outside possibility, 10/9 is pretty much impossible.

Well, what I was trying to say is that if you were tuning a major second by ear, you might hit a ratio that sounds rather in-tune and say "aha! I tuned to a Just major 2nd!" and it might be a 10/9 or a 9/8 or maybe an 8/7 or an 11/10 but you wouldn't be able to tell which. But I would imagine that ratios around the major second area have a strong enough periodicity that they'd given a nice JI buzz. Maybe only in higher registers. But still. I think it's possible in theory.

> > but you'd be hard-pressed to be able to identify which one you'd
> > tuned to! Same thing with 11/8 and 18/13 and 15/11.
>
> Again, 11/8 maybe in a full moon on a Sunday. No way on
> 18/13 or 15/11. Not without using specially prepared timbres
> or something.

Like I said above, you couldn't hit the ratio deliberately and identify it, but one of those ratios would probably stick out as being "Just" if you were sliding from a 4/3 toward a 7/5.

> > I'm frankly amazed that so many people have drawn conclusions
> > about harmony without conducting experiments on triads.
>
> Paul is the first person, to my knowledge, to point out that
> sensory dissonance models like Sethares' fall flat on their
> face when triads and larger chords are considered.

Whatever happened to ol' Paul, anyway? Last I heard of him, he was playing some BP keyboard with Ron Sword at the BP Symposium.

> One should note in these things that understanding consonance
> is somewhere below understanding drain cleaner on society's
> list of priorities, so it's probably not fair to blame
> researchers.

Indeed. I suppose I should be thankful it's been studied at all!

> > I guess my biggest question about these higher-limit ratios is,
> > if you played them on an electric guitar with heavy
> > distortion/fuzz, would they fall apart the way 12-tET's
> > thirds do? I suppose the best measure for that is periodicity.
>
> I think 12-ET 3rds fall apart under distortion because of
> their size, but I'm not a guitarist and I don't have direct
> experience playing various things under distortion.

Then what are you doing with that 22-ET guitar you mentioned earlier?
Seriously though, you can distort other things too, and get the same effect. But perhaps it's just not your "bag." Though I am curious, what do you mean by their "size"?

> > Is there a standard way to calculate periodicity?
>
> Of a specific signal? Yes: autocorrelation. Of a ratio
> played however? Yes: the product of the numerator and
> denominator. :)
>

Cool. I understood something!

-Igs

cityoftheasleep wrote:

> > > it is theoretically possible to tune 10/9 and 9/8 by ear,
> >
> > It is? Have you tried it? In my experience, 9/8 is an
> > outside possibility, 10/9 is pretty much impossible.
>
> Well, what I was trying to say is that if you were tuning a
> major second by ear, you might hit a ratio that sounds rather
> in-tune and say "aha! I tuned to a Just major 2nd!" and it
> might be a 10/9 or a 9/8

There's really no notch as you pull the hammer across these
intervals. There is a faint one for 9/8 on harpsichords, but
you'll be lucky to get it within 5 cents. 10/9 I can't say I
notice a notch at all.

> But I would imagine that ratios around the major second area
> have a strong enough periodicity that they'd given a nice
> JI buzz. Maybe only in higher registers. But still. I think
> it's possible in theory.

Well, why don't you try tuning these?

> > Again, 11/8 maybe in a full moon on a Sunday. No way on
> > 18/13 or 15/11. Not without using specially prepared timbres
> > or something.
>
> Like I said above, you couldn't hit the ratio deliberately and
> identify it, but one of those ratios would probably stick out
> as being "Just" if you were sliding from a 4/3 toward a 7/5.

Try it!

> Whatever happened to ol' Paul, anyway? Last I heard of him,
> he was playing some BP keyboard with Ron Sword at the
> BP Symposium.

He's finally free of the tuning list! His band plays pretty
regularly around the Boston area (12-ET, alas).

> > I think 12-ET 3rds fall apart under distortion because of
> > their size, but I'm not a guitarist and I don't have direct
> > experience playing various things under distortion.
>
> Then what are you doing with that 22-ET guitar you mentioned
> earlier?

Not playing it much. However I was thinking of bringing it
up to oaktown to shown you. But when I do play it, I don't
use distortion (yuck! :p)

> Seriously though, you can distort other things too, and get
> the same effect. But perhaps it's just not your "bag."
> Though I am curious, what do you mean by their "size"?

