Trigger

Here's a dyadic analysis of ED3s 4 to 39. For me any melodic step narrower than 48.188552 cents (apart from a just or tempered unison) is illegal (see my post on "Crows"). ED3s of 40 or higher contain steps narrower than 48.188552 cents.

The harmony intervals between 1/1 and 3/1 that I consider to be good are: 1/1, 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, 13/6, 11/5, 9/4, 16/7, 7/3, 12/5, 5/2, 13/5, 8/3, 11/4, 14/5, 17/6 and 3/1.

Here are the good dyads that appear in ED3 4 to 39 within 6.776 cents (256/255) accuracy...

ED3-4...16/7, 3/1,

ED3-5...5/4, 12/5, 3/1,

ED3-6...6/5, 5/2, 3/1,

ED3-7...7/6, 8/5, 11/5, 3/1,

ED3-8...8/7, 16/7, 3/1,

ED3-9...3/1,

ED3-10...5/4, 12/5, 3/1,

ED3-11...3/1,

ED3-12...6/5, 16/7, 5/2, 3/1,

ED3-13...9/7, 7/5, 5/3, 9/5, 7/3, 11/4, 3/1,

ED3-14...7/6, 8/5, 11/5, 3/1,

ED3-15...5/4, 5/3, 9/5, 12/5, 13/5, 3/1,

ED3-16...8/7, 13/7, 16/7, 14/5, 3/1,

ED3-17...11/7, 13/6, 3/1,

ED3-18...6/5, 5/2, 17/6, 3/1,

ED3-19...9/8, 4/3, 3/2, 2/1, 9/4, 8/3, 17/6, 3/1,

ED3-20...5/4, 11/6, 16/7, 12/5, 17/6, 3/1,

ED3-21...7/6, 8/5, 11/5, 3/1,

ED3-22...9/7, 11/7, 7/3, 3/1,

ED3-23...4/3, 7/5, 13/7, 9/4, 13/5, 3/1,

ED3-24...8/7, 6/5, 11/8, 16/7, 5/2, 3/1,

ED3-25...8/7, 5/4, 13/7, 11/5, 12/5, 11/4, 3/1,

ED3-26...9/7, 7/5, 5/3, 9/5, 7/3, 11/4, 3/1,

ED3-27...4/3, 3/2, 2/1, 13/6, 9/4, 3/1,

ED3-28...9/8, 7/6, 10/7, 8/5, 5/3, 9/5, 11/5, 16/7, 8/3, 3/1,

ED3-29...7/6, 11/7, 11/6, 3/1,

ED3-30...6/5, 5/4, 3/2, 5/3, 9/5, 13/7, 2/1, 12/5, 5/2, 13/5, 3/1,

ED3-31...9/7, 11/8, 10/7, 11/6, 7/3, 13/5, 14/5, 3/1,

ED3-32...8/7, 13/7, 11/5, 16/7, 14/5, 3/1,

ED3-33...8/7, 7/5, 14/5, 3/1,

ED3-34...5/4, 4/3, 10/7, 11/7, 13/6, 9/4, 12/5, 3/1,

ED3-35...7/6, 5/4, 9/7, 3/2, 8/5, 2/1, 11/5, 7/3, 12/5, 3/1,

ED3-36...7/6, 6/5, 7/5, 16/7, 5/2, 17/6, 3/1,

ED3-37...9/8, 10/7, 13/6, 8/3, 11/4, 17/6, 3/1,

ED3-38...9/8, 4/3, 11/8, 3/2, 11/6, 2/1, 9/4, 13/5, 8/3, 11/4, 17/6, 3/1,

ED3-39...5/4, 9/7, 7/5, 11/7, 5/3, 12/7, 7/4, 9/5, 13/7, 11/5, 7/3, 12/5, 13/5, 11/4, 17/6, 3/1,

The ED3 with the most good dyads occurring is 39ED3. If the notes are named 1, 2, 3 etc up to 39 then the tritave (3/1) is 40. Here are the good chords that occur in 39ED3. All of the dyads in each chord are within 6.776 cents of any of the good dyads listed above.

1, 9, 17, 29, 37
1, 9, 21, 29, 37
1, 9, 21, 40
1, 9, 31, 40
1, 10, 19, 31, 40
1, 10, 19, 38
1, 10, 22, 31, 40
1, 10, 22, 38
1, 10, 29, 38
1, 10, 32, 40
1, 13, 21, 29
1, 13, 22, 31
1, 13, 32
1, 13, 35
1, 17, 29, 38
1, 17, 35
1, 17, 38
1, 19, 31, 40
1, 19, 35
1, 19, 37
1, 19, 38
1, 19, 40
1, 20, 29, 38
1, 20, 32, 40
1, 23, 31
1, 23, 32
1, 23, 35

Of course all these chords will be good in all keys. In many cases a chord will sound better when one or more notes are omitted so subsets of the above chords could be used. E.g. some subsets of the 1, 9, 17, 29, 37 chord are...

1, 9, 17, 29
1, 9, 17, 37
1, 9, 29, 37
1, 17, 29, 37

Some subsets of these tetrads are triads.

There are no octaves in 39ED3 which is a good thing because if there were then the octave (2/1) would upstage the tritave (3/1), 2/1 would sound more resolved than 3/1 which sort of defeats the purpose of an ED3. There are also no 3/2s or 4/3s so it should sound more xenharmonic.

