Sunday, April 4, 2021

Easter Sunday Post: Miscellaneous Math and Music

Table of Contents

1. Introduction
2. Lyrics: The Percents Song
3. Lyrics: Sign of the Times
4. Lyrics: All About That Base and Height
5. Lyrics: You Gotta Believe
6. Patterns in the Easter Date
7. Coding the Easter Song in 28EDO
8. More Information on 28EDO
9. Report Cards for EDO's
10. Conclusion

Introduction

Today is Easter Sunday, and this is my first spring break post. I'm using this post to catch up to various topics related to music.  It all begins with lyrics to songs that I performed last month but never posted to the blog until now.

Lyrics: The Percents Song

On Thursday, I sang the Percents Song from Square One TV. Here are the lyrics, followed by the link:

PERCENTS SONG
There's a percentage of people,
Who know a lot about percents.
The rest of the people are simply confused,
They don't understand percents and how they're used.
:Like when you wanna go shopping,
'Cause there's a big sale going on.
It's 30% of the normal price,
To figure the cost you got to think twice.
The number of parts of 100,
Is what a percent is.
Let's take 2: can be said a fraction, 2/100,
Or a decimal .02; that's true.

When you start to notice,
You're gonna see examples of percents.
When you put your money in a savings account,
It earns a percent of the total amount.
Or when you're planning a schedule,
It helps to think about percents.
With 24 hours in every day,
What percent do you work or play?
The number of parts of 100,
Is what a percent is.
Let's take 6: can be said a fraction, 6/100,
Or a decimal .06; that's no trick.

To think abut percents, you got to give your all,
Your own 100%.
If you just take a half, a half's 50%,
It's never gonna make sense!

The number of parts of 100,
Is what a percent is.
Let's take 12: can be said a fraction, 12/100,
Or a decimal .12; good for you.

There's a percentage of people,
Who know a lot about percents.
I hope by now you're not confused,
You know about percents and how they're used.
Now you're one of those people,
Who know a lot about percents...


Lyrics: Sign of the Times

This is a Square One TV song that I performed on Friday. Unlike "Percents Song," I have posted the "Sign of the Times" before -- but not since I introduced the "music" label. So I'll post them again:

Sign Of The Times

Lead vocals by Cris Franco

(something in Spanish)
X is the sign of the times
X es el símbolo de los tiempos
There’s a lot of times going around
A time for lunch, a time for school
But the coolest times going around
Is a multiplication tool
Take a X, we’ll show you where it fits
Not the Brand X, not the Band X
There’s no other corner quite like this
Three times two makes six
X is the sign of the times
X es el símbolo de los tiempos
When you’re going to multiply
Just use the symbol X
Es el símbolo de los tiempos
When you’re adding and your numbers are all the same
You take a shortcut: multiply
Count the number of the numbers you want to gain
That’s your multiplier
X is the sign of the times
X es el símbolo de los tiempos
When you’re going to multiply
Just use the symbol X
Es el símbolo de los tiempos
There’s a multiplication table
Everyone should know by heart
From zero times zero to nine times nine
That’s the place to start
Es el símbolo de los tiempos
We know two times two is four
Three times three is nine
But you can do a whole lot more
With X, the sign of the times
Es el símbolo de los tiempos
Three times four or six times two
Equals one times twelve
Figure out the way it works
And try it for yourself
Es el símbolo de los tiempos
The same is true of four times five
Which equals two times ten
Five times six, fifteen times two
Equal each other again
Es el símbolo de los tiempos
You could keep on going
As long as you’re inclined
With different combinations
Just use the sign of the times
Es el símbolo de los tiempos
Aieie, ha ha ha!
Es el símbolo de los tiempos
(trills)
Es el símbolo de los tiempos
(something in Spanish)
Es el símbolo de los tiempos
(something in Spanish) Aieie! Aieie! Aieie!
Es el símbolo de los tiempos
(something in Spanish)
Es el símbolo de los tiempos
(fade out)
[Notice the ASCII error here -- that A with a tilde should really be an i with an acute accent.]


Lyrics: All About That Base and Height

I'll add a few more lyrics that I've previously posted on the blog, but before I added the "music" label.

