We celebrate 3.14, PI day, with some selections of mathematics, music, and visualization…

Sometimes the results resemble scores, sometimes toys, and sometimes – more rarely – real musical instruments. But part of why I love computing as a window into music is its ability to visualize music’s mathematical beauty.

I happened across this image from Flickr. It’s a chalk pattern on pavement for a children’s game (I’m not actually sure what game). But the math-compelled photographer found in it musical, harmonic intervals. I’ll have to sketch a little Processing and Pd design that plays with this idea. I put it here because someone out there might be inspired to do the same, and this is just ambiguous enough that it could easily lead in dozens of wildly-divergent paths.

I know some of my own students are literally on a beach for spring break and the nerd elite is busy partying in Austin, but, uh, maybe someone out there will file this away for later.

The photographer explains the math:

Also not sure what this game is called, but it contains some interesting mathematical properties. Can you see the oblong numbers (2,6,12,20,30…) in this representation?

Per Mathgym:

Readers who are familiar with the theory of music will recognise the list of oblongs as the intervals in decreasing order of consonance: Octave (1:2), Perfect Fifth (2:3), Perfect Fourth (3:4), Major Third (4:5), Minor Third (5:6), etc. It is Pythagoras who is credited with discovering this mathematical relationship between music and numbers.

This discovery, that the pitch of a note is related to the length of the string which produced it, is credited as being the spark which ignited Pythagoras’ imagination and philosophy. It allowed Pythagoras a glimpse of a whole new order in the Universe, one governed by intellect and logic and capable of the sublimest of pleasures. And a glimpse was all that he needed.

With this discovery, Pythagoras and the Pythagoreans set in train a way of investigation which has proved to be one of the most productive ideas in human history – that mathematics can be used to unravel the mysteries of the Universe.

Now, after those deep thoughts, who’s worked an appetite for some PI pie?

To celebrate what in the US we call 3.14 or PI day, today I’m offering stories that deal with mathematics and circles. First up, an app named for the great philosopher who is credited – even if perhaps ahistorically so – with finding that ratio and ratios in harmonies.

Technology has long introduced innovations that make playing easier for specialists and non-specialists alike. Just ask anyone who plays an instrument like the guitar – frets, and the simplified notation that went with it, go back centuries as a means of allowing more people to make music.

Developer Rob Fielding wants to rethink frets, to bring their disposition and playability closer to the way harmonics work in sound. The creator of the microtonal iPad app Mugician, his next app in development, Pythagoras, offers some fascinating ideas. Forgive me getting a bit theoretical in the prose for those who do speak that language; the videos are always the best way of understanding what’s going on. (The vast majority of even untrained ears have the ability to perceive pitch with astounding accuracy, so you don’t have to be an expert. Usually when people claim to be tone deaf, the problem is that they can’t sing, not that they can’t hear, in my experience.)

I’ll let Rob explain:

Pitch

Pythagoras’ fretless mode uses geometry to mark the harmonically relevant points, not fixed frets. Where the lines intersect with strings, the notes are perfect ratios to each other. This helps you to locate and get to know the useful pitches that are used in world music. [See image, top for a beautiful visualization of how this works. -Ed.] That is explained here:
The Spectrum – Pythagoras’s interface

When you play a chord like a major third, you line up the blue notes to overlap perfectly, and you get shiningly perfect major thirds that way. Same for harmonically correct fifths and fourths. These are the pitches that you hear as overtones when you listen carefully to instruments with lots of sympathetics, etc.

I do want to respond to this one lamentation in Rob’s post: he frets (ahem) that MIDI doesn’t use frequency, and that OSC isn’t well-supported. I actually think MIDI isn’t far off – it just lacks precision. Perception of pitch is complex, but a logarithmic scale (in which 440 sounds one octave higher than 220) is reasonably close to how we hear. And that’s precisely what MIDI gives you; if you just wanted to number the piano, its solution of using a number like 60 for middle C makes perfect sense. (We can overlook for a moment that the definition of MIDI fumbled the octave. The basic idea was still right.)