They're on the small side you know, thirds. Play them with
sawtooths and they're already kinda, not so fresh.

-Carl

you don't *have* to play the bottom fifth...

But this aspect of the conversation is in danger of going into a death
spiral. Obviously I cannot have tried all possible distortions and if
Octavia is doing octave dividing then there is a good reason why its
not working due to critical band issues.

I've written tunes with many bare intervals with lots of distortion -
naked parallel 7th's, parallel naked 9th's, naked parallel tritones
even! - will it work with every possible distortion you can dream up -
well, *perhaps* not. I prefer a brighter sound and not a death metal
EQ mid scoop. A good commercial example of dissonant intervals in
action would be Jimmy Page's Dancing Days from Zeppelin's Houses of
the Holy.

And for extended chords - you can mix in added minor and major 2nds as
well as tritones too.

example 1 on the 7th fret from low to high B F# B D# open B open E
play as an arpeggio

example 2 on the 1st fret from low to high F C F A open B open E play
as an arpeggio

There is more out there of course - though you can't just bang away at
everything - some things take a degree of finesse.

Chris

> Well, some distortions hold up better than others. Ever tried an Octavia or a Boss HyperFuzz? Fuzz effects like these, which IIRC excite the odd harmonics more, totally murder anything other than a root-fifth-octave power-chord.
>
> Also, the maj3's in power chords are really more like 10ths, since they're in the 2nd octave up from the root. And they're also covered up pretty well by the doubled fifth and tripled octave (in a full barre chord). But I digress, some distortions do allow the thirds to survive more or less, though there's a bit of clouding no matter what you do.
>
> -Igs

Thanks for this paper--it very clearly explains some things I've never thought about before.

Caleb

On Jul 26, 2010, at 9:24 PM, Margo Schulter wrote:

