Tuesday, June 27, 2017

Pappas Music Post: The 5- and 7-Note Scales

This is what Theoni Pappas writes on page 178 of her Magic of Mathematics:

"Fourier's discovery makes it possible for these three properties of a sound to be graphically represented and distinct. Pitch is related to the frequency of the curve, loudness to the amplitude and quality to the shape of the periodic function."

Here Pappas is referring to the 19th century French mathematician John Fourier (or Jean-Baptiste, but Pappas anglicizes his name). He has analyzed the musical sound waves caused by vibration in the air, and this can be represented by the sine wave functions that are studied in Pre-Calc and Trig classes.

The fact that pitch is related to frequency and sine waves explains why pitches whose frequencies are in simple whole number ratios sound good together. When two sine functions have frequencies in simple ratios, then they share many maximum and minimum values, which make the two notes sound harmonious, or consonant.

The simplest ratio (after 1/1, the unison, or two of the same note) is 2/1, the octave. Let's use cosine functions to represent our two notes:

y = cos(x)
y = cos(2x)

The maximum values of y = cos(x) are at x = 0, 2pi, 4pi, 6pi, 8pi, and so on. The maximum values of y = cos(2x) also appear at these same x-values, along with x = pi, 3pi, 5pi, 7pi, 9pi, and so on. The fact that all of the first function's maxima are also maxima of the second function make these two notes sound especially consonant.

In fact, we typically give two notes an octave apart the same name, since they sound so good together that we think of the one note as a higher version of the same note. So a note an octave above, or an octave below, a C note is another C note.

The next simplest ratio to consider is 3/2. For reasons that will be made plain later on, the interval between these notes is called a perfect fifth.

y = cos(2x)
y = cos(3x)

These two functions share several maxima -- half of the first function's maxima are shared with a third of the second function's maxima. So the perfect fifth is the next most consonant interval. The note a perfect fifth above a C note happens to be a G note.

If we continue the pattern, we want to consider the ratio 4/3 next. This interval is called a perfect fourth, and the note a perfect fourth above a C note happens to be an F note. We can combine intervals by multiplying their frequencies. So if we combine a perfect fifth with a perfect fourth, the new interval has ratio (3/2)(4/3) = 2/1, the octave.

As I mentioned in my last post, the first person to discover these relationships was Pythagoras. He could have considered higher harmonic ratios, but he stopped here. So any ratio which can be generated by perfect fifths and octaves is called Pythagorean. They have ratios that involve powers of two or three.

Here are some more Pythagorean intervals: if we combine two major fifths, we obtain the interval (3/2)(3/2) = 9/4, which is a major ninth. Since this ratio is larger than an octave and notes an octave apart are equivalent, we typically reduce this interval -- 9/4 = (2/1)(9/8), and so we consider the interval 9/8, which is called a major second or major tone. The note a major second above C happens to be a D note.

If we add another major fifth, we obtain (9/8)(3/2) = 27/16, the major sixth, or A. And if we add another major fifth, we obtain (27/16)(3/2) = 81/32, the major tenth. We can drop this another octave, 81/32 = (2/1)(81/64), and so 81/64 is the Pythagorean major third, or E.

Now let's try to create the simplest possible scale based solely on perfect fifths. Our modern Western scale is based on 12 notes per octave. We write this as 12EDO, 12 equal divisions of the octave.

So a simpler scale might be based on anything from 1EDO to 11EDO, from one to eleven equal divisions of the octave. Some of these are factors of 12, so we can already equally divide the interval just by skipping certain notes in 12EDO.

With 1EDO we have just one note, C. With 2EDO, we have two notes, C and F-sharp (F#) -- and recall that we're trying to obtain the perfect fifth, or G. Even though we can rewrite F# as Gb (G-flat), this is too far removed from G to sound like a perfect fifth. Likewise, with 3EDO we have three notes, C, E, and Ab, and even though we can rewrite Ab as G#, this is too far removed from G to sound like a perfect fifth. The divisions 4EDO and 6EDO also skip the perfect fifth.

In fact, since combining intervals involves multiplying and powers, it's not surprising that dividing intervals involves extracting roots. So we calculate:

G (12EDO's fifth): (2/1)^(7/12) = 1.498
Gb (2EDO's fifth): (2/1)^(1/2) = 1.414
G# (3EDO's fifth): (2/1)^(2/3) = 1.587

In each case, we take the octave (2/1) and raise it to the power that indicates the division of the octave producing our fifth. And only 12EDO's fifth of 1.498 actually rounds off to 1.5 = 3/2, so only this is considered to be a perfect fifth. The fifth of 2EDO is too small, as 1.414 rounds off to 1.4. It is called a "diminished fifth." The fifth of 3EDO is too large, as 1.587 rounds up to 1.6. And so it is called an "augmented fifth."

