Tetrachords and "Munits"

Tetrachords are useful. They are a sequence of melodic steps with a
harmonic ratio on the outside. We might consider other sequences of
steps with other ratios on the outside. I will term such sequences
"munits," short for "melodic unit."

A munit will be denoted by the notation a/b: LsLsLsL, where a/b is a
harmonic ratio called the "framing interval," and LsLsLsLsLsL is some
sequence of step sizes that specifies the munit. I'll use the
following step sizes: s = small, L = large, A = large and audibly more
than twice the size of s (such that chains of munits with lots of A
notes tend to create improper scales). Whether a step is L or A is
likely somewhat subjective.

For example, here are some common munits you probably know very well
that exist in 12-EDO:

4/3: LLs (C D E F)
4/3: LsL (C D Eb F)
4/3: sLL (C Db Eb F)
4/3: sAs (C Db E F)
4/3: LLLLL (C Db D Eb E F)

5/4: LL (C D E)
5/4: sLs (C Db D# E)
5/4: LLLL (C Db D Eb E)

6/5: Ls (C D Eb)
6/5: sL (C Db Eb)
6/5: LLL (C Db D Eb)

2/1: LLsLLLs (C D E F G A B C)

Here are a few that you might not know, but which keep turning up
again and again and again in temperament after temperament. You should
learn all of these:

4/3: LsLs, sLsL (appears in at least Semaphore[9], Blackwood[10], Superpelog[9])
4/3: LLLs, LLsL, LsLL, sLLL (appears in at least Mohajira[10], Mavila[9])
4/3: Lss, sLs, ssL (appears in at least Mohajira[7])
4/3: LLL (appears in at least Porcupine[7], Hedgehog[8])
4/3: LLLL (appears in at least Negri[9])

5/4: Ls (appears in almost any temperament where 81/80 doesn't vanish)
5/4: LLs, LsL, sLL (also appears in almost any temperament where 81/80
doesn't vanish - Porcupine[8], Blackwood[10], Mavila[9], Triforce[9],
etc)
5/4: LLL (appears in at least Negri[9], Beatles[10])

6/5: LL (appears in Mavila[9], Porcupine[7], Mohajira[7], Hedgehog[8], etc)
6/5: LLs, LsL, sLL (appears in Negri[10])

9/8: Ls, sL (appears frickin everywhere)

This is not supposed to be anywhere near an exhaustive list.

Anyway, you surely know intuitively that 5/4 is broken down into two
"whole steps"... in meantone. But do you know intuitively to break it
up into LsL in a million other useful MOS's? Well learn it!

As a last note, I'm not sure that what I'm really trying to get at
here has to do with ratios. Instead, it probably has to do with
categories. For example, If you learn that 6/5 breaks down into any
sort of step pattern, you could probably replace 6/5 with 7/6 and have
the whole thing still be completely intelligible. Same with 5/4 vs
9/7. But hey, it's a start.

-Mike

Also, note to myself (and others) for things that need doing with this

1) Find a way to conduct a more thorough search for what temperaments'
MOS's contain which munits
2) Look at regularly mapped munits, for lovers of harmony. Perhaps
look at munits as Fokker blocks in frame-equivalent space
3) Make a wiki page (after figuring out the above)

Any ideas about the most efficient way to start looking at #1 and #2?