> Hello, all.
>
> To a very rich thread, please let me add a few ideas and a link.
> What I should emphasize is that there's not necessarily any one
> best way to describe temperaments, equal or otherwise.
>
> Furthermore, a ratio need not be tuneable by ear, in my
> view, to be useful in contrapuntal or harmonic as well as
> melodic practice, and possibly also to be a very useful
> landmark on the interval continuum in describing temperaments,
> equal and other.
>
> Here's a paper I wrote a few years and revised with some helpful
> suggestions from David Keenan and Andrew Heathwaite, to whom I
> express my gratitude while emphasizing that the views expressed
> are, of course, my responsibility and not theirs:
>
> <http://www.bestII.com/~mschulter/IntervalSpectrumRegions.txt>
>
> Rational ratios are in my view _one_ helpful landmark -- not
> the only one, or the one always most applicable. However, I
> do find complex as well simple ratios as friendly and welcoming
> landmarks when exploring a familiar region of the spectrum, or
> possibly a new one.
>
> For example, suppose I were writing a brief description of 29-EDO.
> Here are some possible ideas:
>
> * * *
>
> In describing or exploring 29-EDO, we can make the fertile territory
> of this temperament a bit more orderly by looking at three main
> "interval families" available.
>
> The first of these families consists of regular diatonic intervals
> built with chains of from one to six of the near-pure 29-EDO fifths
> (17/29 octave or about 703.45 cents). Since the fifths are less
> than 1.5 cents wide of a pure 3:2 (701.96 cents), these regular
> intervals are quite close to those of Pythagorean tuning using
> these just fifths -- a feature very conducive to some styles,
> and not so conducive to others.
>
> These wide 29-EDO fifths -- extended fifths, as they are sometimes
> called -- result in a kind of accentuated Pythagorean-like tuning
> which can be beautifully apt for the classic 13th-14th century
> styles of Europe, but quite unlikely the meantone or well-tempered
> systems that developed in the 15th-19th centuries. Major thirds
> at around 414 cents (slightly wide of 33:26), and minor thirds at
> 290 cents (just a tad wide of 13:11), very nicely fit a medieval
> style where these complex intervals, like the Pythagorean 81:64
> and 32:27 at 408 and 294 cents, typically resolve to stable
> intervals such as fifths and fourths. For those who relish
> these styles, the near-pure fourths and fifths of 29-EDO contrast
> nicely with the more complex thirds and sixths.
>
> A big asset of 29-EDO for these polyphonic progressions as well
> as pure melody is the regular diatonic semitone or limma
> (2/29 octave) at 83 cents, or somewhere between 22:21 (81 cents)
> and 21:20 (84 cents).
>
> From these regular intervals we move to the next family, augmented
> or diminished intervals formed from chains of 7-11 fifths.
>
> Again, people looking for subtly complex thirds and sixths will
> be happy, while those seeking out the simplest ratios (e.g. 5:4,
> 6:5, 7:6) may be disappointed. A diminished fourth (e.g. C#-F)
> from 8 fifths down or fourths up is 372 cents, a bit wide of
> the large neutral or submajor third at 26:21, while an augmented
> second from 9 fourths up at 331 cents (e.g. F-G#) is just narrow
> of of a small neutral or supraminor third at 63:52 (332 cents).
>
> Along with these thirds, this family also includes a chromatic
> semitone from 7 fifths up (e.g. C-C#), also known as an apotome,
> at 124 cents, a step a bit narrow of 14:13 (128 cents). A
> diminished third at 166 cents from ten fifths down (e.g. C#-Eb),
> precisely twice the size of the usual 83-cent semitone, yields
> a virtually just 11:10.
>
> For this family of intervals, at least two stylistic approaches
> are possible. One approach, for people looking for a harmonic
> approximation of 12-EDO or meantone, is to treat a 372-cent
> or 331-cent third as an inaccurate approximation of 5:4 (386
> cents) or 6:5 (316 cents). In such a context, both the
> notable impurity or complexity of the thirds and the wide
> nature of the 124-cent apotome which here serves in effect
> as a diatonic semitone, can create what is sometimes described
> as a languid or melancholy quality -- different, but not
> neessarily unpleasant.
>
> However, one may alternatively look on these steps and
> intervals as neutral, or at least semi-neutral, possibly
> representing, for example, 26:21, 63:52, and 14:13. In
> some Persian and Turkish contexts, and possibly in
> certain Kurdish ones also, steps of 124 and 165 cents
> may nicely fit the needs of the music -- and likewise
> the semi-neutral thirds, in polyphonic styles.
>
> For example, here is a basic tuning for a Turkish
> interpretation of Makam Huseyni, using a conventional
> note spelling to reflect the chains of fifths:
>
> |-------------| |-------------|
> 0 4 7 12 17 21 24 29
> C# Eb E F# G# Bb B C#
> 0 166 290 497 703 869 993 1200
>
> Makam Huseyni in its basic "textbook" form consists
> of two tetrachords, with a tone between them, each
> having a large neutral second (ideally around 11:10),
> a small neutral second, and an upper tone. It's
> interesting that a small step of 124 cents, found
> here at Eb-E and Bb-B, may occur in certain Persian
> modes.
>
> The third interval family of 29-EDO consists of rather
> complex intervals favored by many of the world's musical
> traditions, here termed _interseptimal_ intervals because
> they occur in the middle region between two septimal
> ratios such as the large 8:7 tone (231 cents) and the
> small 7:6 minor third (267 cents). In 29-EDO, we get
> an intriguing size from 14 fifths up of 248 cents --
> sometimes called a hemifourth, because it's equal
> more or less (here precisely) to half of a fourth.
>
> Hemifourths might suggest a simple septimal ratio
> like 8:7 or 7:6 -- and can delightfully be used
> contrapuntally in some styles as either a very
> wide tone or a very narrow minor third -- but are
> really a breed unto themselves. As it happens,
> the 29-EDO interval of 248 cents is a virtually
> just 15:13, while we also find intervals of
> 455 cents (near 13:10), 745 cents (near 20:13),
> and 952 cents (near 26:15).
>
> These intervals are used in world musics such as
> Balinese or Javanese gamelan, and additionally may
> be introduced into other types of contexts where
> they provide distinctly "xenharmonic" material.
>
> Interseptimal intervals are closely related to
> another member of this third family: the diesis
> of 1/29 octave, or 41 cents, by which 12 fifths
> exceed 7 octaves. This family thus begins with
> a chain of 12 fifths producing the diesis itself,
> and continues as various regular intervals are
> altered by this diesis. Thus a regular 414-cent
> major third plus the diesis, or 497-cent fourth
> minus it, produces a 455-cent interval almost
> exactly at 13:10, somewhere between a usual
> septimal major third at 9:7 (435 cents) and
> a 4:3 fourth. The diesis might, for example,
> be interpreted in JI terms as a 40:39 (44 cents),
> the difference between 13:10 and 4:3.
>
> For the adventurous, the 41-cent diesis may
> sometimes serve as an ultranarrow equivalent
> of a semitone, as may the larger 50-cent step
> in 24-EDO, for example. However, the diesis
> plays a vital role in the structure of the
> tuning for styles where it is unlikely to
> occur as a direct melodic step: for example,
> gamelan modes using the 248 cent hemifourths
> and other interseptimal intervals.
>
> In short, 29-EDO is a versatile system supporting
> an "accentuated Pythagorean" rendition of 13th-14th
> century European music; a family of "semi-neutral"
> intervals very useful for some Near Eastern styles,
> and sometimes applied to get a "different" sound
> for European music that might normally be played
> in a system like meantone, a well-temperament,
> or 12-EDO; and interseptimal intervals fitting
> various musical styles including gamelan as well
> as adventurous xenharmonicism.
>
> * * *
>
> This is just a sample about how I might approach
> an EDO system -- or a regular or irregular temperament
> of some other variety, for that matter. In 31-EDO,
> for example, we would have regular meantone, septimal,
> and neutral families. Knowing where these intervals
> fall on the spectrum, with rational ratios providing
> one set of useful landmarks, is only part of the
> question; what styles the reader of such a description
> is interested in, or might become interested in, is
> another.
>
> Best,
>
> Margo Schulter
> mschulter@...
>
>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> > Whatever happened to ol' Paul, anyway? Last I heard of him,
> > he was playing some BP keyboard with Ron Sword at the
> > BP Symposium.
>
> He's finally free of the tuning list! His band plays pretty
> regularly around the Boston area (12-ET, alas).

