Yahoo Tuning Groups Ultimate Backup makemicromusic Short piece using tempered 2.3.7/5.11/9 tonality diamond

Hi all,

This piece started as an exercise in scale creation. I had four goals
for the scale.

1) Use the 11/9 neutral third.
2) Don't use too many "easy" 5-limit harmonies.
3) Create recognizable fourths and fifths.
4) Avoid impropriety.

I used Scala to generate a tonality diamond using 2, 3, 7/5, and 11/9.
After deleting a few tones to avoid impropriety, I came up with the
following 10-note scale:

! C:\Program Files (x86)\Scala22\2.3.5-7.11-9.diamond.scl
!

10
!
15/14
63/55
11/9
4/3
10/7
3/2
18/11
110/63
28/15
2/1

After studying the interval attribute matrix (thanks to Carl for
pointing that out to me), I decided that I should temper out 243/242
and 1375/1372. That's < -1 5 0 0 -2 | and < -2 0 3 -3 1 |,
respectively.

243/242 makes two neutral thirds equal a perfect fifth (3/2 = 11/9 *
11/9 * 243/242).

243/242 also tempers the difference between taking two steps up from
the 63/55 to the 4/3 (220/189, or 263 cents) and taking two steps up
from the 11/9 to the 10/7 (90/77, or 270 cents), which significantly
smooths out that part of the scale and related intervals.

1375/1372 is the difference between taking three steps up from the
63/55 to the 10/7 (550/441, or 382 cents) and taking three steps up
from the 15/14 to the 4/3 (56/45, or 379 cents). Although I wasn't
intending to have too many 5-limit harmonies, this "accidental 5/4"
still seemed to need tempering to smooth out the scale.

1375/1372 also tempers out the difference between the semitone moving
up from 1/1 to 15/14 (15/14, or 119 cents) and the next semitone from
15/14 to 63/55 (294/275, or 116 cents).

I don't see a temperament name for this combination of commas, so I
don't know what to call it.

This is the resulting scale:

! C:\Program Files
(x86)\Scala22\2.3.5-7.11-9.diamond.tempered_243-242_1375_1372.scl
!

10
!
116.86545
233.73090
350.59635
498.80729
615.67274
701.19271
849.40365
966.26910
1083.13455
1200.00000

This happens to be a subset of 421 EDO. Interestingly, it's not the
patent val: If I understand patent vals,I'm pretty sure the patent val
for 421 EDO is < 421 667 978 1182 1456 |, while this mapping is
something like < 421 667 977 1182 1457 |. (I say "something like"
because Excel gives me slightly different results than Scala gave me.)
I didn't figure out how to use the different mapping myself, though --
that's Scala doing the work using the Temper function. All of the
scale steps are within about 3 cents of pure.

31 EDO would have tempered the same commas, with a standard meantone
compromise of a flatter 697-cent fifth and a more slightly more
accurate neutral third.

41 EDO would also have worked, and been about as accurate as 421 EDO.
Looking at it now, I'm not sure why I shouldn't use 41 EDO instead of
421.

There are three single-step sizes, about 117 cents (most common), 148
cents (only used twice), and one 86-cent step between the 10/7 and the
3/2.

Interestingly, although the 233.7-cent and 966.3-cent intervals are
very close to 8/7 and 7/4, those intervals aren't in the original
scale, and 421 EDO doesn't seem to temper out the difference between,
say, the 7/4 that's not in the scale and the 110/63 that is (the
difference is 441/440, about 4 cents), even with the alternate
mapping. That doesn't matter much to me, since I didn't even have a
pure 7 in the tonality diamond. 41 EDO *does* temper 441/440, so I
could have used that if it were important.

I really like the 0-3-6 (neutral third + perfect fifth) and 0-2-4 (233
cents + perfect fourth) triads.