All About That Base and Height (2D version):

Chorus:
Because you know I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base base base base.
1st Verse:
Yeah, it's pretty clear base times height divided by two,
That equals, equals the area of a triangle. Woo!
'Cause I got those smarts, smarts that all the colleges chase,
And all the right brains in all the right places.
I see the formulas working the area.
We know that stuff is right, come on let's give it a shot!
If you got smart smarts, just let them know,
The height is perpendicular from the bottom to the top.
Pre-Chorus:
Yeah my momma, she told me to worry about your grades,
She said, "Teachers like effort, so you better get all them A's."
You know I won't be no failure up against you all,
So if that's what you're in to, then go ahead and move along. (to Chorus)
2nd Verse:
I'm bringing smartness back! Go ahead and tell them who is
Really right! Naw, I'm just playin'. I know you
Think it's hard! But I'm here to tell you
Base times height is the area of a rectangle and square. (to Pre-Chorus)


All About That Base and Height (3D version):

Chorus:
Because you know I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base and height.
I'm all about that base,
'Bout that base base base base.
1st Verse:
Yeah, it's pretty clear, I really want to,
Calculate your volume, volume, like I'm suppose to do.
'Cause I got that formula that all the students chose.
Just plug in all the right values in all the right places.
See that base! That's the area of the top.
We know the height, come on now make it pop.
If you got your calculator, 'lator, just multiply 'em,
'Cause every cubic inch is perfect from the bottom to the top.
Pre-Chorus:
Hey prisms, cylinders, don't worry about their size,
'Cause students all know the formula to find it right.
You know for the whole volume, it's just V = bh,
And for lateral area, it's L.A. = Ph. (to Chorus)
2nd Verse:
I'm bringing area back! Go ahead with lateral
Area! Naw I'm just playin'
With surface area! Then I have to tell you
First find lateral area then add the bottom and the top. (to Pre-Chorus)

Lyrics: You Gotta Believe

Intro:
Oh yeah, oh, yeah, you gotta believe!
Oh yeah, oh, yeah, you gotta believe!
Oh yeah, oh, yeah, you gotta believe!
Oh yeah, oh, yeah, you gotta believe!

First Verse:
There are too many kids with too few choices,
No one to listen when they raise their voices.
Too often told, be seen and not a word,
Absurd, kick the nice ones to the curb.
Got something on your mind, it can be so much,
Just clear the air, then you can prove you've the right touch,
Gotta know where you're going, gotta know where you've been,
Learn your history in History, know your who, what, where, why, when.
Be proud, stand tall, hold your head up high,
Be a friend to your friends, don't be afraid to cry.
Learn as much as you can, and each one, teach one,
Share your knowledge with your friends, so that you can reach one.
And to those that try to tell you that you never, don't, can't, won't,
Flip the script and prove them wrong.
Believe in yourself and you'll find enough respect,
You know, just keep on keepin' on!

Refrain:
You gotta believe and reach for the sky,
You gotta believe and lift your spirit so high.
You gotta believe, let no one stand in your way,
And your dream will be reality someday.

Second Verse:
When obstacles appear that you can't step around,
Then climb up on over and cruise on down.
When the going gets tough and you're knocked of track,
Another two steps forward, for every one back.
No one said it would be easy, you gotta work with what you've got,
Then when you're seen, you either have to or have not.
It's not what you hold, it's what you've got on the inside,
Knowledge, dignity, self-respect, pride!

(To Refrain)

Third Verse (Walker original, not on video):
So when you're in class and then work seems hard,
Just know that all of you are either smart or almost smart.
But if you slack off, you'll pay the price,
Unless you work a whole lot and make a real sacrifice.
To be a huge success in any class you like,
Just remember, make it easy as riding a bike.
And when it comes time for you to take the big test,
You'll get what you want in life when you always do your best!

(To Refrain)

Outro:
Oh yeah, oh, yeah, you gotta believe!
Oh yeah, oh, yeah, you gotta believe!
Oh yeah, oh, yeah, you gotta believe!
Oh yeah, oh, yeah, you gotta believe!


Patterns in the Easter Date

Three years ago was the last time that I posted on Easter. That day, I posted an Easter song based on the dates changing from year to year. Thus I will reblog this song this year (changing the dates so that it matches Easter this year). The song is written in 28EDO -- not EDL, since it was before I thought about EDL scales (and had just barely figured out how Mocha music works).

