Preinfarction: April 2025

Scansion Templates

3: Girls Girls Girls | Hot Cross Buns | Three Blind Mice

3: The Space Pope | Free Fallin' | Chop Suey | Moon River

5: Autumn In New York | Blinded By The Light | Feliz Navidad | Puttin' On The Ritz | Smoke Gets In Your Eyes

5: Knights In White Satin

5: No, I Won't Back Down

5: Slip Slidin' Away

5: Take My Breath Away

5: The King Of The Road

6: And It's All For Me Grog

6: Caught In A Bad Romance | God Bless America | Hey Jude, Don't Make It Bad |

6: How I Wish You Were Here

6: Please Don't Fear The Reaper | Sitting On A Park Bench | Some Enchanted Evening | Smile Like You Mean It

6: There's Whiskey In The Jar | We're Movin' Right Along

7: Someone To Watch Over Me | Stop Draggin' My Heart Around | It's Only A Paper Moon

7: Rudolph The Red Nosed Reindeer | Somewhere Over The Rainbow | I'm Leaving On A Jet Plane | A La Claire Fontaine

8: All I Want for Christmas Is You | All We Are Is Dust In The Wind | But I Only Have Eyes For You

8: I'm Dreaming Of A White Christmas | Oh What A Beautiful Morning | O Lord, It's Hard To Be Humble

8: June Is Bustin' Out All Over | In The Hills Of Connemara

8: Lots Of Fun At Finnegan's Wake

8: Oh, Is There Not One Maiden Breast?

8: Teenage Mutant Ninja Turtles | Do You Want To Build A Snowman? | How Are Things in Glocca Morra? | Joyful Joyful We Adore Thee | Oh The Shark Has Pearly Teeth Dear

8: Well I Suppose, Anything Goes | I'm Picking Up Good Vibrations

8: The Last Of Barrett's Privateers

8: The Night They Drove Old Dixie Down

8: Till All The Things You Are Are Mine

9: Ba Ba Black Sheep, Have You Any Wool

9: Beautiful Dreamer, Wake Unto Me

9: Blackbird Singing In The Dead Of Night

9: Give Me Some Afro-Cuban Bebop | I'll Never Be Your Beast Of Burden

9: Hush Little Baby, Don't Say A Word

9: I Am A Poor Wayfaring Stranger

9: I Bought Me A Cat, My Cat Pleased Me

9: I Know A Heartache When I See One

9: Just My Rifle, My Pony, And Me

9: Sing Us A Song You're The Piano Man

9: The First Time Ever I Saw Your Face

9: When I Take You Out In The Surrey

9: Whisper Words Of Wisdom, Let It Be

10: And Make A Northwest Passage To The Sea

10: And She's Buying A Stairway To Heaven

10: Buffalo Girls Won't You Come Out Tonight | It's The Most Wonderful Time Of The Year | But I Can't Help Falling In Love With You | Metal Movie Marijuana Meltdown

10: Even If We're Just Dancing In The Dark

10: Gather Up The Pots And The Old Tin Cans

10: I Don't Need To Walk Around In Circles

10: I Eat My Candy With The Pork And Beans

10: I Get Along Without You Very Well

10: I Have Emotional Motion Sickness

10: Night Time On The City Of New Orleans

10: People Say I've Got A Drinkin' Problem

10: Raindrops On Roses, Whiskers On Kittens

10: Summertime And The Living Is Easy

10: The Water Is Wide, I Can Not Cross O'er.

10: There Ain't No Way To Hide Your Lyin Eyes

10: We Will All Go Together When We Go

11: All The Other Kids With The Pumped Up Kicks

11: But Don't Look Back In Anger I Heard Her Say

11: Drop Kick Me, Jesus, Through The Goal Posts Of Life

11: I Gave My Love A Cherry That Had No Stone

11: I Want To Know Have You Ever Seen The Rain

11: If We Make It Through December, We'll Be Fine

11: Now You're Just Somebody That I Used To Know

11: Scarecrow & Fungus They Ran Through A Stoplight

11: The Lord Knows I'm Drinking And Running Around

11: Wearing Out Your Chompers Eatin' Goober Peas

11: You Can Hear The Whistle Blow A Hundred Miles

12: I Been Through The Desert On A Horse With No Name

12: Plenty Of Room At The Hotel California

12: Singing Me And Julio Down By The Schoolyard

12: That's Me In The Spot Light Losing My Religion

12: Yesterday All My Troubles Seemed So Far Away

13: Fly Me To The Moon And Let Me Play Amoong The Stars

13: Hey There, Delilah, What's It Like In New York City?

