Thursday, July 6, 2017

Pappas Music Post: Conclusion

This is what Theoni Pappas writes on page 187 of her Magic of Mathematics:

"Points A and C are speaker locations. Point D is the location of the listener. B is the midpoint of segment AC. Distance BD must be greater than or equal to the distance AB. x is the speakers' distances from the walls -- a minimum of 3 feet."

Here Pappas is describing QSound -- a stereo that produces sound coming in from all directions (in three dimensions) towards the listener.

I knew that QSound is 1994 technology (as this is when Pappas wrote the book), so I'm curious as to whatever happened to QSound. Well, let's find out:

In the early 1990’s, QSound Labs brought effective, practical three-dimensional (3D) audio technology to music and film production studios in the form of a ¼-tonne digital audio processing station called the QSystem. Meanwhile, video gamers enjoyed ground-breaking QSound 3D audio on Capcom™ arcade machines.

Today, QSound technology is embedded in tiny devices such as cell phones and Bluetooth™ headsets, whereas the QSound digital audio algorithm suite has grown considerably.

Ok, so QSound still exists. By the way, notice how QSound works:

The basic QSound 3D building block is a proprietary audio process that creates two or more outputs from each input signal. These outputs incorporate subtle differences in level, frequency content, and timing. Played back on the specific delivery system (headphones or speaker arrangement) for which the algorithm is designed, the effect is to mimic directional cues that a listener would normally receive when listening to an actual sound source at a given location in 3D space.

Ah, so we see how all of the components of sound first discovered by Fourier -- pitch/frequency, loudness/amplitude, and quality -- are modified to make the sound seem to come from different locations in the room. This is similar to how perspective is used to make a 2D image appear 3D.

This is the final page of the "Mathematics Plays Its Music" chapter in Pappas. And so I want to use this post to wrap up what we've been discussing about musical scales.

In the book, Pappas makes a tongue-in-cheek reference to 300EDO as a scale in which each step is about four cents -- the just noticeable difference (JND). We aren't meant to consider 300EDO as an actual scale, but I notice that it has several interesting properties.

First of all, 300 is the LCM of 12 and 50. So 300EDO is the smallest scale that incorporates both our usual scale 12EDO and yesterday's scale 50EDO.

But with each step of 300EDO equal to the JND, we should be able to play any interval and make it sound like just intonation. So a 300EDO player would want to play the fifth of 12EDO at 700 cents and the major third of 50EDO at 384 cents. Notice that the difference between these, the minor third, is part of neither 12EDO nor 50EDO, but at 316 cents, the minor third is almost exactly just.

The 300EDO has a special property -- it's the last scale to temper out the Pythagorean comma. Recall that a scale tempers out the Pythagorean comma if it equates 12 fifths with seven octaves -- that is, if the perfect fifth is exactly 700 cents. Any EDO that's a multiple of 12 -- 24EDO, 36EDO, 48EDO, and so on -- has an interval of 700 cents.

But that 700-cent interval isn't always the perfect fifth. The just perfect fifth is 702 cents -- or to be more accurate, about 701.96 cents. In 300EDO, the next step after 700 cents is 704 cents, and so 701.96 is slightly closer to 700 than 704. But in the next multiple of 12, 312EDO, the next step after 700 cents is 703.85 cents, and 701.96 is a little closer to 703.85 than 700. Thus 703.85 cents is the perfect fifth, and far from tempering out the Pythagorean comma, 312EDO exaggerates it. And so 300EDO is the last EDO to temper out the famous comma.

It's interesting that Pappas cites 4 cents as the just noticeable difference. Some sources give 5-6 cents as the JDN. If we choose 6 cents as the JDN, then 200EDO with its 6-cent steps is interesting:

Notice that 702 cents is a multiple of six, and so 200EDO has a very accurate perfect fifth. In fact, the fifth of 200EDO is better than any lower EDO (from 2 up to 199). On the other hand, the third of 200EDO, like that of 300EDO, is inherited from 50EDO (at 384 cents).

