Counterclockwise – Circle of Fifths or Fourths?

July 20th, 2014

counterclockwise-2048-infinite2048 Infinite – The Circle of Fifths is soon going to have a counterclockwise mode if the Kickstarter is funded. It’s going to sound very different, and here is why:

Going up a fifth repeatedly yields the clockwise circle of fifths. But what about down a fifth? Counterclockwise, of course, but can we really call it the circle of fifths at that point? It really depends on how you look at it. Taking cognition into account really messes the whole thing up.

Whenever we’re working with the raw materials of sounds, music theory should be looked at with a blank slate. No style in mind. Just air reverberating at certain frequencies. Get past all of your personal bias and look at sound the same way a machine would. Pure and cold objectivity (hey, that’s a great description of Stockhausen!).

This is sound we’re talking about! So, let’s listen to the circle of fifths counterclockwise:

Like I said before, I know that an inverted fifth is a fourth, and that there are fourths in the example. But remember; pure and cold objectivity. This means interval classes, not intervals. No tonicization means no hierarchy.

Just because it’s pretty sweet, here’s the tone row in fifths straight down in literal fifths:

Since I’m on a tangent anyway, serialism (the logical conclusion to 12-tone music) is the attempt at eliminating the hierarchy of pitches in music. It is to balance the sonic spectrum. The previous example does not do this. It makes the C at the end the most important note and the note we remember. This does not give us an accurate impression of what the circle of fifths sounds like.

Back to the circle of fifths! Compare the counterclockwise circle of fifths to the clockwise circle of fifths:

Presented in this pure and unbiased form, they sound very similar. Comparable harmonic material results. For all practical theoretical purposes, the counterclockwise circle of fifths is still the same circle of fifths. Just in retrograde.

But then we make music out of it and everything changes. Particularly the bass…which it why everything changes. Here is the Berg style tone row from last time…backwards!  And I mean really really backwards; I forgot to save the MIDI and only had the mp3 file to work with:

Here it is forwards for comparison:

Hear how the middle is a lot crunchier than it is when it’s backwards?

The Counterclockwise Circle of Fifths and Perceived Bass

The bass is quite literally the harmonic foundation and context for everything else you hear in a musical texture. So, the dissonance made out of a counterclockwise circle is welcome because the perceived bass is constantly changing to meet the new sounds. We don’t have to change the bass because our ears do the work or us.

Why do our ears do this? The ratio of the frequencies caused by the notes that make a fifth are 2:3. In other words, the sound waves line up every three oscillations of the higher note. This ratio is easier to understand than the 3:4 ratio of the fourth. Since the fifth’s ratio is simpler, our brains are more drawn to that sound and automatically rearrange any fourth they hear into the simplest ratio. This is possible because the octave above any given note is actually present in the sound via the harmonic series. Our brains reorchestrate sounds into their simplest ratio, and this changes the perceived bass as we go backwards in the circle.

We can also approach it from tonal theory when we are thinking melodically. An ascending melodic fifth (clockwise) ends on the note the ear perceives as least important. This means it sounds unresolved as opposed to the descending fifth (counterclockwise) which ends on the important sounding note (tonic). Here it as ascending:

And now descending, which is the one that sounds resolved and therefore less dissonant (Isn’t the Xylophone sweet):

CounterclockwiseSo counterclockwise fifths are persistently resolving to each other. It puts the music in a state of perpetual resolution. As opposed to clockwise which puts the music in a state of perpetual tension.

Basically, fifths are actually fourths because of some crazy cognition stuff, and going backwards in the circle fixes all dissonance.

So, is it fifths or fourths? If you take cognition into account, it’s fifths. In fact, it’s even more fifths than clockwise. Clockwise is really the awkward direction.

Here is that cool improvisation from last time–backwards! And therefore more resolved:

If all of this stuff sounds jazzy, you are correct. Now listen to Bill Evans.


