Yahoo Tuning Groups Ultimate Backup tuning Classical minor tonality as an example for xenharmonic tonalities

Major tonality (the tonality of music in a major key) is misleadingly simple and easy. If you're trying to look for xenharmonic scales that act like the major scale, you're sure to be disappointed. There are 7 pitch classes in the major scale and traditional music in a major key uses only those 7. All the chords of simple (non-modulating) major chord progressions use only those 7 pitch classes with no alterations. This simplicity makes major tonality *exceptional*.

Consider, on the other hand, what I'm going to call classical minor tonality, as exemplified by a Baroque fugue or a Classical sonata in a minor key. There is no single "minor scale" with 7 pitch classes; instead there are (at least) three different "minor scales" ("natural", "harmonic", and "ascending melodic"), or you could also say that "the minor scale" has at least 9 total pitch classes including alternates. It's not simply a scale with 9 pitch classes on an equal footing though; far from it. If a long run of a scale going up or down appears in a piece of music, it's almost guaranteed to have 7 notes per octave, not 9; the 9 pitch classes really do consist of 5 single notes and 2 pairs of alternates, which rarely appear close together.

Furthermore, a description of classical minor tonality would be incomplete without a listing of the idiomatic chord progressions and melodic figures, and which of the alternate pitch classes they use. The one obvious example is that the minor seventh degree usually appears in certain contexts, like the III chord, while the major seventh degree appears in other contexts, like the V (dominant) chord. This is not to say that augmented III+ chords and minor v chords are forbidden, but their appearance would be remarkable because they are much less idiomatic in classical minor tonality. I hesitate to use the word "function" because it's so overloaded and might give people the wrong idea, but it really is like the minor seventh has a certain "function" and the major seventh a distinct "function".

Complex tonal music from other traditions has very similar concepts. Many maqamat have alternate pitch classes, and analogously to classical minor tonality, they are not used arbitrarily but instead there are more or less flexible idiomatic "rules" about their usage. Often one of a pair of alternates is presented as part of an "ascending" scale and the other as a "descending" variant (although the real situation is clearly more subtle than that). One maqam in particular, maqam Nahawand (http://www.maqamworld.com/maqamat/nahawand.html) is remarkably similar to classical minor tonality, but there are plenty of others that have a similar system of alternates although they bear little resemblance to major or minor scales. I understand that a similar situation exists for Indian ragas. Introductory texts on both maqamat and ragas often emphasize how they are "more than simply scales" because of such related idioms... but classical minor tonality is also more than simply a scale of 7 or 9 notes.

Note that if you modulate from one minor key to another minor key a fifth away (one of the simplest and most common modulations), that's already 11 out of the 12 pitch classes of 12edo. Using only these pitches, you could either write very atonal music that doesn't suggest any tonal center, or very idiomatic tonal music in classical minor tonality. It's not the scale that's important, it's how you use it. (I'm reminded of the time when someone took a Scala file for porcupine[37] and used it to make computer-generated music using all 37 pitch classes indiscriminately. I hope everyone reading this understands why the claim that such music is "in porcupine temperament" is somewhat hilarious.)

So, my suggestion with all this is to avoid using the major scale as an example to compare to xenharmonic scales, and instead to use classical minor tonality as an example and keep in mind that a similar kind of idiom might evolve in tonal xenharmonic music. There is a happy medium in between the two extremes of (1) using only the pitches of a simple scale, and (2) making hyper-chromatic music that treats a large set of pitches as equally central/salient.

An important consequence of this is that computer searches for specific "scales" (simple mathematical *sets* of pitches with no alternates or subordinates defined) may be misguided. Classical major tonality is well modeled as being based on a set of seven pitch classes (the major scale), but classical minor tonality is not. If a computer search won't yield classical minor tonality in meantone temperament (the basis of a vast body of beautiful pieces of music), then I *guarantee* that there's some beautiful form of tonality in porcupine temperament, for example, that it will also fail to find.

This is all basically expanding on something Mike Battaglia said in an insightful post over a year ago: /tuning/topicId_98428.html#98469 , but hopefully presented in a different way that gives people useful ideas.

As for me, if I accomplish nothing else in my life related to xenharmonic music other than creating enough musical examples to establish an idiom for different tonalities (analogous to meantone major and minor) in porcupine temperament, I'll die a happy man.

