Friday, June 30, 2017

Pappas Music Post: The 22-Note Scale

Note: So far, I have edited the following post:

to incorporate Chapter 3 of Eugenia Cheng's Beyond Infinity.

This is what Theoni Pappas writes on page 181 of her Magic of Mathematics:

"As with numeration systems we find these [scales] evolved differently in various civilizations. The ancient Greeks used letters of their alphabet to represent the seven notes of their scale."

And I wrote about those numeration systems in my May 22nd post. Anyway, today's post is about the various musical scales that developed in other parts of the world.

So far, we've already seen scales with five, seven, and twelve notes. But whereas some scales have fewer notes, other have more notes. If there are more than twelve notes in an octave, the interval between each note becomes smaller, and thus we refer to these as microtonal scales.

There is a simple unit to measure the size of a microtonal interval -- the cent. We define the interval between each note of 12EDO, such as C-C# or E-F, to be 100 cents, just as with money, we define a dollar to be 100 cents. Then microtonal music involves intervals of less than 100 cents. The entire octave contains 1200 cents.

What is the smallest possible interval -- that is, the smallest audible interval such that a musician playing the two notes can distinguish between them? Well, let's ask Pappas:

"The trained ear can hear about 300 different sounds in one octave."

Since an octave contains 1200 cents, each note is four cents apart. So the interval of 4 cents is known as the just noticeable difference, also known as a savart, named for a 19th century French physicist.

But even though the JND is 4 cents, don't expect to see 300EDO anytime soon. Pappas writes:

"But to produce a scale with this many notes would be ludicrous, since traditional instruments cannot produce that many notes. For example, if there were 300 notes in an octave, a piano of eight octaves would have 2400 white keys."

Of course, Pappas wrote this in 1994. Believe it or not, there now exist instruments with 205 notes per octave -- that's not quite 300, but probably more than Pappas would anticipate:

But no, today I won't write about 205EDO. Let's look instead at a scale that appears in Pappas:

"In India, music was and is improvised within specific boundaries known as ragas. This octave is divided into 66 intervals called srutis, though in practice there are only 22 srutis, from which two basic seven-note scales are formed."

I haven't been able to find much information about the 66-note scale, so we'll just stick to 22. Here are the 22 notes of the Indian sruti scale, which is an advanced 5-limit scale:

Degree  Ratio      Cents          Indian Name
0           1/1          0                 Chandovati
1           256/243  90               Dayavati
2           16/15      112             Ranjani
3           10/9        182             Raktika
4           9/8          204             Raudri
5           32/27      294             Krodha
6           6/5          316             Vajrika
7           5/4          386             Prasarini
8           81/64      408             Priti
9           4/3          498             Marjani
10         27/20      520             Kshiti
11         45/32      590             Rakta
12         729/512  612             Sandipani
13         3/2          702             Alapini
14         128/81    792             Madanti
15         8/5          814             Rohini
16         5/3          884             Ramya
17         27/16      906             Ugra
18         16/9        996             Ksobhini
19         9/5          1018           Tivra
20         15/8        1088           Kumdvati
21         243/128  1110           Manda
22         2/1          1200           Chandovati

According to Pappas, only seven of these are used to form a scale, just as only seven notes of 12EDO are used to form a scale. She writes that there are two ways to do so, but I've only been able to find one -- and it corresponds to the familiar diatonic scale:

Name     Solfege: Western     Indian     Ratio      Cents
C                             Do              Sa         1/1        0
D                            Re               Re         9/8        204
E                             Mi              Ga         5/4        386
F                             Fa               Ma        4/3        498
G                            Sol              Pa         3/2        702
A                            La               Dha      5/3         884
B                             Ti               Ni         15/8       1088
c                             Do               Sa        2/1        1200

I like how "Re" refers to the same note (9/8) in both Western and Indian solfege. There are also several rhyming sounds: Fa-Ma (4/3), La-Dha (5/3), and Ti-Ni (15/8).

It's interesting to try to assign Western names to all 22 sruti -- and it's also instructive to see how exactly microtonal music works. There is only one unison, 1/1, which we can call C, and there's only one fifth, 3/2, which we can call G.

