Complete generalized system of mode-naming for MOS's and MODMOS's

There are a lot of scales, and no sensible way to name the modes for
all of them. There are two sensible options:

1) something like "5|1" for "5 generators up, 1 generator down"
2) something like "5|1" for "5 'major' intervals, 1 'minor' interval"
(relative to the root. Or perhaps 'large' vs 'small' intervals)

Luckily, the two are actually the same thing - provided we pick the
proper generator! The generator we need to pick to ensure that the
most "up" generators also generates a "most major" mode, in some
sense, will be MOS-specific, and I will call it the "normal generator"
for some MOS. Here are some examples - for now, I will use the word
"major" to denote the larger interval in each class, and "minor" to
denote the smaller, for simplicity - e.g. the perfect fifth is the
"major fifth," and the diminished fifth is the "minor fifth."

Here's meantone[7], where the needed generator is 3/2:
6|0 - LLLsLLs - major interval classes: 2, 3, 4, 5, 6, 7 - minor
interval classes:
5|1 - LLsLLLs - major interval classes: 2, 3, 5, 6, 7 - minor interval
classes: 4
4|2 - LLsLLsL - major interval classes: 2, 3, 5, 6 - minor interval
classes: 4, 7
3|3 - LsLLLsL - major interval classes: 2, 5, 6 - minor interval
classes: 3, 4, 7
2|4 - LsLLsLL - major interval classes: 2, 5 - minor interval classes:
3, 4, 6, 7
1|5 - sLLLsLL - major interval classes: 5 - minor interval classes: 2,
3, 4, 6, 7
0|6 - sLLsLLL - major interval classes: - minor interval classes: 2,
3, 4, 5, 6, 7

Now let's observe mavila[7], where the generator now has to be 4/3
(!!!) instead of 3/2 to generate the same behavior:
6|0 - LssLsss - major interval classes: 2, 3, 4, 5, 6, 7 - minor
interval classes:
5|1 - LsssLss - major interval classes: 2, 3, 4, 6, 7 - minor interval
classes: 5
4|2 - sLssLss - major interval classes: 3, 4, 6, 7 - minor interval
classes: 2, 5
3|3 - sLsssLs - major interval classes: 3, 4, 7 - minor interval
classes: 2, 5, 6
2|4 - ssLssLs - major interval classes: 4, 7 - minor interval classes:
2, 3, 5, 6
1|5 - ssLsssL - major interval classes: 4 - minor interval classes: 2,
3, 5, 6, 7
0|6 - sssLssL - major interval classes: - minor interval classes: 2,
3, 4, 5, 6, 7

This is a GOOD way to do things, firstly because it makes mathematical
sense and is easy to figure out. Secondly, all of the modes on this
continuum are identical to their adjacent neighbors except for the
alteration of one note by an L-s chroma, which makes it easy to see
how one mode modulates into another, and the "most major" modes are at
the top. Lastly, anyone here who's studied modal harmony in depth will
be able to attest to how important the intuitive understanding of both
of these things is for really "getting" modal harmony - both the
generator pattern for the mode in question, as well as how many
"major" or "minor" intervals a scale has.

The latter is actually a bit more important, and it is commonplace for
beginning students of modal harmony to write the diatonic modes down
in order from "most major" to "least major," or "brightest" to
"darkest"; to do so automatically implies a choice of generator and
hence we can kill two birds with one stone. I'm predominantly
self-taught for understanding modal theory, but when I went to UM I
found that everyone there had figured out the same thing I had,
including the teachers: stop thinking about "Dorian" as being "the
major scale starting on D," and start thinking about it like "Aeolian
with a #6," hence placing it in the above continuum. The above setup
notates things into the infinitely more useful generator and
chroma-based paradigm from the getgo.