Stretch Goals for the 2048 Infinite iOS Kickstarter

July 16th, 2014

stretch-goals-2048-infinite-kickstarterThe first two days of the 2048 Infinite Kickstarter have been fantastic! Thanks so much to our (currently) 31 backers who have put us well on our way to making this iOS version of 2048 Infinite – The Circle of Fifths possible. Special thanks to Scott Jon Siegel whose tweet got us an Article at kotaku.com! We ended up having to upgrade our server. Thank you! Technology Tell and Complex Gaming also had kind words to say about 2048 Infinite. Things are going so smoothly that we’ve already decided that we’re going to need some stretch goals.

stretch-goals-upbeat-bird-kickstarter

Stretch goal #1: Upbeat Bird for Android ($700)

We have another game, called Upbeat Bird, that came out only for iPhone back in May. It’s a version of Flappy Bird designed to increase rhythmic accuracy, particularly when playing upbeats. Just like 2048 Infinite, it’s a fun and addictive way to develop a useful musical skill. For another $300 ($700 total) we will be able to get our hands on what we need to put Upbeat Bird out for Android too! This will have a bonus side effect. Rudiment Rock-it will then come out for both Android and iPhone (target date is early August ).

Stretch goal #2: Orchestral Sounds for 2048 Infinite ($1,000)

Caleb (the music guy) wants to design a more complex musical algorithm so the game can play a much larger variety of sounds. Depending on which square the note is formed in, a different instrument will play the pitch. The surrounding harmonic context will play a role in the soundscape as well, resulting in beautiful and lush atmospheric orchestral textures during gameplay. Then Zach will program Caleb’s music into the game.

Thanks so much again for all of the support going into this. We’re super excited to get this iOS version of 2048 Infinite – The Circle of Fifths out into the world!


Kickstarter – Let’s make the 2048 Infinite iPhone App!

July 14th, 2014

Click the download icon to help us get this iPhone app accomplished through Kickstarter!

2048 is a sweet game, and the awesome creator of the game, Gabriele Cirulli made it totally free to use and modify. We saw an opportunity to put a musical twist to it and make a music theory game out of it. But we had to modify it a lot to get it to work with 12 different tiles, and then to make it truly infinite. Once we did that, we realized that it was a music game without sound! Caleb (the pro musician of this duo) put more of his music theory skills to work to helped Zach, the programmer, make it sound delicious. Then we made it for Android, and it’s awesome. Zach has completed over 5 circles in one go, and plans to do even better someday. We love it. It’s a great game.

What’s Next: Kickstarter for the iPhone App!

We’re not done with 2048 Infinite – The Circle of Fifths. There must be an iPhone app, but we’re just not quite equipped to make that happen yet. We were going to wait, but why wait when awesome iPhone owning fans are just itching to see it happen asap. With your help through Kickstarter, we’ll get this app finished way sooner than we thought we were going to.


Score Challenges in 2048 Infinite – The Circle of Fifths

July 13th, 2014
score-2048-infinite-the-circle-of-fifths

Zach’s high score of 532%

2048 Infinite – The Circle of Fifths is obviously based on the popular game, 2048. However, in using the Circle of Fifths, the game becomes infinite, circular, and almost completely different from the original! With the tiles progressing in a circle, the game becomes mesmerizing. This also means that the score has to be calculated differently. As you play, what was once your high tile will eventually be your low tile. To best reflect your progress in the game, the score is shown in a percentage of full circles completed. If your score is 100%, you’ve gotten one full circle! 200%? Two full circles! I’ve (Zach) gotten past 500% myself. How far can you go?

How do we calculate that score percentage?

Let’s look at an example and I’ll show you how we use all the tiles to calculate a score that reflects the progress in the game. You might have the following tiles on the board: B, B, A, F, B, B. Or, listed in a numeric value from counting around the circle of fifths, it would be: 10, 10, 8, 6, 5, 5 (Note: The game actually keeps track of the tiles as numbers like these, and translates them into the appropriate note to display on the screen). One full circle happens when you have made it to the 12th tile. So, with a high tile of 10, the score should be at least 10/12 (83.33%). When we factor in the remaining tiles, the score should be in between 10/12 and 11/12. If we calculate the progress towards a new high tile and add it to 10, we can divide our new number by 12 to get a more accurate score. So let’s use the second highest tile to do that. The progress toward getting a new high tile is the second highest tile divided by the current highest tile. In this case, that is 10/10 (100%). Waitaminute! This leads to a problem: having two of the same highest tiles doesn’t mean you are 100% of the way to a new high tile! Most of the work is done, though, so let’s say that getting two of the same tile is 90% of the work, meaning it should work out to 90% instead of 100%. Our calculation is: 10/10 × 90% = 90%, or 0.9. Add that to 10 to get 10.9 and do 10.9/12 to get a more accurate score: 90.83%.