Keenan

🔗Keenan Pepper <keenanpepper@...>

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
>
> Major tonality (the tonality of music in a major key) is misleadingly simple and easy. If you're trying to look for xenharmonic scales that act like the major scale, you're sure to be disappointed. There are 7 pitch classes in the major scale and traditional music in a major key uses only those 7. All the chords of simple (non-modulating) major chord progressions use only those 7 pitch classes with no alterations. This simplicity makes major tonality *exceptional*.
>
> Consider, on the other hand, what I'm going to call classical minor tonality, as exemplified by a Baroque fugue or a Classical sonata in a minor key. There is no single "minor scale" with 7 pitch classes; instead there are (at least) three different "minor scales" ("natural", "harmonic", and "ascending melodic"), or you could also say that "the minor scale" has at least 9 total pitch classes including alternates. It's not simply a scale with 9 pitch classes on an equal footing though; far from it. If a long run of a scale going up or down appears in a piece of music, it's almost guaranteed to have 7 notes per octave, not 9; the 9 pitch classes really do consist of 5 single notes and 2 pairs of alternates, which rarely appear close together.
>
> Furthermore, a description of classical minor tonality would be incomplete without a listing of the idiomatic chord progressions and melodic figures, and which of the alternate pitch classes they use. The one obvious example is that the minor seventh degree usually appears in certain contexts, like the III chord, while the major seventh degree appears in other contexts, like the V (dominant) chord. This is not to say that augmented III+ chords and minor v chords are forbidden, but their appearance would be remarkable because they are much less idiomatic in classical minor tonality. I hesitate to use the word "function" because it's so overloaded and might give people the wrong idea, but it really is like the minor seventh has a certain "function" and the major seventh a distinct "function".
>
> Complex tonal music from other traditions has very similar concepts. Many maqamat have alternate pitch classes, and analogously to classical minor tonality, they are not used arbitrarily but instead there are more or less flexible idiomatic "rules" about their usage. Often one of a pair of alternates is presented as part of an "ascending" scale and the other as a "descending" variant (although the real situation is clearly more subtle than that). One maqam in particular, maqam Nahawand (http://www.maqamworld.com/maqamat/nahawand.html) is remarkably similar to classical minor tonality, but there are plenty of others that have a similar system of alternates although they bear little resemblance to major or minor scales. I understand that a similar situation exists for Indian ragas. Introductory texts on both maqamat and ragas often emphasize how they are "more than simply scales" because of such related idioms... but classical minor tonality is also more than simply a scale of 7 or 9 notes.
>
> Note that if you modulate from one minor key to another minor key a fifth away (one of the simplest and most common modulations), that's already 11 out of the 12 pitch classes of 12edo. Using only these pitches, you could either write very atonal music that doesn't suggest any tonal center, or very idiomatic tonal music in classical minor tonality. It's not the scale that's important, it's how you use it. (I'm reminded of the time when someone took a Scala file for porcupine[37] and used it to make computer-generated music using all 37 pitch classes indiscriminately. I hope everyone reading this understands why the claim that such music is "in porcupine temperament" is somewhat hilarious.)
>
> So, my suggestion with all this is to avoid using the major scale as an example to compare to xenharmonic scales, and instead to use classical minor tonality as an example and keep in mind that a similar kind of idiom might evolve in tonal xenharmonic music. There is a happy medium in between the two extremes of (1) using only the pitches of a simple scale, and (2) making hyper-chromatic music that treats a large set of pitches as equally central/salient.
>
> An important consequence of this is that computer searches for specific "scales" (simple mathematical *sets* of pitches with no alternates or subordinates defined) may be misguided. Classical major tonality is well modeled as being based on a set of seven pitch classes (the major scale), but classical minor tonality is not. If a computer search won't yield classical minor tonality in meantone temperament (the basis of a vast body of beautiful pieces of music), then I *guarantee* that there's some beautiful form of tonality in porcupine temperament, for example, that it will also fail to find.
>
> This is all basically expanding on something Mike Battaglia said in an insightful post over a year ago: /tuning/topicId_98428.html#98469 , but hopefully presented in a different way that gives people useful ideas.
>
> As for me, if I accomplish nothing else in my life related to xenharmonic music other than creating enough musical examples to establish an idiom for different tonalities (analogous to meantone major and minor) in porcupine temperament, I'll die a happy man.

TL;DR: Don't use the major scale as an example and try to find analogs of it, instead use "being in a minor key" as an example.

Find analogs of the minor tonality system, or natural/harmonic/melodic minor scale complex, or whatever you want to call it.

