**Week 4**

Electronic Music that tries, and fails

15-year old invents ultrasound mosquito larvacide device

The grey album link posted this week on the list is better than the one I had a week or two ago.

It's much better after a few listenings. The first time I thought, "I wish that guy would shut up and let the Beatles play."

But now I can hear beyond the original songs and appreciate the new arrangements, and like them all (though least of all 'justify my thug', there's some actual singing on the vocal track that doesn't match the dubbed-in music)

The with this kind of mix is that the melody supplies the rhythm and the bass line provides the narrative, just the reverse of most older pop music, certainly the White Album.

On the other hand, I can't listen to this for than about an hour without starting to feel irritated.

**#1: Inverse Primes**

- Take the set of prime numbers, including 1 (1,2,3,5,7,9,11...)
- Divide them into 1 to get a set of values between 0 and 1 (0.5, 0.333, 0.2, 0.142857, 0.09...)
- Multiply these values by some frequency (A-440) to get a scale
- Add these numbers to the same frequency (A-440) to get the octave scale above it.

The values are: 880, 660, 587, 528, 503, 480... 440

As the primes increase, the inversions approach 0, so the scale frequencies get closer and closer together.

It's quite euphonious, in a minor kind of way

**#2: Inverse Primes II**

- Take the set of prime numbers, including 1 (1,2,3,5,7,9,11...)
- Divide them into 1 to get a set of values between 0 and 1 (1, 0.5, 0.333, 0.2, 0.142857, 0.09...)
- Multiply these values by some frequency (A-440) to get a scale
- Subtract these numbers from double the same frequency (A-880) to get the octave scale above it.

The values are: 440, 660, 733, 792, 817, 840... 880

As the primes increase, the inversions approach 0, so the scale frequencies get closer and closer together.

In both examples, I pause at the point where the notes get so close that they sound dissonant, in both cases at 1/11

It's a little euphonious, in a minor kind of way

**#3: Inverse Primes III**

Taking just the primes 2, 3, and 5 and combining the two scales above:

n | 1/n | A-440 + A-440/n | 2*A-440 - A-440/n |
---|---|---|---|

1/1 | 1 | 880 | 440 |

1/2 | 0.5 | 660 | 660 |

1/3 | 0.333 | 587 | 733 |

1/5 | 0.2 | 528 | 792 |

Not too bad. The equal-tempered equivalents are:

A (440,440)

C (528,523)

D (587,587)

E (660,659)

F# (733,740)

G (792,784)

A (880,880)

This scale has a sharper minor third, slightly flat fifth, flatter 6th, and sharper 7th.

**#4: birdSong**

The frequencies taken from a sample of a lark sparrow's song.

Snot Wong has some interesting patches for Max/MSP.

There are at least 50 sites that I saw with patches to download, you can google them yourself, but this one is worth listing as well.
He has practically a dissertation written as MSP help files, so you can play along.

This page has more spectral analysis software as well as analyses of whales, human voice, etc.

When I was in college, my job was digital sound engineer, recording and editing language instruction tapes using this brand new tool called ProTools. I used to hang out with the linguistics post-docs and they showed me their work. If I remember correctly, the human voice has three dominant frequencies, with many more less-pronounced ones. So to imitate the human voice, I guess we'd need chords with three pithces. Maybe that why three-note chords sound good to us.

Here's a spectrograph of my voice. Even while trying to speak clearly and evenly, there are lots of harmonics.

Here are the frequencies I found in one bit where I said 'ah' for a few seconds, trying to match C2 on my keyboard: (just looking at frequencies with over 30dB of representation)

144 | D |

268 | C/C# |

402 | G/G# |

526 | C |

670 | E/F |

794 | G/G# |

918 | A/A# |

1052 | C |

1186 | D |

1309 | E |

3010 | F#/G |

3546 | A |

Not at all what I was expecting. The 'C's seem on target, but all the other notes are a surprise. It looks like some kind of 9th chord.

Here's the actual sound

And here it is created by Max (warning: sounds like crap)

So if I ever meet my robot-clone-doppleganger, I'll be able to recognize him by his voice, which clearly, sounds terrible.

Obviously there's a lot more subtlety to the human voice than just raw frequencies.