The three-limit harmonies aren't as abundant as in most scales I've
played with -- six tones have 732-cent "fifths" above them, which of
course means that six also have 467-cent "fourths". Only the 1/1 mode
has both a perfect fourth and a perfect fifth above it. As a result,
when playing with this I've bounced back and forth between split
fifths and split fourths depending on the root of the chord I'm on,
and sometimes doing runs that don't include fifths or fourths at all,
or including the imperfect fourths and fifths more as passing tones
than as harmonies.

Just as there's an "accidental 5/4" in the scale, there are
"accidental 7/6s" as well, which I like.

Here's a simple one-minute piece of music in the scale, for classical
guitar and flute: http://www.freivald.org/~jake/documents/Noon-ish.mp3

I think it might be worth experimenting with the 3/2 mode in order to
capitalize on the 86-cent step as a leading tone, but there are no
good fourths or fifths in that mode, so I haven't tried yet. I still
need my crutches.

I've now spent more time writing about the scale and music than
creating it or composing in it, so I'm just going to send the email.
:)

Regards,
Jake

🔗Michael <djtrancendance@...>

Hi Jake,

First of all, to note, the abundance of 10/7 "real" tri-tones and neutral seconds in your scale are a real treat...and I love it when people step up to make their own "from scratch" scales, nice work! :-)

>"3) Create recognizable fourths and fifths."
I am not sure if I would call the around 463 cent interval in your scale a fourth...but it is still much closer in feel to a fourth and a third.

When I look at the base scale I see an 11/9 and a 28/15...forming the "wolf" fifth that comes from the octave inverse of the above fourth. One of my favorite tricks is taking 11/6 (a perfect fifth over 11/9) and then using it as an efficient neutral seventh...and you could easily "swap" 28/15 for 11/6.

The 18/11 and 15/14 have the same "wolf fifth" forming issue, but the 18/11 also forms that cool 10/7 with 63/55. However swapping the 15/14 with a 19/18 forms a near 14/9 "septimal minor sixth".

The 110/63 has the same issue with 63/55....but 63/55 and 12/7 form a pretty solid perfect fifth...plus the 12/7 also forms a 14/9 between itself and the 4/3 on the
next octave. So one trick is to swap the 110/63 with a 12/7.

Thus....I get the modified version of your scale

19/18 (****was 15/14**********)
63/55

11/9

4/3

10/7

3/2

18/11
12/7 (***was 110/63***)
11/6 (***was 28/15 ****)

Just for grins....I am wondering what your piece done on http://www.freivald.org/~jake/documents/Noon-ish.mp3 ...would sound like re-tuned to this scale?

🔗Ozan Yarman <ozanyarman@...>

An excellent scale demonstrating the profundity of maqam Huzzam as:

3 4 6 7 9 10 12 13 degrees
1 2 1 2 1 2 1 steps
148 202 148 234 117 234 117 cents (consecutive intervals)

with the occasional 966 cent pitch (8th degree) serving as an alteration
for the fifth above the finalis (351 cents).

One can also think of 234, 351, 616, 701 cent pitches altogether as a
wicked Hijaz tetrachord with yet another at 966, 1083, 1317, 1434 cent
pitches forming a disjunct Hijaz tetrachord atop and the wholetone
consigned to a 265 cent interval between 701 and 966 cent pitches. Of
course, "tetrachord" here is warped where the fourth is not pure at all.

I like the piece too!

Good work Jake.

Dr. Oz.