13: If You Don't Give Me Rye Whiskey I Surely Will Die

14: I Was Dancing With My Darling To The Tenessee Waltz

...

Georgian Folk Music

The country of Georgia has some great weird folk music. It often has pretty complex vocal polyphony. It's microtonal (and was even more microtonal before recent western influence). Some people will tell you it's based on a nearly equal heptatonic scale, i.e. 7-EDO. It doesn't sound like 7-EDO to me, but I haven't heard very much. Maybe I'm listening to the wrong sources.

So what is it? How is it tuned? I don't know. Let's figure it out.

Based on some spectral measurements in "Erkomaishvili Dataset: A Curated Corpus of Traditional Georgian Vocal Music for Computational Musicology" by (Rosenzweig, Scherbaum, Shugliashvili, Arifi-Müller, and Müller, 2020), I'd say a reasonable first start at describing the music would be a scale like this:

[P1, M2, AsGrm3, P4, P5, M6, AsGrm7, P8] # [1/1, 10/9, 11/9, 4/3, 3/2, 5/3, 11/6, 2/1]

In relative degrees this is

[10/9, 11/10, 12/11] * 9/8 * [10/9, 11/10, 12/11]

with the repeated tetrachord sized like

[182c, 165c, 151c]

The scale, from the spectral analysis, clearly has a lower major second and major sixth than Pythagorean, and I've simply used the just tunings here. The scale also has neutral 3rds and 7ths. That's my first stab at it. Let's see what other have to say.

"The Georgian Musical System" (Malkhaz Erkvanidze, 2016) describes a Georgian scale that also has a repeated tetrachord structure. My introduction to Georgian folk music was a video of Malkhaz Erkvanidze singing in and directing a trio (https://www.youtube.com/watch?v=FVxD6NB8-CI) so I'm quite curious what he has to say about the genre.

The tetrachord he presents looks like this in cents:

[172c, 154c, 172c]

Let's call this G for the Georgian tetrachord. This is a really weird tetrachord, especially since Georgians use harmonic instruments. If they had inharmonic bells and gongs it wouldn't be so crazy, but they sing and play strings and flutes and other harmonic instruments.

Let's try to figure out a just explanation for the tetrachord. The Georgian tetrachord looks like 7-EDO in-so-much as all of its steps are multiple of 171 cents, which the outer two intervals definitely are, but the middle step is not, and also the Erkvanidze source claims that in some scales the tetrachord appears twice alongside the usual Pythagorean acute major second at 9/8, so that we get an octave scale. How about these for just explanations:

[21/19, 12/11, 209/189] _ [173c, 151c, 174c]

[32/29, 35/32, 116/105] _ [170c, 155c, 172c]

I haven't worked much with 19-limit just intonation or beyond. I never thought there was much use for it. I hope Georgian music isn't 19-limit (or 29-limit). That would be kind of crazy. Or maybe it would be fun. I consider both of these tetrachords above to be perceptually indistinguishable from the Georgian tetrachord of Erkvanidze.

What's the just tuning of the full scale that has [G + AcM2 + G] as its tetrachord structure? If we use the 19-limit intonation of the G tetrachord, we get this scale:

[1/1, 21/19, 252/209, 4/3, 3/2, 63/38, 378/209, 2/1]

If we use the 29-limit intonation, we get this scale:

[1/1, 32/29, 35/29, 4/3, 3/2, 48/29, 105/58, 2/1]

I think the first one is ugly for its high complexity 3rd interval ratio and the second one is ugly for its high prime limit, but what can you do?

The paper goes on to describe some other scales that can be made with the Georgian tetrachord. We could have a scale that doesn't repeat at the octave like

[G + G + AcM2 + G + G]

This spans

P4 + P4 + AcM2 + P4 + P4 = P8 + Grm7 = Grm14.

I almost wonder if I'm misunderstanding the paper's notation, such that the notated numbers aren't cents of frequency ratios, since the author simply uses [Bb, Eb] for the key signature in his staff notation, but that seems unlikely since the G tetrachord spans

[172c + 154c + 172c] = 498c

just like a justly tuned P4.

...

Here's an idea! The ratio of 172c to 154c is very close to 8/7. If we want an EDO to represent both of these frequency ratios, we can do fairly well if it has a step around

172c/8 = 21.5 cents

154c/7 = 22 cents.

This means 55-EDO or 56-EDO. I think 56-EDO makes more sense. I don't think Georgian folk music is actually based on 56-EDO, but 56-EDO does a good job of representing Erkvanidze's tetrachord.