39 notes per tritave is a bit unwieldy so I did a bit of pruning. My formula for melody is

2/x + 2/y

Put any just interval (n and d < 256) through the formula and if the result >=2 you have a Super Major Melodic interval (I know that the term "Super Major" has another meaning in relation to chords). If the result is between 1.0 and 1.9999 you have a Blue Major melodic interval. Between 0.5 and 0.9999 you have a Blue Minor melodic interval. Between 0.25 and 0.4999 you have an Ultra Minor. All of the above are, to me, "sweet". Between 0.125 and 0.2499 you have a "tolerable" (but not sweet) interval. Less than 0.125 the intervals are "intolerable".

For me all of the melodic intervals in a scale should be "tolerable or better" but the notes paired with the tonic (which is more important than the other notes) should be sweet (i.e. value 0.25 or higher).

I identified 14 notes in 39ED3 that, when paired with the tonic, were not "sweet" melodically (i.e. they weren't within 6.776 cents of a "sweet" note). If the keys in 39ED3 are named 1 to 39 then the notes that are not sweet (when paired with 1/1) are: 2, 3, 5, 11, 15, 16, 18, 25, 26, 27, 30, 34, 36 and 39.

So I cut out these 14 notes and here's the 25 key scale I got...

Trigger
0.0,
146.304,
243.84,
292.608,
341.377,
390.145,
438.913,
536.449,
585.217,
633.985,
780.289,
877.825,
926.593,
975.362,
1024.13,
1072.9,
1121.67,
1316.74,
1365.51,
1463.04,
1511.81,
1560.58,
1658.11,
1755.65,
1804.42,
1901.95.

This scale has only 25 keys per tritave and is thus more manageable and all chords played should relate very well to the tonic. I call it "Trigger".

If the notes in Trigger are named 1, 2, 3 etc up to 25 then the tritave is 26.

Here are the good chords that occur in Trigger if the lowest note is in the key of 1. These are the same chords as those listed above but the numbers have been changed to conform to the 25 keys of Trigger. Try omitting various combinations of notes in each chord. I haven't worked out all the good chords where the lowest notes are higher than 1 yet. If anyone shows any interest I might work them out but it's a time consuming job.

Good Trigger chords where lowest note is 1...

1, 6, 11, 19, 24
1, 6, 14, 19, 24
1, 6, 14, 26
1, 6, 20, 26
1, 7, 12, 20, 26
1, 7, 12, 25
1, 7, 15, 20, 26
1, 7, 15, 25
1, 7, 19, 25
1, 7, 21, 26
1, 9, 14, 19
1, 9, 15, 20
1, 9, 21
1, 9, 23
1, 11, 19, 25
1, 11, 23
1, 11, 25
1, 12, 20, 26
1, 12, 23
1, 12, 24
1, 12, 25
1, 12, 26
1, 13, 19, 25
1, 13, 21, 26
1, 16, 20
1, 16, 21
1, 16, 23

John.

--- In tuning@yahoogroups.com, "john777music" <jfos777@...> wrote:
> ED3-39...5/4, 9/7, 7/5, 11/7, 5/3, 12/7, 7/4, 9/5, 13/7, 11/5, 7/3, 12/5, 13/5, 11/4, > 17/6, 3/1,
>
> The ED3 with the most good dyads occurring is 39ED3.

This is what is known as "Triple Bohlen-Pierce", since it subdivides every step of 13-ED3 (the original Bohlen-Pierce scale is 13-ED3) into 3 equal parts. The first octave of it is somewhat like 25-EDO (step-size of exactly 48 cents), but as you get higher the difference mounts quickly and you get lots of good intervals that 25-EDO does not really approximate at all. Triple BP was first proposed by Paul Erlich, I think. Ron Sword has built a few guitars in triple BP and is quite the fan of it. It

Also of note: 19-ED3 is indistinguishable from 12-EDO, and 30-ED3 is very similar to 19-EDO but with a very slight stretch, and is also related to Wendy Carlos' Beta scale.

-Igs

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> > ED3-39...5/4, 9/7, 7/5, 11/7, 5/3, 12/7, 7/4, 9/5, 13/7, 11/5, 7/3, 12/5, 13/5, 11/4, > 17/6, 3/1,

Sorry, also meant to add: triple BP allows for the extension of BP harmony into higher limits, since you can play a full 3:5:7:9:11:13:15 septad in really excellent tuning; in fact, this maybe the simplest rank-1 temperament to achieve such a chord with such great accuracy.

-Igs

Thanks Igs,

I didn't know 39ED3 had been looked at before. Trigger is different though as it omits 14 of the 39 keys of 39ED3.

Did you get my email offlist about the good chords that occur in my subset of 22EDO?

John.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "john777music" <jfos777@> wrote:
> > ED3-39...5/4, 9/7, 7/5, 11/7, 5/3, 12/7, 7/4, 9/5, 13/7, 11/5, 7/3, 12/5, 13/5, 11/4, > 17/6, 3/1,
> >
> > The ED3 with the most good dyads occurring is 39ED3.
>
> This is what is known as "Triple Bohlen-Pierce", since it subdivides every step of 13-ED3 (the original Bohlen-Pierce scale is 13-ED3) into 3 equal parts. The first octave of it is somewhat like 25-EDO (step-size of exactly 48 cents), but as you get higher the difference mounts quickly and you get lots of good intervals that 25-EDO does not really approximate at all. Triple BP was first proposed by Paul Erlich, I think. Ron Sword has built a few guitars in triple BP and is quite the fan of it. It
>
> Also of note: 19-ED3 is indistinguishable from 12-EDO, and 30-ED3 is very similar to 19-EDO but with a very slight stretch, and is also related to Wendy Carlos' Beta scale.
>
> -Igs
>