Even so, he was recently explaining starling temperament to Andrew Heathwaite.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> Gene> > "with Carl's apparent idea that around 27 cents and under
> > > (according to harmonic entropy) is when two ratios sound
> > > indistinguishable...correct?"
> >
> > Correct?
>
> Carl>"Certainly not."
>
>...
> So Carl, what is your idea so far as harmonic entropy (you used the terms "27 cents" and "indistinguishable" in the same sentence to describe what you meant before, but now seem to be denying correlation between those terms and your take on harmonic entropy)?

As far as I recall, Carl did not use either of those terms, even separately. He talked of the product of numerator and denominator being >70, and this was in answer to a question about complexity.

Dear Margo,

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> Here's a paper I wrote a few years and revised with some helpful
> suggestions from David Keenan and Andrew Heathwaite, to whom I
> express my gratitude while emphasizing that the views expressed
> are, of course, my responsibility and not theirs:
>
> <http://www.bestII.com/~mschulter/IntervalSpectrumRegions.txt>

Thank you for providing such a superbly-written and most helpful resource! Your writing is clear, concise, and (dare I say) elegant, and this paper is just what I needed.

What fascinates me a great deal is the fact that the "nobly intoned" mediants are essentially local maxima of harmonic entropy. Also, the fact that there is no noble mediant in the major second range is quite telling of...something. I feel as if I'm on the brink of revelation regarding harmony and musical perception, but I can't quite get over the last barrier to "enlightenment". There is something going on with the number phi, both in harmony and melody. I look at Wilson's "scale tree" and notice that something special is happening when, in an MOS scale, L:s = phi...but I can't put my finger on what it is. Some great pattern is staring me in the face, if I could but bring my focus to it. More meditation is in order.

Thank you, though, for giving me another piece of the puzzle!

-Igs

(concerning Margo's spectral regions article) Impressive, it really does go
in depth about how all the tonal classes listed in Scala are named as such
mathematically, how they derive from both musical history and use in modern
scale systems, how certain sections "blur" between others (esp. noble mediants),
and how far the boundaries of influence of each region stretch (much like in the
whole "harmonic entropy" theory).

Igs>"There is something going on with the number phi, both in harmony and
melody"
I get the feeling it represents a state of ambiguity, which can be viewed as
more or less resolved depending on the listener. It can actually make it seem
more resolved in some ways IE "it blends a bit with every other tone around it
having little bias, thus making it 'predictably ambiguous' " while less in
others IE "it's not optimized for any one tonal class".
PHI seems to me like the color white: it acts like a strong mix of colors (IE
maximum red + green + blue) and thus works with everything, but it's not
especially good or showing defined flavor/bias with anything either.