...

I can't help but wonder if the Erkvanidze came up with the Georgian tetrachord by saying, "our music sounds like 7-edo, but I want to describe it with a tetrachord structure; so which relative step of our scale can I mistune so that most of the frequency ratios are 1200/7 cents and one of them is 1200 * log_2(9/8) cents?". Maybe not. But it sure would parsimoniously explain the origin of a tetrachord that makes no sense unless you go to 19-limit.

On the other hand, I've heard that [C, F, G] is a very prototypical chord in Georgean music, i.e. Csus4 and Fsus2, and that kind of scans if they don't have particularly consonant 2nd, 3rd, 6th, or 7th intervals but they do have a nice P4 and P5.

...

Looking at figure 12 in "Analysis of the Tbilisi State Conservatory Recordings of Artem Erkomaishvili in 1966" by Scherbaum et al 2017, I would have guessed that the Georgian scale had a neutral sixth and a major 7th

[P1, M2, n3, P4, P5, n6, M7, P8]

instead of a major 6th and neutral seventh that I saw in "Erkomaishvili Dataset: A Curated Corpus of Traditional Georgian Vocal Music for Computational Musicology". Also in this source the Georgian intonation of P4 is shown to be a little sharp, like 18 cents over just. We can resolve the weirdness about which of 6th and 7th degree is neutral pretty easily: Erkvanidze tells us that Georgian music uses both disjunct and conjuct scales with the tetrachord repeated, i.e.

[G + T + G] or [G + G + T]

So the previous scale could be analyzed as

[10/9, 11/10, 12/11] * [10/9, 11/10, 12/11] * 9/8

and everything works out. Except for the slightly sharpened fourth, but I don't really care about that.

I think my superparticular tetrachord, [10/9, 11/10, 12/11], thus explains spectral data from multiple ethnographic papers, and it looks nicer than the 19-limit Georgian chord, and it's not a tetrachord I've seen in Arabic or Turkish or Persian music. It has a [Major, neutral, neutral] sound like a Arabic or medieval Ottoman rast, but those have an intonation more like

[9/8, 11/10, 320/297] _ [204c, 165c, 129c]

so my version of the Georgian tetrachord

[10/9, 11/10, 12/11] _ [182c, 165c, 151c]

has more equal frequency ratio sizes. Not quite 7-EDO equal, but audibly much closer to equal than Rast.

Now, I'm not positive that this is closer to the intonation of Georgian folk music than Erkvanidze's tetrachord. But it's beautiful and I hope it is. We've got three consecutive super particular ratios from the harmonic series spanning a fourth. In the disjunct tetrachord, with 9/8 below the upper tetrachord, we've got four consecutive super particulars. Makam Saba has a similar thing going on

[11/10, 12/11, 13/12, 15/13]

but not including 10/9, and one of Ben Johnston's scales has a similar thing going on, but in reverse

[12/11, 11/10, 10/9, 9/8, 16/15, 15/14, 14/13, 13/12]

But the harmonic series isn't in reverse, so why not do it forwards? I think it's a very beautiful tetrachord.

The next thing for me to do is to listen to some hundred year old recordings of Georgian folk music and compare the sound to both my Georgian tetrachord and Erkvanidze's Georgian tetrachord.

...

Musical Wavelength Space

Suppose you've got a fretless string instrument and you want to place markers on the neck to help you find frequency ratios over open string frequencies.

If you place a marker far from the sound hole so that 1/9 of the string is close to the tuning pegs and 8/9 of the string is free to vibrate when you pluck or bow or strum near the sound hole, the excited tone will have a frequency 9/8 over the frequency of the open string. If we put a marker there, I'll say that the marker is at 1/9 of the string length, meaning that 8/9 of the string is left free to vibrate.

The string length 8/9 and the frequency ratio 9/8 are reciprocal fractions, and this isn't a coincidence. Delightfully, the function that transforms 1/9 into 9/8 is also an involution, i.e. it can transform in both ways.

y = x / (x - 1)

frequency_ratio = (string_division) / (string_division - 1)

string_division = (frequency_ratio) / (frequency_ratio - 1)

If we divide the string into simple units, what simple frequency ratios do we get? Here's a little table

