24 Limit JI and more

24 Limit Just Intonation

My intention here is to promote two of my tunings (Raven Temperament and Raven JI, a just tuning) and my latest book (The Mathematics of Music and Raven Temperament). I also outline some of my ideas here that I think are both new and good and deserving of a mention on this group.

I begin with some of my ideas that relate to just intonation and the construction of just scales which some here may find interesting. I am not a strict just intonation man and my favorite scale is not a just scale but JI is the foundation of my approach. That is, over the last four or more years my method for constructing new tunings has always been to create a just scale first, and then temper (adjust) most of the notes (by not more than +/-6.7758 cents or 256/255) to achieve better melody and/or more good harmony intervals within +/-6.7758 cents accuracy. What follows applies to musical tones with a regular harmonic series (i.e. the frequencies of the partials are very close to x, 2x, 3x, 4x etc. and the intensities of the partials are very close to y, y/2, y/3, y/4 etc.) unless stated otherwise.

Good Just Harmony Intervals
The following is my method for identifying good (sweet) harmony intervals which I think works better than Harmonic Entropy. Consider a just interval 5/3. The first four harmonics of the 3 are 3, 6, 9, 12 and the first four harmonics of the 5 are 5, 10, 15, 20. (I considered the first 1024 harmonics of both notes in a harmony interval in the program I wrote to quantify the harmony (overall concordance) of a harmony interval but I'm just using four harmonics here to illustrate the method more simply.) I treat each harmonic as a single note so we have 8 notes: 3, 6, 9, 12, 5, 10, 15 and 20. If each of the 8 notes is paired with each of the other seven notes we get 28 distinct pairs of notes (intervals) which could be compared to sine wave intervals as they have no overtones. I think that concordance has to do with periodicity and dissonance has to do with beating. If two sine wave notes (no overtones) correspond to x /y (x>=y) then for me the periodicity (concordance) value is 1/x + 1/y. Note that x and y should *not* be simplified because of periodicity (e.g. 15/9 could be simplified to 5/3 but should be entered into the formula *without* simplification, as 1/15 + 1/9). The dissonance value is y/x if y/x is less than or equal to 0.9375 (15/16). If y/x is greater than 0.9375 (a very narrow interval) then the dissonance value should be (1 - y/x)*15. So for each of the 28 pairs in this example the concordance and dissonance values are worked out and the overall dissonance (beating) value is subtracted from the overall concordance (periodicity) value. Also the result for each pair must be `weighted' according to the intensity of the weaker harmonic of the two notes in each pair. If the pair is a/b and `a' is a 2nd harmonic and `b' is a 4th harmonic then the `b' is the weakest (quietest) note in the pair and the result (concordance - dissonance) should be divided by the number of the weaker harmonic (in this case 4). This idea is similar to the idea of a chain being only as strong as its weakest link. When the values for all 28 pairs are added together a constant value `z' should be added to the result but I haven't worked out what z is yet (this is explained in detail in my latest book). Regardless of whether or not I know what z is I can certainly arrange intervals in decreasing order of concordance and by listening I can choose a cut off point above which I deem harmony intervals to be good and below which harmony intervals should be bad. Here are the harmony intervals I consider to be good over a one octave range. The number in the middle is the width of the interval in cents and the number on the right is the relative strength value of the interval, relative to a zero point I chose in a listening test.

1/1 0.0 93.83
9/8 203.9 0.92
8/7 231.2 2.61
7/6 266.9 4.80
6/5 315.6 7.81
5/4 386.3 12.12
9/7 435.1 2.12
4/3 498.0 19.11
11/8 551.3 0.20
7/5 582.5 6.66
10/7 617.5 1.58
3/2 702.0 32.23
11/7 782.5 1.20
8/5 813.7 5.81
5/3 884.4 16.67
12/7 933.1 0.87
7/4 968.8 9.43
9/5 1017.6 5.20
11/6 1049.4 2.42
13/7 1071.7 0.43
2/1 1200.0 67.21

All of these intervals occur at least once per octave in my Raven Temperament tuning within +/-6.7758 cents (256/255) accuracy. 6.7758 cents is my maximum deviation (when tempering) from a just interval. I have a list of 90 good, just harmony intervals between 1/1 and 24/1 inclusive which I think is exhaustive and complete. For me if an interval is expressed as x/y and if x and/or y is greater than 24 then the interval is not good (both in melody and in harmony) in a "just" sense. The 32/1 (five octaves) interval, in harmony, does not sound dissonant to my ears, but it does sound weak, too weak for me, so I rule it out. So for me the widest legal just interval, in melody or in harmony is 24/1.

Note that there are some just harmony intervals that I give a strength value greater than zero (similar to the list above) but they contain a number higher than 24. For me these intervals are not dissonant but rather they are weak, too weak for me (e.g. 32/1) so I won't use them.

Good (24 Limit) Just Melodic Intervals
It seems to me that the strength value of a just melodic interval, x/y, is 1/x + 1/y (I use 2/x + 2/y in my book for reasons explained in the book but this does not affect the strength order of the intervals). Also I think that if x and/or y is greater than 24 then the interval is not good.

What about 29/27? This sounds acceptable in melody but contains a 29 (greater than 24). It turns out that 29/27 is within 4.3 cents of 15/14. So in the context of 24 limit JI I would not consider 29/27 to be a good "just" interval but it is very close to a good just interval, close to 15/14 and close enough to be acceptable if you're not strict.

My understanding of convention is that any interval x/y can be called "just" if both x and y are integers regardless of the size of x or y (e.g. 1024/1023 is a just interval). I make a further distinction, that if an interval, in melody, x/y, is just *and* x and y are both less than 25 then the interval is not only just but also "24 Limit Just" and therefore, for me, good. So in melody there are exactly 180 different "24 Limit Just" intervals where x and y are less than 25. I have found that in harmony (if I'm right) there are exactly 90 "24 Limit Just" harmony intervals (listed in the book).

Some think that melodically anything goes. I disagree. I think that a good example of a melodic interval that I consider to be too weak or sour is 721 cents. 721 cents is 19 cents above 3/2 and is 19 cents below the next "24 Limit Just" melodic interval, 23/15. Try it yourself.

37 cents is another good example of a sour melodic interval. 37 cents is half way between 256/255 (a tempered unison) and 24/23 (the narrowest, apart from a unison, legal melodic 24 Limit Just interval) tempered by minus 6.7758 cents.

