Yahoo Tuning Groups Ultimate Backup tuning Plausibly "Just" Triads

/tuning/files/IgliashonJones/JI%20Triads.pdf

I've been searching the harmonic series for triads that would form good bases for subgroup temperaments, and this is what I've come up with. I based this list on the following rules of thumb:

1. No octave inversions--all chords listed are the simplest (Tenney Height-wise) voicings.

2. No primes above 13.

3. No harmonics above 32; because of 1 and 2, this actually restricts to 27.

4. At least two dyads in the triad must be 15 odd-limit or lower.

I'm posting this because I want to know if you guys think these restrictions are reasonable for what to consider "Just". They seem reasonable to me, but I'd love to hear any objections.

-Igs

🔗Carl Lumma <carl@...>

"cityoftheasleep" <igliashon@...> wrote:

>I based this list on the following rules of thumb:
>
> 1. No octave inversions--all chords listed are the simplest
> (Tenney Height-wise) voicings.
> 2. No primes above 13.
> 3. No harmonics above 32; because of 1 and 2, this actually
> restricts to 27.
> 4. At least two dyads in the triad must be 15 odd-limit or lower.
> I'm posting this because I want to know if you guys think these
> restrictions are reasonable for what to consider "Just". They
> seem reasonable to me, but I'd love to hear any objections.

Sounds reasonable to me. -C.

🔗Michael <djtrancendance@...>

Igs>"2. No primes above 13."

To me, the buck generally stops at 11 for "prime limit" of chords...with only the exceptions of 13/9 and 13/7 IE 7:10:13. Anything with 13/8 or 13/11 in particular, I've found, is virtually always "too sour", and 8:11:13 becomes nightmare-ish. :-D.

>"3. No harmonics above 32; because of 1 and 2, this actually restricts to 27. "

Sounds just about perfect...see my JI program (calculates the JI chord closest to any given chord)...I generally get the best results by typing in 30 or under as the "limit" parameter.

>"4. At least two dyads in the triad must be 15 odd-limit or lower."

15 sounds too far to me. Actually I would say 9-limit is about where things start tipping: generally chords even within my scales only have one 15-limit ratio and at least 2 7-or-lower-limit dyads, but I can see 9-limit being OK in some cases.

🔗genewardsmith <genewardsmith@...>

🔗lobawad <lobawad@...>

I think this is a completely reasonable, and mainstream-friendly, method of approaching and list of "plausibly Just" intervals.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> /tuning/files/IgliashonJones/JI%20Triads.pdf
>
> I've been searching the harmonic series for triads that would form good bases for subgroup temperaments, and this is what I've come up with. I based this list on the following rules of thumb:
>
> 1. No octave inversions--all chords listed are the simplest (Tenney Height-wise) voicings.
>
> 2. No primes above 13.
>
> 3. No harmonics above 32; because of 1 and 2, this actually restricts to 27.
>
> 4. At least two dyads in the triad must be 15 odd-limit or lower.
>
> I'm posting this because I want to know if you guys think these restrictions are reasonable for what to consider "Just". They seem reasonable to me, but I'd love to hear any objections.
>
> -Igs
>

🔗bigAndrewM <bigandrewm@...>

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:

> I've been searching the harmonic series for triads that would form good bases for subgroup temperaments, and this is what I've come up with. I based this list on the following rules of thumb:

Suppose we take on of your triads, and assume octave equivalence. We can divide by the least integer, getting two JI intervals which are equivalent to your three integers. Then we can remove factors of two, by octave equivalence. Then we can add 2 to the two intervals, again by octave equivalence. Then we can find the normal list corresponding to this list, which gives the subgroup your triad generates in a canonical form. Doing this to each of your 72 triads gives 49 subgroups:

[[2, 3], [2, 5], [2, 5/3], [2, 3, 5], [2, 3, 7], [2, 3, 11], [2, 3, 13], [2, 3, 7/5], [2, 3, 11/5], [2, 3, 13/5], [2, 3, 13/7], [2, 5, 7], [2, 5, 11], [2, 5, 13], [2, 5, 7/3], [2, 5, 11/3], [2, 5, 13/3], [2, 7, 11], [2, 7, 13], [2, 7, 13/3], [2, 9, 5], [2, 9, 7], [2, 9, 11], [2, 9, 13], [2, 9, 15], [2, 9, 21], [2, 11, 13], [2, 15, 7], [2, 15, 11], [2, 15, 13], [2, 5/3, 7/3], [2, 5/3, 9/7], [2, 5/3, 11/3], [2, 5/3, 11/9], [2, 5/3, 13/3], [2, 5/3, 13/9], [2, 7/3, 11/3], [2, 7/3, 13/3], [2, 7/5, 11/5], [2, 7/5, 13/5], [2, 7/5, 15/13], [2, 9/5, 9/7], [2, 9/5, 13/9], [2, 9/7, 11/9], [2, 9/7, 13/9], [2, 11/5, 13/5], [2, 11/7, 13/7], [2, 15/7, 15/11], [2, 15/11, 15/13]]