Today is Easter. Last year, Easter was on April 12th. Next year, the holiday will fall on April 17th. A long time ago when I was young, I often thought of the changing Easter date like a bunny (the Easter bunny, of course) hopping his way around the calendar months of March and April.

I once noticed that the Easter dates follow a sort of pattern. For example, Easter fell on April 6th in 1980, the year I was born. When I had a part-time job at UCLA, two girls worked with me, and both were born on April 6th, 1980 (so there was a party of both of them the same day). As it turns out, Easter hasn't fallen on April 6th since -- and it won't again until their 62nd birthday!

I'm writing about patterns in the date because I'm wondering whether, just like the digits of repeating decimals, we could create a new song -- an Easter song, if you will -- out of the pattern in the dates.

Let's create a very simple Calendar Reform on the fly here. Actually, this calendar will be identical to the Gregorian Calendar, except it has a four-day week rather than a seven-day week. We'll simply call the four days of the week Oneday, Twoday, Threeday, and Fourday. This is a perpetual calendar in that March 1st will always fall on a Oneday -- which is convenient not only because Easter always falls in either March or April, but it allows us to make February 28th and 29th both blank days (and these are needed since 4 * 91 = 364).

We know that Easter always falls on a Sunday using the seven-day week. But what day of the week does Easter fall using the four-day calendar? For example, if March 1st is Oneday, then yesterday, March 31st, would be Threeday, and so today is Fourday. To make it simple, we might think of today as March 32nd, so we must reduce 32 mod 4.

Of course, Easter is defined to be Sunday of the seven-day week, not the four-day week. So we might expect the holiday to fall on each of the four days of the week equally. Well, let's find out by calculating a few Easters, starting with the first spring after the creation of the blog and extending through the 2020's:

April 5th, 2015 (March 36th = Fourday)
March 27th, 2016 (Threeday)
April 16th, 2017 (March 47th = Threeday)
April 1st, 2018 (March 32nd = Fourday)
April 21st, 2019 (March 52nd = Fourday)
April 12th, 2020 (March 43rd = Threeday)
April 4th, 2021 (March 35th = Threeday)
April 17th, 2022 (March 48th = Fourday)
April 9th, 2023 (March 40th = Fourday)
March 31, 2024 (Threeday)
April 20th, 2025 (March 51st = Threeday)
April 5th, 2026 (March 36th = Fourday)
March 28th, 2027 (Fourday)
April 16th, 2028 (March 47th = Threeday)
April 1st, 2029 (March 32rd = Fourday)

Of 15 Easters, seven fall on Threeday and eight on Fourday, with none on Oneday or Twoday. That we can invent a four-day week on the fly and then have a decade and a half's worth of Easters land on only two of the four days seems more like a pattern than a coincidence.

In fact, I discovered the pattern a few years earlier, when I look at the Easters during the decade before I created this blog:

March 27th, 2005
April 16th, 2006
April 8th, 2007
March 23rd, 2008
April 12th, 2009
April 4th, 2010
April 24th, 2011
April 8th, 2012
March 31st, 2013
April 20th, 2014

Of 10 Easters, seven are in April -- and notice that the April dates are 4, 8, 12, 16, 20, and 24. So all of these are multiples of four. We can consider March 31st to be April 0th -- and zero is also a multiple of four. The other two March dates, the 27th and 23rd, are four and eight days before April 0th, so the pattern continues.

Thus all ten Easters are on the same day of the four-day week! In order to avoid negative April dates, we've been converting these dates to positive March dates. The latest April date, the 24th, is converted to March 55th, and all ten Easters fall on Threeday. So during this stretch, we don't even have any Fourday Easters, much less Oneday or Twoday Easters. And we have an entire quarter century without any Oneday or Twoday Easters.