--
✩ ✩ ✩
www.ozanyarman.com

Jake Freivald wrote:
> Hi all,
>
> This piece started as an exercise in scale creation. I had four goals
> for the scale.
>
> 1) Use the 11/9 neutral third.
> 2) Don't use too many "easy" 5-limit harmonies.
> 3) Create recognizable fourths and fifths.
> 4) Avoid impropriety.
>
> I used Scala to generate a tonality diamond using 2, 3, 7/5, and 11/9.
> After deleting a few tones to avoid impropriety, I came up with the
> following 10-note scale:
>
> ! C:\Program Files (x86)\Scala22\2.3.5-7.11-9.diamond.scl
> !
>
> 10
> !
> 15/14
> 63/55
> 11/9
> 4/3
> 10/7
> 3/2
> 18/11
> 110/63
> 28/15
> 2/1
>
> After studying the interval attribute matrix (thanks to Carl for
> pointing that out to me), I decided that I should temper out 243/242
> and 1375/1372. That's< -1 5 0 0 -2 | and< -2 0 3 -3 1 |,
> respectively.
>
> 243/242 makes two neutral thirds equal a perfect fifth (3/2 = 11/9 *
> 11/9 * 243/242).
>
> 243/242 also tempers the difference between taking two steps up from
> the 63/55 to the 4/3 (220/189, or 263 cents) and taking two steps up
> from the 11/9 to the 10/7 (90/77, or 270 cents), which significantly
> smooths out that part of the scale and related intervals.
>
> 1375/1372 is the difference between taking three steps up from the
> 63/55 to the 10/7 (550/441, or 382 cents) and taking three steps up
> from the 15/14 to the 4/3 (56/45, or 379 cents). Although I wasn't
> intending to have too many 5-limit harmonies, this "accidental 5/4"
> still seemed to need tempering to smooth out the scale.
>
> 1375/1372 also tempers out the difference between the semitone moving
> up from 1/1 to 15/14 (15/14, or 119 cents) and the next semitone from
> 15/14 to 63/55 (294/275, or 116 cents).
>
> I don't see a temperament name for this combination of commas, so I
> don't know what to call it.
>
> This is the resulting scale:
>
> ! C:\Program Files
> (x86)\Scala22\2.3.5-7.11-9.diamond.tempered_243-242_1375_1372.scl
> !
>
> 10
> !
> 116.86545
> 233.73090
> 350.59635
> 498.80729
> 615.67274
> 701.19271
> 849.40365
> 966.26910
> 1083.13455
> 1200.00000
>
> This happens to be a subset of 421 EDO. Interestingly, it's not the
> patent val: If I understand patent vals,I'm pretty sure the patent val
> for 421 EDO is< 421 667 978 1182 1456 |, while this mapping is
> something like< 421 667 977 1182 1457 |. (I say "something like"
> because Excel gives me slightly different results than Scala gave me.)
> I didn't figure out how to use the different mapping myself, though --
> that's Scala doing the work using the Temper function. All of the
> scale steps are within about 3 cents of pure.
>
> 31 EDO would have tempered the same commas, with a standard meantone
> compromise of a flatter 697-cent fifth and a more slightly more
> accurate neutral third.
>
> 41 EDO would also have worked, and been about as accurate as 421 EDO.
> Looking at it now, I'm not sure why I shouldn't use 41 EDO instead of
> 421.
>
> There are three single-step sizes, about 117 cents (most common), 148
> cents (only used twice), and one 86-cent step between the 10/7 and the
> 3/2.
>
> Interestingly, although the 233.7-cent and 966.3-cent intervals are
> very close to 8/7 and 7/4, those intervals aren't in the original
> scale, and 421 EDO doesn't seem to temper out the difference between,
> say, the 7/4 that's not in the scale and the 110/63 that is (the
> difference is 441/440, about 4 cents), even with the alternate
> mapping. That doesn't matter much to me, since I didn't even have a
> pure 7 in the tonality diamond. 41 EDO *does* temper 441/440, so I
> could have used that if it were important.
>
> I really like the 0-3-6 (neutral third + perfect fifth) and 0-2-4 (233
> cents + perfect fourth) triads.
>
> The three-limit harmonies aren't as abundant as in most scales I've
> played with -- six tones have 732-cent "fifths" above them, which of
> course means that six also have 467-cent "fourths". Only the 1/1 mode
> has both a perfect fourth and a perfect fifth above it. As a result,
> when playing with this I've bounced back and forth between split
> fifths and split fourths depending on the root of the chord I'm on,
> and sometimes doing runs that don't include fifths or fourths at all,
> or including the imperfect fourths and fifths more as passing tones
> than as harmonies.
>
> Just as there's an "accidental 5/4" in the scale, there are
> "accidental 7/6s" as well, which I like.
>
> Here's a simple one-minute piece of music in the scale, for classical
> guitar and flute: http://www.freivald.org/~jake/documents/Noon-ish.mp3
>
> I think it might be worth experimenting with the 3/2 mode in order to
> capitalize on the 86-cent step as a leading tone, but there are no
> good fourths or fifths in that mode, so I haven't tried yet. I still
> need my crutches.
>
> I've now spent more time writing about the scale and music than
> creating it or composing in it, so I'm just going to send the email.
> :)
>
> Regards,
> Jake
>
>
> ------------------------------------
>
> Yahoo! Groups Links
>
>
>
>