Even outside MIDI, a numbering system like that in MIDI – mapping pitch space to a logarithmic scale to make them match intuitively what we hear – is not uncommon. The problem is that MIDI doesn’t have a rational way of dealing with what happens in between the notes, as it used integers for efficiency. Take MIDI’s logarithmic scale and set floating-point numbers (numbers with a decimal place, like 60.5 instead of 60), and you have a pretty decent solution. You could still, if you didn’t want integers to represent 12-tone equal-tempered pitch, apply different scales and modes. But I think if you wanted a decent way of communicating note values, unless I’m really missing something, sending floating point numbers that default to a 12-TET logarithmic scale can’t be too bad. I understand that most instruments don’t yet respond in any standardized way, but I refuse to believe this is an intractable problem. I’m happy to discuss in comments. Heck, if we just got Max and Pd patchers to agree on something, I’d be pleased.

On to another very cool idea:

Octave Rounding

Pythagoras is using octave rounding in its latest incarnation when you press the “Auto” button for the octave switch. What this means is simply that it doesn’t care about what octave a note is in, it will pick the closest octave to the last played note. This allows for astounding feats of arpeggiation and pentatonic scales – even when playing fretless. Here is the more popular video:

And here is the improvement upon it the next day (much less viewed video) where you can slide up or down a fourth:

This octave rounding is an idea I implemented a few years ago in my Samchillian derivative called Xstrument. (Both Xstrument and Mugician are open source projects on github). This idea is very applicable to 2 octave keyboards as well.

Here is the idea with a trivial Pd program:

It’s really great stuff. As it happens, I’ve been exploring new geometries for music making myself, interested along similar lines. And musical inventor Roger Linn has had a lot of things to say about it lately, too, including his respect for Rob’s work.

So, I’d love to have a discussion. What interesting interfaces have you seen for music? Are there any you find playable in practical circumstances? And why can’t we just solve this issue of how to transmit pitch information between software and hardware once and for all? (I don’t yet know how HD-MIDI will address the issue; that’ll be interesting to see.)

And don’t miss Rob’s blog:
http://rrr00bb.blogspot.com/

Finally, here’s Jordan Rudess rocking out with Mugician, Rob’s (currently-available) app.

slicedbread, on behalf of The Covert Operators, has released a free Windows utility that generates “endless harmonic oscillators” for Ableton Live’s Simpler and Sampler instruments. (Since this was a released, a Mac build has been made available, as well; see link below.) Even if you don’t intend to use the utility directly, pay attention – The Covert Ops already have a sample pack up full of oscillators, and you can bet the presence of this utility means more will come. (Even Robert Henke was impressed on the forums.)

Live 6 introduced the file format for “Ableton Meta Sounds.” Bjorn Vayner is currently breaking down how the format works, but the short upshot is that you can make oscillator sources that won’t alias for sound design in Simpler and Sampler. The AMS File Utility does more, too – export tunings (even microtuned stuff), and make oscillator variations. It’s sampling for people who like synthesis. In fact, not only is it fun to make additive synthesis-style oscillators dragging individual harmonics, but it’s a total breeze to change the offset and make equal-tempered stuff, negative scales, and other tunings.

Description on the forums:

AMS File Utility for Microtonal/Traditional Tunings

And from the very awesome Covert Operators site, some of the behind-the-scenes action, plus the Mac build (updated with additional links!):

Meta Files: Uncovering the .ams format, Part 1

Meta Files: Uncovering the .ams format, Part 2

Meta Files: Uncovering the .ams format, Part 3

Mac OS X Meta Application

Thanks for reminding us of this, Tony. I’m a bit behind on all this, but better late than never. Since I am lagging, has anyone made some AMS packs since this came out in September?