Since 200EDO has such an accurate perfect fifth, we're inclined to use Pythagorean notation, where C-G-D-A-E are each a perfect fifth apart, and so C-E is the Pythagorean major third (81/16 or 408 cents) rather than the just major third (5/4 or 386 cents).

In 12EDO, E represents both 5/4 and 81/16, as 12EDO is a meantone tuning. On the other hand, 22EDO is not meantone, and so we must decide which third we mean by the note E.

To qualify as a meantone tuning, a scale must equate 10/9 and 9/8. It is not sufficient for a scale to have a tone between 10/9 and 9/8 to be considered meantone. For example, 48EDO has the same 200-cent 9/8 as 12EDO, but 10/9 is 175 cents (and 5/4 is 375 cents), and so 48EDO is not considered to be a meantone tuning. The "meantone" actually isn't the "mean" (or average) between the two tones, but rather is the average between the unison (1/1) and the major third (5/4). Only when 5/4 is divided into two equal tones does the scale qualify as meantone. Meantone scales always temper out the syntonic comma 81/80.

-- 12EDO tempers out both Pythagorean and syntonic commas. E, Fb, and major 3rd are enharmonic.
-- 300EDO tempers out the Pythagorean comma only. E, Fb are enharmonic (at 400 cents), but neither of these is the major third (at 384 cents).
--50EDO tempers out the syntonic comma only. E and major 3rd are enharmonic, but neither of these is the same as Fb (at 432 cents).

We see that 200EDO tempers out neither comma. Since the note E represents the Pythagorean major third and the syntonic comma isn't tempered out, E (at 408 cents) is not the major third. Meanwhile, E is not the same as Fb -- in a Pythagorean tuning, E and Fb differ by the Pythagorean comma.

And so an interesting question is, exactly which note is Fb in 200EDO? Since this is a flat note, we use a circle of fourths instead of fifths: C-F-Bb-Eb-Ab-Db-Gb-Cb-Fb.

So Fb is eight fourths away from C. Since each fifth is 702 cents, each fourth is 498 cents, so eight fourths is 3984 cents. Reducing this by three octaves (at 1200 cents each) gives us 384 cents, so we conclude that Fb is 384 cents.

Hey, that sounds familiar -- 384 cents is the major third in 200EDO. So we conclude that, while E is not the major third and E is not Fb, Fb really is the major third!

What's going on here? Notice that there are two intervals called "comma" here -- the Pythagorean and syntonic commas. The Pythagorean comma is about 24 cents and the syntonic comma is 22 cents -- so these two commas are about the same size. Some tunings, like 200EDO, take full advantage of the near equality of these two commas. The note E is a Pythagorean major third, but instead of lowering it by a syntonic comma, we lower it by a Pythagorean comma to obtain Fb.

And so in 200EDO, C-Fb-G is a major third. Also, C-D#-G is a minor third in 200EDO. Notice that this only happens in tunings that don't distinguish betwen 81/80 and the Pythagorean comma -- that is, they temper out the difference between the two commas (called a "schisma"). Both 12EDO and 200EDO temper out the schisma, and so C-Fb-G is major and C-D#-G is minor in both. On the other hand, 19EDO doesn't temper out the schisma, and so C-Fb-G isn't major (it's supermajor), and likewise C-D#-G isn't minor (it's subminor) in 19EDO.

Meanwhile, if you prefer five cents for the JDN, here's a link to 240EDO:

One way to generate music in alternate scales is to take 12EDO music and naively convert it to 17EDO, 19EDO, or a higher scale. After all, we've seen that the notes in these other scales are often named the same as in 12EDO, just with different enharmonicities.