The association of each triad to the corresponding subgroup is as follows:

[3, 4, 5]: [2, 3, 5]
[4, 5, 7]: [2, 5, 7]
[4, 6, 7]: [2, 3, 7]
[5, 6, 7]: [2, 5/3, 7/3]
[5, 6, 9]: [2, 3, 5]
[5, 7, 9]: [2, 9/5, 9/7]
[5, 8, 9]: [2, 9, 5]
[6, 7, 9]: [2, 3, 7]
[6, 8, 9]: [2, 3]
[6, 7, 11]: [2, 7/3, 11/3]
[6, 8, 11]: [2, 3, 11]
[6, 9, 11]: [2, 3, 11]
[6, 10, 11]: [2, 5/3, 11/3]
[7, 8, 9]: [2, 9, 7]
[7, 8, 11]: [2, 7, 11]
[7, 9, 11]: [2, 9/7, 11/9]
[7, 10, 11]: [2, 7/5, 11/5]
[7, 8, 13]: [2, 7, 13]
[7, 9, 13]: [2, 9/7, 13/9]
[7, 10, 13]: [2, 7/5, 13/5]
[7, 11, 13]: [2, 11/7, 13/7]
[7, 12, 13]: [2, 7/3, 13/3]
[8, 9, 11]: [2, 9, 11]
[8, 10, 11]: [2, 5, 11]
[8, 9, 13]: [2, 9, 13]
[8, 10, 13]: [2, 5, 13]
[8, 11, 13]: [2, 11, 13]
[8, 12, 13]: [2, 3, 13]
[8, 9, 15]: [2, 9, 15]
[8, 10, 15]: [2, 3, 5]
[8, 11, 15]: [2, 15, 11]
[8, 12, 15]: [2, 3, 5]
[8, 13, 15]: [2, 15, 13]
[8, 14, 15]: [2, 15, 7]
[9, 10, 13]: [2, 9/5, 13/9]
[9, 10, 15]: [2, 3, 5]
[9, 11, 15]: [2, 5/3, 11/9]
[9, 13, 15]: [2, 5/3, 13/9]
[10, 11, 13]: [2, 11/5, 13/5]
[10, 12, 13]: [2, 5/3, 13/3]
[10, 11, 15]: [2, 3, 11/5]
[10, 12, 15]: [2, 3, 5]
[10, 13, 15]: [2, 3, 13/5]
[10, 14, 15]: [2, 3, 7/5]
[11, 12, 15]: [2, 5, 11/3]
[11, 13, 15]: [2, 15/11, 15/13]
[11, 14, 15]: [2, 15/7, 15/11]
[12, 13, 15]: [2, 5, 13/3]
[12, 14, 15]: [2, 5, 7/3]
[12, 13, 21]: [2, 7, 13/3]
[12, 14, 21]: [2, 3, 7]
[12, 16, 21]: [2, 3, 7]
[13, 14, 21]: [2, 3, 13/7]
[13, 15, 21]: [2, 7/5, 15/13]
[14, 15, 21]: [2, 3, 7/5]
[14, 16, 21]: [2, 3, 7]
[14, 18, 21]: [2, 3, 7]
[14, 15, 25]: [2, 5/3, 9/7]
[14, 20, 25]: [2, 5, 7]
[15, 18, 25]: [2, 5/3]
[15, 21, 25]: [2, 5/3, 7/3]
[15, 20, 27]: [2, 3, 5]
[15, 25, 27]: [2, 3, 5]
[16, 18, 21]: [2, 9, 21]
[16, 20, 25]: [2, 5]
[16, 18, 27]: [2, 3]
[16, 24, 27]: [2, 3]
[18, 20, 25]: [2, 9, 5]
[18, 20, 27]: [2, 3, 5]
[18, 22, 27]: [2, 3, 11]
[20, 22, 25]: [2, 5, 11]
[20, 24, 27]: [2, 9, 15]

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

>
Suppose we take on of your triads, and assume octave equivalence. We can divide by the least integer, getting two JI intervals which are equivalent to your three integers. Then we can remove factors of two, by octave equivalence. Then we can add 2 to the two intervals, again by octave equivalence. Then we can find the normal list corresponding to this list, which gives the subgroup your triad generates in a canonical form. Doing this to each of your 72 triads gives 49 subgroups:
>

Hey, that's pretty neat! Thanks, Gene. A question, though: in cases where multiple triads collapse to the same subgroup, will it be the case that different temperaments of that subgroups will have different complexity orderings of the different triads? For example, will all temperaments of [2, 3, 7] give (say) 12:14:21, 12:16:21, 14:16:21, and 14:18:21 in the same order of complexity? If so, then that really narrows down what I need to look for.