[Non-text portions of this message have been removed]

🔗Jake Freivald <jdfreivald@...>

Mike said:
> First of all, to note, the abundance of 10/7 "real" tri-tones
> and neutral seconds in your scale are a real treat...

Interesting set of sounds, right? Stickman recommended Munir Bachir on a
list (this list?), and I liked the neutral intervals in that music, so I
thought I should play with them a little bit. At this point, I'd say
everyone should. :) Interestingly, I don't find them to be "neutral" -- they
seem very consonant to me, not "major" per se but mostly happy.

> and I love it when people step up to make their own "from
> scratch" scales, nice work! :-)

Thanks. All part of the learning process. It's like learning a city: I don't
feel like I know it until I've walked its streets, and preferably gotten a
little lost along the way.

> "3) Create recognizable fourths and fifths."
> I am not sure if I would call the around 463 cent interval in your
> scale a fourth

I don't. :) The scale does have *some* perfect fourths and fifths in it,
though, and that's all I was going for.

My tools are all based around 12-note scales (I haven't sussed out how to
manage microtonal scales in Lilypond yet), and I find that I can't have
decent fourths and fifths on every step unless I stick to traditional (12
EDO, meantone, Pythagorean) scales; those scales are fine, but not what I'm
trying to play with right now. That said, I wanted at least a few P4/P5s in
there.

> When I look at the base scale I see an 11/9 and a 28/15...forming [snip]

Interesting ideas here. I'll play with 11/9 and 11/6 a little bit -- that's
an interesting set of sounds, with or without a fifth in between.

> Just for grins....I am wondering what your piece done
> on http://www.freivald.org/~jake/documents/Noon-ish.mp3
> ...would sound like re-tuned to this scale?

Sure, easy enough. I wasn't sure what you wanted, so I've rendered it twice,
in pure JI and tempered to 41 EDO. The most common commas seem to be
243/242, 441/440, and 540/539, and 41 EDO tempers all three.

http://www.freivald.org/~jake/documents/noon-ish_mikeMod-just.mp3<http://www.freivald.org/%7Ejake/documents/Noon-ish.mp3>
http://www.freivald.org/~jake/documents/noon-ish_mikeMod-41EDO.mp3<http://www.freivald.org/%7Ejake/documents/Noon-ish.mp3>

There's a definite difference when using your scale vs. the original, but I
haven't listened closely enough to see whether the tempered scale is
significantly different from the just scale. I'll withhold further comment
until you've heard these.

Regards,
Jake

[Non-text portions of this message have been removed]

🔗Jake Freivald <jdfreivald@...>

🔗Michael <djtrancendance@...>

Jake>"Interestingly, I don't find them to be "neutral" -- they
seem very consonant to me, not "major" per se but mostly happy."