We can add the third highest tile into the mix, too. Using the second highest tile let us get that 10.9 we used, so let’s use the next tile to add another decimal place onto that and get more precise! The next tile is 8. We can divide 8 by 10, the second highest tile, to get the progress towards getting a new second highest tile. But is that right? To be more accurate, we should calculate it in relation to the current low tile the game is giving. This also means we can just ignore the low tiles on the board because they are not really part of the progress. They’re just handed to you! The low tile at this point is 5, so the calculation is (8-5)/(10-5) × 90% = 0.54. To add it as the next decimal place to the previous number (our 10.9), we do 0.54/10 to get 0.054. 10.9 + 0.054 = 10.954. We can do this with all the remaining tiles to make our number as precise as we can.

That’s a summary of what the game does to calculate the score. Here’s how it looks as the game does everything:

Set of tiles: 10, 10, 8, 6, 5, 5

Progress towards new high tile:
(10−5)÷(10−5)×(9÷10) ⇒ 0.9
+
(8−5)÷(10−5)×(9÷100) ⇒ 0.054
+
(6−5)÷(8−5)×(9÷1000) ⇒ 0.003
= 0.957

Add value of current highest tile:
10.957

Get the score!
10.957 ÷ 12 = 91.31%

After the next move, there is a new high tile. These are the tiles:
F, A, F, B, B, B
Or, in numbers:
11, 8, 6, 5, 5, 5

Here’s the score calculations for that set of tiles:

Set of tiles: 11, 8, 6, 5, 5, 5

Progress towards new high tile:
(8−5)÷(11−5)×(9÷10) ⇒ 0.45
+
(6−5)÷(8−5)×(9÷100) ⇒ 0.03
= 0.48

Add value of current highest tile:
11.48

Get the score!
11.48 ÷ 12 = 95.67%


Tone Row – The Circle of Fifths

July 8th, 2014

tonal-tone-rowThe Circle of Fifths can be considered a tone row. Cool, huh? One of the defining concepts of the tonal system when written in a score turns out to be atonal. It’s an oxymoron, really. A tonal tone row. Atonal tonality. Consonant dissonance. A never-ending resolution. The beauty of the paradox is unending.

Two Sides of the Tonal Tone Row Spectrum

The amount of tonality you hear really depends on the rate at which you hear the notes and how many are played at a time. To start out the audio examples, listen to a clear presentation of the circle of fifths as a tone row:

Now, before anyone decides to be a Smart Aleck, yes I know that when you invert a fifth it turns into a fourth and that my example has fourths. But since this is atonal music, we are dealing in pitch classes, not in intervals. :-) I win.

Since the first example is played rapidly, it is perceived as sounding random. Yet at the same time it is symmetrical; perfectly ordered and balanced. That’s the touch of tonality leaking into the sound. We just can’t fully get away from it!

Let’s try a bunch of notes together without using orchestration to change how it’s heard.

Pretty much the same effect. Now lets try changing the orchestration and the number of notes we hear at one time. We can trick the ear into hearing tonality.

We were certainly able to drive that F to resolve to the E. Not very tonal, but we were able to get enough emphasis on a single pitch the make some of the original serialists not approve (Webern, anyway). Let’s see if we can sound like Berg by using rhythm to reach into tonality.

Still Serialism, but we are certainly getting dangerously close to tonality! Now let’s do a little repetition and break from the serial idea entirely.

I could listen to stuff like that all day! All 12 pitches were used many times using the contour of the circle of fifths, but we were definitely in C. The reason it sounds so different is that the pitches are separated into musical ideas over time instead of just thrown at the ear all at once. It’s also why we can’t call it serialism, but it’s still the circle of fifths! And now for a little hop to the total opposite side of the style spectrum.

And that’s how to serialize tonality. I hope you enjoyed my little tone row based compositions!

music-theory-2048-infinite

To hear the circle of fifths paradox over a very long period of time, try 2048 Infinite – The Circle of Fifths. It’s as fun as it sounds.