Constructing Just Tunings
Consider this just tuning (my own Blue Just Tuning)...
1/1, 15/14, 9/8, 6/5, 5/4, 4/3, 7/5, 3/2, 8/5, 5/3, 9/5, 15/8, 2/1. It looks and sounds pretty good but it has one huge flaw: it contains melodic just intervals x/y where x and or y is greater than 24 which for me are illegal. For example pair 5/3 with 9/8 and you get 40/27 (680.4 cents). 680.4 cents is more 17 cents away from the nearest 24 Limit Just interval: 22/15 (663.0 cents).

It seems to me that it is impossible to create a 24 Limit Just tuning that has more than seven or eight notes per octave where all the notes relate to each other according to a 24 Limit Just interval (at least over a one octave range between 1/1 and 2/1). Here's the best that I could do...

I started with 1/1, 4/3, 3/2, 5/3, 2/1 because 4/3, 3/2, 5/3 and 2/1 are very strong melodically when paired with 1/1. Next I looked for the strongest (when paired with 1/1) just note that goes with all the other notes according to 24 Limit Just intervals and I got 5/4. Then 7/4 and finally 7/6. So the scale I got (called Raven JI) was:

1/1, 7/6, 5/4, 4/3, 3/2, 5/3, 7/4, 2/1.

Any note here, paired with any other note (over the one octave range) produces a 24 Limit Just interval.

You cannot add another just note to this scale without introducing an interval that is not 24 Limit (i.e. contains a 25 or higher) which would be illegal for me. The implication here is that if the 24 Limit idea is correct then it is impossible to build 24 limit just scales that have more than 7 or 8 notes per octave where all the melodic intervals (at least over a one octave range, between 1/1 and 2/1 inclusive) are 24 Limit Just intervals. In other words if there is a just tuning you like that has more than 7 or 8 notes per octave then it will have to contain at least one just melodic interval that is not 24 Limit and therefore (for me anyway) the just tuning must be illegal.

There are a few compositions in Raven JI by Chris Vaisvil on my web site...

http://www.johnsmusic7.com

The compositions do not sound extra special or spectacular (with regard to my tuning and not Chris' playing) but mathematically (at least over a one octave range and melodically) they should be pure, according to my current understanding.

Below is a list of just chords in Raven JI where the tonic (1/1) is assigned the key of D. The root note of each chord is often *not* the lowest note. I have a method for identifying the root note of a chord which is outlined in the book.

D 0.0 cents 1/1
E 266.9 cents 7/6
F 386.3 cents 5/4
G 498.0 cents 4/3
A 702.0 cents 3/2
B 884.4 cents 5/3
C 968.8 cents 7/4
D 1200.0 cents 2/1

The tonic is D. The chords listed below, and any subset of a chord listed should all be good. The root note of each chord is often not the lowest note in the chord.

D,A,D,F,A,C,D 2:3:4:5:6:7:8
D,G,B,D,E,G,A,D 3:4:5:6:7:8:9:12
D,G,B,D,E,G,B,D 3:4:5:6:7:8:10:12
D,F,A,C,D,F,A,C,D 4:5:6:7:8:10:12:14:16
D,E,G,A,D 6:7:8:9:12
D,E,G,B,D,E,G 6:7:8:10:12:14:16
D,G,A,D,A,D 6:8:9:12:18:24
D,B,D,F,A,D 6:10:12:15:18:24
D,B,D,F,B,D 6:10:12:15:20:24

E,C,E,C,E 2:3:4:6:8
E,G,A,D 7:8:9:12
E,G,B,D,E,G 7:8:10:12:14:16
E,A,D,C 7:9:12:21
E,D,E,C 7:12:14:21

F,B,F,B,F 3:4:6:8:12
F,A,C,D,F,A,C,D 5:6:7:8:10:12:14:16
F,A,D,F,A,C,D,F 5:6:8:10:12:14:16:20

G,D,G,B,D,E,G 2:3:4:5:6:7:8
G,B,D,E,G,A,D 4:5:6:7:8:9:12
G,B,D,E,G,B,D,E,G 4:5:6:7:8:10:12:14:16
G,B,D,A,D,F 4:5:6:9:12:15
G,B,D,B,D,F 4:5:6:10:12:15
G,A,D,A,D 8:9:12:18:24

A,D,F,A,C,D,F,A 3:4:5:6:7:8:10:12
A,C,D,F,A,C,D 6:7:8:10:12:14:16

B,F,B,F,B 2:3:4:6:8
B,D,E,G,A,D 5:6:7:8:9:12
B,D,E,G,B,D,E,G 5:6:7:8:10:12:14:16
B,D,G,B,D,E,G,A 5:6:8:10:12:14:16:18
B,D,G,B,D,E,G,B 5:6:8:10:12:14:16:20
B,D,A,D,F,A 5:6:9:12:15:18
B,D,B,D,F,A 5:6:10:12:15:18
B,D,B,D,F,B 5:6:10:12:15:20

C,E,C,E,C 3:4:6:8:12
C,D,F,A,C,D 7:8:10:12:14:16

Why I don't use generators or commas...

I don't know what a val or a monzo or a wedgie is but I know a little bit about commas and generators and I explain here why I don't use them. I'm not saying there's anything wrong with using commas and generators and they certainly "work" but I don't think using them is the best or only way to do it. I think my way, outlined below, is better.

For me the melodic aspect of a tuning is much more important than the harmony (chords) aspect. It's no good having a lot of great chords if you can't make a decent chord progression or melody.

My favourite 12TET scale is the pentatonic minor. If you play in E then the notes are E, G, A, B, D, E. When I improvise using this scale and when I play G or A or B or D I feel like I'm going somewhere. When I play an E I feel like I've arrived. It's the same with the white keys on a piano, the C key will always sound the most resolved. So for me the tonics of a scale (1/1 and 2/1) are much more important than the notes in between. I want a scale where the notes are as strong as possible when paired melodically with both 1/1 and 2/1 (with 1/1 being more important than 2/1 because 1 is smaller than 2).

The only advantage I can see to using chains of intervals is the ability to modulate. It seems to me that using a chain of generators the notes in the chain closest to the starting note can also be used as tonics. If the starting note is called 0 (zero) then the 6 notes in the chain going down from and up from 0 could be written as -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6.