25.0 -> 25/24 _ 71c

24.5 -> 49/47 _ 72c

24.0 -> 24/23 _ 74c

23.5 -> 47/45 _ 75c

23.0 -> 23/22 _ 77c

22.5 -> 45/43 _ 79c

22.0 -> 22/21 _ 81c

21.5 -> 43/41 _ 82c

21.0 -> 21/20 _ 84c

20.5 -> 41/39 _ 87c

20.0 -> 20/19 _ 89c

19.8 -> 99/94 _ 90c

19.6 -> 98/93 _ 91c

19.5 -> 39/37 _ 91c

19.4 -> 97/92 _ 92c

19.2 -> 96/91 _ 93c

19.0 -> 19/18 _ 94c

18.8 -> 94/89 _ 95c

18.6 -> 93/88 _ 96c

18.5 -> 37/35 _ 96c

18.4 -> 92/87 _ 97c

18.2 -> 91/86 _ 98c

18.0 -> 18/17 _ 99c

17.8 -> 89/84 _ 100c

17.6 -> 88/83 _ 101c

17.5 -> 35/33 _ 102c

17.4 -> 87/82 _ 102c

17.2 -> 86/81 _ 104c

17.0 -> 17/16 _ 105c

16.8 -> 84/79 _ 106c

16.6 -> 83/78 _ 108c

16.5 -> 33/31 _ 108c

16.4 -> 82/77 _ 109c

16.2 -> 81/76 _ 110c

16.0 -> 16/15 _ 112c

15.8 -> 79/74 _ 113c

15.6 -> 78/73 _ 115c

15.5 -> 31/29 _ 115c

15.4 -> 77/72 _ 116c

15.2 -> 76/71 _ 118c

15.0 -> 15/14 _ 119c

14.8 -> 74/69 _ 121c

14.6 -> 73/68 _ 123c

14.5 -> 29/27 _ 124c

14.4 -> 72/67 _ 125c

14.2 -> 71/66 _ 126c

14.0 -> 14/13 _ 128c

13.8 -> 69/64 _ 130c

13.6 -> 68/63 _ 132c

13.5 -> 27/25 _ 133c

13.4 -> 67/62 _ 134c

13.2 -> 66/61 _ 136c

13.0 -> 13/12 _ 139c

12.8 -> 64/59 _ 141c

12.6 -> 63/58 _ 143c

12.5 -> 25/23 _ 144c

12.4 -> 62/57 _ 146c

12.2 -> 61/56 _ 148c

12.0 -> 12/11 _ 151c

11.8 -> 59/54 _ 153c

11.6 -> 58/53 _ 156c

11.5 -> 23/21 _ 157c

11.4 -> 57/52 _ 159c

11.2 -> 56/51 _ 162c

11.0 -> 11/10 _ 165c

10.8 -> 54/49 _ 168c

10.6 -> 53/48 _ 172c

10.5 -> 21/19 _ 173c

10.4 -> 52/47 _ 175c

10.2 -> 51/46 _ 179c

10.0 -> 10/9 _ 182c

9.9 -> 99/89 _ 184c

9.8 -> 49/44 _ 186c

9.7 -> 97/87 _ 188c

9.6 -> 48/43 _ 190c

9.5 -> 19/17 _ 193c

9.4 -> 47/42 _ 195c

9.3 -> 93/83 _ 197c

9.2 -> 46/41 _ 199c

9.1 -> 91/81 _ 202c

9.0 -> 9/8 _ 204c

8.9 -> 89/79 _ 206c

8.8 -> 44/39 _ 209c

8.7 -> 87/77 _ 211c

8.6 -> 43/38 _ 214c

8.5 -> 17/15 _ 217c

8.4 -> 42/37 _ 219c

8.3 -> 83/73 _ 222c

8.2 -> 41/36 _ 225c

8.1 -> 81/71 _ 228c

8.0 -> 8/7 _ 231c

7.9 -> 79/69 _ 234c

7.8 -> 39/34 _ 238c

7.7 -> 77/67 _ 241c

7.6 -> 38/33 _ 244c

7.5 -> 15/13 _ 248c

7.4 -> 37/32 _ 251c

7.3 -> 73/63 _ 255c

7.2 -> 36/31 _ 259c

7.1 -> 71/61 _ 263c

7.0 -> 7/6 _ 267c

6.9 -> 69/59 _ 271c

6.8 -> 34/29 _ 275c

6.7 -> 67/57 _ 280c

6.6 -> 33/28 _ 284c

6.5 -> 13/11 _ 289c

6.4 -> 32/27 _ 294c

6.3 -> 63/53 _ 299c

6.2 -> 31/26 _ 305c

6.1 -> 61/51 _ 310c

6.0 -> 6/5 _ 316c

5.9 -> 59/49 _ 322c

5.8 -> 29/24 _ 328c

5.7 -> 57/47 _ 334c

5.6 -> 28/23 _ 341c

5.5 -> 11/9 _ 347c

5.4 -> 27/22 _ 355c

5.3 -> 53/43 _ 362c

5.2 -> 26/21 _ 370c

5.1 -> 51/41 _ 378c

5.0 -> 5/4 _ 386c

4.9 -> 49/39 _ 395c

4.8 -> 24/19 _ 404c

4.7 -> 47/37 _ 414c

4.6 -> 23/18 _ 424c

4.5 -> 9/7 _ 435c

4.4 -> 22/17 _ 446c

4.3 -> 43/33 _ 458c