🔗gedankenwelt94 <gedankenwelt94@...>

Dear Keenan,

thanks for your post, I very much appreciate what you suggest here! :)

--- In tuning@yahoogroups.com, "Keenan Pepper" <keenanpepper@...> wrote:
> There are 7 pitch classes in the major scale and traditional music in a major key uses only those 7.

This is how it's usually taught, and probably not far from the truth. However, there's a scale called 'harmonic major' which "found occasional use during the common practice era" (quoting wikipedia), and looks like this:

C D E F G Ab B C
1:1 9:8 5:4 4:3 3:2 8:5 15:8 2:1

It can be seen as a major scale with a minor sixth, or as a harmonic minor scale with a major third. Harmonically, it consists of two major chords stacked on top of a minor chord, which makes it the inversion of the harmonic minor scale (which is a major chord stacked on top of two minor chords), and can be used in a very similar way. Functionally, it's a major scale with a iv (instead of a IV), just like the harmonic minor scale replaces the iv by a IV.

Now throw in the melodic major scale, and you can use the major scale in a way similar to common practice of minor tonality:

C D E F G Ab Bb C
1:1 9:8 5:4 4:3 3:2 8:5 9:5 2:1

Melodic major is the inversion of (ascending) melodic minor, and - when implemented as a MODMOS - just another mode of the (ascending) melodic minor scale, btw.

As I noted in another topic, the (ascending) melodic minor scale (as a MODMOS) can be generalized easily:

If you have an n-note MOS (1), extend the generator chain to n + 2 notes (2), and remove the second note from each end of the chain (3):

(1) F C G D A E B
(2) Bb F C G D A E B F#
(3) Bb - C G D A E - F#

The result is a very simple MODMOS / PERMOS with 2 specific intervals per generic interval, except on the generator level, where it is 3.

- Gedankenwelt

🔗gedankenwelt94 <gedankenwelt94@...>

🔗Margo Schulter <mschulter@...>

Dear Keenan,

Thank you for a great discussion of tonality, modality, and the
importance of different inflections in expressive music!

Your mention of Maqam Nahawand let me to the idea of offering a
few quick examples in a 12-note Near Eastern tuning that might
easily map to a conventional keyboard. As it happens, this
12-note set is strictly proper -- but the main point is to
illustrate a few of the modal inflections relevant to your post.

First, the tuning:

! met24c-cs12-archytan-maqam_cup.scl
!
Constant Structure, tempered subdivision of Archytas Chromatic
12
!
68.55468
207.42187
288.86719
357.42187
496.28906
564.84375
703.71093
772.26562
911.13281
992.57812
1061.13281
2/1

Taking the 1/1 as C, we have this 12-note keyboard mapping.
Note that an ASCII "d" shows an Arab half-flat, while a "+" shows
an Arab half-sharp. In this tuning, these signs generally lower
or raise a note by around 1/3 tone, while a flat lowers by around
2/3 tone. Certain notes might function as either a flat or a
half-sharp, as we'll see.

69 289 565 772 992
C+/Db Eb Gb/F+ Ab/G+ Bb
C D Ed F G A Bd C
0 207 357 496 704 911 1061 1200

As it happens, Maqam Nahawand -- somewhat analogous to a Western
minor tonality -- is intimately related to Maqam Rast. So let's
first look at Rast, often called the "fundamental scale" of Arab
music, and then Nahawand, both of which typically occur on C, so
that modulating between them is very common. More generally,
maqamat (the Arabic plural form of maqam) sharing the same usual
final or resting note, here C, have a certain "family affinity"
which permits freely modulating from one to another.

In this keyboard mapping, we find a "textbook" form of modern
Rast on the seven diatonic keys:

207 150 139 207 207 150 139
C D Ed F G A Bd C
0 207 357 496 704 911 1061 1200
|------- Rast ------| |-------- Rast -------|

Generally Rast has two tetrachords -- here disjunct, with a step
of around 9:8 in the middle of the octave -- each with a lower
tone around 9:8, and a larger neutral second followed by a
smaller one. Exact tunings vary a great deal in the Near East,
with this one perhaps best expressed in JI terms as 39:44:48:52
or 1/1-44/39-16/13-4/3 (0-209-359-498 cents).