Does this mean that Oneday and Twoday Easters are impossible, just as Monday and Tuesday Easters are impossible in the seven-day week? Well, let's look at a chart:

1967-1984: Oneday and Twoday Easters
1985-1990: All Twoday Easters
1991-2004: Twoday and Threeday Easters
2005-2014: All Threeday Easters
2015-2028: Threeday and Fourday Easters
2029-2034: All Fourday Easters
2035-2048: Fourday and Oneday Easters
2049-2058: All Oneday Easters

So all four days are possible, but they appear in clusters. There are stretches with many Twoday Easters and stretches without them. My friends' Easter birth on April 6th, a Oneday, occurred just before a long stretch without any Oneday Easters. This is why they still have 24 more years left before they can celebrate another Easter birthday.

The following link provides us an explanation for this phenomenon:

http://www.webexhibits.org/calendars/calendar-christian-easter.html

Suppose you know the Easter date of the current year, can you easily find the Easter date in the next year? No, but you can make a qualified guess. If Easter Sunday in the current year falls on day X and the next year is not a leap year, Easter Sunday of next year will fall on one of the following days: X-15, X-8, X+13 (rare), or X+20. If Easter Sunday in the current year falls on day X and the next year is a leap year, Easter Sunday of next year will fall on one of the following days: X-16, X-9, X+12 (extremely rare), or X+19. (The jump X+12 occurs only once in the period 1800-2200, namely when going from 2075 to 2076.) If you combine this knowledge with the fact that Easter Sunday never falls before 22 March and never falls after 25 April, you can narrow the possibilities down to two or three dates.

Let's look at those lists again:

Non-leap: X - 15, X - 8, X + 13, X + 20
Leap: X - 16, X - 9, X + 12, X + 19

Of these eight possible "jumps" (bunny hops?), four of them are multiples of four, and none of them are two away from a multiple of four. Since most years don't have February 29th and the +13 jump is given as "rare," the three jumps -15, -8, +20 are the most common -- and two of these three jumps are multiples of four. During the stretch from 2005-2014, every time the -15 jump would have occurred, a February 29th conspired to make it a -16 jump, which is a multiple of four. And since the most common non-multiple of four jump is -15 (which is equivalent to +1 mod 4, not -1), the Easters tend to progress forward through the four-day week rather than backward.

It's also possible to see why these eight are the only possible jumps. Recall the definition of Easter as given at the above link:

Easter Sunday is the first Sunday after the first full moon after vernal equinox.

This means that we are taking a solar year (from equinox to equinox) and rounding it to an integer multiple of lunar months (from full moon to full moon), and then rounding the result off to an integer multiple of weeks (from Sunday to Sunday). There are three binary choices to make:

  • Do we round the year up or down to an integer multiple of months?
  • Do we round the result up or down to an integer multiple of weeks?
  • Is there or is there not a Leap Day?
With three binary choices, there are only 2^3 or 8 possibilities -- and all of them are listed.

A solar year is either 12 or 13 months. (The 13th month or Leap Month often appears in lunisolar calendars such as the Hebrew Calendar -- the ultimate inspiration for Passover and Easter.) A year with 12 lunar months is about 354 days, which rounds to either 50 weeks or 51 weeks. As a solar year has 365 days, 50 weeks is 15 days short of a year (-15 jump) and 51 weeks is 8 days short (-8). A year with 13 lunar months is about 384 days, which rounds to either 54 weeks or 55 weeks. Then 54 weeks is 13 days long of a year (+13 jump) and 55 weeks is 20 days long of a year (+20). Also, notice that 354 days is about halfway between 50 and 51 weeks, while 384 days is much closer to 55 weeks than to 54 weeks. This is why the 54-week jump (+13) is "rare." The calculations for Leap Day are based on taking the year to be 366 days rather than 365.

Now that we finally know the pattern of the Easter dates, let's make them into an Easter song. When I first discovered the Easter pattern years ago, I wanted to make the common jumps correspond to consonant intervals. For example, maybe -8 would be a major third, while -15 is a perfect fifth. But then -16, another bunny hop, would be an augmented fifth (two major thirds) -- although this might sound as a minor sixth, which is consonant, depending on the situation. (Recall what I wrote about train whistles in an earlier post -- they sound like dissonant augmented triads.)

In the end, I wasn't able to make the Easter dates into a coherent song. But I came up with this idea years ago, back before I knew about scales other than 12EDO. Perhaps it's possible to make an Easter song after all, just in another scale besides 12EDO.