On 27 July 2010 05:12, cityoftheasleep <igliashon@...> wrote:

> Well, what I was trying to say is that if you were
> tuning a major second by ear, you might hit a
> ratio that sounds rather in-tune and say "aha!  I
> tuned to a Just major 2nd!" and it might be a
> 10/9 or a 9/8 or maybe an 8/7 or an 11/10 but
> you wouldn't be able to tell which.  But I would
> imagine that ratios around the major second area
> have a strong enough periodicity that they'd given
> a nice JI buzz.  Maybe only in higher registers.
> But still.  I think it's possible in theory.

I hope you'd be able to tell an 8/7 from an 11/10. There's about 44
cents between them. If that's inaudible we may as well give up this
microtonality business altogether.

Graham

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
> ... Now minor thirds are a difference story. But major 12 et 3rds are a part of many power chords....

Hi Chris,
I am confused - I thought that power chords were defined as doh-soh-doh? i.e. no 3rd is struck (but one may be heard :-)

Steve M.

Hi Steve,

I can understand your confusion - I did not know that the definition
of a power chord changed since I used guitar chord charts in the early
70's as I taught myself guitar. In the dark ages a common power chord
was the same as the E major at the nut but with a bar - say for
instance at the 5th fret it would be an a major. I still use this and
also the equivalent to A major at the nut. With lots of distortion.

Those fifths I see listed as "power chords" today are really "power
dyads"......

Just as an aside
There are instances where Toni Iommi of Black Sabbath works in minor
and minor + minor 7th chords - (see Cornucopia from the album Volume 4
by Black Sabbath). That particular progression goes as I used to play
it at the 3rd fret G Gm C C m7. Though I wonder if I turned
distortion setting from Sabbath to Carcass if it would still work as
well. And I will admit BS cheated in production by layering a bright
relatively clean guitar over the heavily distorted deep one.

Chris

chris said ... Now minor thirds are a difference story. But major 12
et 3rds are a part of many power chords....

Hi Chris,
I am confused - I thought that power chords were defined as
doh-soh-doh? i.e. no 3rd is struck (but one may be heard :-)

Steve M.

My point wasn't that they're indistinguishable (that's surely a ludicrous thing to claim), but that if you were trying to tune a "major second" Justly by ear, you wouldn't be able to confidently claim which ratio you'd tuned. You'd probably land between 10/9 and 9/8, but I mentioned 11/10 and 8/7 because they are outside possibilities of what someone might hear as a major second.

I tried tuning some major seconds by ear the other day on guitar, though sadly I did not have my keyboard handy to "check my work", and I sure found it to be a troubling range. I did find a few "notches" that felt like JI, one of which was below 8/7 and the other above 12/11 (I was tuning my 16-EDO guitar, so using those frets as reference points)...but who knows what I actually tuned? I'll try again when I have some time, and see if I can't measure what I tune to using a retunable keyboard.

-Igs

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> On 27 July 2010 05:12, cityoftheasleep <igliashon@...> wrote:
>
> > Well, what I was trying to say is that if you were
> > tuning a major second by ear, you might hit a
> > ratio that sounds rather in-tune and say "aha!  I
> > tuned to a Just major 2nd!" and it might be a
> > 10/9 or a 9/8 or maybe an 8/7 or an 11/10 but
> > you wouldn't be able to tell which.  But I would
> > imagine that ratios around the major second area
> > have a strong enough periodicity that they'd given
> > a nice JI buzz.  Maybe only in higher registers.
> > But still.  I think it's possible in theory.
>
> I hope you'd be able to tell an 8/7 from an 11/10. There's about 44
> cents between them. If that's inaudible we may as well give up this
> microtonality business altogether.
>
>
> Graham
>

cityoftheasleep wrote:

> I tried tuning some major seconds by ear the other day on
> guitar, though sadly I did not have my keyboard handy to
> "check my work", and I sure found it to be a troubling range.
> I did find a few "notches" that felt like JI, one of which
> was below 8/7 and the other above 12/11 (I was tuning my
> 16-EDO guitar, so using those frets as reference points)...
> but who knows what I actually tuned? I'll try again when
> I have some time, and see if I can't measure what I tune to
> using a retunable keyboard.

There is nothing better than hands-on experience.
You should try to get a tuner that displays cents offsets
so you can see what you tuned.

-Carl

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
> ... I did not know that the definition of a power chord changed since I used guitar chord charts in the early 70's as I taught myself guitar. ...