Of course, if you're following this, then you're now understanding
that you've actually understood modal harmony all along, and are now
playing chord progressions on your piano that morph from Lydian to
Ionian to Mixolydian and back, thus revealing my hidden agenda to
teach you guys jazz harmony without you all realizing it ;)

We can also thus describe the MODMOS's as follows:

Ascending melodic minor - 5|1 b3 (or 3|3 #6#7, etc)
Harmonic minor - 2|4 #7 (or 5|1 b3b6, etc)
Harmonic major - 5|1 b6 (or 2|4 #3#7, etc)

-Mike

Correction - there is one issue that remains - the issue of whether
diminished[8] is something like 1|0 or 4|0. That is, are we describing
the number of generators per period, or per equivalence interval?

A hint can come from Paul's Pajara scales. SPM is either going to be
Pajara 2|2 #8 or Pajara 4|4 #8, depending on what we pick above. Since
the #8 is referring to a number within the equivalence interval, and
NOT the period, it might make more sense to say Pajara 4|4 #8, and
have it mean the equivalence interval all around.

I guess it's that Pajara 2|2 is better from the generator paradigm,
and Pajara 4|4 is better from the major/minor paradigm. Which one
dominates? Any thoughts? Pajara 4|4 does accurately reflect that there
are 4 "major" intervals, and 4 "minor" intervals, which is nice. On
the other hand, there's something that I do like about Pajara 2|2 #8,
because from a generator standpoint, Pajara 4|4 requires the musician
to divide by 2, which is unnecessary.

Any thoughts? Hopefully I can get Carl to start philosophizing here,
because I'm going in circles and need to cut myself off to do work...

-Mike

On Fri, Aug 5, 2011 at 6:11 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
> There are a lot of scales, and no sensible way to name the modes for
> all of them. There are two sensible options:
>
> 1) something like "5|1" for "5 generators up, 1 generator down"
> 2) something like "5|1" for "5 'major' intervals, 1 'minor' interval"
> (relative to the root. Or perhaps 'large' vs 'small' intervals)
>
> Luckily, the two are actually the same thing - provided we pick the
> proper generator! The generator we need to pick to ensure that the
> most "up" generators also generates a "most major" mode, in some
> sense, will be MOS-specific, and I will call it the "normal generator"
> for some MOS. Here are some examples - for now, I will use the word
> "major" to denote the larger interval in each class, and "minor" to
> denote the smaller, for simplicity - e.g. the perfect fifth is the
> "major fifth," and the diminished fifth is the "minor fifth."
>
> Here's meantone[7], where the needed generator is 3/2:
> 6|0 - LLLsLLs - major interval classes: 2, 3, 4, 5, 6, 7 - minor
> interval classes:
> 5|1 - LLsLLLs - major interval classes: 2, 3, 5, 6, 7 - minor interval
> classes: 4
> 4|2 - LLsLLsL - major interval classes: 2, 3, 5, 6 - minor interval
> classes: 4, 7
> 3|3 - LsLLLsL - major interval classes: 2, 5, 6 - minor interval
> classes: 3, 4, 7
> 2|4 - LsLLsLL - major interval classes: 2, 5 - minor interval classes:
> 3, 4, 6, 7
> 1|5 - sLLLsLL - major interval classes: 5 - minor interval classes: 2,
> 3, 4, 6, 7
> 0|6 - sLLsLLL - major interval classes: - minor interval classes: 2,
> 3, 4, 5, 6, 7
>
> Now let's observe mavila[7], where the generator now has to be 4/3
> (!!!) instead of 3/2 to generate the same behavior:
> 6|0 - LssLsss - major interval classes: 2, 3, 4, 5, 6, 7 - minor
> interval classes:
> 5|1 - LsssLss - major interval classes: 2, 3, 4, 6, 7 - minor interval
> classes: 5
> 4|2 - sLssLss - major interval classes: 3, 4, 6, 7 - minor interval
> classes: 2, 5
> 3|3 - sLsssLs - major interval classes: 3, 4, 7 - minor interval
> classes: 2, 5, 6
> 2|4 - ssLssLs - major interval classes: 4, 7 - minor interval classes:
> 2, 3, 5, 6
> 1|5 - ssLsssL - major interval classes: 4 - minor interval classes: 2,
> 3, 5, 6, 7
> 0|6 - sssLssL - major interval classes: - minor interval classes: 2,
> 3, 4, 5, 6, 7
>
>
> This is a GOOD way to do things, firstly because it makes mathematical
> sense and is easy to figure out. Secondly, all of the modes on this
> continuum are identical to their adjacent neighbors except for the
> alteration of one note by an L-s chroma, which makes it easy to see
> how one mode modulates into another, and the "most major" modes are at
> the top. Lastly, anyone here who's studied modal harmony in depth will
> be able to attest to how important the intuitive understanding of both
> of these things is for really "getting" modal harmony - both the
> generator pattern for the mode in question, as well as how many
> "major" or "minor" intervals a scale has.
>
> The latter is actually a bit more important, and it is commonplace for
> beginning students of modal harmony to write the diatonic modes down
> in order from "most major" to "least major," or "brightest" to
> "darkest"; to do so automatically implies a choice of generator and
> hence we can kill two birds with one stone. I'm predominantly
> self-taught for understanding modal theory, but when I went to UM I
> found that everyone there had figured out the same thing I had,
> including the teachers: stop thinking about "Dorian" as being "the
> major scale starting on D," and start thinking about it like "Aeolian
> with a #6," hence placing it in the above continuum. The above setup
> notates things into the infinitely more useful generator and
> chroma-based paradigm from the getgo.
>
> Of course, if you're following this, then you're now understanding
> that you've actually understood modal harmony all along, and are now
> playing chord progressions on your piano that morph from Lydian to
> Ionian to Mixolydian and back, thus revealing my hidden agenda to
> teach you guys jazz harmony without you all realizing it ;)
>
> We can also thus describe the MODMOS's as follows:
>
> Ascending melodic minor - 5|1 b3 (or 3|3 #6#7, etc)
> Harmonic minor - 2|4 #7 (or 5|1 b3b6, etc)
> Harmonic major - 5|1 b6 (or 2|4 #3#7, etc)
>
> -Mike
>