-Igs

🔗cityoftheasleep <igliashon@...>

Well, it's just a starting-point, really. I needed to trim down the humongous range of possible triads within the first 32 harmonics. I wanted to err on the "generous" side of plausible. I'm going to go through and actually play the questionable triads and see if they do what they're supposed to as Just triads.

8:11:13 is definitely a tough case, I have tried it and I'm a bit on the fence with it. It seems pretty weak, and is very sensitive to mistuning (thus may not be suitable for tempering). It might be good to add a rule of "no more than one prime above 7", i.e. eliminating all chords with both a 13 and an 11 in them. I don't have any problem with 8:13, though, and I think 8:10:13 sounds pretty good. Yeah, it's on the border, but I think the 4:5 gives it just enough of a boost.

Also, I left out the rule of "no dyads closer than 15:16". This may still be too close, the cut-off may be more like 13:14 or 12:13, but I did use that rule in generating this list.

I'll see what my ears tell me, and when I find some examples that trouble me, I'll post them here to put them to a vote. I do expect the list to shrink a bit after listening tests, though.

-Igs

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
>
> Igs>"2. No primes above 13."
>
> To me, the buck generally stops at 11 for "prime limit" of chords...with only the exceptions of 13/9 and 13/7 IE 7:10:13. Anything with 13/8 or 13/11 in particular, I've found, is virtually always "too sour", and 8:11:13 becomes nightmare-ish. :-D.
>
>
> >"3. No harmonics above 32; because of 1 and 2, this actually restricts to 27. "
>
> Sounds just about perfect...see my JI program (calculates the JI chord closest to any given chord)...I generally get the best results by typing in 30 or under as the "limit" parameter.
>
>
> >"4. At least two dyads in the triad must be 15 odd-limit or lower."
>
> 15 sounds too far to me. Actually I would say 9-limit is about where things start tipping: generally chords even within my scales only have one 15-limit ratio and at least 2 7-or-lower-limit dyads, but I can see 9-limit being OK in some cases.
>

🔗Mike Battaglia <battaglia01@...>

On Wed, Mar 9, 2011 at 1:29 PM, cityoftheasleep <igliashon@...> wrote:
>
> Well, it's just a starting-point, really. I needed to trim down the humongous range of possible triads within the first 32 harmonics. I wanted to err on the "generous" side of plausible. I'm going to go through and actually play the questionable triads and see if they do what they're supposed to as Just triads.

In terms of these being the only triads that should get the "just"
moniker, I disagree. I like triads like 8:10:14:19, to name a simple
example. Then you have 14:17:21, which is a 17-limit superminor chord
that is lower in complexity than 14:18:21. And besides those examples,
check out some of the 17-limit examples in the "semitonal" thread I
made a few weeks ago, which I came up with without thinking within the
prime-limit paradigm at all.

On the other hand, in terms of these being a good list of JI subgroups
to get us started, I do agree.

-Mike

🔗Michael <djtrancendance@...>

Igs>"8:11:13 is definitely a tough case, I have tried it and I'm a bit on the
fence with it. It seems pretty weak, and is very sensitive to
mistuning (thus may not be suitable for tempering)."

Yep, that's the chord I mentioned as being relatively low odd-limit, but strikingly sour. To me, the 13/11 in there and the 13/8 even more so tear it apart to a great deal despite the chord's being fairly "periodic" far as being 13-odd-limit and no higher.

Come I think of it, to myself at least, the most sour dyads ever...20/11, 16/11, 13/10, 17/13 , 13/11, 49/33 (wolf fifth), 18/13, 20/13, 21/13, and 13/8 are pretty horrid and tend to tear apart chords regardless of their limit. As you can see I'm not a big fan of 13 odd limit and find 11 limit significantly better...in fact when a 13-odd-limit dyad works...I find it coincides with a much lower limit dyad (IE acts more like a tempered dyad than its own stable entity).
In fact even higher odd-limit dyads tend to sound better...as the tend to be closer to the "field of attraction" of lower-limit dyads that work than the above dyads.

If you need an easy way to figure out which chords won't work...I would recommend looking for the above dyads...specifically the 13-odd-limit ones.