My understanding is that 0 is the best choice as tonic but you could use -1 or +1 as a tonic as well. In other words you can modulate, change from one key to another. I never liked key changes in music and if I want to play a microtonal scale in a different key then I just retune my midi keyboard accordingly. With a microtonal guitar I'm sure it's feasible to build a pedal that will tune up or down the notes played so as to conform with a certain chosen key.

I have a list of notes that go well melodically with 1/1 and 2/1. So when I build a scale I can find exactly what notes I want without having to fool around with generators and commas. For example I start with 1/1 and 2/1 and select notes in between. 2/1 is the strongest note melodically (apart from a unison) when paired with 1/1. Melodically 4/3 and 3/2 are very strong when paired with both 1/1 and 2/1 and I wouldn't touch a scale that has no fourth (4/3), fifth (3/2) or octave (2/1) within +/-6.7758 cents accuracy. Also, for me 5/3 makes a major melodic interval when paired with 1/1 and 6/5 makes a major melodic interval when paired with 2/1. For the last year or two all the scales I worked on began with 1/1, 6/5, 4/3, 3/2, 5/3, 2/1 (tempered later) and at this point I don't think I would ever use a tuning that did not contain these six notes within +/-6.7758 cents (256/255) accuracy.

I made a comprehensive list of seventy, 24 limit, just (when paired melodically with both 1/1 and 2/1) notes and when I build a new tuning I choose notes from this list that are strong when paired melodically with both 1/1 and 2/1. The notes I choose depend on what I want. My Raven Temperament tuning contains *all* of the harmony intervals an octave or less wide (listed above and within +/-6.7758 cents accuracy) and the notes were chosen to achieve this. Another tuning does not have *all* the harmony intervals listed but it might have more good harmony intervals overall (even though some of the good harmony intervals in my list do not occur).

I always temper these just scales to maximize the number of good harmony intervals (within 6.7758 cents or 256/255) and to avoid bad melodic intervals (e.g. 721 cents and 37 cents).

I could go on at length but I've said enough for now. Here's a challenge for the experts in the group. I set out to build a scale that has these three properties...

1. *All* of the 21 intervals in my list of good harmony intervals above must occur at least once per octave within +/-6.7758 cents of just.

2. The scale contains a 4/3 (IV) and a 3/2 (V) within +/-6.7758 cents accuracy and if a major chord is defined as 2:3:4:5:6:8 then all the intervals in the I (1/1) Major, IV (4/3) Major and V (3/2) Major chords are good within +/-6.7758 cents accuracy.

3. No sour melodic notes (at least between 1/1 and 2/1 occur.

Here are the ranges that I think are sour melodically (e.g. any melodic interval between 6.7759 and 66.9048 inclusive should be illegal)

6.7759 - 66.9048
223.4626 - 224.3982
273.6468 - 274.5824
304.2889 - 308.8654
322.4172 - 323.9854
372.6013 - 379.5378
393.0896 - 397.6661
477.5568 - 491.2691
504.8209 - 516.5430
543.7267 - 544.5420
570.1582 - 575.7363
589.2881 - 590.2237
609.7763 - 610.7119
669.8251 - 695.1791
708.7309 - 733.2297
771.6918 - 775.7161
802.3339 - 806.9104
820.4622 - 823.4774
866.2243 - 877.5828
891.1346 - 904.0144
925.4176 - 926.3532
952.9710 - 962.0500
975.6018 - 980.9708
1002.8659 - 1010.8204
1024.3722 - 1028.2199
1041.7717 - 1042.5870
1078.4777 - 1081.4928
1133.0952 - 1193.2241

I identified all of the 24 limit just melodic intervals an octave or less wide and then calculated the ranges that are not within 6.7728 cents of any interval in the list.

The scale I worked out has these three properties and is called Raven Temperament (version 2).

The challenge is can anyone here build a scale with 12 notes per octave that has all three properties listed above using chains of generators? I'll bet it can't be done.

John O'Sullivan

http://www.johnsmusic7.com

On Wed, Jan 16, 2013 at 5:32 PM, john777music <jfos777@...> wrote:
>
> Here are the ranges that I think are sour melodically (e.g. any melodic
> interval between 6.7759 and 66.9048 inclusive should be illegal)
>
...
> 223.4626 - 224.3982

16-EDO has an interval in it which is 225 cents. This is less than a
cent away from your 224.3982, but then again the entire range of
sourness you list above is less than a cent in extent. So is 225 cents
OK, or no?

> The challenge is can anyone here build a scale with 12 notes per octave
> that has all three properties listed above using chains of generators? I'll
> bet it can't be done.

If your specific desiderata are exactly 12 notes per octave, and that
you want every one of those intervals to appear once, then I also
doubt it can be done. Generated scales are typically better if your
desiderata are:

1) you don't care about every "good" interval appearing exactly
"once", but rather you care about "the best intervals" appearing "as
much as possible" (e.g. it doesn't matter if 14/11 never makes an
appearance if we're getting seven 5/4's and three 3/2's and a bunch of
7/4's)

2) you don't limit yourself to exclusively 12-note scales (e.g.
15-note scales are fine), but instead have a general goal of getting
the most "good" intervals in as small a scale as possible

3) You don't impose a hard 6.8 cent cutoff, but instead your goal is
to have the whole temperament as accurate as possible, with the most
emphasis placed on tuning "better" intervals accurately

If in addition to that, you care about melodic and scalar
considerations, it's hard to get away from scale systems with two
generators.

-Mike

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

> If in addition to that, you care about melodic and scalar
> considerations, it's hard to get away from scale systems with two
> generators.

Why? I think I'll put up some rank four keenanismic (385/384) elf scales for you to chew on, and you can tell me what's wrong with them.

--- In tuning@yahoogroups.com, "john777music" wrote:

>For me if an interval is expressed as x/y and if x and/or y is greater than 24 then the interval is not good (both in melody and in harmony) in a "just" sense.

This terminology is going to confuse people, as there are already two other usages of "limit" which conflict with it--prime limit and odd limit. You might call it weil limit since you are limiting weil height.

> It seems to me that it is impossible to create a 24 Limit Just tuning that has more than seven or eight notes per octave where all the notes relate to each other according to a 24 Limit Just interval (at least over a one octave range between 1/1 and 2/1).

There will have to be some upper bound.

> 1/1, 7/6, 5/4, 4/3, 3/2, 5/3, 7/4, 2/1.
>
> Any note here, paired with any other note (over the one octave range) produces a 24 Limit Just interval.