If we start with a 12EDO song in the key of C major, it should be easy to convert this to 19EDO, since the C major scale is C-D-E-F-G-A-B-C in both 12EDO and 19EDO. But notice that most modern songs don't stick to a single scale. We often refer to this as "jazzing up" a song, but even songs in genres other than jazz tend away from a single major scale towards atonality.

For the recent Fourth of July holiday, I checked out a music book from the library: All American Patriotic Songbook by John L. Haag (dated 1996). Even if we look only at the songs written in the key of C major, we'll find plenty of sharps and flats in most songs.

The first song in the book is "Alexander's Ragtime Band," by Irving Berlin. Ragtime is a form of jazz, and so we expect plenty of accidentals. This song starts in the key of C major before switching to F major. But in the first line (which is in C major), we see the notes D#, F#, G#, and Bb. That's four out of the five black keys right there.

The next song is "Alabama Jubilee" by Jack Yellen. This song is in the key of C major, and yet it contains D#, F#, and G# in the first line. We skip to the next C major song, "America, I Love You," and this song contains the notes C#, D#, G#, and Bb in the first line. "Battle Hymn of the Republic" is the first song that actually sticks only to the notes of C major. Again, we think back to the Google Fischinger player that could only play C, D, E, and G. But even if we added F, A, and B, we still couldn't play most of the C major songs in the book.

The question I'm asking is, if we're converting this to 19EDO, which note should we play for the notes which aren't in C major? For example, G# and Ab are interchangeable in 12EDO but not in 19EDO, so it makes a difference which note we play. We could just convert the notes naively -- that is, play the note G# if it is written as G#, and Ab if it is written as Ab. But it could be the case that playing the opposite note might actually sound better in 19EDO.

In general, there are three reasons why black keys appear in a modern C major song:

-- as a secondary dominant. This was the main reason for non-diatonic notes in classical music -- the note F# may appear as part of a D7 chord, since D7 is the dominant of G, which is itself the dominant of C major.
-- as a secondary leading tone. The note right before a G might be F#, even without a D7 chord. In this case, the F# often appears as a sixteenth note, last note of a triplet, or other downbeat note.
-- as part of a symmetrical chord. In 12EDO, this could be an augmented chord, but more likely it would be a diminished seventh chord. So we might see F# as part of an F#dim7 chord.

One song that appears in my songbook is "Meet Me in St. Louis, Louis." Notice that I changed this to "Meet Me in Pomona, Mona" as part of music break in my class leading up to the field trip to the LA County Fair in Pomona.

In this book, the song is written in the key of C major. Yet there are plenty of accidentals, and for all three reasons listed above:

-- There are plenty of accidentals. The line "We will dance the 'Hoochee Koochee'" is played with a series of secondary dominants: E7-A7-D7-G7-C. The first three of these require sharps.
-- There are several leading tones as well. The line "So he said 'Where can Flossie be at?'" contains an A# between two B's in the melody: G-A-B-A#-B-c-B-A-d.
-- There are two diminished seventh chords: Ebdim7 and F#dim7.

Notice that Ebdim7 and F#dim7 are actually the same chord in 12EDO -- the difference is that the note in the name of the chord is the bass note. But F#dim7 is spelled out as F#-A-C-D#, while Ebdim7 is spelled out as Eb-F#-A-C. Neither of this is strictly correct -- the dim7 chord should only contain minor thirds, so F#dim7 should be F#-A-C-Eb, while Ebdim7 is Eb-Gb-Bbb-Dbb -- that is with two double flats. (I notice that there are guitar chords shown in the score, and both Ebdim7 and F#dim7 have the exact same fingering on the guitar.)

None of this makes any difference in 12EDO. The real problem is in trying to convert the song to 19EDO, where D# and Eb are not enharmonic and the dim7 chord is not symmetrical.