Same here. If anything I find them a bit "weaker" than major intervals, but also softer and, in a weird way, more forgiving.
----------------

Now about the music...

http://www.freivald.org/~jake/documents/noon-ish_mikeMod-just.mp3<http://www.freivald.org/%7Ejake/documents/Noon-ish.mp3>

http://www.freivald.org/~jake/documents/noon-ish_mikeMod-41EDO.mp3<http://www.freivald.org/%7Ejake/documents/Noon-ish.mp3>

Funny, my scale your piece NOT in 41EDO sounds sadder...almost "down a key/semitone"...but more relaxed and a bit less sharp/assertive in mood to my ears.

The parts that really threw me around a bit were at
1) 24-27 seconds
2) 40-45 seconds

Those were, perhaps, the only parts in either version that sounded unstable/vague enough in mood that I didn't think I could trick a 12TET listener into thinking it was "in key". They sounded (to my ear) a fair deal less grating in my "just" scale, but also with a good deal more vagueness and less inflection (IE they sounded more soft/"neutral", so to speak). Dare I guess...perhaps they would work better in 31EDO than 41EDO (my scales tend to round best to 31EDO)...

>"I find that I can't have decent fourths and fifths on every step unless I stick to traditional (12 EDO, meantone, Pythagorean) scales"

Right, you have to give up that whole "circle of fifths" pattern and replace the fifths/fourths with something merely "fifth/fourth-ish". Which, to me, makes things like 14/9 and 22/15 "substitute fifths" and their resulting 9/7 and 15/11 "substitute fourths" (even though 9/7 is not technically a fourth and 14/9 is technically a "septimal minor sixth") ideal candidates.

There are many fascinating options when you try to take combinations of 3/2, 14/9, 22/15 and chain them to nearly meet the octave and then temper slightly to make them fit the octave...Mohajira being an obvious one.

Perhaps I'm oversimplifying here, but a think a good basic gateway into Middle Eastern music (IE your example of Munir Bachir) is to simply take a couple of fifths (maximally spaced apart is often best) and replace them with diminished fifths nearing 22/15, forming a bunch of wonderful neutral intervals for that happy/relaxed neutral mood.

🔗Ozan Yarman <ozanyarman@...>

🔗Jake Freivald <jdfreivald@...>

First, to Dr. Oz: I saw several items about "Iranian" music in the brief
search I did on the maqam Huzzam. I clearly wasn't paying enough attention,
because there's ample material out there showing that the maqam Huzzam is
Turkish (and perhaps Arabic?), and I apologize for any inadvertant rudeness.
My ignorance is, unfortunately, very great.

Next, to Mike:

> Funny, my scale your piece NOT in 41EDO sounds sadder...almost
> "down a key/semitone"...but more relaxed and a bit less sharp/
> assertive in mood to my ears.

The second note in the piece -- which is a common note throughout -- is the
9th note in the scale. It's 110/63 (965 cents) in my scale and 12/7 (933
cents) in yours, so it's definitely "down a semitone". I like it less than
the original. It tough to objectively understand yourself, but I think the
965-cent note is flat enough to be a little "in your face", but not
FLAT-flat, if you know what I mean. The 933-cent note is just more (or less,
rather) than my ear wants to hear.

The other two replacements you made -- 19/18 (94 cents) replacing 15/14 (119
cents) and 11/6 (1049 cents) replacing 28/15 (1080 cents) are barely played,
and usually in the background, so they'd make less of a difference to the
piece overall.

> The parts that really threw me around a bit were at
> 1) 24-27 seconds
> 2) 40-45 seconds

Interestingly, those don't strike me as unstable or vague.

I also don't think this piece as written feels "12-EDO" to me, and I wonder
what 12-equal people (is there a term for that? 12-harmonicists? Microtonal
muggles?) would think. The first two notes of the melody drop from 1200
cents to 850, which is a warning shot to everyone that this isn't a normally
tuned piece. Then it bounces back up to the octave and down to the perfect
fifth, to make you comfortable that I won't violate your sense of tonality
too much. The flute is, generally speaking, more "normal" in the notes it
chooses than the guitar, which carries the neutral and septimal minor
chords; however, the guitar is a mellow instrument with a smooth texture,
which avoids making the chords sound too harsh. Also, the chord roots are
1/1, 3/2, 4/3, and 11/9, which makes for a diatonic-sounding structure with
some xenharmony thrown in.

