Speaking of summer, today's post is all about the top three posts on my blog in the past year, according to hit count. All three of them are from last summer. (That's why I posted my fourth most popular post yesterday, since it's a Geometry post.)
My #1 post is all about number bases ():
In this post I link to Dozens Online, the one forum devoted to number bases -- and I believe that the high hit count is due to that forum reciprocating the link to this blog. Actually, since then Dozens Online has migrated to another forum, so let me give the new link:
My #2 post is all about Lee Canter's book on classroom management ():
Hmm, I'll be teaching a summer school class where classroom management will definitely
be important. I ought to return to Canter's book at least once before that class begins.
My #3 post is all about microtonal music ():
And of course, I'm currently returning to music in alternate scales as I prepare to compose some math songs for my summer class. But at the time I made this post, it was before I determined that Mocha music is based on EDL's, not EDO's. And so I wrote so much in that post about EDO's -- for example, converting the song "Meet Me in Pomona, Mona" (or "St Louis, Louis") to 19EDO. This was in anticipation of finding the Mocha emulator and playing microtonal music in it. Since then, I discovered that Mocha music is based on EDL, and so everything I wrote about EDO is irrelevant (especially scales with very tiny step sizes, like 200EDO).
I wasn't going to make another music post this week, but of course I can't resist it after subbing in a music class today.
By the way, I'm trying to stay away from converting existing songs from 12EDO to 12EDL or any other Mocha-playable scale. Converting a song written in 12EDO to any other scale -- even another EDO like 19EDO -- is like speaking another language with an accent. A 12EDO song written in 19EDO is an English speaker trying to speak French -- the song sounds better in the original 12EDO than in 19EDO. Instead, it's better to compose a new song in the new 19EDO scale -- this is French spoken by a native speaker. The rule of thumb of all music is that it needs to sound good.
In fact, here's a YouTube link to the same song, first in 19EDO, then in 12EDO:
Notice what the 19EDO guitarist, Gareth Evans, writes:
A comparison of 19-tone equal temperament and 12-tone equal temperament. The piece is an exerpt of 'Seigneur Dieu ta pitie' by Guillaume Costeley which was originally composed in 19-TET. The first version is played on a 19-TET guitar and the second version is a 12-TET approximation.
And in the comments, it's pointed out that this is one of the rare times that 19EDO sounds better than 12EDO. It's because the song is originally composed in 19EDO. Tunes usually sound best in their original scale. And so if I want to use 12EDL, I shouldn't compose 12EDO songs to 12EDL -- instead I should compose new 12EDL tunes.
Then again, converting songs into new scales is a good exercise in order to demonstrate the power of the new scale. Once we see what 12EDO songs sound like in the new scale, we can figure out how to compose tunes in the new scale. So let me try to convert some of the songs I hear played in my classes today to the scale of the week, 12EDL.
In second period band, one girl is playing a French horn. The song is "Symphonic Suite from Jurassic World." This song is written in the key of F major, so we don't expect it to sound good in 12EDL, where the primary triad is minor. But there is a section ("Jurassic World Suite") that is written in F minor instead of F major. The section from measure 113 to measure 144 is playable in 12EDL.
By the way, I notice that while the F major parts are written in 4/4 time, the F minor parts have an unusual time signature, 3/8. I assume that the eighth notes in 3/8 time should be played half as long as the quarter notes in 4/4 time (for if the eighth notes in 3/8 were to be played as one beat of 4/4, they would have been written in 3/4 time, not 3/8).
Thus I'll use 2 as the length of the eighth note in the SOUND command. The entire measure is a dotted quarter note of length 6. Many of the notes are very long, consisting of dotted quarter notes tied across several measures. This explains why many of the notes will have lengths of more than 6. There are also several rests, which I indicate in the DATA lines with a 0 followed by the length of the empty FOR loop used to produce the rest (where 200 = eighth note rest).