So now let's try 5EDO, five equal divisions of the octave. This is a division that can't be played on any Western instruments. We can let three of those divisions represent our fifth:

5EDO's fifth: (2/1)^(3/5) = 1.516

This is a bit wider than the usual fifth of 12EDO. It's just about the widest that a fifth can be and still sound plausibly like a perfect fifth. Thus, the simplest scale with perfect fifths is 5EDO.

Pappas writes:

"The Chinese used a pentatonic (five note) scale."

So we see that the ancient Chinese may have used the five-note scale that we are developing. It's possible to connect the pentatonic scale to the Pythagorean intervals we found above:

Note     Interval             Notes of 5EDO
C          1/1 = 1              2^(0/5) = 1
D          9/8 = 1.125       2^(1/5) = 1.1487
E          81/64 = 1.2656  2^(2/5) = 1.3195
G          3/2 = 1.5           2^(3/5) = 1.5157
A          27/16 = 1.6875  2^(4/5) = 1.7411
c           2/1 = 2              2^(5/5) = 2

Even though the perfect fifth of 5EDO isn't too far off, the third and sixth of 5EDO are noticeably distant from the ratios 81/64 and 27/16. Indeed, the major third of 5EDO sounds more like a perfect fourth than a major third.

Most likely, the Chinese didn't actually use an EDO scale at all, but instead they just used the Pythagorean intervals directly. This is called "just intonation."

Here is the Chinese scale written again. This time, I used the actual Chinese names of the notes as given at the following website:


Note          Interval
Gong         1/1
Shang        9/8
Jue             81/64
Zhi             3/2
Yu              27/16
Gong         2/1

In general, notes in just intonation sound more harmonious. Equal divisions are used only so that we can convert songs from one key to another. So a song in equal division C can be converted into any other key, but a song in just intonation C can only be played in the key of C.

According to the following link, the Chinese also used a seven-note scale:


In 7EDO, there is a version of a perfect fifth using the fourth scale degree:

7EDO's fifth: (2/1)^(4/7) = 1.486

This is a bit narrower than the usual fifth of 12EDO. It's just about the narrowest that a fifth can be and still sound plausibly like a perfect fifth. The second simplest scale with perfect fifths is 7EDO.

The seven-note Chinese scale is based on Pythagorean just intonation, and is obtained by adding two more perfect fifths. A perfect fifth above E (81/64) is B (243/128), and a perfect fifth above B, lowered by an octave, is 729/512. This note is so far above the usual F that it's usually written as F#.

The link above also mentions the Xiazhi scale, where the fourth note of the scale is F (4/3), as well as the Qingyu scale, which has both F as the fourth note and the note a perfect fourth above F as the seventh note (16/9). This note is so far below the usual B that it's usually written as Bb.

Let's look at the Xiazhi scale in more detail:

Note    Interval
C         1/1
D         9/8
E         81/64
F         4/3
G        3/2
A        27/16
B        243/128
c         2/1

The author (Lena) writes, "However, there is still a marked difference between this and the Western major scale. When this scale is used in Chinese music, the skeletal basis for the melody is still the [Pythagorean -- dw] pentatonic scale."

And we can see why she writes this -- in my last post I gave the value of the note E is 5/4, but in this scale the note E is 81/64. So this scale isn't identical to the just major scale from my last post, even though both are essentially the major scale.

The seven-note scale is the origin of the name "perfect fifth" -- it's the fifth note of the scale. Indeed, let's name all of the intervals in the above chart:

Note    Interval     Name
C         1/1             perfect unison
D         9/8             major second
E         81/64         major third
F         4/3              perfect fourth
G        3/2              perfect fifth
A        27/16          major sixth
B        243/128      major seventh
c         2/1              perfect octave

The inclusive counting used in interval names often leads to confusion -- for example, a major second plus a major second is a major third -- (9/8)(9/8) = 81/64 -- even though 2 + 2 = 4, not 3. Similarly, a major second plus a perfect fourth is a perfect fifth, even though 2 + 4 = 6, not 5.

Before we leave Lena's website, I notice that she mentions an interesting five-note scale:

"The neutral pentatonic scale is made up of the notes 5 7 1 2 4 of the diatonic scale with the 7 a little lower and the 4 a little higher. There is now the presence of three intervals [which] are smaller than a major third but bigger than a minor third, giving this scale its particular flavor."

The author starts this scale with 5, which indicates the note G. But notice that by "the 7 a little lower," she does not mean Bb, and by "the 4 a little higher," she does not mean F#.