So it's far enough from 12-EDO without being obnoxious, which is
more-or-less what I was going for. Your mileage may vary, of course, and I'm
always interested in alternate opinions.

> Dare I guess...perhaps they would work better in 31EDO than
> 41EDO (my scales tend to round best to 31EDO)...

I did a version of the song in 31EDO, and it doesn't change the basic
difference between your scale and mine. Of course, the fourths and fifths
are less pure, and my ears are starting to get to the point where I'll
notice the difference. Also, the 94-cent 19/18 gets mapped to 116 cents,
which is very close to 15/14. Although you won't notice the difference in
this piece, you might notice it elsewhere.

http://www.freivald.org/~jake/documents/noon-ish_mikeMod-31EDO.mp3

Regards,
Jake

[Non-text portions of this message have been removed]

🔗Michael <djtrancendance@...>

Jake>"It's 110/63 (965 cents) in my scale and 12/7 (933 cents) in yours, so it's definitely "down a semitone". I like it less than the original. It tough to objectively understand yourself, but I think the 965-cent note is flat enough to be a little "in your face", but not

FLAT-flat, if you know what I mean. The 933-cent note is just more (or less,

rather) than my ear wants to hear."

Agreed again, the Just version of my version of your scale is less punchy sounding due to feeling "down a semitone".

>"I did a version of the song in 31EDO, and it doesn't change the basic

difference between your scale and mine...The first two notes of the melody drop from 1200 cents to 850"
How is this possible? My scale includes the perfect 2/1 octave as its period...so anything that's originally an octave should round to the octave. :-S

>"Also, the 94-cent 19/18 gets mapped to 116 cents,

which is very close to 15/14. Although you won't notice the difference in

this piece, you might notice it elsewhere."

116 cents? I calculated the closest note in 31EDO to 19/18 is 77.413 cents, not the next tone in 31EDO of 116.1295 cents. Granted, 77.4 cents still is fairly far from 94-cents...but is still a fair deal closer than 116 cents.
Again this begs the question how are you mapping these notes (on the surface, the method seems far less than ideal)?

>"I did a version of the song in 31EDO, and it doesn't change the basic difference between your scale and mine. Of course, the fourths and fifths are less pure, and my ears are starting to get to the point where I'll notice the difference."

Again I am confused...as I have known 31EDO for having very good pure fifths, what are you mapping the fifth to in 31EDO?
-----------------
If you understand the "tempering"/rounding scheme I'm using to put my scale in 31EDO (which is apparently a good deal different than yours)...I would be interested in seeing you post a version of the song re-tuned to that.

[Non-text portions of this message have been removed]

🔗Jake Freivald <jdfreivald@...>

I said:
> > I did a version of the song in 31EDO, and it doesn't change the basic
> > difference between your scale and mine...The first two notes of the
> > melody drop from 1200 cents to 850

Mike replied:
> How is this possible? My scale includes the perfect 2/1 octave as its
> period...so anything that's originally an octave should round to the octave.

Sorry, I said the wrong thing, and we're miscommunicating as well. :)

In the original, as I've constructed the piece in LilyPond, the first two notes of the melody are C and Bb. Those tones correspond to 2/1 and 110/63, or 1200 and 966 cents. (I said 850, which was incorrect; I was thinking of the A eighth note that comes on the upbeat.) In your scale, the Bb is 12/7, so the melody goes from 1200 cents to 933 cents.

Make more sense? Same issue, just on the wrong step of the scale.

> > Also, the 94-cent 19/18 gets mapped to 116 cents, which is very close
> > to 15/14. Although you won't notice the difference in this piece, you
> > might notice it elsewhere.
>
> 116 cents? I calculated the closest note in 31EDO to 19/18 is 77.413
> cents, not the next tone in 31EDO of 116.1295 cents.