So what might be a good EDO for the Easter song? We notice that there are 35 possible Easters (from March 22nd to April 25th), so maybe 35EDO would work. Or we could use 34EDO instead, so that the notes for March 22nd and April 25th are an octave apart. Another possibility is 31EDO -- this is a commonly used EDO for 11-limit music (as explained in an earlier post). Also, since March has 31 days, it means that Easters on the same day of the month (March 22nd and April 22nd, up to March 25th and April 25th) are an octave apart.

But the Easter date patterns we found above involve two key numbers -- 7 and 4. And so the best EDO for an Easter song would be 7 * 4, or 28EDO. Indeed, 28EDO is the only EDO such that the inversion of a possible bunny hop is another bunny hop (such as -15 and +13, -9 and +19). And so 28EDO it is.

Coding the Easter Song in 28EDO

We know that music on Mocha is based on EDL (equal divisions of length) rather than EDO (equal division of the octave). Thus trying to play an EDO on Mocha is unnatural -- and indeed, we can only approximate 28EDO on the emulator.

I wrote earlier that the EDO's up to 12 (the "macrotonal EDO's") sound quite well on Mocha, but the accuracy drops off quickly past 12EDO. The multiples of four (16EDO, 20EDO, 24EDO, 28EDO) are slightly better than non-multiples of four. Once we reach 31EDO, the odd EDO's are marginally better than the even EDO's -- in reality, all of them are very inaccurate. (This is another situation where 16-bit Atari music shines -- although Atari music is also based on EDL, we'd be able to approximate EDO's better on Atari than on Mocha.)

As it turns out, Degree 210 (Sound 51) -- the root note of the New 7-Limit Scale -- is also a good root note for a 28EDO scale. To create the scale start with Degree 210 and divide by the 28th root of two for each step until we reach 105, one octave above Degree 210. Here's the resulting scale:

Step  Degree  Sound
0       210        51
1       205        56
2       200        61
3       195        66
4       190        71
5       186        75
6       181        80
7       177        84
8       172        89
9       168        93
10     164        97
11     160        101
12     156        105
13     152        109
14     148        113
15     145        116
16     141        120
17     138        123
18     134        127
19     131        130
20     128        133
21     125        136
22     122        139
23     119        142
24     116        145
25     113        148
26     110        151
27     108        153
28     105        156

Some of these are more accurate than others. These include:


Step  Degree  Sound
0       210        51
3       195        66
6       181        80
9       168        93
10     164        97
11     160        101
12     156        105
20     128        133
24     116        145
25     113        148
28     105        156

These can indicate how close some of the steps of 28EDO are to just intervals. For example, that Step 3 corresponds to Degree 195 tells us that this interval from Steps 0 to 3 represents the ratio 210/195, which reduces to 14/13 (a 13-limit interval).

One important interval is that from Steps 0 to 9. The ratio 210/168 reduces to 5/4, a major third. In fact, 28EDO approximates a just major third to within one cent -- better than any simpler EDO. And furthermore, -9 is one of the Easter bunny hops, so here I achieve my original goal of making the valid Easter jumps correspond to consonant intervals.

The reason that we see EDO's like 24 and 31 more often than 28 -- despite its accurate major third -- is that 28EDO perfect fifth is inaccurate (and fifths are more important than thirds). In fact, the perfect fifth of 28EDO is the same as that of 7EDO -- Step 16. This is confirmed by the presence of Degree 141 in the above chart for Step 16 -- had Step 16 been closer to a just 3/2, it would have been listed as Degree 140 (210/140 = 3/2), not Degree 141.

I've mentioned 7EDO in previous posts -- there was apparently an ancient Chinese scale (Qingyu) based on five notes of a 7EDO scale. We found out that this scale fails to distinguish between major and minor intervals.

In fact, the Xenharmonic website often gives 7EDO a special name -- whitewood. It refers to the idea of removing all of the black keys on a piano, leaving only the white keys. Without black keys, the interval from C-D is the same as that from E-F -- seven equal intervals add up to the octave.


Another name given at Xenharmonic for EDO's like 7 and 28 is "perfect EDO." This is because there are no "major" or "minor" intervals, only "perfect" intervals.