>"It might be good to add a rule of "no more than one prime above 7",

Good general idea, though I might raise that to 9 and also note that certain 11-limit dyads can be put together to sound stable IE 1/1 11/9 22/15 with 6/5, 11/9, and 22/15 (inversion of 15/11)...which has two 11-limit dyads.

>"Also, I left out the rule of "no dyads closer than 15:16". This may
still be too close, the cut-off may be more like 13:14 or 12:13, but I
did use that rule in generating this list."

   Interesting idea...although I'd say, even for triads, you can get under 15/16...you just have to have another dyad that's very large and stable to balance it out. IE 1/1 16/15 4/3 =
15:16:20 with a nice smooth 5/4 to help stabilize the 16/15.
   I'd even say...you can really push the critical band for any one dyad, so far as you stretch far from it in the others. And I am thinking the chords 9:10:11 and 10:11:12 represent about as far as people could likely take so far as clustering of notes in triads.

🔗genewardsmith <genewardsmith@...>

🔗john777music <jfos777@...>

Igs,

here's my take on just triads.

Identify all the just intervals an octave or less wide that you consider to be good in harmony. Then when you make a triad if the three intervals that occur in the triad are good then the triad should be *good*.

Here's my list of good intervals...

9/8, 8/7, 7/6, 6/5, 5/4, 9/7, 4/3, 11/8, 7/5, 10/7, 3/2, 11/7, 8/5, 5/3, 12/7, 7/4, 9/5, 11/6, 13/7, 2/1.

John.

🔗Carl Lumma <carl@...>

🔗Michael <djtrancendance@...>

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> In terms of these being the only triads that should get the "just"
> moniker, I disagree. I like triads like 8:10:14:19, to name a simple
> example. Then you have 14:17:21, which is a 17-limit superminor chord
> that is lower in complexity than 14:18:21. And besides those examples,
> check out some of the 17-limit examples in the "semitonal" thread I
> made a few weeks ago, which I came up with without thinking within the
> prime-limit paradigm at all.

14:17:21 does not sound Just to me. It sounds like an out-of-tune 10:12:15. 14:18:21 is higher in complexity but is composed of simpler dyads--6:7, 7:9, 2:3. 14:17:21 has 14:17, 17:21, and 2:3. This makes more of a difference than the triadic complexity. I tried a ton of 17- and 19-limit chords, and while they did sound "nice" they don't blend as well as those composed of simpler dyads. Also, whatever weak identity they might have, they tolerate very little tempering--if any at all.

Justness does seem to have some gray areas, but I'm trying to stay out of them.

> On the other hand, in terms of these being a good list of JI subgroups
> to get us started, I do agree.

Fair enough.

-Igs

🔗cityoftheasleep <igliashon@...>

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

> Yes, good point. I certainly don't agree that odd numbers
> over 13 have any advantage to primes > 13. Igs: just use odds.

Dyadically-speaking, I agree. Triadically, I disagree strongly.

When a triad is composed of simple dyads, that seems to exert some sort of effect on its concordance. You can break odds down so long as they are not primes. This is why chords like 16:18:21 and 16:20:25 have any concordance at all. I say this having spent a good deal of time playing around with the harmonic series. 16:19:25 sounds much weaker and more discordant than 16:20:25.

What I'm not sure of is how high the odds can get before the triadic complexity outweighs the simplicity of the component dyads.

I really believe there are multiple factors when it comes to triadic concordance--critical band, component dyads, and prime-limit (to say nothing of entropy) all seem to exert varying degrees of influence.

-Igs

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
>
> > Yes, good point. I certainly don't agree that odd numbers
> > over 13 have any advantage to primes > 13. Igs: just use odds.
>
> Dyadically-speaking, I agree. Triadically, I disagree strongly.
>
> When a triad is composed of simple dyads, that seems to exert
> some sort of effect on its concordance.

This reminds me that it might be a good time to send you a
listening test I made a while back. Would you like to give
it a try?

-Carl

🔗cityoftheasleep <igliashon@...>

🔗Michael <djtrancendance@...>

Igs>"14:17:21 does not sound Just to me. It sounds like an out-of-tune 10:12:15. "
21/17 is darn near the rather sour 16/13...and 17/14 fairly far from 11/9, close enough to 6/5 (minor third) to be confused with it, but far enough from 6/5 where it, like Igs said, just sounds like an out-of-tune minor chord.
To make a more "stable" version of that chord, I would suggest simply taking 11/9 * 11/9 = approximately 3/2, which would be pretty much a neutral triad (neither major nor minor...far enough from either IMVHO to be distinct and stable as its own entity).