More precisely, this is the usual modern form of the _ascending_
Maqam Rast. In descending, the seventh step is often lowered from
a largish neutral seventh to a minor seventh, which gives us this
form:

207 150 139 207 207 81 207
C D Ed F G A Bb C
0 207 357 496 704 911 993 1200
|------- Rast ------| |------ Nahawand -----|

Lowering the seventh step not only underscores the descending
motion back toward C, the final or resting note (sometimes also
known as the tonic by analogy to 17th-19th century Western
tonality), but introduces the new element of a melodic semitone,
contrasting with the usual tones and neutral second steps of
Rast. The upper tetrachord on G, tone-semitone-tone, is called
Nahawand, also the name of Maqam Nahawand which features this as
its "root" tetrachord above the final.

Here is one possible form of ascending Nahawand on C, with Arab
understandings varying a great deal:

207 81 207 207 69 289 139
C D Eb F G Ab Bd C
0 207 289 496 704 772 1061 1200
|----- Nahawand ----| |------- Hijaz -------|

This interpretation of ascending Nahawand has a lower Nahawand
tetrachord, a middle tone, and for its upper tetrachord one of
the many forms of Hijaz. The safest general description of Hijaz
focuses on its large middle step appreciably larger than a 9:8
tone, and often somewhere between 8:7 and 32:27 or so. Here the
middle step is identical to a regular minor at around 13:11,
although a smaller size around 7:6, for example, might be
considered more stylish.

In this interpretation of Nahawand, the Hijaz tetrachord has a
small semitone, a large step around 13:11, and an upper neutral
second step around 13:12 (69-289-139 cents). Note that the
neutral seventh step is identical to that of ascending Rast.
Many current versions of Nahawand prefer a higher minor or
neutral sixth step and a major rather than neutral seventh; but
the interpretation above has an interesting affinity with Rast.

Looking at the lower Nahawand tetrachord, we see that only one of
the four notes is changed from Rast: a third step at 289 rather
than 357 cents. In the upper Hijaz tetrachord, similarly, only
the second step, lowered from a major to a minor sixth, differs
from the Rast tetrachord on G in the ascending form of that
maqam.

A common descending form of Nahawand has minor sixth and seventh
steps, comparable to a Western natural minor. The upper or Kurdi
tetrachord, semitone-tone-tone, is like the European Phrygian
mode:

207 81 207 207 69 220 207
C D Eb F G Ab Bb C
0 207 289 496 704 772 993 1200
|----- Nahawand ----| |------- Kurdi -----|

Still another possibility, however, occurs in ascending to the
final or its octave, especially when making a cadence. Here we
can use both the major sixth and neutral seventh steps of Rast
for a kind of "melodic Nahawand," as it were. For example, when
cadencing on the final, I might descend from F to C, and then
play a kind of coda, A-Bd-C:

207 150 139 207 81 207
[G] A Bd C D Eb F
704 911 1061 0 207 289 496
|...................|--------------------|
"Rast" Nahawand

Many contemporary performers might prefer for this idiom A-B-C
with a major rather than neutral seventh, which would make the
usage closer to that of a Western melodic minor. However, I find
a special charm in the major sixth and neutral seventh as a
common element shared between this form of Nahawand and Rast.

I might also mention another maqam typically played on C which in
this 12-note tuning exhibits a stylized intonation of a Hijaz or
related genus with a smaller interval than the usual minor
third. This is Maqam Nakriz, which seems to combine some aspects
of Maqam Rast and Maqam Nahawand, and can serve as a nice
variation on either or both:

207 81 276 139 207 81 207
C D Eb F+ G A Bb C
0 207 289 565 704 911 993 1200
|---------- Nakriz ----------|--- Nahawand ------|

Nakriz (or Nikriz) in this interpretation has a tone and semitone
(like Nahawand), and then a large interval like that of Hijaz,
here 276 cents or somewhat larger than 7/6, followed by a smaller
neutral second to complete the fifth (207-81-276-139 cents). To
this lower pentachord is added an upper Nahawand tetrachord,
which provides a pleasant contrast.

Like Nahawand, Nikriz in my practice may have a neutral seventh
step below the final, and possibly a major sixth step below that
(A-Bd-C). Current Arab practice often favors a major rather
neutral seventh step as a raised leading tone for both maqamat.

As I've cautioned, not only the fine points of intonation, but
the more general nature of certain inflections -- whether
ascending Nahawand has a neutral or major seventh, or whether the
lower pentachord of Nakriz has a minor third and low tritone or a
neutral third and higher tritone -- can vary, in theory and
practice!

My purpose here, however, is to show how different maqamat can
interact, with the case of Rast and Nahawand being an especially
interesting one.

With many thanks,

Margo