Of course, 28EDO does have a major third (Step 9) in addition to 7EDO's "perfect third." In other words, 28EDO is what we get if we cross a perfect EDO with a just major third.

This indicates what our Easter song will sound like. We'll hear essentially 7EDO music during the stretches when Easter falls on the same day of the four-week (like 2005-2014), then slowly more and more major thirds (and minor sixths) appear. Eventually we'll hit another stretch (like 2029-2034) with the same 7EDO scale transposed up a 28EDO step.

Let's program this song in Mocha. We'll begin by coding the 28EDO scale:

NEW
10 DIM S(35)
20 FOR X=1 TO 35
30 S(X)=INT(210/2^((X-4)/28)+.5)
40 NEXT X

Don't forget to use the up arrow for the exponentiation ^ symbol. This sets up a 28EDO scale, where Note 4 is Degree 210 and Note 28 is Degree 105. The idea is that Note 1 will correspond to March 22nd and Note 35 is April 25th, so that the earliest Easters play the lowest notes (which are the largest degrees). 

Now for the trickiest part -- calculating an Easter date. We begin by asking the user for a year:

50 INPUT Y

Notice the definition of the Remainder or mod function given:

Remainder(x|y) (or x mod y) means the remainder when you divide x by y. It is never negative, and is defined in terms of the [] operation as follows:
Remainder(x|y) = x - y[x/y]
In Mocha, [] is the INT operation, so we write:

R=X-Y*INT(X/Y)

This is actually the same remainder function we used in the repeating decimals song earlier. It would be easier if the mod function had a single symbol, like % in C and C++. But unfortunately, BASIC doesn't have such a symbol built in!

Let's proceed with the algorithm mentioned at the above link. For example, at the above link we see:

G = year mod 19

In the program, we write this as

60 G=Y-19*INT(Y/19)

Here is the rest of the algorithm:

70 C=INT(Y/100)
80 HH=C-INT(C/4)-INT((8*C-13)/25)+19*G+15
90 H=HH-30*INT(HH/30)
100 I=H-INT(H/28)*(1-INT(29/(H+1))*INT((21-G)/11))
110 JJ=Y+INT(Y/4)+I+2-C-INT(C/4)
120 J=JJ-7*INT(JJ/7)
130 L=I-J+7

You might notice that the link above gives L = I - J, not L = I - J + 7. But let's see why I included it:

L is the number of days from 21 March to the Sunday on or before the Paschal full moon (a number between -6 and 28)

In other words, L is the number of days from March 21st to Palm Sunday. We might as well add 7 to make this the number of days from March 21st to Easter Sunday. And then this becomes a number between 1 and 35, which we can enter directly into Mocha:

140 SOUND 261-S(L),4

The song is supposed to continue on to the following year to computer the next Easter:

150 Y=Y+1
160 GOTO 60

We can begin this song at any year, such as 2018, and hear a different part of the song. I suppose that purists should begin the song with 1583 -- the first spring of the Gregorian Calendar.

Just as with the repeating decimals, this song repeats -- but according to the link above, Easter repeats only after 5.7 million years!

The link also states that Julian (Orthodox) Easter repeats only after 532 years. The link doesn't state this, but the first lines (calculating I and J that we skipped over) are actually for Julian Easter. (But the way, this year Orthodox Easter is a week after Gregorian Easter on April 8th.) This we can code in Mocha by deleting some lines:

DEL 70-80
90 II=19*G+15
100 I=II-30*INT(II/30)
110 JJ=Y-INT(Y/4)+I

This song thus repeats after 532 notes. (Notice that 533 isn't a prime, much less a full reptend prime, but 541 is indeed a full reptend prime. So the Orthodox Easter song is similar in length to the song for the repeating decimal 1/541.)

More Information About 28EDO

Here's a link to the 28EDO page at the Xenharmonic website:


28edo, a multiple of both 7edo and 14edo (and of course 2edo and 4edo), has a step size of 42.857 cents. It shares three intervals with 12edo: the 300 cent minor third, the 600 cent tritone, and the 900 cent major sixth. Thus it tempers out the greater diesis 648:625. It does not however temper out the 128:125 lesser diesis, as its major third is less than 1 cent flat (and its inversion the minor sixth less than 1 cent sharp). It has the same perfect fourth and fifth as 7edo. It also has decent approximations of several septimal intervals, of which 9/7 and its inversion 14/9 are also found in 14edo.