>"Justness does seem to have some gray areas, but I'm trying to stay out of them."

To me there are three definite classes here: minor, neutral, and major...and getting stuck in the center of those (IE in so called "gray areas" like you mentioned, such as the midpoint between neutral and minor thirds) means a very unstable-sounding chord hovering between "fields of attraction" (term borrowed from Harmonic Entropy).

🔗Michael <djtrancendance@...>

Igs>"I really believe there are multiple factors when it comes to triadic
concordance--critical band, component dyads, and prime-limit (to say
nothing of entropy) all seem to exert varying degrees of influence."

Awesome, we need more people like this who look at acoustics with an open mind toward influence of all theories instead of a view that "theory x always beats theory y".

>"What I'm not sure of is how high the odds can get before the triadic
complexity outweighs the simplicity of the component dyads. "

   Excellent question, in my opinion. To me it seems to be (like you said) around harmonic 30 or so on the harmonic series.
Though, in general, I find if 70% or so (or more) of the dyads that form a chord are 7-odd-limit or lower on average (say, one 5-limit dyad, one 9-limit, one 11-limit)...the overall chord should be fairly stable. I view triadic complexity as a nice way to simplify things further once you get something that works well with component dyad complexity IE a way to "slightly tweak it"...rather than a preferred way to make chords from scratch.
   Sure you can use it to make chords from scratch, but then how many not-so-huge scales will be able to estimate it well? I guess you could say that's why I stick to dyads when making scales initially...and then tweak them slightly to form better triads.

🔗Mike Battaglia <battaglia01@...>

On Wed, Mar 9, 2011 at 6:00 PM, cityoftheasleep <igliashon@...> wrote:
>
> 14:17:21 does not sound Just to me. It sounds like an out-of-tune 10:12:15.

I would contest that 10:12:15 is the one true minor chord to begin
with. But, assuming that it is, what about 10:12:15:17? What about
10:12:15:19? What about 10:12:14:17? How about 14:17:19:21, or
14:15:17:19:21? I'm sure if you play around you'll be able to come up
with some just chord that involves 17 or 19.

Back when I was involved with 72-tet, I tried to explore 19-limit
otonal chords. Things like 2:3:5:7:9:11:13:15:17:19, or
4:5:7:11:13:17:19, or maybe just 8:10:14:17:19.

> 14:18:21 is higher in complexity but is composed of simpler dyads--6:7, 7:9, 2:3. 14:17:21 has 14:17, 17:21, and 2:3. This makes more of a difference than the triadic complexity.

OK, but I think some of this is your subjective opinion, as I'm more
of a fan of 10:13:15 than 14:18:21. I was defending something like
13-tet as being a "valid" tuning system that roughly approximates JI
in the Grove thread because although the error is too high for a lot
of people, it's a subjective judgement that should not define
terminology or anything concrete about music theory. In this case I
make the same argument from the other side of the fence.

> I tried a ton of 17- and 19-limit chords, and while they did sound "nice" they don't blend as well as those composed of simpler dyads. Also, whatever weak identity they might have, they tolerate very little tempering--if any at all.

I agree that they tolerate very little tempering, as higher ratios
seem to be much more sensitive to mistuning insofar as their "locking"
into the perception of a VF is concerned.

> Justness does seem to have some gray areas, but I'm trying to stay out of them.

I suggest calling these chords something like "simple just intonation"
vs "extended JI," analogous to how impressionistic harmony is
sometimes called "extended harmony" and so on. I think that redefining
the entire "just" moniker to just this type of thing is a bit too
restrictive. 4:5:6:7:8:9:10:11:12:13:14:15:16:17:18:19:20:21:22:23
definitely sounds harmonic and "just" to me.

-Mike

🔗cityoftheasleep <igliashon@...>

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I would contest that 10:12:15 is the one true minor chord to begin
> with.

So would I. But I would not contest its similarity with 14:17:21.

> But, assuming that it is, what about 10:12:15:17? What about
> 10:12:15:19? What about 10:12:14:17? How about 14:17:19:21, or
> 14:15:17:19:21? I'm sure if you play around you'll be able to come up
> with some just chord that involves 17 or 19.

Those are tetrads. At the level of tetrads, higher primes seem to become more sensible. I'm not contesting this. At the level of triads, I really think prime 13 is the cut-off.

> OK, but I think some of this is your subjective opinion, as I'm more
> of a fan of 10:13:15 than 14:18:21.

I'm not talking about which triads I "like better". I'm talking about which triads I hear as more coherent and less beating.