Don’t get too excited. Gibson Guitar is not, in fact, introducing a fully robotic guitar. Or a creepy robot doll that plays a guitar. Nor are they shipping you a handsome (male/female/your choice) robot assistant who will follow you around and tune your guitar for you. Too bad. But they are launching a robotic, self-tuning guitar on December 7. And most importantly, it comes in a limited-edition frost blue paint retro-robotic job, which even as a non-guitarist, I have to admit is super hot. So, what’s robotic about it? Its tuning system:

In addition to its automated tuning and alternate/open tuning functions, the Gibson Robot Guitar offers a unique Intonation function, which guides even the most tweak-phobic player through the simple steps of achieving perfect intonation on this revolutionary instrument. No tools or external tuners or other gadgets are needed other than a small screwdriver and the Robot Guitar’s own Master Control Knob (MCK). The guitar itself “talks you through” the entire process, resulting in a correctly intonated guitar in a fraction of the time it takes even a professional guitar tech to do the same job.

My favorite line was this, from the press release:

“Since the dawn of the instrument, musicians have come to accept the guitar’s imperfections and lack of tonal precision as necessary evils. Onstage and off, guitarists have fought to stay in tune. Every music lover and performer has had to suffer through the show—halting, mood-killing atonal droning of a loudly amped guitar being brought into tune.”

Or, erm, not brought into tune, as the case may be. (Jump!)

I read that initially as “every music lover … had to suffer through the show,” and the whole passage makes it sound like pretty much everybody just has to suffer guitars. Boy, am I ever a Keyboard player.

In all seriousness, the good news here is that this instrument really opens up the possibilities of open and alternative tunings, which make a terrific difference in sound even untrained ears may be aware of. And that’s not an “experimental” thing; it’s part of the history of the music. Gibson does a nice job of summarizing that on their page, and has an interview with the inventor, as well.

The connection to this site, aside from “robot” in the name — tuning is important, and it’s something I’d love to see soft synth makers make more accessible to lay people (hint: make it easier to adjust tuning in the interface, do some more interesting presets, and even think about controllers).

Robot Guitar Product Site with manual, demo videos, and background [Gibson Guitar]

No word yet on whether Van Halen will be endorsing this. Gibson: might want to send them one, just in case. (Hey, I had to get one cheap shot in.)

Pray that one day your onstage train wrecks will get this much analysis. Mine tend to involve only free produce — not always fresh, sadly.

Wilfred Fumbly says, just so you don’t have to bother him about it any more:

I took it off for a couple reasons. First, I’m pretty busy with work and found having to respond/answer/defend was taking up a lot of time and mental energy.

Secondly, I began to question my original theory. At the time, it seemed like a reasonably theory because the song isn’t exactly in C#… it’s close. So a sample rate screw up seemed a good explanation and the math seemed to work in its favour too. I began to doubt the theory though, as time went on and VH continued to play the song at other shows in the same weird key.

It’s possible that they simply adapted to a screwed up computer file and my theory was right all along. But there’s no way to know that. Apparently VH has played the song in C, C#, and D on various tours… but I didn’t know that at the time.

So, rather than mislead people into thinking I knew the answer for sure… I took the video response off.

Cuz I’m not sure anymore.

Or maybe the whole event is the result of a discontinuity in time, produced by the unnatural attempt to keep Van Halen’s career alive, as The Master suggests in comments.

What’s amazingly tiresome about this argument is the esoteric tweak-freakish strong-minded opinions toward one cause or another.
Here’s the bottom line. The mighty Van Halen is nothing but a dream, a dream best left remembered. To go out on tour in their current state, and with the current line-up, is a complete waste of time. Shame on them, and shame on anyone who promotes their ridiculous conceited behavior. I’m glad someone one was out of tune. Who cares if it was Eddie or the keyboard. They sucked, they sounded horrible, and people paid a LOT of money for it. Shame shame shame! If they all got on the same page way back in the day, they might have had a better chance of staying together as a band. If Eddie quit drinking and started smoking weed, or if David Lee Roth did the opposite, then they might still be together. You can’t have a pot head and a drunk competing with each other for control of a band like that. They both need to be on the same page chemically.

See? Everyone has a theory.

Thanks, Wilfred Fumbly. (video’s gone now … more in a moment)

So, the original theory holds: most likely a sample rate issue. Well, unless Van Halen is really old school, run their backing tracks on reel-to-reels, and had that set to the wrong speed. Sample rates it is.