Ah! Thanks for explaining this, RavenJI is the result of an optimization procedure: find the largest JI periodic scale with the weil height of all intervals within the octave less than 25. I wonder if it really is unique and optimal?

--- In tuning@yahoogroups.com, "genewardsmith" wrote:
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia wrote:
>
> > If in addition to that, you care about melodic and scalar
> > considerations, it's hard to get away from scale systems with two
> > generators.
>
> Why? I think I'll put up some rank four keenanismic (385/384) elf scales for you to chew on, and you can tell me what's wrong with them.
>

Since John gave a 7 and a 12 note scale I guess I'll do the same. Lots of other elves, plus a definition, can be found here:
http://xenharmonic.wikispaces.com/Elves

! elfkeenanismic7.scl
!
Keenanismic tempered [8/7, 5/4, 4/3, 3/2, 8/5, 7/4, 2] = cross_7, 284et tuning
7
!
232.39437
384.50704
498.59155
701.40845
815.49296
967.60563
2/1

! elfkeenanismic12.scl
!
Keenanismic tempered [12/11, 8/7, 6/5, 5/4, 4/3, 11/8, 3/2, 8/5, 5/3, 7/4, 11/6, 2], 284et tuning
12
!
152.11268
232.39437
316.90141
384.50704
498.59155
549.29577
701.40845
815.49296
883.09859
967.60563
1047.88732
2/1

Gene wrote:

> > 1/1, 7/6, 5/4, 4/3, 3/2, 5/3, 7/4, 2/1.
> >
> > Any note here, paired with any other note (over the one octave
> range) produces a 24 Limit Just interval.
>
> Ah! Thanks for explaining this, RavenJI is the result of an
> optimization procedure: find the largest JI periodic scale with
> the weil height of all intervals within the octave less than 25.
> I wonder if it really is unique and optimal?

The largest such scales are 12 13 14 15 16 17 18 19 20 21 22 23
and its subharmonic inverse, correct?

Hard to be sure, as "Weil height" is not defined on the wiki,
mathworld, or wikipedia.

> This terminology is going to confuse people,

-Carl

John wrote:

> It's the same with the white keys on a piano, the C key will always sound
the most resolved.

I'll start by saying that I like the scales John has identified (Blue; just
and tempered; Raven, just and tempered; the 31 EDO subset). They're all
useful and consonant and all that good stuff. Above all, John, you should
do what you like. However, I was struck by this line in your latest missive.

I've pretty much proved to myself that this isn't the case. The trivial
case is audible when I'm playing in Am. C simply doesn't sound the most
resolved when I'm playing in Am. That leads me to believe that what sounds
most resolved will depend on what notes you emphasize, what orders you play
them in, etc.

I'm a noodler, so I took that idea a while back and tried to take it as far
as I could in 12 EDO (which would control for any xenharmonic weirdness of
other tunings). I started noodling around in B Locrian on my piano (i.e.,
all of the white keys, but starting on B). There's no doubt that a final B
note can sound more resolved than a final C note.

Here's one way I make it happen.

* I start by droning a bit with the left hand on a low B, maybe playing a
B-D dyad. Then in the right hand I play melodies that start on B, that
emphasize D, and that mostly (but not completely) avoid F. I don't
necessarily avoid C, although it tends to be part of little ornaments like
B-C-B-A-B. I'm essentially playing a melody over a Bm with an
only-occasionally-sounded b5.

* After I've gotten that sound in my ear, I progress down a major third to
G major (which feels *really* good, but not more "resolved" -- I haven't
lost the fact that I'm in a mode of B), down a second to F major (usually
emphasizing the sharp 4th), back up a second to G, and then back to B.

* I do that for a few minutes, spending probably as much time on Bmb5 as I
do on the combination of F and G. Melodies in the right hand continue
throughout.

* Then, while in the B-based section, I start doing downward runs in the
right hand. A high B to B an octave below sounds perfectly resolved, even
with the flat fifth. But then I boost it one note, and start making runs
from C to C an octave below using the same timing -- and the C *forces* me
to play the B. Forces. I mean, the run simply isn't resolved unless I drop
a half-step to B.

* And, lest you think it sounds that way because I'm playing a B in the
left hand rather than a C: I sometimes move my left hand up to a C at the
beginning of the C-to-C run. For the first tens of seconds it most
certainly does not sound resolved, but rather like I've made a great leap
to a different key. (If I play in C long enough, moving back to B will
sound funny in the same way, of course.) In the same way, if I use my B
drone while playing a C to C run, and then play a C major chord only when I
land on the final C -- well, it sounds like aliens have landed. It doesn't
even really sound "major".

I keep thinking I should record or render something like this so people can
hear what it sounds like. It's really remarkable.

Perhaps, John, you're too absolutist when it comes to saying that
"such-and-such a note sounds most resolved". I tend to distrust those kinds
of comments because they don't fit my experience. A given note may sound
more or less resolved in a different context, and given your tendency to
isolate individual dyads in listening tests, you may not be able to account
for that in the way you create scales.

Regards,
Jake

--- In tuning@yahoogroups.com, "Carl Lumma" wrote:
>
> Gene wrote:
>
> > > 1/1, 7/6, 5/4, 4/3, 3/2, 5/3, 7/4, 2/1.
> > >
> > > Any note here, paired with any other note (over the one octave
> > range) produces a 24 Limit Just interval.
> >
> > Ah! Thanks for explaining this, RavenJI is the result of an
> > optimization procedure: find the largest JI periodic scale with
> > the weil height of all intervals within the octave less than 25.
> > I wonder if it really is unique and optimal?
>
> The largest such scales are 12 13 14 15 16 17 18 19 20 21 22 23
> and its subharmonic inverse, correct?

I stand corrected and I'm shocked that I made such a huge oversight. For the last few months I've been working on 7 limit notes (7 limit that is when pairing notes with 1/1) because I found that higher prime limits yielded few or no good chords with more than 2 notes. So if you omit 13/12, 17/12, 19/12 and 22/1 (11 limit) then you get the seven note scale I proposed: 1/1, 7/6, 5/4, 4/3, 3/2, 5/3, 7/4, 2/1. Even so, the scale Carl proposed is much better than mine and is possibly the largest and best 24 limit just scale.