20 FOR X=1 TO 38
30 READ D,T
40 IF D>0 THEN SOUND 261-N*D,T ELSE FOR I=1 TO T:NEXT I
50 NEXT X
60 DATA 0,200,6,2,8,2,6,18
70 DATA 0,200,6,2,8,2,10,4,11,14
80 DATA 0,200,6,2,8,2,7,4,6,4,6,2
90 DATA 8,2,6,10,6,2,8,24,0,2400
100 DATA 8,2,8,2,8,2,8,2,8,2,8,2,8,1
110 DATA 8,1,8,2,8,2,8,2,8,2,8,2,8,2
120 DATA 0,2200,10,6,10,6,9,6
Notice that even though an F natural minor scale contains the note Db, the note D appears in measure 122 (Degree 7). This actually fits the 12EDL scale. I decided to stop at measure 144 because 145 contains the note Db, which belongs in F natural minor.
At the top of the score is written "F HORN 2." Here the F doesn't stand for "French" (as in French horn) -- it means the note F. The horn is called a transposing instrument -- if we play what appears to be a C major scale in the score, it sounds like a F major scale. This is known as Concert F. Thus the F major and F minor scales in the horn score actually sound like Concert Bb major and minor.
The first line of the program tells Mocha which of the 15 possible EDL scales to play. Here I chose N=15, which makes the root note green Bb. I chose it because the girl is playing the horn in Concert Bb major/minor. If you prefer to play the tune in F minor as written in the score, then we change the first line to N=10, which makes the root green F.
In fact, let's look at the 12EDL in all three keys -- white A, green F, and green Bb:
The 12EDL scale in white A (N=1):
Degree Ratio Note
12 1/1 white A
11 12/11 amber B
10 6/5 green C
9 4/3 white D
8 3/2 white E
7 12/7 red F#
6 2/1 white A
The 12EDL scale in green F (N=10):
Degree Ratio Note
120 1/1 green F
110 12/11 amber-green G
100 6/5 deep green Ab
90 4/3 green Bb
80 3/2 green C
70 12/7 greenish D
60 2/1 green F
The 12EDL scale in green Bb (N=15):
Degree Ratio Note
180 1/1 green Bb
165 12/11 amber-green C
150 6/5 deep green Db
135 4/3 green Eb
120 3/2 green F
105 12/7 greenish G
90 2/1 green Bb
We see that to change the tonic to a green note like green F or green Bb, we must add "green" to the Kite color of each note. Thus "white" becomes "green," "green" becomes "deep green," and "red" becomes "greenish."
We also see that changing keys has nothing to do with the Circle of Fifths. In fact, perfect fifths (3/2) aren't quite as important as perfect twelfths/tritaves (3/1). Given any note, we can always lower the note by an octave or tritave since we can always multiply the Degree by 2 or 3 -- provided, of course, that we don't fall below Mocha range (Degree 260). On the other hand, odd numbers can't be halved (to whole numbers), and so odd degrees can't be raised an octave. Lowering by a perfect fifth entails lowering by a tritave and then raising an octave, so only even degrees can be lowered by a perfect fifth. Raising a perfect fifth entails raising a tritave and then lowering by an octave. Only degrees divisible by 3 can follow that first step.
One of the string classes opens by practicing the G major scale. But then they switch to another song. The first three notes are F, G, Ab, and so this sounds like another candidate to convert to 12EDL.
The song is called "007: Through the Years." I check out the score, and see that the notes F, G, Ab begin at measure 144 (section "Live and Let Die"). But then the next three notes are C-D-Eb, which doesn't fit since there is no 12EDL scale beginning at C (at least not
green C, that is).
This demonstrates the limits of the 12EDL, a very simple scale with only six notes. Then again, it's awkward to convert band music to Mocha anyway -- bands contain many different instruments, yet Mocha can't play harmony at all. I can't be sure, for example, that the horn girl is even playing the melody of her song. In the strings class, the violinists are clearly playing the melody. A bassist is playing repeated F notes during this section.
By the way, the rhythm of this song is especially tricky. In measure 65, there's actually a triplet of 32nd notes. Three of these make a sixteenth note, and so technically the three notes are 48th notes. We could declare each 48th note to be of length 1 in SOUND, but this is quickly followed by a whole note, which would have to be length 48. Here's what that part would look like using the PLAY command:
This is a dotted eighth note (L8.) followed by the 32nd note triplet (L48) consisting of the three notes Db, Eb, F (as Mocha uses a dash "-" for "flat). Then there's an eighth note, and finally the whole note F. String players definitely have to play some very fast notes!