> I was defending something like
> 13-tet as being a "valid" tuning system that roughly approximates JI
> in the Grove thread because although the error is too high for a lot
> of people, it's a subjective judgement that should not define
> terminology or anything concrete about music theory. In this case I
> make the same argument from the other side of the fence.

But I agree about 13-TET. You're making a different argument here. Again, I am looking for triads (TRIADS, 3 notes, NOT tetrads or anything higher) that sound INDISPUTABLY Just. Not "nice" or "pleasant" or "happy", simply "beatless" and "coherent" and "insensitive enough to mistuning that they can be tuned with an error of at least 8 to 10 cents without "turning into" a different chord". Tell me, Mike: if you flatten a 17/14 by 10 cents, does it still sound like 17/14 and not a slightly-sharp 6/5? If you sharpen it by 10 cents, does it not start to sound like an 11/9?

> I suggest calling these chords something like "simple just intonation"
> vs "extended JI," analogous to how impressionistic harmony is
> sometimes called "extended harmony" and so on. I think that redefining
> the entire "just" moniker to just this type of thing is a bit too
> restrictive. 4:5:6:7:8:9:10:11:12:13:14:15:16:17:18:19:20:21:22:23
> definitely sounds harmonic and "just" to me.

Let's keep this to triads, okay? Look: I'm not trying to draw a line that says what is definitely not Just. I'm trying to draw a line of what definitely IS Just. This admits that chords off the list might still sound Just, but it also means that no chord on the list does NOT sound Just. So give me a break, okay?

-Igs

🔗Mike Battaglia <battaglia01@...>

On Thu, Mar 10, 2011 at 12:24 PM, cityoftheasleep
<igliashon@...> wrote:
>
> So would I. But I would not contest its similarity with 14:17:21.

You said that you only want to deal with triads, but my original
reason for bringing this example up was that interesting things start
happening when you throw numbers into this chord that lie between 14
and 21, like 14:15:17:19:21 or something. Then you start to hear a
real difference.

> > OK, but I think some of this is your subjective opinion, as I'm more
> > of a fan of 10:13:15 than 14:18:21.
>
> I'm not talking about which triads I "like better". I'm talking about which triads I hear as more coherent and less beating.

There are at least three perceptual phenomena that occur when a
low-complexity just chord is played:
1) Beatlessness
2) The appearance of a phantom VF (although it may not necessarily be
the GCD of the chord)
3) Periodicity buzz

First it seemed like you were concerned about periodicity stuff in
that 14:17:21 sounded like a mistuned 10:12:15, which puts us into
more of an HE-based frame. Now you're talking about beatlessness. The
sensation of periodicity buzz alone is something that may enable you
to distinguish 14:17:21 from 10:12:15. But it seems like you're
concerned about the first two, and you want to prioritize #1.

In that case, I still hear 10:13:15 as more coherent than 14:18:21. I
don't hear much beating at all in either chord. 14:18:21 sounds like a
mistuned 4:5:6, just like 14:17:21 sounds like a mistuned 10:12:15.
The identity of 14:18:21 stems in part from its being a "false major
chord" sounds dissonant. 10:13:15, on the other hand, sounds like the
major third has now become so sharp that it's something new; it's a
beautifully hollow chord in which the ultramajor third is neither a
mistuned 5/4 nor a mistuned 4/3, while retaining a few characteristics
of both. In addition, I hear periodicity buzz more clearly from
10:13:15 than I do from 14:18:21. So I would call 10:13:15 more "just"
sounding than 14:18:21, or at least just as "just" sounding.

> But I agree about 13-TET. You're making a different argument here. Again, I am looking for triads (TRIADS, 3 notes, NOT tetrads or anything higher) that sound INDISPUTABLY Just. Not "nice" or "pleasant" or "happy", simply "beatless" and "coherent" and "insensitive enough to mistuning that they can be tuned with an error of at least 8 to 10 cents without "turning into" a different chord". Tell me, Mike: if you flatten a 17/14 by 10 cents, does it still sound like 17/14 and not a slightly-sharp 6/5? If you sharpen it by 10 cents, does it not start to sound like an 11/9?

How about 11:14:17? Is that just? How about 9:13:17?

> Let's keep this to triads, okay? Look: I'm not trying to draw a line that says what is definitely not Just. I'm trying to draw a line of what definitely IS Just. This admits that chords off the list might still sound Just, but it also means that no chord on the list does NOT sound Just. So give me a break, okay?

That wasn't clear when I first read your message.