>
> Hard to be sure, as "Weil height" is not defined on the wiki,
> mathworld, or wikipedia.
>
> > This terminology is going to confuse people,

The "24" in "24 Limit JI" is neither a prime nor is it odd so there shouldn't be any confusion.

John.

>
> -Carl
>

On Thu, Jan 17, 2013 at 9:06 AM, genewardsmith <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia wrote:
>
> > If in addition to that, you care about melodic and scalar
> > considerations, it's hard to get away from scale systems with two
> > generators.
>
> Why? I think I'll put up some rank four keenanismic (385/384) elf scales
> for you to chew on, and you can tell me what's wrong with them.

There's obviously nothing absolutely "wrong" with them, but what
exceptional melodic or scalar properties do they have that I can
harness which are comparable to what MOS has?

They're fine scales, but at least given the 7-note one which you laid
out here, I don't think that they're going to do lots of things that
(for instance) the meantone[7] scale system or lead to a really easily
comprehensible substitute for the diatonic scale which gives you easy
modulation, a simple chromatic scale, the obvious perceptual quality
of various harmonic ratios partitioning into different generic
interval classes (which imo is what epimorphicity is really all
about), nice smooth "stepwise" melodic runs, etc.

And what about MODMOS's? If I were to build an entire piece of music
off of this scale, is there an accompanying scale system and a way of
making chromatic alterations that I can use to get different related
sounds?

Whether any of those things are important depends on what you care
about. None of these are really deal-breakers, and you can't use them
to say anything is "wrong" with your elf scales. It just depends on
what you want. There are plenty of MOS's which are improper too, and I
don't use them in the same way that I'd use, for instance,
porcupine[7]; I typically treat them more like auxiliary scales which
can be used to create a certain effect. I can see your scale being
very useful in that sense, but I think I'd be more likely to play with
keenanismic Fokker blocks as my main center, modulating with them as I
want, and then I'd jump to this elf every so often to capture this
specific sound whenever I want.

Here's another question: would C D Eb F Gb Ab A B C be a
meantone/8-EDO elf? If so, then I'd probably treat your scale in
keenanismic temperament the same way that I treat the above scale in
meantone. I'd probably play in meantone most of the time, but then I
might play the above scale over a diminished chord as a sort of
embellishment, or perhaps I'd play the mode starting over Ab on top of
an Ab7 chord; this is basically what we do all the time in jazz,
except instead of diminished[8] we're using the above meantonized
version. I'm not sure if the above is exactly an elf, but I'd probably
treat your scale in the context of keenanismic temperament the same
way. (I need to learn more about keenanismic Fokker blocks before I
say anything else though).

Of course, all of this could be a failure of imagination on my part,
but for what it's worth, that's how I'd be likely to do things. So now
I ask you, what are the -melodic- advantages that elves are supposed
to have?

> This terminology is going to confuse people, as there are already two
> other usages of "limit" which conflict with it--prime limit and odd limit.
> You might call it weil limit since you are limiting weil height.

It's been called integer-limit before.

-Mike

On Thu, Jan 17, 2013 at 9:06 AM, genewardsmith <genewardsmith@...>
wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia wrote:
>
> > If in addition to that, you care about melodic and scalar
> > considerations, it's hard to get away from scale systems with two
> > generators.
>
> Why? I think I'll put up some rank four keenanismic (385/384) elf scales
> for you to chew on, and you can tell me what's wrong with them.

I guess my reservations about it can be summed up as follows: we're
not going to stay on this one elf scale forever. Is this just a scale,
or does it lead to an accompanying entire system of scales, like
meantone[7] has a whole system of MODMOS's, and of modulating to
parallel modes which have lots of common tones, and so on? Are there
things like MODMOS's I can use to get different related sounds, and if
so, what's the way you advocate constructing them?

If I modulate, I expect that the fact that there are like 6
incommensurate chroma changing at once, and in different ways for
different scale degrees which aren't very correlated with one another,
that it will be too much for me to ever reach the complete and utter
absolute mental comprehension of the scale that I really crave. How do
I deal with this? You're a mathematician, and I'm not; cognizing that
much stuff going on at once is not my forte. Am I supposed to train to
keep track of that much information all changing at once, or should I
stop trying to comprehend and approach the whole thing more passively,
or what? Should I think of all of the chromas together as being
intonational variants of the same specific interval, and in general
should I be grouping intervals together like that?

I have no idea, but for MOS I don't even have to think about these
questions. There's one chroma and modulation gives you common tones
everywhere. That isn't to say that elves are bad, it's just to say
that I don't know what techniques I can use with them, and I don't
know what scalar or melodic properties I can take advantage of with
them.

-Mike

On Thu, Jan 17, 2013 at 2:43 PM, Carl Lumma <carl@...> wrote:
>
> Hard to be sure, as "Weil height" is not defined on the wiki,
> mathworld, or wikipedia.

Weil height is defined here: http://xenharmonic.wikispaces.com/Height

Some background on why the name is so elusive to find:

In math circles, it's often just called "height", or is presented as a
canonical example of a height function with no further qualifying
adjective. I was originally calling it "Farey height," but Gene found
this paper here where the authors refer to it explicitly as "Weil
height": http://www.math.rochester.edu/people/faculty/ttucker/Papers.dir/Equi.pdf.
They further define a generalization for algebraic number fields which
isn't relevant to us.

The terminology is a bit screwed up in the same way that the
terminology around the "interior product" is. If you look at the
differential topology literature, they have a definition in which you
can only take the interior product of what we'd call a 1-val and a
multimonzo of any grade. If you look at the geometric/clifford algebra
literature, they have a definition in which the interior product is
taken between two multimonzos and the dual space is left out of it,
because they assume the existence of an inner product, so that they
view operations <12 19 28|-4 4 -1> being the same as |12 19 28> · |-4
4 -1> (which in abstract algebra terms means they view there as being
an isomorphism between the space of monzos and the space of vals).

In both cases, I think Gene's approach to naming was correct, since we
can't force people in different fields to get their acts together and
agree on a terminology, so all we can do is try to pick the most
sensible solution ourselves. It does require something of a rosetta
stone for us to make sense out of the rest of the mathematical
literature, or to communicate with other mathematicians, however.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia wrote:
>
> On Thu, Jan 17, 2013 at 2:43 PM, Carl Lumma wrote:
> >
> > Hard to be sure, as "Weil height" is not defined on the wiki,
> > mathworld, or wikipedia.
>
> Weil height is defined here:
> http://xenharmonic.wikispaces.com/Height

Ah, I just saw a bunch of blanks before. My browser is a
bit overextended at the moment. The math takes several
seconds to load.