The act of converting EDL scales to EDO and vice versa is tricky. We see that the simplest EDL scales, 2EDL to 10EDO, convert to 12EDO like clockwork. But 12EDL is tricky because it's the first one that requires the prime factor 11. And all EDL's past 12 will also require 11, so they'll all be slightly off in 12EDO. Is it possible to find other EDO's that can represent 12EDL better?
Well, let's have Mocha find the EDO scales for us:
10 INPUT "EDL"; L
30 A=A+1: E=0
40 FOR X=L/2+1 TO L-1
80 NEXT X
90 IF E<B THEN B=E:PRINT A;
100 GOTO 30
Here's what this program does -- we enter the EDL that we wish to approximate in an EDO scale, such as 12. The computer then searches all EDO's for the best approximation to the key intervals of that EDL (for example, for 12 we search 12/11, 12/10, 12/9, 12/8, 12/7) in each of the EDO's. An error term E is generated. If E beats the best error B so far, then the EDO number A is printed.
The computer will always start with 1, since 1EDO is by definition better than all smaller EDO's (vacuously true). Usually a few smaller EDO's are printed -- for example, 2 might appear if the 600-cent tritone is close enough to a ratio. Most of the time we want to find the next EDO better than 12, so we can ignore all EDO's until we get past 12. This is an infinite loop, so it will run forever unless we press the Esc (BREAK) key.
Here are the EDO's generated by the program for 4EDL through 12EDL:
4EDL: 1. 2, 5, 12, 41, 53, 306, ...
6EDL: 1, 2, 3, 4, 5, 7, 12, 19, 53, 171, ...
8EDL: 1, 2, 3, 5, 10, 31, 171, ...
10EDL: 1, 2, 3, 4, 6, 12, 19, 53, 72, 99, 171, ...
12EDL: 1, 3, 4, 5, 7, 19, 22, 24, 31, 41, 72, 270, ...
We recall that 4EDL contains only the perfect fourth in addition to the octave. And so the 4EDL list contains those EDO's that have a good approximation of the perfect fourth (or its inversion, the perfect fifth). Of course, 12EDO shows up in this list. The next two EDO's that approximate the fifth after 12EDO are 41EDO and 53EDO.
We notice that 12EDO appears on the 10EDL list. This is because 12EDO is excellent at representing the 10EDL scale. But 12EDO is missing from the 12EDL list. This is because 12EDO isn't good at representing the prime 11 needed for amber B.
Ignoring the small trivial EDO's, the first EDO in the 12EDL list is 19 -- and of course, 19EDO is the first one I mentioned in my #3 post from last summer. Actually, we don't think of 19EDO as representing 11 too well, but at least it's better than 12EDO. Other EDL's such as 22 and 31 are considered to represent the 11-limit better. And 24EDO is just 12EDO with quarter tones added between all the semitone steps. These quarter tones are clearly better at represent such in-between notes such as amber B.
Here is another video about 19EDO:
Here the 19EDO keyboardist, Jonathan Glasier, demonstrates many intervals, including both 7-limit (in the second video) and 11-limit (in the third video). But I post only the first video here.
This means that if you have a microtonal instrument/player that uses EDO's, and you wish to play along with some of the EDL music that I post on this website, then you can try using one of these EDO's. Here is a 12EDL scale starting on A in 19EDO:
The 12EDL scale in 19EDO A
Degree Ratio Note
12 1/1 A
11 12/11 Bb
10 6/5 C
9 4/3 D
8 3/2 E
7 12/7 Gb
6 2/1 A
We notice that 19EDO solves the amber B/Bb problem -- we definitely want Bb. And notice that the red F# becomes Gb -- which is not enharmonic to F# -- in 19EDO.
Last month, I tried to create a New 11-Limit Scale in Mocha. This attempt was a failure -- it might have contained many jade and amber triads, but music can't be composed in it.
In the end, the 12EDL scale is the only New 11-Limit Scale in Mocha that works. This scale doesn't contain a jade or amber triad at all -- instead it's a scale that makes sense both melodically and harmonically. The 12EDL scale is said to include the entire tonality diamond in the 11 odd-limit. This means that it contains all of the simple ratios of whole numbers where the largest odd number appearing is 11.