And a quick conjecture before I retire for the afternoon -

For any MOS xLys, if x > y, the "normal generator" as I'm calling it
will be the version that's greater than half the period. If y > x,
then the version will be the version that's smaller than half the
period. Remember, this is the generator that will make it so that

Examples:
2L3s - generator is the 4/3 (meantone[5], mavila[5])
5L2s - generator is the 3/2 (meantone[9])
2L5s - generator is the 4/3 (mavila[7])
7L2s - generator is the 3/2 (mavila[9])
1L7s - generator is the 9/5 (porcupine[7])
7L1s - generator is the 10/9 (porcupine[8])
2L8s - generator is the 4/3 (pajara[10])

I think this will be trivial to prove, because the basic goal is to
concentrate as many L intervals as possible at the beginning of the
scale. I've only sort of half-assedly considered it so far and will
have to nitpick later, though.

And for the record, I'm leaning towards Pajara 2|2 #8 for standard
pentachordal major. I'll leave Ryan Avella to comment on the rest of
it, because he's getting into this really deep concept about whether
we want the notation to reflect someone "designing a scale" (e.g.
putting it together with generator patterns) or "interpreting a scale"
(e.g. reading it and evoking something they already know). I'll leave
you all to talk about the use cases for this notation and come back
later, when I'm not drowning in work...