28edo can approximate the 7-limit subgroup 2.27.5.21 quite well, and on this subgroup it has the same commas and tunings as 84edo. The temperament corresponding to orwell temperament now has a major third as generator, though as before 225/224, 1728/1715 and 6144/6125 are tempered out. The 225/224-tempered version of the augmented triad has a very low complexity, so many of them appear in the MOS scales for this temperament, which have sizes 7, 10, 13, 16, 19, 22, 25.

Another subgroup for which 28edo works quite well is 2.5.11.19.21.27.29.39.


The chart at the link doesn't do this, but I like the idea of calling the 7EDO intervals "perfect," one step above perfect "major," and one step below perfect "minor." In other words, "major" is what the chart calls "up," and "minor" is what the chart calls "down." We might as use the name "augmented" for "double-up" and "diminished" for "double-down."

Then we can call Step 9 a "major third," which is what we expect for a interval near 5/4. Then Step 7, the "minor third," is the same as the minor third of 12EDO.

Meanwhile, many "augmented" and "diminished" represent septimal intervals -- for example, Step 2 is 21/20, Step 6 is 7/6, Step 10 is 9/7, and Step 14 is 7/5. Some of the "perfect" intervals correspond to undecimal intervals -- Step 4 is the perfect (neutral) second 11/10, while Step 8 is the perfect (neutral) third 11/9.

By this definition, two perfect fifths don't add up to a major second (instead of an octave). In other words, 9/8 is inconsistent. This happens because the whitewood perfect fifth isn't accurate, so two different perfect fifths are needed to reach 9/8. The circle of fifths clearly doesn't work in 28EDO the way it does in 12EDO.

In fact, this inconsistency results in some strange sounds when one tries to play a chord progression, such as in the Eagles' Hotel California:


In the Easter song, this chord drift is a feature, not a bug.

Report Cards for EDO's

Recall that the focus for Mocha music should be EDL's, not EDO's. While macrotonal EDO's (from 7 to 11) can be approximated well in Mocha, microtonal EDO's like 28EDO can't. The code above is close to 28EDO (at least much closer than a standard 12EDO instrument can approach it), but there are audible differences between it and true 28EDO. This is why it's not recommended to compose a song in a microtonal EDO and expect it to sound correct in Mocha. (Again, I only wrote the Easter song in 28EDO because it was before I knew about EDL.)

Still, since those who wish to learn about alternate scales naturally look at EDO's first, it's worth looking at the various EDO's to see which ones are better than others.

For example, it's well known that in our standard 12EDO scale, the prime (harmonic) Degrees sound progressively worse in 12EDO as we move from Degree 2 to Degree 11. Degree 2, the octave (2/1) in of course exact (the O in EDO). Degree 3, the perfect fifth (3/2) is very close. Degree 5, the major third (5/4), should be 386.3 cents, but it is 400 cents in 12EDO. Degree 7, the harmonic seventh (7/4), should be 968.8 cents, but it is 1000 cents in 12EDO. Degree 11 (11/8) should be 551.3 cents, but it is 600 cents in 12EDO.

Hmm -- 386.3 vs. 400, 968.8 vs. 1000, 551.3 vs. 600. The digits 86.3, 68.8, 51.3 show us why major thirds sound good in 12EDO, 11/8's sound terrible, and 7/4's somewhere in between. But to my teacher eyes (especially after singing a percents song), those look like percents -- that is, grades. So the major third in 12EDO should get a grade of 86.3% or B, a 7/4 a 68.8% or D, and 11/8 a 51.3% or F.

The octave is exact, so it gets a grade of 100%. As for the perfect fifth, we see that 702 cents is fairly close to 700 cents, so we wouldn't want the grade to be 02%. Instead, we take the subharmonic -- the perfect fourth, 498 cents. Then the grade becomes 98%, a solid A. In other words, we choose either the harmonic or subharmonic that sounds sharp in the EDO so that a grade between 50% and 100% is reached, and then map that percentage to a grade.