-C.

--- In tuning@yahoogroups.com, "Carl Lumma" wrote:
>
> Gene wrote:
>
> > > 1/1, 7/6, 5/4, 4/3, 3/2, 5/3, 7/4, 2/1.
> > >
> > > Any note here, paired with any other note (over the one octave
> > range) produces a 24 Limit Just interval.
> >
> > Ah! Thanks for explaining this, RavenJI is the result of an
> > optimization procedure: find the largest JI periodic scale with
> > the weil height of all intervals within the octave less than 25.
> > I wonder if it really is unique and optimal?
>
> The largest such scales are 12 13 14 15 16 17 18 19 20 21 22 23
> and its subharmonic inverse, correct?

What is the subharmonic inverse of this scale? I don't know what the term means.

>
> Hard to be sure, as "Weil height" is not defined on the wiki,
> mathworld, or wikipedia.
>
> > This terminology is going to confuse people,
>
> -Carl
>

--- In tuning@yahoogroups.com, Mike Battaglia wrote:
http://www.math.rochester.edu/people/faculty/ttucker/Papers.dir/Equi.pdf.

http://www.math.rochester.edu/people/faculty/ttucker/Papers.dir/Equi.pdf

--- In tuning@yahoogroups.com, "john777music" wrote:

> > The largest such scales are 12 13 14 15 16 17 18 19 20 21 22 23
> > and its subharmonic inverse, correct?
>
> What is the subharmonic inverse of this scale? I don't know
> what the term means.

You take the reciprocals, basically. If 1/1 5/4 3/2 2/1 is
a harmonic scale, the subharmonic version is 1/1 4/3 8/5 2/1.
In Scala, you can do this to a scale with the following
command sequence: invert, reverse, normalize.

In the case, it is

!
Subharmonics 12-24, normalized to the octave
12
!
16/15
8/7
16/13
4/3
32/23
16/11
32/21
8/5
32/19
16/9
32/17
2/1
!

(The above can also be saved as an ASCII file with a .scl
extension and loaded directly into Scala.)

Subharmonic scales were described by Partch and others.
See http://en.wikipedia.org/wiki/Subharmonics
and http://en.wikipedia.org/wiki/Tonality_diamond
for example.

-Carl

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

> > Why? I think I'll put up some rank four keenanismic (385/384) elf scales
> > for you to chew on, and you can tell me what's wrong with them.
>
> I guess my reservations about it can be summed up as follows: we're
> not going to stay on this one elf scale forever. Is this just a scale,
> or does it lead to an accompanying entire system of scales, like
> meantone[7] has a whole system of MODMOS's, and of modulating to
> parallel modes which have lots of common tones, and so on? Are there
> things like MODMOS's I can use to get different related sounds, and if
> so, what's the way you advocate constructing them?

My choice of keenanismic elves was adventitious; it just happened that I was about to put them up as rank four examples. However, since you ask it strikes me it might be a good way to make use of the theory of keenanismic tablets which I worked out but never used save for an uninteresting example.

Thanks.

The subharmonic scale you sent me contains four 32s so it's not "24 Limit".

I asked Manuel Op de Coul if the 12 13 14 ... 23 24 scale is new and he said it's in the archives. Does it have a name? Anyone here know?

John.

--- In tuning@yahoogroups.com, "Carl Lumma" wrote:
>
> --- In tuning@yahoogroups.com, "john777music" wrote:
>
> > > The largest such scales are 12 13 14 15 16 17 18 19 20 21 22 23
> > > and its subharmonic inverse, correct?
> >
> > What is the subharmonic inverse of this scale? I don't know
> > what the term means.
>
> You take the reciprocals, basically. If 1/1 5/4 3/2 2/1 is
> a harmonic scale, the subharmonic version is 1/1 4/3 8/5 2/1.
> In Scala, you can do this to a scale with the following
> command sequence: invert, reverse, normalize.
>
> In the case, it is
>
> !
> Subharmonics 12-24, normalized to the octave
> 12
> !
> 16/15
> 8/7
> 16/13
> 4/3
> 32/23
> 16/11
> 32/21
> 8/5
> 32/19
> 16/9
> 32/17
> 2/1
> !
>
> (The above can also be saved as an ASCII file with a .scl
> extension and loaded directly into Scala.)
>
> Subharmonic scales were described by Partch and others.
> See http://en.wikipedia.org/wiki/Subharmonics
> and http://en.wikipedia.org/wiki/Tonality_diamond
> for example.
>
> -Carl
>

On Fri, Jan 18, 2013 at 11:48 AM, john777music <jfos777@...> wrote:
>
> Thanks.
>
> The subharmonic scale you sent me contains four 32s so it's not "24
> Limit".

Your scale is also not 24-limit then, assuming it ever repeats for
more than one octave. For instance, your 13/7 becomes 26/7 an octave
up, which is no longer in the 24-limit.

> I asked Manuel Op de Coul if the 12 13 14 ... 23 24 scale is new and he
> said it's in the archives. Does it have a name? Anyone here know?

It's usually just called harmonics 12-24.

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia wrote:
>
> On Fri, Jan 18, 2013 at 11:48 AM, john777music wrote:
> >
> > Thanks.
> >
> > The subharmonic scale you sent me contains four 32s so it's not "24
> > Limit".
>
> Your scale is also not 24-limit then, assuming it ever repeats for
> more than one octave. For instance, your 13/7 becomes 26/7 an octave
> up, which is no longer in the 24-limit.

I said in the first post in this thread that I was talking about a one octave range. Here's what I said...

:It seems to me that it is impossible to create a 24 Limit Just tuning that has more than seven or eight notes per octave where all the notes relate to each other according to a 24 Limit Just interval (at least over a one octave range between 1/1 and 2/1).

So the scale I got (called Raven JI) was:
1/1, 7/6, 5/4, 4/3, 3/2, 5/3, 7/4, 2/1.
Any note here, paired with any other note (over the one octave range) produces a 24 Limit Just interval."

>
> > I asked Manuel Op de Coul if the 12 13 14 ... 23 24 scale is new and he
> > said it's in the archives. Does it have a name? Anyone here know?
>
> It's usually just called harmonics 12-24.

Thanks.