"When you ascend by a cycle of justly tuned perfect fifths (ratio 3:2), leapfrogging 12 times, you eventually reach a note around seven octaves above the note you started on, which, when lowered to the same octave as your starting point, is 23.46 cents higher than the initial note. This interval, 531441:524288 or approximately 1.0136:1, is called a Pythagorean comma.

This interval has serious implications for the various tuning schemes of the chromatic scale, because in western music, 12 perfect fifths and seven octaves are treated as the same interval. Equal temperament, today the most common tuning system used in the west, gets around this problem by flattening each fifth by a twelfth of a pythagorean comma (2 cents), thus giving perfect octaves."

51^{st} root of 2 | = 1.0136839 |

Pythagorean comma | = 1.0136432647705078125 |

52^{nd} root of 2 | = 1.01341899 |

What does this mean?

It means you smell.

The amazing thing about Pythyagorus and the other ancients is that they did their **long division using fucking Roman Numerals!**

There are a few fundamental irrational numbers:
pi = 3.14159265

e = 2.71828183

the golden ratio = 1.61803399

Early confusion over the exact value of pi was because people assumed it had to result from a basic ratio, such as 22/7.

I have a theory, that there is a basic irrational number that determines fifths in music that is not based on a fraction such as 3/2 and results in a euphonious scale that includes no comma.

It can't be pi, e, or the golden mean, because they all sound terrible (I tried them all and pi sounds best, but still bad)

In order to eliminate the comma, the ratio has to be around 1.49.

Now, I tried 3/2 (1.359140915) which sounds okay, actually, and pi/2 (1.5707963267948966192313216916398) isn't as bad as it could be.

Cent

"The cent is a logarithmic measure of relative pitch or intervals. 1200 cents are equal to one octave, and an equally tempered semitone is equal to 100 cents. The formula to determine the value in cents between two notes with frequencies a and b.

The measure was developed by A. J. Ellis around the 1870s, and was published in his edition of Hermann von Helmholtz's On the Sensations of Tone. It has since become the standard way of measuring intervals in equal temperament systems or for comparison with equal temperament systems."

So the comma (logarithmic) is equal to about a quarter of a half-note (linear)

This JavaScript (I think) shows the comma for A-440

Going up by fifths, you wind up at 57088.388671875, but going up by octaves you get 56320.

If you go down by octaves from the first value, you get 446.0030364990234375.

Since A# = 466.1637596Hz, and the difference between A and A# = 26.1637596Hz, a comma here of 6 cycles per second is significant.

So, my proposed new irrational number is:

This page shows how the intervals stay the same all the way up the chain. It's so close to 1.5 that you don't hear the difference between say, 100Hz and 149.8Hz

And now. Oh shit. I look at the chart with the 12-tone equal temperament and see my number already there, as the 'fixed' ratio for a perfect 5th.

Oh well. I guess that's what they were talking about.

Let's a try a kind of fibonacci sequence, using the ratios between numbers for our scale:

1 | 1/0 | - |

1 | 1/1 | 1 |

2 | 2/1 | 2 |

3 | 3/2 | 1.5 |

5 | 5/3 | 1.67 |

8 | 8/5 | 1.6 |

13 | 13/8 | 1.625 |

21 | 21/13 | 1.615 |

34 | 34/21 | 1.619 |

55 | 55/34 | 1.617 |

89 | 89/55 | 1.618 |

144 | 144/89 | 1.618 |

233 | 233/144 | 1.618 |

It looks good. It even includes the basic pleasing intervals of 3/2 (fifth), 5/3 (sixth), and 8/5 (minor 6th)

The assymptotes to the golden ratio, so I guess this scale is just a single octave with infinitely many notes in teh area where sixths normally are.

Unfortunately, there is no equivalents of 2nds, 3rds, 4ths, or 7ths. Unless...