For example, the 13th harmonic 13/8 is 840.53 cents, so we use the 13th subharmonic 16/13, 359.47 cents, to obtain a grade of 59.47%, which is also an F (but very close to D-). It's up to you to decide how to round grades, whether 59.5% rounds to 60%, as well as plus and minus grades (so that the grade for 7/4, 68.8%, could be written as D+).

So here is the report card for 12EDO:

12EDO Report Card:
Degree 3: 98.0% A
Degree 5: 86.3% B
Degree 7: 68.8% D
Degree 11: 51.3% F
Degree 13: 59.4% F
Degree 17: 95.0% A
Degree 19: 97.5% A

So our standard scale gets progressively worse from 2 to 11, then gets better as we reach 17 and 19.

To get report cards for other EDO's, we consider "relative cents" -- each step of the EDO is divided into 100 cents, so that one cent of 7EDO equals four cents of 28EDO. For example, in 7EDO, the size of the perfect fourth (4/3) in 7EDO relative cents is:

log(4/3) / log(2) * 700 = 290.5 rel cents (a grade of 90.5% or A-).

Let's try some macrotonal EDO's first:

7EDO Report Card:
Degree 3: 90.5% A-
Degree 5: 74.6% C
Degree 7: 65.1% D
Degree 11: 78.3% C
Degree 13: 90.3% A-
Degree 17: 61.2% D
Degree 19: 73.5% C

9EDO Report Card:
Degree 3: 73.5% C
Degree 5: 89.7% B+
Degree 7: 73.3% C
Degree 11: 86.5% B
Degree 13: 69.6% D+
Degree 17: 78.7% C
Degree 19: 76.8% C

Because these are relative cents, different EDO's get different grades even if they have identical approximations for the same interval. Thus Degree 5 gets a grade of B+ in 9EDO and a B in 12EDO, even though the 5/4 major third is exactly one-third of an octave in each scale. In other words, 9EDO is relatively better for its step size at approximating 5/4.

Now me move to the microtonal side:

16EDO Report Card:
Degree 3: 64.0% D
Degree 5: 84.9% B
Degree 7: 91.7% A
Degree 11: 64.9% D
Degree 13: 79.2% C+
Degree 17: 60.0% D-
Degree 19: 96.6% A

19EDO Report Card:
Degree 3: 88.5% B
Degree 5: 88.3% B
Degree 7: 66.0% D
Degree 11: 72.9% C
Degree 13: 69.1% D+
Degree 17: 66.1% D
Degree 19: 71.0% C

21EDO Report Card:
Degree 3: 71.5% C
Degree 5: 76.0% C
Degree 7: 95.4% A
Degree 11: 64.8% D
Degree 13: 70.9% C-
Degree 17: 83.6% B
Degree 19: 79.3% C+

24EDO Report Card:
Degree 3: 96.0% A
Degree 5: 72.6% C
Degree 7: 62.3% D
Degree 11: 97.3% A
Degree 13: 81.0% B
Degree 17: 90.0% A-
Degree 19: 95.0% A

This is the quarter-tone scale. It inherits good grades for Degrees 3, 17, 19 from 12EDO and adds on Degrees 11 and 13 via half-sharps/flats. Again, Degree 5 gets only a C grade because it doesn't improve on 12EDO's major third despite its smaller step size.

And here's the one we want for our Easter song:

28EDO Report Card:
Degree 3: 62.1% D
Degree 5: 98.6% A
Degree 7: 60.5% D-
Degree 11: 86.4% B
Degree 13: 61.2% D
Degree 17: 55.1% F
Degree 19: 94.1% A

We know that 28EDO has an excellent major third, hence its A grade for Degree 5. The other Degree for which 28EDO gets a good grade is Degree 11, with a B grade. It has the same perfect fifth as 7EDO and 21EDO -- its D grade for Degree 3 reflects the fact that it can't improve from those lower EDO's.

I'll do one more report card -- the ever-popular 31EDO:

31EDO Report Card:
Degree 3: 86.6% B
Degree 5: 97.9% A
Degree 7: 97.1% A
Degree 11: 75.7% C
Degree 13: 71.3% C
Degree 17: 71.1% C
Degree 19: 68.5% D

Conclusion

I hope you enjoyed your Easter holiday. I will probably make one more post during the spring break period, with the topic to be something other than music.
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