4.2 -> 21/16 _ 471c

4.1 -> 41/31 _ 484c

4.0 -> 4/3 _ 498c

3.9 -> 39/29 _ 513c

3.8 -> 19/14 _ 529c

3.7 -> 37/27 _ 545c

3.6 -> 18/13 _ 563c

3.5 -> 7/5 _ 583c

3.4 -> 17/12 _ 603c

3.3 -> 33/23 _ 625c

3.2 -> 16/11 _ 649c

3.1 -> 31/21 _ 674c

3.0 -> 3/2 _ 702c

2.9 -> 29/19 _ 732c

2.8 -> 14/9 _ 765c

2.7 -> 27/17 _ 801c

2.6 -> 13/8 _ 841c

2.5 -> 5/3 _ 884c

2.4 -> 12/7 _ 933c

2.3 -> 23/13 _ 988c

2.2 -> 11/6 _ 1049c

2.1 -> 21/11 _ 1119c

2.0 -> 2/1 _ 1200c

Now, there's no real reason to use decimal as string divisors. You could just as easily place a marker at 3.1 of a the string length as you could at 22/7 of the string length, but I wanted to see how this looked. I've also hidden any frequency ratios with numerators more than 99.

There are some conspicuous absences on this list. Like the justly tuned minor sixth, 8/5, or the justly tuned major seventh, 15/18, and the justly tuned minor seventh, 9/5. Our involution function certainly has less precision near the octave, so we might try using another decimal digit at the high end:

3.0 -> 3/2 _ 702c

2.96 -> 74/49 _ 714c

2.95 -> 59/39 _ 717c

2.92 -> 73/48 _ 726c

2.9 -> 29/19 _ 732c

2.88 -> 72/47 _ 738c

2.85 -> 57/37 _ 748c

2.84 -> 71/46 _ 751c

2.8 -> 14/9 _ 765c

2.76 -> 69/44 _ 779c

2.75 -> 11/7 _ 782c

2.72 -> 68/43 _ 793c

2.7 -> 27/17 _ 801c

2.68 -> 67/42 _ 809c

2.65 -> 53/33 _ 820c

2.64 -> 66/41 _ 824c

2.6 -> 13/8 _ 841c

2.56 -> 64/39 _ 858c

2.55 -> 51/31 _ 862c

2.52 -> 63/38 _ 875c

2.5 -> 5/3 _ 884c

2.48 -> 62/37 _ 894c

2.45 -> 49/29 _ 908c

2.44 -> 61/36 _ 913c

2.4 -> 12/7 _ 933c

2.36 -> 59/34 _ 954c

2.35 -> 47/27 _ 960c

2.32 -> 58/33 _ 976c

2.3 -> 23/13 _ 988c

2.28 -> 57/32 _ 999c

2.25 -> 9/5 _ 1018c

2.24 -> 56/31 _ 1024c

2.2 -> 11/6 _ 1049c

2.16 -> 54/29 _ 1076c

2.15 -> 43/23 _ 1083c

2.12 -> 53/28 _ 1105c

2.1 -> 21/11 _ 1119c

2.08 -> 52/27 _ 1135c

2.05 -> 41/21 _ 1158c

2.04 -> 51/26 _ 1166c

2.0 -> 2 _ 1200c

This got us our minor seventh, but not the other two chromatic 5-limit frequency ratios. It turns out those don't have finite decimal representations: A string divisor of 15/7 gives us a frequency ratio of 15/18, and a string divisor of 8/3 gives us a frequency ratio of 8/5.

I kind of like this? We've found a procedure which privileges a different set of ratios and intervals compared to normal just intonation.

It's not much of a difference: the 2.68 divisor gives us a frequency ratio of 67/42, which is a perceptually indistinguishable 5 cents flat of 8/5. The sounds are still there. But I still think it's a neat sound space. Maybe don't use the high precision divisors if you want a really distinct sound?

...

Labels

Scansion Templates

Georgian Folk Music

Musical Wavelength Space