Lets' add a column to allow division by **2** numbers behind. Of course, then we end up with intervals such as 3/1, which is outside the range of an octave, so let's also divide everything by 2, to keep it in scope:

1 | 1/0 | - | 1/0 | - | 1/0 | - |

1 | 1/1 | 1 | 1/0 | - | 1/0 | - |

2 | 2/1 | 2 | 2/1 | 2 | 2/2 | 1 |

3 | 3/2 | 1.5 | 3/1 | 3 | 3/2 | 1.5 |

5 | 5/3 | 1.67 | 5/2 | 2.5 | 5/4 | 1.25 |

8 | 8/5 | 1.6 | 8/3 | 2.67 | 8/6 | 1.33 |

13 | 13/8 | 1.625 | 13/5 | 2.6 | 13/10 | 1.3 |

21 | 21/13 | 1.615 | 21/8 | 2.625 | 21/16 | 1.3125 |

34 | 34/21 | 1.619 | 34/13 | 2.615 | 34/26 | 1.307 |

55 | 55/34 | 1.617 | 55/21 | 2.619 | 55/42 | 1.309 |

89 | 89/55 | 1.618 | 89/34 | 2.617 | 89/68 | 1.308 |

144 | 144/89 | 1.618 | 144/55 | 2.618 | 144/110 | 1.309 |

233 | 233/144 | 1.618 | 233/89 | 2.618 | 233/178 | 1.309 |

The first new column levels off at the golden mean + 1. I didn't know that would happen. And of course the second levels off to half that.

We could keep messing around with this forever, I suppose, but let's now order our scale:

1/1 | 1 |

5/4 | 1.25 |

13/10 | 1.3 |

... | (gold+1)/2 |

8/6 (4/3) | 1.33 |

3/2 | 1.5 |

8/5 | 1.6 |

... | gold |

13/8 | 1.625 |

5/3 | 1.67 |

2/1 | 2 |

So the lowest note above the root is a major third, then some meanderings around half of one plus the golden ratio (which would be like a diminished 4th, I guess), then a 4th, then a 5th, then a collection around the golden mean, which is around a major 6th. So, no equivalents of 2nds or 7ths.

Let's add another column:

number | ratio | decimal | ratio2 | decimal | 'repaired' ratio2 | decimal | ratio3 | decimal |

1 | 1/0 | - | 1/0 | - | 1/0 | - | 1/0 | - |

1 | 1/1 | 1 | 1/0 | - | 1/0 | - | 1/0 | - |

2 | 2/1 | 2 | 2/1 | 2 | 2/2 | 1 | 2/0 | - |

3 | 3/2 | 1.5 | 3/1 | 3 | 3/2 | 1.5 | 3/1 | 3 |

5 | 5/3 | 1.67 | 5/2 | 2.5 | 5/4 | 1.25 | 5/1 | 5 |

8 | 8/5 | 1.6 | 8/3 | 2.67 | 8/6 | 1.33 | 8/2 | 4 |

13 | 13/8 | 1.625 | 13/5 | 2.6 | 13/10 | 1.3 | 13/3 | 4.33 |

21 | 21/13 | 1.615 | 21/8 | 2.625 | 21/16 | 1.3125 | 21/5 | 4.2 |

34 | 34/21 | 1.619 | 34/13 | 2.615 | 34/26 | 1.307 | 34/8 | 4.25 |

55 | 55/34 | 1.617 | 55/21 | 2.619 | 55/42 | 1.309 | 55/13 | 4.23 |

89 | 89/55 | 1.618 | 89/34 | 2.617 | 89/68 | 1.308 | 89/21 | 4.23 |

144 | 144/89 | 1.618 | 144/55 | 2.618 | 144/110 | 1.309 | 144/34 | 4.23 |

233 | 233/144 | 1.618 | 233/89 | 2.618 | 233/178 | 1.309 | 233/55 | 4.23 |

It looks like we can't go on infintely after all. Even if we fix the column by finding the ratios an octave down, we get mostly the same values

I'll drop the values that hover around the assymptotes, and we get: (bold indicates the 'pure' tones that we got on the first pass, the other ones can be optional)

1/1 | 1 |

5/4 | 1.25 |

(gold+1)/2 | 1.309 |

3/2 | 1.5 |

gold | 1.618 |

5/3 | 1.67 |

2/1 | 2 |

There are no numbers in the ratios larger than 5. That seems clean and pure.

And again with sample frequencies and equal-termpered equivalents:

1/1 | 1 | 440 | A |

5/4 | 1.25 | 550 | C# |

(gold+1)/2 | 1.309 | 576 | C#/D |

3/2 | 1.5 | 660 | E |

gold | 1.618 | 712 | F/F# |

5/3 | 1.67 | 735 | F# |

2/1 | 2 | 880 | A |

The ugliest notes are the gold and (gold+1)/2. With the others we can make a decent A major chord.