Thanks Chris, no problem. I should say that I know very little about rock (or pop or jazz) but in my limited listening I had never noticed that the 3rd was missing, until I read about it. The wikipedia entry for "power chord" says it can sound like major or minor depending on context. I guess this means it is a psychoacoustical effect, rather than a 3rd actually generated in the instrument/ sound system (does anyone here know)? I assume that the 5th is tempered, so that would make the 3rd impure too?

Steve M.

This version of Greenday doing I Fought the Law bothers me because they play power chords going 5-4-3-1 in F C-Bb-A-F, and the A sounds like a major chord.

http://www.youtube.com/watch?v=_TOVkiBE2r4

Sounds bad to me.

caleb

On Jul 29, 2010, at 11:38 AM, martinsj013 wrote:

> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
> > ... I did not know that the definition of a power chord changed since I used guitar chord charts in the early 70's as I taught myself guitar. ...
>
> Thanks Chris, no problem. I should say that I know very little about rock (or pop or jazz) but in my limited listening I had never noticed that the 3rd was missing, until I read about it. The wikipedia entry for "power chord" says it can sound like major or minor depending on context. I guess this means it is a psychoacoustical effect, rather than a 3rd actually generated in the instrument/ sound system (does anyone here know)? I assume that the 5th is tempered, so that would make the 3rd impure too?
>
> Steve M.
>
>

When I was learning guitar in the early '90s, a power chord was unambiguously (at least in the guitar magazines of the time) a 3-note chord, consisting of root-fifth-octave. It was most popular among punk bands of the era, since (lacking a third) it required very little musical understanding to employ, and is the most stable "triad" (if you can call it that) in 12-equal and thus will sound fine no matter how much distortion you use. Popular distortion pedals of the day, like the Boss DS-1 and the Electro-Harmonix Big Muff, tend to render more complex chords into a less-intelligible mush. Hence, in genres that rely on heavy distortion, the harmony tends to be very simple.

-Igs

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@> wrote:
> > ... I did not know that the definition of a power chord changed since I used guitar chord charts in the early 70's as I taught myself guitar. ...
>
> Thanks Chris, no problem. I should say that I know very little about rock (or pop or jazz) but in my limited listening I had never noticed that the 3rd was missing, until I read about it. The wikipedia entry for "power chord" says it can sound like major or minor depending on context. I guess this means it is a psychoacoustical effect, rather than a 3rd actually generated in the instrument/ sound system (does anyone here know)? I assume that the 5th is tempered, so that would make the 3rd impure too?
>
> Steve M.
>

Interesting on the idea of major / minor perception. I can't say I've *ever*
had that perception from a 5th.

The fifth in 12 equal is almost dead on (700 cents in 12 versus 702 in pure)
so I don't think that is an issue. The major third is quite a bit sharp in
12 equal - sharp enough that I can distinguish the JI major third from the
12 equal major third on my fretless guitar easily.

Chris

On Thu, Jul 29, 2010 at 11:38 AM, martinsj013 <martinsj@...> wrote:

>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> > ... I did not know that the definition of a power chord changed since I
> used guitar chord charts in the early 70's as I taught myself guitar. ...
>
> Thanks Chris, no problem. I should say that I know very little about rock
> (or pop or jazz) but in my limited listening I had never noticed that the
> 3rd was missing, until I read about it. The wikipedia entry for "power
> chord" says it can sound like major or minor depending on context. I guess
> this means it is a psychoacoustical effect, rather than a 3rd actually
> generated in the instrument/ sound system (does anyone here know)? I assume
> that the 5th is tempered, so that would make the 3rd impure too?
>
> Steve M.
>
>
>

looks to me that all of the chords are major and are of two types - if in
the nut position E major and A major.

I agree at that extended section something sounds out - but this is a live
recording and strings do go out of tune.

And I don't know about the rest of the guitarists here - I find that is I
tune my guitar to 12 equal exact with a tuner I then go back and make
adjustments - especially to the B string - to make it sound better.