-Mike

On Fri, Aug 5, 2011 at 6:27 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
> Correction - there is one issue that remains - the issue of whether
> diminished[8] is something like 1|0 or 4|0. That is, are we describing
> the number of generators per period, or per equivalence interval?
>
> A hint can come from Paul's Pajara scales. SPM is either going to be
> Pajara 2|2 #8 or Pajara 4|4 #8, depending on what we pick above. Since
> the #8 is referring to a number within the equivalence interval, and
> NOT the period, it might make more sense to say Pajara 4|4 #8, and
> have it mean the equivalence interval all around.
>
> I guess it's that Pajara 2|2 is better from the generator paradigm,
> and Pajara 4|4 is better from the major/minor paradigm. Which one
> dominates? Any thoughts? Pajara 4|4 does accurately reflect that there
> are 4 "major" intervals, and 4 "minor" intervals, which is nice. On
> the other hand, there's something that I do like about Pajara 2|2 #8,
> because from a generator standpoint, Pajara 4|4 requires the musician
> to divide by 2, which is unnecessary.
>
> Any thoughts? Hopefully I can get Carl to start philosophizing here,
> because I'm going in circles and need to cut myself off to do work...
>
> -Mike
>
>
>
> On Fri, Aug 5, 2011 at 6:11 PM, Mike Battaglia <battaglia01@gmail.com> wrote:
>> There are a lot of scales, and no sensible way to name the modes for
>> all of them. There are two sensible options:
>>
>> 1) something like "5|1" for "5 generators up, 1 generator down"
>> 2) something like "5|1" for "5 'major' intervals, 1 'minor' interval"
>> (relative to the root. Or perhaps 'large' vs 'small' intervals)
>>
>> Luckily, the two are actually the same thing - provided we pick the
>> proper generator! The generator we need to pick to ensure that the
>> most "up" generators also generates a "most major" mode, in some
>> sense, will be MOS-specific, and I will call it the "normal generator"
>> for some MOS. Here are some examples - for now, I will use the word
>> "major" to denote the larger interval in each class, and "minor" to
>> denote the smaller, for simplicity - e.g. the perfect fifth is the
>> "major fifth," and the diminished fifth is the "minor fifth."
>>
>> Here's meantone[7], where the needed generator is 3/2:
>> 6|0 - LLLsLLs - major interval classes: 2, 3, 4, 5, 6, 7 - minor
>> interval classes:
>> 5|1 - LLsLLLs - major interval classes: 2, 3, 5, 6, 7 - minor interval
>> classes: 4
>> 4|2 - LLsLLsL - major interval classes: 2, 3, 5, 6 - minor interval
>> classes: 4, 7
>> 3|3 - LsLLLsL - major interval classes: 2, 5, 6 - minor interval
>> classes: 3, 4, 7
>> 2|4 - LsLLsLL - major interval classes: 2, 5 - minor interval classes:
>> 3, 4, 6, 7
>> 1|5 - sLLLsLL - major interval classes: 5 - minor interval classes: 2,
>> 3, 4, 6, 7
>> 0|6 - sLLsLLL - major interval classes: - minor interval classes: 2,
>> 3, 4, 5, 6, 7
>>
>> Now let's observe mavila[7], where the generator now has to be 4/3
>> (!!!) instead of 3/2 to generate the same behavior:
>> 6|0 - LssLsss - major interval classes: 2, 3, 4, 5, 6, 7 - minor
>> interval classes:
>> 5|1 - LsssLss - major interval classes: 2, 3, 4, 6, 7 - minor interval
>> classes: 5
>> 4|2 - sLssLss - major interval classes: 3, 4, 6, 7 - minor interval
>> classes: 2, 5
>> 3|3 - sLsssLs - major interval classes: 3, 4, 7 - minor interval
>> classes: 2, 5, 6
>> 2|4 - ssLssLs - major interval classes: 4, 7 - minor interval classes:
>> 2, 3, 5, 6
>> 1|5 - ssLsssL - major interval classes: 4 - minor interval classes: 2,
>> 3, 5, 6, 7
>> 0|6 - sssLssL - major interval classes: - minor interval classes: 2,
>> 3, 4, 5, 6, 7
>>
>>
>> This is a GOOD way to do things, firstly because it makes mathematical
>> sense and is easy to figure out. Secondly, all of the modes on this
>> continuum are identical to their adjacent neighbors except for the
>> alteration of one note by an L-s chroma, which makes it easy to see
>> how one mode modulates into another, and the "most major" modes are at
>> the top. Lastly, anyone here who's studied modal harmony in depth will
>> be able to attest to how important the intuitive understanding of both
>> of these things is for really "getting" modal harmony - both the
>> generator pattern for the mode in question, as well as how many
>> "major" or "minor" intervals a scale has.
>>
>> The latter is actually a bit more important, and it is commonplace for
>> beginning students of modal harmony to write the diatonic modes down
>> in order from "most major" to "least major," or "brightest" to
>> "darkest"; to do so automatically implies a choice of generator and
>> hence we can kill two birds with one stone. I'm predominantly
>> self-taught for understanding modal theory, but when I went to UM I
>> found that everyone there had figured out the same thing I had,
>> including the teachers: stop thinking about "Dorian" as being "the
>> major scale starting on D," and start thinking about it like "Aeolian
>> with a #6," hence placing it in the above continuum. The above setup
>> notates things into the infinitely more useful generator and
>> chroma-based paradigm from the getgo.
>>
>> Of course, if you're following this, then you're now understanding
>> that you've actually understood modal harmony all along, and are now
>> playing chord progressions on your piano that morph from Lydian to
>> Ionian to Mixolydian and back, thus revealing my hidden agenda to
>> teach you guys jazz harmony without you all realizing it ;)
>>
>> We can also thus describe the MODMOS's as follows:
>>
>> Ascending melodic minor - 5|1 b3 (or 3|3 #6#7, etc)
>> Harmonic minor - 2|4 #7 (or 5|1 b3b6, etc)
>> Harmonic major - 5|1 b6 (or 2|4 #3#7, etc)
>>
>> -Mike
>>
>