In fact, one interesting part of the song is at the beginning of the Hoochee Koochee dance. There is a leading tone D-D#-E sequence. But the chords at this point are awkward -- right between a G7 chord and the secondary dominant sequence E7-A7-D7-G7-C is an F7 chord. Most likely, the intent of the F7 is to bridge the gap from G7 to E7. But the D# note is being played over the F7 chord, which ought to be spelled F-A-C-Eb. If we change this to D#, then it becomes F-A-C-D# -- which is actually a F harmonic 7th chord in 19EDO! Still, having an F harmonic 7th chord between the G and E dominant 7th chords may be awkward.

Here's how I would convert "St. Louis" to 19EDO:

-- I'd leave the 7th chords as dominant 7ths. Yes, I know that the harmonic 7th is more consonant, but dom 7th aren't supposed to be consonant -- its dissonance allows the dominant 7th to resolve to the tonic, as in G7-C. If the G7 in G7-C is dominant, then all such 7ths should be. (Instead, the harmonic 7th should be reserved for jazz songs in which all the chords are harmonic 7ths, including the tonic.)
-- Leading tones should remain as diatonic semitones, as in B-C. One might argue that a true leading tone in 19EDO should be something like Cb-C. But if the leading tone in a C major scale is a diatonic semitone B-C, then all such leading tones should be. The first appearance of a leading tone in the main melody is A# (B-A#-B, listed above), so I'd play it as written. (In class, I often played my "Pomona" parody on my guitar in the key of G major, where this becomes F#-E#-F#. Of course on my 12EDO guitar I ended up playing F#-F-F#, but the leading tone to F# remains E# in 19EDO.)
-- Diminished 7ths are the trickiest to resolve. It depends on what role they are serving.

Sometimes diminished 7th chords act like dominant 7ths in certain situations. In this song, the Ebdim7 chord appears in the sequence C-Ebdim7-G. We notice that if we spelled Ebdim7 as D#dim7 instead, the resulting chord D#-F#-A-C has three notes in common with D7 (F#-A-C), which is the actual secondary dominant to G. So I'd play it as D#-F#-A-C in 19EDO.

Meanwhile, the F#dim7 chords in the song are part of the sequence F-F#dim7-C. If we spell F#dim7 correctly as F#-A-C-Eb, then it has two notes in common (A-C) with the preceding F chord. So I'd play it as F#-A-C-Eb in 19EDO.

The trickiest part for me to figure out is the D# played over the F7 chord. Using the leading tone rule it should remain D#, but using the dominant 7th rule it should be Eb -- it just so happens that D# and Eb are the same note in 12EDO, but not 19EDO. But notice that F7 isn't really a secondary dominant leading to its tonic (which would be Bb), but just a transition G7-F7-E7. I see no reason why we can't use E#7 instead, so that the dominant 7th would be D# (E#-Gx-B#-D#). In other scales such as 31EDO, the interval G-E (a descending minor 3rd) can be divided evenly in half, with the note halfway in between being F^ (F half-sharp). So we might use G7-F^7-E7 instead, and the dominant 7th over F^ would be Ev (E half-flat, as in F^-A^-C^-Ev).

There's also another point in the score where D# appears. At "it's too slow for me here," there is another leading tone sequence D-D#-E, but the chord sequence is F-Ab7-C. An argument could be made that we should play F#dim7 instead, since we've established the F-F#dim7-C riff earlier. We could also try play a chord like Ab-C-D#-F#, so that C and F# match F#dim7 from earlier but the D# fits the leading tone sequence D-D#-E. But this isn't any established chord.

The ultimate decision for this and all other chords is -- what sounds right? After all, music is meant to be heard. Perhaps future musicians accustomed to 19EDO already know which notes and chords to play in this situation.

This is the final music post. I've squeezed in so many music posts in order to follow the music chapter in Pappas -- more posts than I usually write during a week in summer.

And so once again I'm taking some time off from blogging -- as in a week or two. I'll continue to edit the spring break posts to accommodate Eugenia Cheng, but otherwise I'm not touching this blog.

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