Thus we can find such intervals as 11/10 (between Bd and C), 9/8 (between C and D). It also contains more exotic intervals such as 14/9 and 16/11 -- these are the tonality diamond because they contain no odd number greater than 11 (even though even numbers greater than 11 appear). The interval 14/9 appears between F#> and D, and the interval 16/11 appears between E and Bd. Yes, neither of these fit inside the A-A octave, but they're still part of the tonality diamond.
The Xenharmonic website is disappearing in a few months. I wish to preserve some of the information here (since it's the source material for much of what I write). The website focuses on EDO's, not EDL's, but I'll post info about EDO's that are related to EDL's, as we just found out 19EDO is. Here's the link for 19EDO (Caution -- this will be a dead link after July):
In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the scale derived by dividing the octave into 19 equally large steps. Each step represents a frequency ratio of the 19th root of 2, or 63.16 cents. It is the 8th prime edo, following 17edo and coming before 23edo.
Interest in this tuning system goes back to the sixteenth century, when composer Guillaume Costeley used it in his chanson Seigneur Dieu ta pitié of 1558. Costeley understood and desired the circulating aspect of this tuning; in 1577 music theorist Francisco de Salinas in effect proposed it. Salinas discussed 1/3-comma meantone, in which the fifth is of size 694.786 cents; the fifth of 19-et is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which fails to close by less than a cent, so that his suggestion is effectively 19-et. In the nineteenth century mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as 50 equal temperament(summary of Woolhouse's essay).
The most salient characteristic of 19-et is that, having an almost just minor third and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for meantone temperament. It is also a suitable for magic/muggles temperament, because five of its major thirds are equivalent to one of its twelfths. For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is 31 equal temperament. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; 41 equal temperament more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic.
As a means of extending harmonyBecause 19 EDO allows for more blended, consonant harmonies than 12 EDO does, it can be a much better candidate for using alternate forms of harmony such as quartal, secundal, and poly chords. William Lynch suggests the use of seventh chords of various types to be the fundamental sonorities with a triad deemed as incomplete. Higher extensions involving the 7th harmonic as well as other non diatonic chord extensions which tend to clash in 12 EDO blend much better in 19 EDO.
Chord Names [Kite's colors, otonal/utonal explained here -- dw]
All 19edo chords can be named using conventional methods, expanded to include augmented and diminished 2nd, 3rds, 6ths and 7ths. Here are the blue, green, yellow and red triads:
|color of the 3rd||JI chord||notes as edosteps||notes of C chord||written name||spoken name|
|blue||6:7:9||0-4-11||C Ebb G||C(b3) or C(d3)||C flat-three or C dim-three|
|green||10:12:15||0-5-11||C Eb G||Cm||C minor|
|yellow||4:5:6||0-6-11||C E G||C||C major or C|
|red||14:18:27||0-7-11||C E# G||C(#3) or C(A3)||C sharp-three or C aug-three|
0-6-11-15 = C E G Bbb = C,bb7 or C,d7 = C double-flat-seven or C major dim-seven or C add dim-seven = 4:5:6:7
0-5-11-15 = C Eb G A# is Cm,#6 or Cm,A6 = C minor sharp-six or C minor aug-six = 1/(4:5:6:7) = 1/1 - 6/5 - 3/2 - 12/7
The last two chords illustrate how the 15\19 interval can be considered as either 7/4 or 12/7, and how 19edo tends to conflate otonal and utonal septimal ratios.
Articles [links that hopefully won't be dead after July -- dw]
- 19-Tone Theory and Applications by Hubert S. Howe Jr. Permalink
- Tunings for 19 Tone Equal Tempered Guitar by William A. Sethares Permalink
- Microtonalism by Bailey, Morrison, Pearson and Parncutt Permalink
- 19-tone equal-temperament and 1/3-comma meantone - Encyclopedia of Microtonal Music Theory Permalink
My next post will be on Monday. That post should definitely be music-free (unless I suddenly find myself subbing a music class again) -- instead I'll start the SBAC review. (But finding number bases, classroom management, and microtonal music in my top three posts tells me that this has definitely expanded beyond a Geometry blog!)