That's correct, the *closest* note is, but if you use the 19-limit patent val for 31, which is
< 31 49 72 87 107 115 127 132 |, 19/18 maps to 116 cents, not 77.

Now that I think of it, though, 19/18 is the only 19-limit interval in the modified scale, so I guess it wouldn't hurt to use the val
< 31 49 72 87 107 115 127 131 | instead. (I just reduced the 19 mapping by 1.) Someone who knows the fundamentals of the math better might be willing to weigh in on whether it makes a difference in any other way, but if I understand correctly, it shouldn't.

> > Again this begs the question how are you mapping these notes (on the
> > surface, the method seems far less than ideal)?

I'm going off of what I learned in this thread:
/tuning/topicId_98950.html#98950

At this point, I'm pretty sure it's right. Are you using a mapping technique, or just rounding?

> > I did a version of the song in 31EDO, and it doesn't change
> > the basic difference between your scale and mine. Of course,
> > the fourths and fifths are less pure, and my ears are starting
> > to get to the point where I'll notice the difference.
>
> Again I am confused...as I have known 31EDO for having very good
> pure fifths, what are you mapping the fifth to in 31EDO?

It *is* very good: 696.77 cents, only about five cents flat. It's just on the verge of what I can hear, and only in certain circumstances -- sometimes I won't notice at all. But in this case, maybe because of the other flattened notes, it just seems a touch under.

> If you understand the "tempering"/rounding scheme I'm using to put
> my scale in 31EDO (which is apparently a good deal different than
> yours)...I would be interested in seeing you post a version of the
> song re-tuned to that.

I think this is the scale you recommended:

19/18
63/55
11/9
4/3
10/7
3/2
18/11
12/7
11/6
2/1

...which, tempered to 31 EDO, and using the alternate mapping for 19/18, gives me this:

77.41935
232.25806
348.38710
503.22581
619.35484
696.77419
851.61290
929.03226
1045.16129
1200.00000

Yes?

I don't think you will hear any significant difference between that and what I rendered in 31 EDO before, because the only difference is the 77-cent interval, which is only used once, as one of the falling 16th notes at about 29 seconds in.

Regards,
Jake

🔗Michael <djtrancendance@...>

Me>>"I calculated the closest note in 31EDO to 19/18 is 77.413 cents, not the next tone in 31EDO of 116.1295 cents.

Jake>"That's correct, the *closest* note is, but if you use the 19-limit

patent val for 31, which is < 31 49 72 87 107 115 127 132 |, 19/18 maps to 116 cents, not 77."
I think I had this argument with Gene about patent vals ages ago when designing a program to find optimum scales under 31EDO... And this is a shining example of the difference between how I think notes should be remapped in both re-tuning and tuning theory in general (closest error) vs. how patent vals map them.

>"At this point, I'm pretty sure it's right. Are you using a mapping technique, or just rounding?"
Rounding to the nearest number. Which, oddly enough, I've found works quite well...especially in cases where the rounding difference is about 14 cents or less near a simple interval (IE 4/3, 5/4, 6/5, 7/4...) or less than 8 cents from any extended-just interval (IE 16/9, 11/9, 13/9)... Maybe I just have weird ears that thinks closer rounded ratios (far as error) sound more alike than patent val mappings...

>"I don't think you will hear any significant difference between that and

what I rendered in 31 EDO before, because the only difference is the

77-cent interval, which is only used once, as one of the falling 16th

notes at about 29 seconds in."
Ah ok, so it's not much of an issue anyhow as "99%" of the notes in that piece are the same.

[Non-text portions of this message have been removed]

🔗Jake Freivald <jdfreivald@...>

I am responding here to make sure that people who are interested on MMM see the answer, because I think it's very important. I'm also cc'ing the tuning list, because I think any further discussion should take place there.