But there are four each of seconds, thirds, fourths, sixths, and sevenths. Well, notice that we already have seven notes in our just major scale, so we only need 15 more notes. Using C-D-E-F-G-A-B-c to refer to the just major scale is called Johnston notation.

Let's try to name the thirds. We already have the major third 5/4, which we call E. Now the minor third is 6/5, and so we want to call this note Eb (or E-flat). In Johnston notation, the flat symbol means to lower a note by 25/24, called a chromatic semitone. We see that this works because:

(6/5)(25/24) = 5/4.

Now we have only two thirds left -- 32/27 and 81/64. Let's look at 81/64 first. We notice that:

(9/8)(9/8) = 81/64.

Now 9/8 is a major second, and we know that two major seconds are a major third. Actually, we've seen before that 81/64 is a Pythagorean (or 3-limit) major third, but it's not the same as our usual Ptolemaic (or 5-limit) major third 5/4. In fact, we have:

(9/8)(10/9) = 5/4.

The interval 9/8 is often called a "major tone," while 10/9 is called a "minor tone." So a major tone plus a minor tone equals a major third.

(10/9)(81/80) = 9/8.

The difference between the two tones is 81/80, which has a special name -- syntonic comma. We already have another interval called a "comma" -- the Pythagorean comma. As it turns out, these two commas are about the same size (syntonic = 22 cents, Pythagorean = 24 cents) -- the difference between them is within the just noticeable difference. In general, any small interval around the size of the Pythagorean or syntonic commas may be called a "comma."

Finally, 32/27, is a Pythagorean minor third. We add a syntonic comma and we get a just minor third:

(32/27)(81/80) = 6/5.

In 12EDO, we don't distinguish between the major and minor tones. In other words, 12EDO is said to temper out the syntonic comma -- that is, reduce it to a unison. The tone in 12EDO is 200 cents, which is smaller than the major tone (204 cents) and larger than a minor tone (182 cents). Therefore we call it a "meantone."

But our 22-note system is not a meantone system. The major and minor tones are distinct, and so we have two major thirds and two minor thirds separated by the syntonic comma. In Johnston notation, we use + to raise a note by a syntonic comma, and - to lower it by that comma. Then we have:

Note     Ratio      Cents          Name
Eb-       32/27      294             Pythagorean minor third
Eb        6/5          316             Minor third
E          5/4          386             Major third
E+        81/64      408             Pythagorean major third

As it turns out, the sixths and sevenths follow the same pattern. The lowest 6th (7th) is a Pythagorean minor 6th (7th), then the minor 6th (7th), then the major 6th (7th), and the highest is a Pythagorean minor 6th (7th), which is easy to remember. The notes are Ab-, Ab- A, A+, Bb-, Bb, B, and B+.

But there are problems when we try to name the seconds and fourths. If the seconds followed the same patterns as the thirds, then D would be 10/9 and D+ would be 9/8. But in the just major scale, the note D is 9/8, not 10/9. This means that D- would be 10/9. The next lower second, 16/15, is a chromatic semitone below D-, so it would be Db-. And then the lowest second, 256/243, is a syntonic comma below Db-, so it's Db--. Likewise, the fourths must be named F, F+, F#+, and F#++, since the lowest fourth is the perfect fourth 4/3, which must be F.

I don't necessarily like having to name notes Db-- and F++ in what should be a simple system. There is another system of notation, called HE notation. In this system, the bare notes without + or - symbols are the Pythagorean intervals. Here are what the seconds look like in HE notation:

Note     Ratio       Cents          Name
Db        256/243  90               Pythagorean minor second
Db+      16/15      112             Minor second
D-         10/9        182             Major second
D          9/8           204             Pythagorean major second

(Confusing, the interval 10/9 is both a "major second" and a "minor tone.") In fact, the following names of the seconds are more common:

Note     Ratio       Cents          Name
Db        256/243  90               Pythagorean limma
Db+      16/15      112             Diatonic semitone
D-         10/9        182             Minor tone
D          9/8           204            Major tone

The fourths are even trickier to name. We want to call 4/3 a "perfect fourth," but we don't want to call any fourth a "minor fourth" -- especially not any interval wider than a perfect fourth. Don't forget that in this system, all intervals without a + or - symbol are Pythagorean, so that C-E is a Pythagorean major third, not a just major third. (This is the price we pay to avoid Db-- and F#++.)