-Mike

🔗Juhani <jnylenius@...>

Hi,

I did a subjective listening test. I played each triad on your list on my Tonal Plexus keyboard, making note of the chords that I hear "clicking in place" and fuse when I play them on the correct keys on the instrument, instead of nearby keys that mistune the pitches by ca. 6 cents or more. The default tuning of the instruments is itself a temperament (205tet) but no interval is more than ca. 3 cents off. For some chords I used a just tuning (less than 0.1 cents off) but mostly I simply played around with the default 205tet. I used an overtone-rich, non-vibrato reed-like patch from a Korg M3 as the sound.

Here's what I was able to hear and what I feel I can tune by ear:

WIth the exception of maybe 12:16:21, 15:18:25 and 16:20:25, I don't hear the kind of "special" difference in the triads listed in the third column whether they're in tune or not. Predictably, I can easily hear if a 3:2 or 5:4 in them is mistuned but I can't hear that tuning the 21's, 25's or 27's accurately would make these rather complex chords sound stable or "just".
So I think I would leave the identities 21, 25 and 27 out of my own list of possible just triads.
I do hear a tiny difference in even the most complex triads on the list but one that I don't feel is enough for the kind of tunable-by-ear, special relationship that I understand by the term 'just'.
I also had trouble with many of the triads involving 15, notably 10:13:15, 12:14:15, 10:14:15 and others. Even so, I would probably keep 15-limit triads on my list.
From the triads in first column, I had difficulties in hearing the ones involving both 7 and 11, or both 7 and 13, click into place when tuned. 7:8:11 is difficult for me, as well as 7:12:13.

I'm not able to tune 17-limit or 19-limit triads by ear.

So for my own ears, I would put the odd-limit down to 15. There are higher-limit triads the justness of which I'm able to hear, to be sure. On the other hand, even many 15-limit triads are rather tricky for me to hear, so I could also make a selection of reliable just triads that would have only about 35 triads.

Juhani

🔗genewardsmith <genewardsmith@...>

🔗Juhani <jnylenius@...>

The keyboard layout is Pythagorean; it's based on 41tet, and each of the 41 zones is divided into 5 parts, ca. 5.85cents each, this being thought of as an average JND (Just Noticeable Difference). Aaron Hunt explains his rationale behind it on his website:
http://www.h-pi.com/glossary.html http://www.h-pi.com/theory/huntsystem1.html

In practice, the clear Pythagorean logic of the keyboard, with double and triple sharps and flats, makes it rather easy to find any given interval. It so happens that in 205tet 7/4 falls right between two keys (but it's stlll less than 3 cents off). This means that the interval between the syntonic and the septimal comma is tempered out, as both commas are usually played as four keysteps. Similarly, the difference between 32:27 and 13:11 is tempered out.

Anyway, any tuning can be mapped to the keyboard, and as septimal intervals are a bit off, as I said, I tried some of the chords that have them with a more precise tuning.

Juhani

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Juhani" <jnylenius@> wrote:
>
> >The default tuning of the instruments is itself a temperament (205tet)
>
> Do you know why that is?
>

🔗genewardsmith <genewardsmith@...>

🔗Juhani <jnylenius@...>

Meaning, 5 steps for septimal comma instead of 4? Yes, I often do that and it sounds good.

The 3165/3125 is the difference between which ratios?

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Juhani" <jnylenius@> wrote:
>
> > Anyway, any tuning can be mapped to the keyboard, and as septimal intervals are a bit off, as I said, I tried some of the chords that have them with a more precise tuning.
>
> You can use the flatter 7 instead, in which case the system tempers out 3136/3125 instead of 5120/5103; that sort of thing can be convenient. In this case, it makes for a hemithirds tuning.
>

🔗Carl Lumma <carl@...>

Thank you, Juhani, for posting this excellent account of this
experiment, which would be nearly infeasible for anyone not in
possession of a rare instrument like the 'Plexus. -Carl