John.

>
> -Mike
>

--- In tuning@yahoogroups.com, "john777music" wrote:
>
>
>
> --- In tuning@yahoogroups.com, Mike Battaglia wrote:
> >
> > On Fri, Jan 18, 2013 at 11:48 AM, john777music wrote:
> > >
> > > Thanks.
> > >
> > > The subharmonic scale you sent me contains four 32s so it's not "24
> > > Limit".
> >
> > Your scale is also not 24-limit then, assuming it ever repeats for
> > more than one octave. For instance, your 13/7 becomes 26/7 an octave
> > up, which is no longer in the 24-limit.

Sorry, I thought you were referring to my just scale (Raven JI). My Raven Temperament scale is also 24 limit (within 6.7758 cents accuracy) but only over a one octave range. I state this in my book.

>
> I said in the first post in this thread that I was talking about a one octave range. Here's what I said...
>
> :It seems to me that it is impossible to create a 24 Limit Just tuning that has more than seven or eight notes per octave where all the notes relate to each other according to a 24 Limit Just interval (at least over a one octave range between 1/1 and 2/1).
>
> So the scale I got (called Raven JI) was:
> 1/1, 7/6, 5/4, 4/3, 3/2, 5/3, 7/4, 2/1.
> Any note here, paired with any other note (over the one octave range) produces a 24 Limit Just interval."
>
> >
> > > I asked Manuel Op de Coul if the 12 13 14 ... 23 24 scale is new and he
> > > said it's in the archives. Does it have a name? Anyone here know?
> >
> > It's usually just called harmonics 12-24.
>
> Thanks.
>
> John.
>
> >
> > -Mike
> >
>

To Jake

--- In tuning@yahoogroups.com, Jake Freivald wrote:
>
> John wrote:
>
> > It's the same with the white keys on a piano, the C key will always sound
> the most resolved.
>
> I'll start by saying that I like the scales John has identified (Blue; just
> and tempered; Raven, just and tempered; the 31 EDO subset). They're all
> useful and consonant and all that good stuff. Above all, John, you should
> do what you like. However, I was struck by this line in your latest missive.
>
> I've pretty much proved to myself that this isn't the case. The trivial
> case is audible when I'm playing in Am. C simply doesn't sound the most
> resolved when I'm playing in Am.

If Major is defined as strong and happy and Minor is weaker or sad then by definition the C major scale should be stronger/happier than the A minor scale. Sure, over a one octave range between A and the next A then using *only* these eight white keys the bottom and top A keys will sound the most resolved and the C key in between won't. However over a greater range (e.g. three or more octaves using only white keys) it seems to me that the C keys are the most resolved sounding. I should also say that I don't think that using A as the tonic when playing white keys only should be illegal, I just think it's inferior to C. Thanks for positive comments on my scales.

John.

That leads me to believe that what sounds
> most resolved will depend on what notes you emphasize, what orders you play
> them in, etc.
>
> I'm a noodler, so I took that idea a while back and tried to take it as far
> as I could in 12 EDO (which would control for any xenharmonic weirdness of
> other tunings). I started noodling around in B Locrian on my piano (i.e.,
> all of the white keys, but starting on B). There's no doubt that a final B
> note can sound more resolved than a final C note.
>
> Here's one way I make it happen.
>
> * I start by droning a bit with the left hand on a low B, maybe playing a
> B-D dyad. Then in the right hand I play melodies that start on B, that
> emphasize D, and that mostly (but not completely) avoid F. I don't
> necessarily avoid C, although it tends to be part of little ornaments like
> B-C-B-A-B. I'm essentially playing a melody over a Bm with an
> only-occasionally-sounded b5.
>
> * After I've gotten that sound in my ear, I progress down a major third to
> G major (which feels *really* good, but not more "resolved" -- I haven't
> lost the fact that I'm in a mode of B), down a second to F major (usually
> emphasizing the sharp 4th), back up a second to G, and then back to B.
>
> * I do that for a few minutes, spending probably as much time on Bmb5 as I
> do on the combination of F and G. Melodies in the right hand continue
> throughout.
>
> * Then, while in the B-based section, I start doing downward runs in the
> right hand. A high B to B an octave below sounds perfectly resolved, even
> with the flat fifth. But then I boost it one note, and start making runs
> from C to C an octave below using the same timing -- and the C *forces* me
> to play the B. Forces. I mean, the run simply isn't resolved unless I drop
> a half-step to B.
>
> * And, lest you think it sounds that way because I'm playing a B in the
> left hand rather than a C: I sometimes move my left hand up to a C at the
> beginning of the C-to-C run. For the first tens of seconds it most
> certainly does not sound resolved, but rather like I've made a great leap
> to a different key. (If I play in C long enough, moving back to B will
> sound funny in the same way, of course.) In the same way, if I use my B
> drone while playing a C to C run, and then play a C major chord only when I
> land on the final C -- well, it sounds like aliens have landed. It doesn't
> even really sound "major".
>
> I keep thinking I should record or render something like this so people can
> hear what it sounds like. It's really remarkable.
>
> Perhaps, John, you're too absolutist when it comes to saying that
> "such-and-such a note sounds most resolved". I tend to distrust those kinds
> of comments because they don't fit my experience. A given note may sound
> more or less resolved in a different context, and given your tendency to
> isolate individual dyads in listening tests, you may not be able to account
> for that in the way you create scales.
>
> Regards,
> Jake
>

To Gene

--- In tuning@yahoogroups.com, "genewardsmith" wrote:
>
>
>
> --- In tuning@yahoogroups.com, "john777music" wrote:
>
> >For me if an interval is expressed as x/y and if x and/or y is greater than 24 then the interval is not good (both in melody and in harmony) in a "just" sense.
>
> This terminology is going to confuse people, as there are already two other usages of "limit" which conflict with it--prime limit and odd limit. You might call it weil limit since you are limiting weil height.

As I said in an earlier post the "24" in "24 Limit JI" is neither prime nor odd so there shouldn't be any confusion.