The author doesn't give ratios, so we can't quite be sure what she means by "the 7 a little lower." But as I think about it, I actually believe that this is based on a true equal division 7EDO scale! Like Lena, let's start the scale on note 5 (or G) and try building a 7EDO scale on it. For contrast, let's also build a Qingyu scale on G:

Note    Interval             Notes of 7EDO    Name
G         1/1 = 1              2^(0/7) = 1           perfect unison
A         9/8 = 1.125       2^(1/7) = 1.1041  (not used in pentatonic scale)
B         81/64 = 1.2656 2^(2/7) = 1.219    neutral third
C         4/3 = 1.3333     2^(3/7) = 1.3459  perfect fourth
D         5/4 = 1.5           2^(4/7) = 1.486    perfect fifth
E         27/16 = 1.6875 2^(5/7) = 1.6407  (not used in pentatonic scale)
F         16/9 = 1.7778    2^(6/7) = 1.8114   neutral seventh
G         2/1 = 2              2^(7/7) = 2            perfect octave

As I mentioned above, the fifth and fourth of 7EDO are (barely) close enough to just intonation to
retain the names "perfect fifth" (and fourth). But the third of 7EDO is too narrow to be called a major third (the interval G-B), yet too wide to be called a minor third (the interval G-Bb, which would be 32/27 or 1.1852), and so it's a neutral third. Likewise, the seventh of 7EDO is too narrow to be called a major seventh (the interval G-F#, which would be 243/128 or 1.8984), yet too wide to be called a minor seventh (the interval G-F), and so it's a neutral seventh. This explains the word "neutral" in the name "neutral pentatonic scale." So it appears that the first equal division scale was actually 7EDO in ancient China, not 12EDO.

The usual 12EDO scale arguably contains both a pentatonic and a heptatonic scale. The heptatonic scale lies on the white keys of a piano (C-D-E-F-G-A-B-C) while the pentatonic scale lies on the black keys. We can start the major pentatonic scale on the note called F# (F#-G#-A#-C#-D#-F#), which we can also write using flats on Gb (Gb-Ab-Bb-Db-Eb-Gb).

But let's return to the major pentatonic scale on C (C-D-E-G-A-C). This is similar to the scale used by the Google Fischinger player from my last post, except that the Fischinger player also omits the note A in addition to F and B. In that post, I was a little frustrated that I couldn't play the notes F, A, and B, and so I couldn't play any songs that include any of these three notes.

Recall that some intervals, such as the perfect fifth, are considered to be consonant because they form a simple ratio, 3/2. But there are some intervals that are not consonant, but are called "dissonant," because the notes sound bad together.

One of these is the tritone. Recall that a "tone" is a major second -- the interval C-D. The three intervals F-G, G-A, and A-B are each tones, so F-B contains three tones -- a tritone. This interval has the ugly-looking ratio 729/512 rather than a simple interval like 3/2, and so it is dissonant -- so much that it was once called "the Devil in music." The other dissonant interval is the semitone, which is smaller than a tone ("half" a tone). Each sharp or flat raises or lowers a note by a semitone, so C-C# is a semitone. In the diatonic scale, there are two intervals, E-F and B-C, that are semitones since they are smaller than a tone. The ratios for both E-F and B-C are 256/243, which is the second worst ratio (after the tritone's 729/512) in the diatonic scale, and so it is also dissonant.

Google assumed that most users of the Fischinger player aren't musicians. Such users would actually start stringing random notes together rather than try to play actual songs. And so to make the random notes sound more consonant, Google eliminated the possibility of playing dissonant notes.

In order to avoid the tritone (F-B), we can leave out both F and B. And since F and B are included in both of the semitones (E-F and B-C), omitting F and B also eliminates the semitones.

And so the pentatonic scale avoids the two most dissonant intervals, the tritone and semitone. In fact, the pentatonic scale is the largest scale within 12EDO that avoids both of them -- as soon as we try to add a sixth note, either a tritone or a semitone must appear. (This is easily proved -- we want to avoid semitones, so the smallest possible step is a whole tone. Since each whole tone is two degrees of 12EDO, we must have only whole tones if we want to squeeze in six notes. But the whole tone scale contains tritones. QED)

The Fischinger player avoids the note A, even though it's part of the pentatonic scale. I assume that A is avoided in order to gain even more consonance -- of the four remaining notes C, D, E, and G, three of them form the major triad C-E-G. I suppose that D is retained so that at least there could be some semblance of melody with C-D-E in addition to the harmony of C-E-G.

This is why I compared this to the traditionalist debate in my last post. With only the four notes C, D, E, and G, even non-musicians can create something that sounds good. But I'm more familiar with music, and musicians would want to play songs that contain more notes, but Google holds them back in order to suit the non-musicians. Likewise, according to traditionalists, Common Core holds back stronger students in math (by not encouraging eighth grade Algebra I) in order to suit the weaker students who aren't interested in early Algebra I.

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