The next trick will be to do it again using the E-660 as the root:

1/1 | 1 | 660 | E |

5/4 | 1.25 | 825 | G# |

(gold+1)/2 | 1.309 | 864 | G#/A |

3/2 | 1.5 | 990 | B |

gold | 1.618 | 1,068 | C |

5/3 | 1.67 | 1,102 | C# |

2/1 | 2 | 1,320 | E |

Combining these tables and halving some of the values, we get the full monty:

440 | A |

495 | B |

534 | C |

550 | C# |

576 | C#/D |

660 | E |

712 | F/F# |

735 | F# |

825 | G# |

880 | A |

And just for overkill:

1/1 | 1 | 990 | B |

5/4 | 1.25 | 1237.5 | D# |

(gold+1)/2 | 1.309 | 1295.91 | D#/E |

3/2 | 1.5 | 1485 | F# |

gold | 1.618 | 1601.82 | G/G# |

5/3 | 1.67 | 1653.3 | G# |

2/1 | 2 | 1980 | B |

To produce: (cleaning out redundancies and adjacents) The standard frequencies are in the right column

440 | A | 440 |

495 | B | 493.88 |

534 | C | 523.25 |

550 | C# | 554.37 |

576 | D | 587.33 |

619 | D# | 622.25 |

660 | E | 659.26 |

712 | F | 698.46 |

735 | F# | 739.99 |

801 | G | 783.99 |

825 | G# | 830.61 |

880 | A | 880 |

I'm still missing an A#.

Maybe this shouldn't count. All I've done really is replicate a version of a just tuning that surely existed hundreds of years ago. But play it in mixolydian or phrygian, and then you've got something!

Cornell's Ornithology Lab (birds.cornell.edu) is probably the best of its kind in the world (and a beautiful facility nestled in Sapsucker Woods in Ithaca, NY if you're ever upstate). They have a free gallery/museum/exloratorium with lots of exhibits in which you can, for example, manipulate the wave forms of different bird songs. They've made the software available on their site, under the Bioacoustics Research Program

I downloaded Raven which has a 10-minute timeout on the license, and doesn't allow saving. They also have an app called Canary on the same page which is mac only, so I didn't try it.

Raven come with some good audio files.

For example, here is a Lark Sparrow

And here is a Canyon Wren

And here is a Bowhead Whale

Sped up 2x, 4x, 8x, 16x

The 4x and 8x sounds like bird calls, and the 16x sounds more like an insect.

For comparison, there is also a Spotted Hyena, and perhaps the most interesting, a Bearded Seal

Many animals seem to use a lot of rising and falling pitches, rather than single ones.

Also, there is no sound envelope with these sounds, playing them backwards sounds pretty much the same as forwards - there is no attack, just quick fade in and quick fade out.

You should also be aware of Silbo Gomero, a human language based entirely on whistling, used in the Canary Islands.
Here's an example with three people communicating.
The langugage supposedly has eight 'elements' which would correspond to phonemes, I suppose. I'm guessing it uses duration and rising vs. falling pitch as some of them, as opposed to absolute pitch.

Here's a translation of the above conversation

"Hey, Servando!" "What?" "Look, go tell Julio to bring the castanets." "OK. Hey, Julio!" "What?" "Lili says you should go get the kids and have them bring the castanets for the party." "OK, OK, OK."

Using Raven to run a spectrum analysis on the Lark Sparrow recording, I find these dominant frequencies (paired with conventional equal-tempered note equivalents)

2,412 | D/D# |

2,584 | D#/E |

2,670 | E/F |

4,134 | B/C |

4,221 | C/C# |

4,306 | C/C# closer to C# |

4,996 | D# |

5,082 | D#/E |

5,426 | E/F |

6,460 | G/G# |

9,388 | D |

And some weaker overtones at | |

10,900 | E/F |

16,350 | B/C |

At first glance I look for doublings to indicate octaves, 2,412 * 2 = 4,824, which is close to the derived value of 4,996, but still almost an entire half-step below.

Similarly, 2,584 doubles to 5,168, which is near 5,082, but about a third-step above.

A little closer is 5,426 which doubles to 10,852, very close to 10,900.