I've also seen a system where the intervals are named after colors, where "green" means minor and "yellow" means major. The advantage is that we can now have a "green fourth" and a "yellow fourth" as opposed to the awkward "minor fourth" and "major fourth."

In this system, all Pythagorean intervals are "white." For seconds and sixths, the larger interval is called "white" and the smaller called "small white." For the other intervals, the smaller interval is called "white" and the larger called "large white." So now we have:

Note     Ratio       Cents         Name
Db        256/243  90              Small white second
Db+      16/15      112            Green second
D-         10/9        182            Yellow second
D          9/8          204            White second
Eb        32/27       294            White third
Eb+      6/5          316             Green third
E-         5/4          386             Yellow third
E          81/64      408             Large white third

You may be wondering why there are two Db notes instead of calling one of them C#. The reason is that in these systems, C#-F# and Db-Gb must each be a perfect fourth, in the same way that C-F and D-G are perfect fourths. But neither Db- nor Db has a sruti that is a perfect fourth above it:

Db: (256/243)(4/3) = 1024/729
Db+: (16/15)(4/3) = 64/45

In order to maintain the circle of fifths (or fourths), only F# can be given a sharp name -- all the other black keys are flat. (Of course, if you find this confusing, we can stick to the original Indian names.)

Notice that like the Chinese scale, the Indian srutis are based on just intonation. It is possible to try assigning an equal division, 22EDO, to these srutis. Some of the 22EDO notes are close to just intonation while others aren't:

Note     Ratio       Cents         22EDO Cents     Difference     Solfege
C          1/1          0                 0                         0                    do
Db        256/243  90              55                        -35                di
Db+      16/15      112            109                      -3                  ra
D-         10/9        182            164                      -18                ru
D          9/8          204            218                      +14                re
Eb        32/27      294             273                     -21                ma
Eb+      6/5          316             327                     +11               me
E-         5/4          386             382                     -4                  mi
E          81/64      408             436                     +28               mo
F           4/3          498            491                      -7                  fa
F+        27/20      520             545                     +25                fu
F#-       45/32      590             600                     +10                fi
F#         729/512  612            655                      +43               su
G          3/2          702            709                      +7                 sol
Ab        128/81    792            764                      -28                 lo
Ab+      8/5          814            818                      +4                 le
A-         5/3          884            873                      -11                la
A          27/16      906            927                      +21                li
Bb        16/9        996             982                     -14                 ta
Bb+      9/5          1018          1036                    +18                tu
B-         15/8        1088          1091                    +3                  ti
B          243/128  1110           1145                    +35               da
c           2/1          1200           1200                   0                    do

We can see that the most accurate intervals in 22EDO are the just diatonic semitone (16/15) and the just major third (5/4). The sum of these two intervals is the perfect fourth (4/3 -- notice how the errors combine: 3 + 4 = 7 cents), and so it and its inversion, the perfect fifth (3/2) are also quite accurate.

On the other hand, the least accurate intervals are the Pythagorean limma (256/243) and the Pythagorean tritone (729/512). Then again, I wrote that these are already the two most dissonant intervals anyway -- so they might actually sound better in 22EDO.

Here is a link to a website that discusses 22EDO and other alternate tunings.

Finally, it's one thing to read about the 22-sruti Indian scale, but music is meant to be heard. So here is a video in which srutis are played:

In this video, Dr. Oke plays an F major triad in both 12EDO and the sruti scale. He points out that the sruti F triad is based on just intonation (ratio 4-5-6 or 100-125-150) and sounds more consonant than the 12EDO version. He also demonstrates how to play 22 notes when there are only 12 actual keys per octave on his keyboard.

My original intention wasn't to start out talking about music and switching to traditionalists, but somehow I developed that habit.

For my latest traditionalists post, I wish to discuss a special math program at a certain school district here in Southern California. I didn't know of its existence until I interviewed with the district during my long search for a new teaching job.

The district is Pasadena, and the program is Math Academy. Here's a link to an article about it:

Math Academy is a super-accelerated program. The students currently in the program are in middle school, as the first cohort hasn't made it to high school yet.