Mike said:
> I think I had this argument with Gene about patent vals ages
> ago when designing a program to find optimum scales under 31EDO...
> And this is a shining example of the difference between how I
> think notes should be remapped in both re-tuning and tuning theory
> in general (closest error) vs. how patent vals map them.

To clarify one thing: The patent val may not always be the val you want; in the current "shining example" case, for example -- which is really a best case for you, rather than a good example generally -- changing one number in the val will give you a better mapping for this specific scale.

Respectfully, though, I think you're wrong to only focus on the error of a individual intervals. Whatever makes you happy is fine, of course, but you're losing more than you're gaining.

Here's why:

----------

1. Using a subset of an EDO to represent a JI scale always introduces error. Error isn't always bad, of course; it's this error that lets us temper out a specific comma, for instance. How that error gets applied, however, is critical.

---

2. When you round intervals to their nearest EDO steps, you try to minimize the error for each interval independently. Because you're considering each interval independently, there's no mathematical relationship between the number of EDO steps for one interval vs. another. In other words, this isn't a regular temperament.

As a result, "31 EDO tempers out 352/351" isn't true for your scale. It can't be, because the primes 2, 3, 11, and 13 aren't getting mapped to the EDO. In fact, the notion of commas becomes *meaningless*. If you do get relationships that look like an interval has been tempered out, they're *coincidental* -- but *not* a necessary result of the process.

---

3. When you use a formal mapping to determine which EDO steps represent a JI interval, you are applying the error in a consistent way.

When given a collection of monzos that represent your intervals, you use the same val for the EDO to create homomorphisms. (In other words, the single val <V| is used with each interval's monzo |M> to create <V|M>, which tells you the number of EDO steps you should use to represent each interval.)

Because these homomorphisms are based on the primes of the intervals AND the way these primes get mapped to the EDO, this *is* a regular temperament. You *can* say "31 EDO tempers out 352/351" because that's a necessary result of using the mapping process to get the EDO steps for each interval.

This sounds complex, but it's really not. I have a simple spreadsheet that does the job for me.

----------

In the current case, you had only one 19-limit interval in the scale. Because of that, we could adjust the 19-limit val without affecting any other intervals. It's therefore a best-possible-case scenario for the "rounding" approach vs. the "mapping" approach, because the rounding doesn't really do any damage to the relationships between the EDO intervals, and you ended up with a scale that *could* result from a val; however, that's *coincidental*, not *necessary*. On the other hand, if you had had other 19-limit intervals, choosing the rounded number for one versus the mapped number for another would have resulted in having an irregular temperament, and all notions of "such-and-such comma is tempered out" would have flown out the window.

You may not like the fact that regular temperaments can give you the "wrong" interval, i.e., one that you wouldn't have picked if you were rounding. I didn't either, which is why I thought I was doing something wrong -- but I wasn't. As Graham succinctly explained, "When you choose a regular temperament, you give an inherent error to each prime interval (mapping of a prime number). Those errors accumulate when you look at smaller intervals. After a point, you'll always [have] something with a higher error than the step size you're dealing with." Perfectly said.

Thus using mappings (i.e., vals) to create a scale has a significant theoretical advantage over rounding for the general case. It *is* Paul Erlich's "Middle Path" between JI and EDOs, whereas rounding is not. If you like things like Porcupine and Mavila, you need to pay attention to the mappings, and not just rounding.

Obviously, you don't have to follow the theory, and you can use any theory (or no theory!) to create scales that make you happy. Also, if you have a physical instrument that can't step outside of a certain EDO, you may use what you can despite the fact that it's not theoretically the right way to go. But the mapping theory is clearly superior in most cases to the rounding theory.

To sum up: If you care about tempering commas, you should use the mapping method. Also, it seems to me that the mapping method is an important part of tuning theory because it retains the relationships of the primes in the intervals, whereas rounding is not because it does not.

Regards,
Jake

Short piece using tempered 2.3.7/5.11/9 tonality diamond

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