The measurements on these can't be precise, so I may be allowed to assume some fudge factor.

Remarkably, although the pitches above fall in between the standard note frequencies, they do so regularly and consistently. Also, the majority of notes fall between C and F, regardless of which octave it's in.

The 12^{th} root of 2 is 1.05946309. If we look at the two lower frequencies above, the difference is 1.0713101160862354892205638474295 - close but different.

If I divide adjacent frequencies, I get numbers such as 1.191, 1.160, 1.041, 1.033, 1.021, 1.020, 1.017. None are the same, but they're all in the same order of magnitude and there seems to be hovering around 1.020.

Now, Google tells me that the 32^{nd} root of 2 is 1.02189715.

Let's build a scale, using the base Lark Sparrow note of 2,412 as our fundamental (with a little recorrecting). It holds up pretty well,
for example, 4134 * 1.02189715 is 4224.5228181, quite close to the 4221 that I got from the spectral analysis

2,412 |

2464.815 |

2518.788 |

2573.942 |

2640.582 |

2728.465 |

2788.211 |

2849.264 |

... |

4045.416 |

4134 |

4224.522 |

4313.427 |

4400.289 |

4496.642 |

... |

4888.946 |

4996 |

5105.398 |

5193.281 |

5306.999 |

5423.207 |

5544.813 |

... |

6321.575 |

6460 |

6601.455 |

... |

9186.834 |

9388 |

9593.570 |

... |

10666.43 |

10900 |

11138.67 |

... |

15999.65 |

16350 |

16708.01 |

On this page I made a JavaScript that spits out all values based on the above scale, starting at each different value from the spectrograph (using the first few).

Some are close, but I'm doubting whether it's right.

This page has a good Java applet to let you combine notes and see their waveforms.

I've been Googling, looking for the piece that used an 11th interval to suggest the braying of a donkey. "Donkey+11th" didn't yield much, nor did "donkey+11th+interval". "11th" by itself produced, of course, thousands of references to 'September 11th' I tried synonyms for 'donkey' and let's just say that while "11th+ass" got lots of results, the fruit it yielded was less sweet than I had hoped.

Let's do the Fibonacci (one 'n', two 'c's) thing again, shall we? But this time using Prime numbers.

Here are the first 8 (did I miss any?) 2, 3, 5, 7, 11, 13, 17, 19

n | 1/n | A-440 + A-440/n | Equal-Tempered Equivalent |
---|---|---|---|

1/1 | 1 | 880 | A |

1/2 | 0.5 | 660 | E |

1/3 | 0.333 | 587 | D |

1/5 | 0.2 | 528 | C |

1/7 | 0.142857 | 503 | B/C |

1/11 | 0.09 | 480 | A#/B |

1/13 | 0.076923 | 474 | A#/B |

1/17 | 0.0588235294117647 | 466 | A# |

1/19 | 0.052631578947368421 | 463 | A# |

Not bad. Levels off to A#, and eventually to A.

n | n/n_{-1} | decimal | n/n_{-2} | decimal |
---|---|---|---|---|

1 | - | - | - | - |

2 | 2/1 | 2 | - | - |

3 | 3/2 | 1.5 | 3/1 | 3 |

5 | 5/3 | 1.667 | 5/2 | 2.5 |

7 | 7/5 | 1.4 | 7/3 | 2.333 |

11 | 11/7 | 1.57 | 11/5 | 2.2 |

13 | 13/11 | 1.18 | 13/7 | 1.857 |

17 | 17/13 | 1.308 | 17/11 | 1.54 |

19 | 19/17 | 1.118 | 19/13 | 1.462 |

23 | 23/19 | 1.211 | 23/17 | 1.353 |

29 | 29/23 | 1.261 | 29/19 | 1.526 |

31 | 31/29 | 1.069 | 31/23 | 1.348 |

37 | 37/31 | 1.194 | 37/29 | 1.276 |

41 | 41/37 | 1.108 | 41/31 | 1.323 |

43 | 43/41 | 1.049 | 43/37 | 1.162 |

Clearly we have a leveling again as the ratios approach 1.
But we want to have a ratio as low as the 12th root of 2.
Looking at the Mersiennes, it looks like we may need all numbers up to 40 or so.

But now I have too many. This one's too complicated.