--- In tuning@yahoogroups.com, "Juhani" <jnylenius@...> wrote:
>
> Hi,
>
> I did a subjective listening test. I played each triad on your
> list on my Tonal Plexus keyboard, making note of the chords that
> I hear "clicking in place" and fuse when I play them on the
> correct keys on the instrument, instead of nearby keys that
> mistune the pitches by ca. 6 cents or more. The default tuning
> of the instruments is itself a temperament (205tet) but no
> interval is more than ca. 3 cents off. For some chords I used
> a just tuning (less than 0.1 cents off) but mostly I simply
> played around with the default 205tet. I used an overtone-rich,
> non-vibrato reed-like patch from a Korg M3 as the sound.
>
> Here's what I was able to hear and what I feel I can tune by ear:
>
> WIth the exception of maybe 12:16:21, 15:18:25 and 16:20:25, I
> don't hear the kind of "special" difference in the triads listed
> in the third column whether they're in tune or not. Predictably,
> I can easily hear if a 3:2 or 5:4 in them is mistuned but I can't
> hear that tuning the 21's, 25's or 27's accurately would make
> these rather complex chords sound stable or "just".
> So I think I would leave the identities 21, 25 and 27 out of my
> own list of possible just triads.
> I do hear a tiny difference in even the most complex triads on
> the list but one that I don't feel is enough for the kind of
> tunable-by-ear, special relationship that I understand by the
> term 'just'.
> I also had trouble with many of the triads involving 15, notably
> 10:13:15, 12:14:15, 10:14:15 and others. Even so, I would
> probably keep 15-limit triads on my list.
> From the triads in first column, I had difficulties in hearing
> the ones involving both 7 and 11, or both 7 and 13, click into
> place when tuned. 7:8:11 is difficult for me, as well as 7:12:13.
>
> I'm not able to tune 17-limit or 19-limit triads by ear.
>
> So for my own ears, I would put the odd-limit down to 15. There
> are higher-limit triads the justness of which I'm able to hear,
> to be sure. On the other hand, even many 15-limit triads are
> rather tricky for me to hear, so I could also make a selection
> of reliable just triads that would have only about 35 triads.
>
> Juhani
>

🔗genewardsmith <genewardsmith@...>

🔗cityoftheasleep <igliashon@...>

Hi Juhani,

--- In tuning@yahoogroups.com, "Juhani" <jnylenius@...> wrote:

> Here's what I was able to hear and what I feel I can tune by ear:
>
> WIth the exception of maybe 12:16:21, 15:18:25 and 16:20:25, I don't hear the kind of > "special" difference in the triads listed in the third column whether they're in tune or not. > Predictably, I can easily hear if a 3:2 or 5:4 in them is mistuned but I can't hear that
> tuning the 21's, 25's or 27's accurately would make these rather complex chords sound > stable or "just".

What did you think of 14:18:21 and 16:18:21?

My hypothesis is that the 21's, 25's, and 27's don't sound Just on their own, but if there are two other dyads in the chord that sound Just, the overall chord will sound Just despite the un-Just dyad. Did you find this to be the case?

> So I think I would leave the identities 21, 25 and 27 out of my own list of possible just
> triads.

Presumably excepting the three exceptions you mentioned?

> I do hear a tiny difference in even the most complex triads on the list but one that I
> don't feel is enough for the kind of tunable-by-ear, special relationship that I
> understand by the term 'just'.

Noted. I'll check these myself and see if I agree.

> I also had trouble with many of the triads involving 15, notably 10:13:15, 12:14:15,
> 10:14:15 and others. Even so, I would probably keep 15-limit triads on my list.

I suspected as such...anything with a 14:15 in it seemed questionable. 13:15's a bit iffy too.

> From the triads in first column, I had difficulties in hearing the ones involving both 7
> and 11, or both 7 and 13, click into place when tuned. 7:8:11 is difficult for me, as well > as 7:12:13.

These also don't surprise me. 11's and 13's are definitely a bit iffy in general.

> I'm not able to tune 17-limit or 19-limit triads by ear.

I've yet to meet someone who can!

> So for my own ears, I would put the odd-limit down to 15. There are higher-limit triads > the justness of which I'm able to hear, to be sure. On the other hand, even many 15-
> limit triads are rather tricky for me to hear, so I could also make a selection of reliable
> just triads that would have only about 35 triads.

Odd-limit is just a rule of thumb...probably the thing to do is to just pick out the ones from this list that work and not care about their odd-limit. 12:14:21 is a piece of a 12:14:18:21 subminor 7th chord, which is 1/1-7/6-3/2-7/4, so it seems like a special case where the 21 should actually work in spite of itself.

-Igs

🔗Juhani <jnylenius@...>

> What did you think of 14:18:21 and 16:18:21?

I find that 14:18:21 is sensitive to mistuning of the 2:3 (14:21) and beating/ coarseness disappears when that's in place but the precise tuning of the middle note doesn't make much difference in stability.

16:18:21 is pretty dissonant (with 6:7 as the simplest dyad) and thus rather difficult but it's true that when the 6:7 is in tune, I hear some stability appearing.

> Odd-limit is just a rule of thumb...probably the thing to do is to just pick out the ones from this list that work and not care about their odd-limit. 12:14:21 is a piece of a 12:14:18:21 subminor 7th chord, which is 1/1-7/6-3/2-7/4, so it seems like a special case where the 21 should actually work in spite of itself.

Yeah, it does.

Juhani