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> Correction - there is one issue that remains - the issue of whether
> diminished[8] is something like 1|0 or 4|0. That is, are we describing
> the number of generators per period, or per equivalence interval?
>
> A hint can come from Paul's Pajara scales. SPM is either going to be
> Pajara 2|2 #8 or Pajara 4|4 #8, depending on what we pick above. Since
> the #8 is referring to a number within the equivalence interval, and
> NOT the period, it might make more sense to say Pajara 4|4 #8, and
> have it mean the equivalence interval all around.

Well in Blackwood 5L5s, for example, we all know that there are 10 pitch classes (other than the octave) because 5+5=10. So there is no need to communicate the number of pitch classes in this new generator notation.

Instead, we could think of it as a ratio. Pajara 2|2 gives the amount of generators necessary to construct the scale, but it also gives a ratio: 2 major intervals for every 2 minor intervals. We already know that there is 10 notes in Pajara, and that there is a 50-50 distribution of major-minor intervals, so that there must be 5 major intervals and 5 minor.

4|4 gives the same ratio as 2|2, but it needs to be divided by 2 first in order to give the correct amount of generators. Therefore I see it as inferior. If there are any other advantages to calling it 4|4 as opposed to 2|2, please let me know, because I am still having trouble trying to wrap my mind around the whole concept.

Ryan

--- In tuning-math@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
> And for the record, I'm leaning towards Pajara 2|2 #8 for standard
> pentachordal major. I'll leave Ryan Avella to comment on the rest of
> it, because he's getting into this really deep concept about whether
> we want the notation to reflect someone "designing a scale" (e.g.
> putting it together with generator patterns) or "interpreting a scale"
> (e.g. reading it and evoking something they already know). I'll leave
> you all to talk about the use cases for this notation and come back
> later, when I'm not drowning in work...

I'll go into more detail for anyone reading.

Basically there are two ways to notate standard pentachordal major: 2|2 #8, and 4|4 #8. The first one gives the number of generators up and down per period, whereas the second one gives the total number of generators per octave. The reason it is ambiguous is because the period is not the octave.

Lets make a list of Pros and Cons for each system then. I will call the former "Construction Catalyst" and the latter "Interpretation Catalyst."

Construction Catalyst:
Pros: Makes scale construction simpler by reflecting the true generator patterns, and gives a <ratio> of Major-to-Minor pitch classes
Cons: Requires previous knowledge of the amount of periods in the octave

Interpretation Catalyst:
Pros: Gives the <exact> number of Major and Minor pitch classes in an octave
Cons: Does not accurately reflect the number of generators per period

I am currently in favor of the Construction Catalyst because the Pros outweigh the Cons, in my humble opinion. I'd like to hear more from Mike, and from anyone else interested as well.

Ryan