Chris

On Thu, Jul 29, 2010 at 11:53 AM, caleb morgan <calebmrgn@...> wrote:

>
>
> This version of Greenday doing I Fought the Law bothers me because they
> play power chords going 5-4-3-1 in F C-Bb-A-F, and the A sounds like a major
> chord.
>
> http://www.youtube.com/watch?v=_TOVkiBE2r4
>
> Sounds bad to me.
>
> caleb
>
>
>
> On Jul 29, 2010, at 11:38 AM, martinsj013 wrote:
>
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> > ... I did not know that the definition of a power chord changed since I
> used guitar chord charts in the early 70's as I taught myself guitar. ...
>
> Thanks Chris, no problem. I should say that I know very little about rock
> (or pop or jazz) but in my limited listening I had never noticed that the
> 3rd was missing, until I read about it. The wikipedia entry for "power
> chord" says it can sound like major or minor depending on context. I guess
> this means it is a psychoacoustical effect, rather than a 3rd actually
> generated in the instrument/ sound system (does anyone here know)? I assume
> that the 5th is tempered, so that would make the 3rd impure too?
>
> Steve M.
>
>
>
>

I never learned guitar, but when I was learning to play keys
with my guitarist friends in the early '90s, a power chord was
unambiguously a 3-note chord consisting of root-fifth-octave. :)

-Carl

cityoftheasleep wrote:

> When I was learning guitar in the early '90s, a power chord was
> unambiguously (at least in the guitar magazines of the time) a
> 3-note chord, consisting of root-fifth-octave.

I'm surprised that you are saying punk is heavily distorted - most punk I've
heard uses more classic rock level distortion - or cleaner - and with good
midrange and high end response. I can't see how a 3rd would be taboo in that
level of distortion and that EQ.

Now... if you are talking death / black metal - I can certianly understand
where a third is a problem. As you say complex chords do become mush in that
scenario - I certainly agree

BUT I heartily disagree that genres that have heavy distortion *have* to
have simple harmony - there are other intervals, as I stated earlier, that
*do* work with heavy distortion, at least to my ears.

By the way I believe they were selling these (Electro-Harmonix Big Muff,)
when I was a kid. :-)

My fav distortion pedal is MXR distortion + - because it does NOT color my
tone and I can still adjust the tone with my guitar.

chris

On Thu, Jul 29, 2010 at 1:04 PM, cityoftheasleep <igliashon@...>wrote:

>
>
> When I was learning guitar in the early '90s, a power chord was
> unambiguously (at least in the guitar magazines of the time) a 3-note chord,
> consisting of root-fifth-octave. It was most popular among punk bands of the
> era, since (lacking a third) it required very little musical understanding
> to employ, and is the most stable "triad" (if you can call it that) in
> 12-equal and thus will sound fine no matter how much distortion you use.
> Popular distortion pedals of the day, like the Boss DS-1 and the
> Electro-Harmonix Big Muff, tend to render more complex chords into a
> less-intelligible mush. Hence, in genres that rely on heavy distortion, the
> harmony tends to be very simple.
>
> -Igs
>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, "martinsj013"
> <martinsj@...> wrote:
> >
> > --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@> wrote:
> > > ... I did not know that the definition of a power chord changed since I
> used guitar chord charts in the early 70's as I taught myself guitar. ...
> >
> > Thanks Chris, no problem. I should say that I know very little about rock
> (or pop or jazz) but in my limited listening I had never noticed that the
> 3rd was missing, until I read about it. The wikipedia entry for "power
> chord" says it can sound like major or minor depending on context. I guess
> this means it is a psychoacoustical effect, rather than a 3rd actually
> generated in the instrument/ sound system (does anyone here know)? I assume
> that the 5th is tempered, so that would make the 3rd impure too?
> >
> > Steve M.
> >
>
>
>

Well, 20 years is a long time in pop music Carl. I'm not challenging that
things have changed.

Chris

On Thu, Jul 29, 2010 at 1:42 PM, Carl Lumma <carl@...> wrote:

>
>
> I never learned guitar, but when I was learning to play keys
> with my guitarist friends in the early '90s, a power chord was
> unambiguously a 3-note chord consisting of root-fifth-octave. :)
>
> -Carl
>
>
> cityoftheasleep wrote:
>
> > When I was learning guitar in the early '90s, a power chord was
> > unambiguously (at least in the guitar magazines of the time) a
> > 3-note chord, consisting of root-fifth-octave.
>
>
>

The distortion makes the 5th partial a little stronger.

perhaps:

2,3,4--sum tones = 5,6,7

difference tones = 1,2

Or, with piano, I just added distortion to my piano sound, and the 5th partial is pretty strong.