-Mike

On Tue, Dec 20, 2011 at 8:24 AM, Mike Battaglia <battaglia01@...> wrote:
> Tetrachords are useful. They are a sequence of melodic steps with a
> harmonic ratio on the outside. We might consider other sequences of
> steps with other ratios on the outside. I will term such sequences
> "munits," short for "melodic unit."
>
> A munit will be denoted by the notation a/b: LsLsLsL, where a/b is a
> harmonic ratio called the "framing interval," and LsLsLsLsLsL is some
> sequence of step sizes that specifies the munit. I'll use the
> following step sizes: s = small, L = large, A = large and audibly more
> than twice the size of s (such that chains of munits with lots of A
> notes tend to create improper scales). Whether a step is L or A is
> likely somewhat subjective.
>
> For example, here are some common munits you probably know very well
> that exist in 12-EDO:
>
> 4/3: LLs (C D E F)
> 4/3: LsL (C D Eb F)
> 4/3: sLL (C Db Eb F)
> 4/3: sAs (C Db E F)
> 4/3: LLLLL (C Db D Eb E F)
>
> 5/4: LL (C D E)
> 5/4: sLs (C Db D# E)
> 5/4: LLLL (C Db D Eb E)
>
> 6/5: Ls (C D Eb)
> 6/5: sL (C Db Eb)
> 6/5: LLL (C Db D Eb)
>
> 2/1: LLsLLLs (C D E F G A B C)
>
> Here are a few that you might not know, but which keep turning up
> again and again and again in temperament after temperament. You should
> learn all of these:
>
> 4/3: LsLs, sLsL (appears in at least Semaphore[9], Blackwood[10], Superpelog[9])
> 4/3: LLLs, LLsL, LsLL, sLLL (appears in at least Mohajira[10], Mavila[9])
> 4/3: Lss, sLs, ssL (appears in at least Mohajira[7])
> 4/3: LLL (appears in at least Porcupine[7], Hedgehog[8])
> 4/3: LLLL (appears in at least Negri[9])
>
> 5/4: Ls (appears in almost any temperament where 81/80 doesn't vanish)
> 5/4: LLs, LsL, sLL (also appears in almost any temperament where 81/80
> doesn't vanish - Porcupine[8], Blackwood[10], Mavila[9], Triforce[9],
> etc)
> 5/4: LLL (appears in at least Negri[9], Beatles[10])
>
> 6/5: LL (appears in Mavila[9], Porcupine[7], Mohajira[7], Hedgehog[8], etc)
> 6/5: LLs, LsL, sLL (appears in Negri[10])
>
> 9/8: Ls, sL (appears frickin everywhere)
>
> This is not supposed to be anywhere near an exhaustive list.
>
> Anyway, you surely know intuitively that 5/4 is broken down into two
> "whole steps"... in meantone. But do you know intuitively to break it
> up into LsL in a million other useful MOS's? Well learn it!
>
> As a last note, I'm not sure that what I'm really trying to get at
> here has to do with ratios. Instead, it probably has to do with
> categories. For example, If you learn that 6/5 breaks down into any
> sort of step pattern, you could probably replace 6/5 with 7/6 and have
> the whole thing still be completely intelligible. Same with 5/4 vs
> 9/7. But hey, it's a start.
>
> -Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Tetrachords are useful. They are a sequence of melodic steps with a
> harmonic ratio on the outside. We might consider other sequences of
> steps with other ratios on the outside. I will term such sequences
> "munits," short for "melodic unit."

Munits are awesome. I love munits. Do you say "em unit" or "myoonit" or what?

> A munit will be denoted by the notation a/b: LsLsLsL, where a/b is a
> harmonic ratio called the "framing interval," and LsLsLsLsLsL is some
> sequence of step sizes that specifies the munit. I'll use the
> following step sizes: s = small, L = large, A = large and audibly more
> than twice the size of s (such that chains of munits with lots of A
> notes tend to create improper scales). Whether a step is L or A is
> likely somewhat subjective.
>
> For example, here are some common munits you probably know very well
> that exist in 12-EDO:
>
> 4/3: LLs (C D E F)
> 4/3: LsL (C D Eb F)
> 4/3: sLL (C Db Eb F)
> 4/3: sAs (C Db E F)
> 4/3: LLLLL (C Db D Eb E F)
>
> 5/4: LL (C D E)

> 5/4: sLs (C Db D# E)

This one, which in meantone is 9/7 sLs, is really important because when you hear it you know exactly where you are in harmonic minor. It has great tonicizing power

> 5/4: LLLL (C Db D Eb E)
>
> 6/5: Ls (C D Eb)
> 6/5: sL (C Db Eb)
> 6/5: LLL (C Db D Eb)
>
> 2/1: LLsLLLs (C D E F G A B C)
>
> Here are a few that you might not know, but which keep turning up
> again and again and again in temperament after temperament. You should
> learn all of these:
>
> 4/3: LsLs, sLsL (appears in at least Semaphore[9], Blackwood[10], Superpelog[9])

This one is nuts! Especially the variant 4/3 sLLs. You hear that and instantly think "tritone", so it really messes with your head when it's 4/3.

> 4/3: LLLs, LLsL, LsLL, sLLL (appears in at least Mohajira[10], Mavila[9])
> 4/3: Lss, sLs, ssL (appears in at least Mohajira[7])
> 4/3: LLL (appears in at least Porcupine[7], Hedgehog[8])
> 4/3: LLLL (appears in at least Negri[9])
>
> 5/4: Ls (appears in almost any temperament where 81/80 doesn't vanish)

I suggest that "Ls" is a uniquely weak munit, because any interval can be divided into two unequal parts, in any temperament. All other sequences have at least two of the same kind of step, so there's more structure to them.

> 5/4: LLs, LsL, sLL (also appears in almost any temperament where 81/80
> doesn't vanish - Porcupine[8], Blackwood[10], Mavila[9], Triforce[9],
> etc)
> 5/4: LLL (appears in at least Negri[9], Beatles[10])
>
> 6/5: LL (appears in Mavila[9], Porcupine[7], Mohajira[7], Hedgehog[8], etc)

I find this one to be very recognizable and pleasant. It's an important munit in pelog[5], in which it appears only once. I guess the same thing holds for mohajira[7].

> 6/5: LLs, LsL, sLL (appears in Negri[10])

And don't forget 6/5 sLs, which is in godzilla[9] and is totally nuts.