Here is the level of math the students are taking:

-- 6th grade: Algebra I
-- 7th grade: Geometry/Algebra II/Pre-Calc combined
-- 8th grade: AP Calculus

The school at which the program was piloted is a K-8 school (just like my old school), and the students there began the program in fifth grade:

-- 5th grade: Algebra I
-- 6th grade: Geometry/Algebra II/Pre-Calc combined
-- 7th grade: AP Calculus AB
-- 8th grade: AP Calculus BC???

This is very similar to the accelerated BASIS charter program in Arizona and other states. The difference is that, while the top BASIS students also begin Algebra I in fifth grade, there are two levels of math between Algebra I and Calculus (based on the Saxon curriculum), not just one. So the top students at BASIS begin Calculus in 8th, not 7th, grade. Then again, BASIS accelerates students in all subjects, while Math Academy focuses, as its name implies, only on math.

We know why traditionalists want to push 8th grade Algebra I and senior year Calculus. They believe that students can't be competitive in STEM majors at top universities unless they've completed Calculus prior to their arrival there. But we look at what the founder of Math Academy says:

There are several advantages, which I’ll go into, but one of the most important is that of increased optionality. For kids to have any hope of surviving a STEM field at an elite university, they need elite-level training. The reality is that their chances of succeeding at something like math or physics at a place like Stanford, having only taken up through AP Calculus, are simply not very good.
[emphasis mine]

So according to the founder, a student who has "only" taken Calculus in high school won't be able to succeed in a STEM field at Stanford. Contrast this with most traditionalists, who would be happy to see their students taking "only" Calculus. He proceeds:

In fact, it’s not much different than expecting to play basketball at UCLA having only played the sport on your high school’s junior varsity team. These schools select the top students from around the world and those kids have all done considerably more than what’s offered in the standard high-school curriculum. In fact, a mathematician from Caltech recently told me that their undergraduate admissions committee looks for candidates who’ve already published original mathematical research, which of course, is lightyears beyond the kind of math that kids do in high school.

Wow, high school students are expected to have done "original mathematical research"? I have a Masters Degree in math and I've never done original research, which I associate with candidates for doctoral degrees. And this is just to get into freshman university classes!

Meanwhile, recent posts written by actual traditionalists are focusing on the lower grades. Here is a Barry Garelick post that has drawn many comments:

This 2014 story published in the Atlanta Journal Constitution reports the shocking news that teaching first-grade students math using the dreaded worksheets, and traditional modes of education was more effective than ” group work, peer tutoring or hands-on activities that use manipulations, calculators, movement and music.”

As usual, I agree with the traditionalists regarding the lower grades. It's usually the older students, not the first graders, who ask questions like "Why do we have to learn this stuff?" when handed a traditional worksheet.

As usual, co-author SteveH has posted several comments. Here is one of them:

My son loved worksheets and flash cards. He loved having skills. When I mentioned worksheets to his Kindergarten teacher, I thought she was going to call the police.
Hands-on group projects in class can only work (even though it’s not the best use of time) for those students who get the skills at home. Duh, that’s how it works in music. The best are created from private lessons and individual skill/musicality “homework.” In school, they have group band or orchestra where the engaging fun is had. However, the only musicians to get into All-State are ones who had private lessons. When my son played in our State’s Honors Recital (after audition selection by musicians), they finally allowed the listing of the private lessons teacher in the program, not the high school music teacher. This is how El Sistema works. Kids from the barrios get private lessons at their own level starting from an early age. They can and do then compete with kids from affluent families for the top regional and country orchestras. It’s all about content skills and knowledge from an early age. All schools have to do is to allow for some opt-in direct math teaching classes and they will quickly see amazing results. My old high school was the best in music in Connecticut because we had this sort of process that started in the lower grades.
Long ago, I came to the conclusion that this was all about their academic turf. What they believe is all about them, not the subject or skills. That’s why it changes in high school. Students have teachers with subject area turf.
Notice that both the Math Academy founder and SteveH use analogies from other subjects (sports to math) to explain why hard work at a high level is effective. I found out the hard way that it's a bad idea to tell students those analogies when they ask "Why do we have to learn this stuff?" But it's okay when the recipients of the message are adults on a blog rather than students in a classroom.

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