Power chords are more major than minor because of the overtones?

caleb

On Jul 29, 2010, at 1:42 PM, Carl Lumma wrote:

> I never learned guitar, but when I was learning to play keys
> with my guitarist friends in the early '90s, a power chord was
> unambiguously a 3-note chord consisting of root-fifth-octave. :)
>
> -Carl
>
> cityoftheasleep wrote:
>
> > When I was learning guitar in the early '90s, a power chord was
> > unambiguously (at least in the guitar magazines of the time) a
> > 3-note chord, consisting of root-fifth-octave.
>
>

> When I was learning guitar in the early '90s, a power chord was
> unambiguously (at least in the guitar magazines of the time) a
> 3-note chord, consisting of root-fifth-octave.

Which is also how I learned to play power chords. Although I have seen
people (IE Nirvana, I believe) use an "inverted" C5 F5 C6 variant...then again
all that does is make the intervals 4th then 5th instead of 5th then 4th. Oddly
enough, my ears like the inverted version better...

Well, I dragged out my Super Fuzz pedal (70's vintage) and *with* the amp's
distortion (Crate IIR) and played around with some intervals. Lo and behold
- a Just major third sounds to me as nice and beatless as a fourth or fifth.
So... JI metal anyone? Or perhaps Lucy Tuned metal? There is a thought....

Here is a pic of my pedal

http://www.univox.org/pics/effects/superfuzz_red.gif

And honestly I see no reason the following dyads can't be used - minor 7th,
minor 9th - of course they are not as "clean" as a octave or fifth or fourth
- but to me usable and adds variety.

Chris

On Thu, Jul 29, 2010 at 1:52 PM, caleb morgan <calebmrgn@...> wrote:

>
>
> The distortion makes the 5th partial a little stronger.
>
> perhaps:
>
> 2,3,4--sum tones = 5,6,7
>
> difference tones = 1,2
>
> Or, with piano, I just added distortion to my piano sound, and the 5th
> partial is pretty strong.
>
> Power chords are more major than minor because of the overtones?
>
> caleb
>
>
> On Jul 29, 2010, at 1:42 PM, Carl Lumma wrote:
>
>
>
> I never learned guitar, but when I was learning to play keys
> with my guitarist friends in the early '90s, a power chord was
> unambiguously a 3-note chord consisting of root-fifth-octave. :)
>
> -Carl
>
> cityoftheasleep wrote:
>
> > When I was learning guitar in the early '90s, a power chord was
> > unambiguously (at least in the guitar magazines of the time) a
> > 3-note chord, consisting of root-fifth-octave.
>
>
>
>

Lo and behold
> - a Just major third sounds to me as nice and beatless as a fourth or fifth.
> So... JI metal anyone? Or perhaps Lucy Tuned metal? There is a thought....

I made an experimental four string guitar-like instrument with a 19-tone subset of 50-edo partly with the intention of testing out overdriven triads. Problem is, I've been too lazy to electrify it so far.

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Well, I dragged out my Super Fuzz pedal (70's vintage) and *with* the amp's
> distortion (Crate IIR) and played around with some intervals. Lo and behold
> - a Just major third sounds to me as nice and beatless as a fourth or fifth.
> So... JI metal anyone? Or perhaps Lucy Tuned metal? There is a thought....

Niiiice pedal! I'm jealous. And I also agree, when I'm tuning my open strings, sometimes I just tune to pure fourths, and more than a few times I've tuned a Just major third (5/4) and thought it was a fourth!

In 16-EDO, I use 11/6 (neutral seventh) and 7/4 (harmonic seventh) as power chords. Works pretty well, if you can believe it!

-Igs

By power chords do you mean naked dyads?

in that case I certainly think I can believe it!

On Fri, Jul 30, 2010 at 12:23 AM, cityoftheasleep
<igliashon@...>wrote:

>
>
> --- In tuning@yahoogroups.com <tuning%40yahoogroups.com>, Chris Vaisvil
> <chrisvaisvil@...> wrote:
> >
> > Well, I dragged out my Super Fuzz pedal (70's vintage) and *with* the
> amp's
> > distortion (Crate IIR) and played around with some intervals. Lo and
> behold
> > - a Just major third sounds to me as nice and beatless as a fourth or
> fifth.
> > So... JI metal anyone? Or perhaps Lucy Tuned metal? There is a
> thought....
>
> Niiiice pedal! I'm jealous. And I also agree, when I'm tuning my open
> strings, sometimes I just tune to pure fourths, and more than a few times
> I've tuned a Just major third (5/4) and thought it was a fourth!
>
> In 16-EDO, I use 11/6 (neutral seventh) and 7/4 (harmonic seventh) as power
> chords. Works pretty well, if you can believe it!
>
> -Igs
>
>
>