>
> > It seems to me that it is impossible to create a 24 Limit Just tuning that has more than seven or eight notes per octave where all the notes relate to each other according to a 24 Limit Just interval (at least over a one octave range between 1/1 and 2/1).
>
> There will have to be some upper bound.
>
> > 1/1, 7/6, 5/4, 4/3, 3/2, 5/3, 7/4, 2/1.
> >
> > Any note here, paired with any other note (over the one octave range) produces a 24 Limit Just interval.
>
> Ah! Thanks for explaining this, RavenJI is the result of an optimization procedure: find the largest JI periodic scale with the weil height of all intervals within the octave less than 25. I wonder if it really is unique and optimal?
>

According to Manuel Raven JI is unique but as Carl pointed out it's not optimal. The 12 to 24 harmonics (a superset of Raven JI) seems to be optimal for 24 Limit JI over a one octave range.

John.

--- In tuning@yahoogroups.com, "john777music" wrote:

> The subharmonic scale you sent me contains four 32s so it's
> not "24 Limit".

Sorry, I goofed. Here's the correct version:

!
Subharmonics 12-24
12
!
24/23
12/11
8/7
6/5
24/19
4/3
24/17
3/2
8/5
12/7
24/13
2/1
!

-Carl

This is significant, a complement to 24 Limit Rainbow (the name I'm giving the 12 to 24 harmonic series). Thanks for showing me this. From my point of view I think that these two scales are as good as it gets for JI (according to my own chosen criteria, others will want different things), at least over the one octave range.

John.

--- In tuning@yahoogroups.com, "Carl Lumma" wrote:
>
> --- In tuning@yahoogroups.com, "john777music" wrote:
>
> > The subharmonic scale you sent me contains four 32s so it's
> > not "24 Limit".
>
> Sorry, I goofed. Here's the correct version:
>
> !
> Subharmonics 12-24
> 12
> !
> 24/23
> 12/11
> 8/7
> 6/5
> 24/19
> 4/3
> 24/17
> 3/2
> 8/5
> 12/7
> 24/13
> 2/1
> !
>
> -Carl
>

On Thu, Jan 17, 2013 at 10:13 PM, genewardsmith
<genewardsmith@...> wrote:
>
> My choice of keenanismic elves was adventitious; it just happened that I
> was about to put them up as rank four examples. However, since you ask it
> strikes me it might be a good way to make use of the theory of keenanismic
> tablets which I worked out but never used save for an uninteresting example.

OK, fair enough. I'm still curious how MODelfs might work though. I
was thinking about it earlier, and elves are very much in line with
some completely unrelated ideas about notation I've had recently, in
that they're ways to consider what scales someone who's used to
temperament "A" might like if they start playing in temperament "B".
It's a good step towards a full understanding of the ways that
different tuning systems interrelate.

Also, this is going to get totally buried in the onslaught of replies
on XA about Orwell, but since it's more relevant here, I was really
curious if you could tell me if these are Orwell elf scales; the
simplest JI transversals are both epimorphic under <7 11 16 20 24|.
Those transversals are:

1/1 11/10 7/6 11/8 3/2 8/5 7/4 2/1
1/1 11/10 7/6 11/8 3/2 8/5 15/8 2/1

I think that neither will be elves directly, since 16/15 is less
complex that 11/10 in orwell, but maybe they'll be modes of elves or
something. I came up with these scales from Andrew's "tonal" orwell
composition here:

https://soundcloud.com/andrew_heathwaite/demo-track-no-mo

I've been literally playing this piece on repeat for like five hours
now, while I work, and now I can wrap my head around what's going on.
He's using the LsLsLsLss mode of orwell[9], but he's left the 9/7 out.
So a transversal of what he's using is 1/1 11/10 7/6 11/8 3/2 8/5 7/4
15/8 2/1.

He does something special with that 7/4 and 15/8. he'll often omit the
15/8 for long stretches at a time and play this melodic figure: 7/4
2/1 11/5 2/1 7/4 2/1 7/4 3/2, where he passes over the 15/8 and goes
right from 7/4 to 2/1. Since the 15/8 all but disappears for a while
when this happens, my brain pretty much doesn't even hear it as being
there anymore; this makes the 8/7 sound like a type of step, or a type
of "second" or whatever, rather than a type of "third" where it passes
over a note.

So you get the following scale

1/1 11/10 7/6 11/8 3/2 8/5 7/4 2/1

then sometimes he throws the 7/4 away and puts 15/8 back, yielding

1/1 11/10 7/6 11/8 3/2 8/5 15/8 2/1

these are both epimorphic under <7 11 16 20 24|.

So I find it interesting that Andrew deliberately threw that 9/7 away,
and then treated the 7/4 and 15/8 in such a way as to obtain these
scales which are clearly perceptually organized by <7 11 16 20 24|
into generic interval classes. Maybe a good way to learn new
temperaments is to start with a <7 11 16 ...| elf as a stepping stone,
and then branch out from there to the MOS's.

-Mike

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

> 1/1 11/10 7/6 11/8 3/2 8/5 15/8 2/1
>
> these are both epimorphic under <7 11 16 20 24|.
>
> So I find it interesting that Andrew deliberately threw that 9/7 away,
> and then treated the 7/4 and 15/8 in such a way as to obtain these
> scales which are clearly perceptually organized by <7 11 16 20 24|
> into generic interval classes. Maybe a good way to learn new
> temperaments is to start with a <7 11 16 ...| elf as a stepping stone,
> and then branch out from there to the MOS's.

As I mentioned on FB, the elf is 11/10 7/6 11/8 16/11 12/7 15/8 2, which is epimorphic. Transpose up by 11/8 and reduce to the least complex orwell interval, and you get 7/6 9/7 11/8 3/2 8/5 15/8 2. This isn't epimorphic. This can happen as the 11-limit patent val doesn't support orwell.

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

> As I mentioned on FB, the elf is 11/10 7/6 11/8 16/11 12/7 15/8 2, which is epimorphic. Transpose up by 11/8 and reduce to the least complex orwell interval, and you get 7/6 9/7 11/8 3/2 8/5 15/8 2. This isn't epimorphic. This can happen as the 11-limit patent val doesn't support orwell.

If we look for transversals of least complexity for orwell which are of Graham complexity 7 and epimorphic, we get:

[11/10, 7/6, 11/8, 3/2, 8/5, 15/8, 2] and
[11/10, 9/7, 11/8, 3/2, 8/5, 15/8, 2] starting from 0

None starting from -1

[11/10, 7/6, 11/8, 16/11, 8/5, 15/8, 2] and
[11/10, 7/6, 11/8, 16/11, 12/7, 15/8, 2] starting from -2

None starting from -3, -4 -5, -6 or -7

All four of these scales are elves of a sort.