> 9/8: Ls, sL (appears frickin everywhere)
>
> This is not supposed to be anywhere near an exhaustive list.
>
> Anyway, you surely know intuitively that 5/4 is broken down into two
> "whole steps"... in meantone. But do you know intuitively to break it
> up into LsL in a million other useful MOS's? Well learn it!
>
> As a last note, I'm not sure that what I'm really trying to get at
> here has to do with ratios. Instead, it probably has to do with
> categories. For example, If you learn that 6/5 breaks down into any
> sort of step pattern, you could probably replace 6/5 with 7/6 and have
> the whole thing still be completely intelligible. Same with 5/4 vs
> 9/7. But hey, it's a start.

No, this is all wrong. You don't need categories for munits; munits ARE categories. Well, at least part of them.

I think a category consists of:
* A set of compatible munits.
* A very rough interval size.
* A vague expectation of harmonic quality. This at least includes consonance vs dissonance, but probably does not include specific VFs at all.

Keenan

On Tue, Dec 20, 2011 at 10:35 AM, Keenan Pepper <keenanpepper@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> >
> > Tetrachords are useful. They are a sequence of melodic steps with a
> > harmonic ratio on the outside. We might consider other sequences of
> > steps with other ratios on the outside. I will term such sequences
> > "munits," short for "melodic unit."
>
> Munits are awesome. I love munits. Do you say "em unit" or "myoonit" or what?

myoonit, like "munition."

> > 5/4: sLs (C Db D# E)
>
> This one, which in meantone is 9/7 sLs, is really important because when you hear it you know exactly where you are in harmonic minor. It has great tonicizing power

Yes, and it's also in diminished[8] and in the altered scale. It has a
few interpretations.

> > 4/3: LsLs, sLsL (appears in at least Semaphore[9], Blackwood[10], Superpelog[9])
>
> This one is nuts! Especially the variant 4/3 sLLs. You hear that and instantly think "tritone", so it really messes with your head when it's 4/3.

> > 5/4: Ls (appears in almost any temperament where 81/80 doesn't vanish)
>
> I suggest that "Ls" is a uniquely weak munit, because any interval can be divided into two unequal parts, in any temperament. All other sequences have at least two of the same kind of step, so there's more structure to them.

You should strengthen it by learning 5/4: LsL below. Ls sounds
"subminor," but then the next L makes it sound major. The whole 225
cents in 16-edo being ambiguous thing is captured in a bunch of
temperaments and is the characteristic sound of the 5/4: LsL trichord.

> > As a last note, I'm not sure that what I'm really trying to get at
> > here has to do with ratios. Instead, it probably has to do with
> > categories. For example, If you learn that 6/5 breaks down into any
> > sort of step pattern, you could probably replace 6/5 with 7/6 and have
> > the whole thing still be completely intelligible. Same with 5/4 vs
> > 9/7. But hey, it's a start.
>
> No, this is all wrong. You don't need categories for munits; munits ARE categories. Well, at least part of them.
>
> I think a category consists of:
> * A set of compatible munits.
> * A very rough interval size.
> * A vague expectation of harmonic quality. This at least includes consonance vs dissonance, but probably does not include specific VFs at all.

I agree in general with this list, but I note that this list does not
imply your statement "you don't need categories for munits." This is
because the last thing you mentioned - "a vague expectation of
harmonic quality" - does not necessarily have anything to do with
concordance at all, but may still have to do with learned factors. A
set of learned factors that comprises a harmonic quality is itself
just another layer of categorization.

For example, in the categorical experiments, all of the munits stayed
the same except that 5/4 became 9/7 and 6/5 became 7/6, and we already
know that the harmonic qualities didn't break. Additionally, over the
course of these experiments, the concordance of the major thirds
varied from relatively low at 5/4 to rather high at around 350 cents,
yet some aspect of the "harmonic quality" stayed the same. Hence, the
"major third" category itself implies a "harmonic" relationship with
the tonic, in addition to a melodic one, that is apparently
independent of its intonation.

Additionally, for the very sharp tunings, the augmented fourth ended
up being closer to 3/2 than the actual generator, yet it still sounded
unstable and in need of resolution. And around 22-EDO, the augmented
second became a 5/4, but that didn't change the harmonic quality. This
shouldn't be too much of a surprise because the German sixth chord,
intoned as a near perfect 4:5:6:7 in quarter comma meantone, is one of
the least stable chords around.

Thus, the last item in your list - "a vague expectation of harmonic
quality" - is itself a function of categorical perception. So the
"harmonic quality" that one is expecting is still a function of many
things other than the intonation, many of which are probably learned.
Something deeper is at work.

What I do know is that this mysterious "harmonic quality" is what I'm
learning to reassociate with these munits, whatever it is. As a
heuristic guess it may related to the same thing that caused your
perception of pelog to change when you learned that there were empats
everywhere.

-Mike

On Tue, Dec 20, 2011 at 10:35 AM, Keenan Pepper <keenanpepper@...> wrote:
> > 6/5: LLs, LsL, sLL (appears in Negri[10])
>
> And don't forget 6/5 sLs, which is in godzilla[9] and is totally nuts.

This one appears in Blackwood[10] and a bunch of random MOS's in 16-EDO as well.

-Mike