More importantly, we’ve definitively proven Eddie is a “great guitar player,” which I know is what was really bothering everybody about this clip. He demonstrates this greatness with true vigor, by playing as loudly as possible for five minutes completely out of tune with the backing track and the vocals (which were matching the backing track) as if he’s completely deaf. If you had any doubts about what a true Guitar Hero is, now you know. (And yeah, unfortunately, I do think that really was his only choice. Guess the techs couldn’t get the clock rate set back to normality.)

Speaking of Guitar Hero / Rock Band: Activision / Harmonix, if you’re listening, I think you know what my request for an Easter Egg in your game would be.

Updated: The video is gone. So now we can not only speculate about what happened to Van Halen, but what happened to the video. Perhaps WilfredFumbly noticed that, while the keyboard part in Greensboro was pitched higher than the original album recording, so were other gigs on the tour. That means the guitar is far from absolved. And it lends new credence to my “Wormhole Theory.” Maybe Eddie’s guitar was temporarily replaced with one from the past, in which the song was in a different key, or even an alternate universe where this is in tune.

Okay. I got nothing.

So what happens when you’re Van Halen, the last song in your set list is the million-seller “Jump” with its synthesizer-keyboard opening…and the recording you’re using to play back the synth is accidentally run at 48K instead of 44.1K?

What happens is exactly this (recorded in Greensboro, NC four days ago)

Yes, just as disturbing as the possibility that your backing tracks could spontaneously introduce a new tuning system in front of thousands of fans, hundreds of thousands more may know about it within days on YouTube. Techs everywhere just felt a chill go up their spine. Erm, once they stopped crying they were laughing so hard.

If you enjoyed this musical performance, get ready for Guitar Hero IV: Quartertone! Charles Ives Expansion Pack!

And Van Halen, the American Festival of Microtonal Music here in New York may be able to welcome you soon.

Now, truly, no one has an excuse for playing an out-of-tune guitar. howtotuneaguitar.org features a Web-based interface for tuning, and step-by-step instructions in case you’re really a newcomer. The tuner itself goes well beyond the basics, with support for Standard, Drop D, Open C, Open G, Open D, Open G, Half Step Down, Full Step Down, Open E or Admiral tunings. Powered by Flash, you can trigger looped recordings of the strings one by one.

The site has chords, scales, lessons, and a forum; you can even see blog-style entries, like one dedicated exclusively to the “D suspended-four” chord. Is the day coming soon when chords themselves will have their own blogs? (phyrigianmode.blogspot.com?)

Guitar Tuner @ HowToTuneAGuitar

Nigeria Orders First Million OLPC Laptops at vnunet.com, which also has two videos of working prototypes; via worldchanging

For more background on the project, see The Laptop Crusade, from this month’s Wired.

See also our previous story, Creative, Networked Music Making on $100 One Laptop Per Child, which brought some interesting debate on all sides of this issue.

Beyond the idea of giving millions of children new access to computing, there’s a separate mission that’s come up: how to create useful music software for children. As covered in that previous article, a team of developers is working on new music software called TamTam that will have two lives: one, as creative musical software for the OLPC hardware, and a second, as open source software anyone can run. That means that even if you don’t agree with OLPC’s aims and implementation, TamTam could still have potential running on used laptops here in the US. (Given the problems of toxic computer waste, I’m just as interested in how we can recycle computers without short-shrifting children that receive them.)

Reader Nat Lécaudé, who initially brought TamTam to our attention, talks about working on the project:

I got involved through the University of Montreal where I am a student. Jean Piché, my teacher is the author of Cecilia, a sound manipulation program based around Csound. Since we have a lot of Csound knowledge and that the OLPC runs Csound as it’s audio engine, people from MIT asked Jean to form a team to do a music software for the OLPC.

Regarding different tunings, we’re still not sure how it will be done but it will definitely be there. We were thinking about defining a global tuning and that the notes would snap to the good spot on a grid. Lots of solutions for this, we’ll have to wait a little to see how it will be handled. Since in Gamelan for example, each instrument is slightly detuned to create a vibrato effect, we will need a mechanism to detune a specific track slightly. That would be I guess, the most complex scenario.

Ed.: There’s actually a lot to this point. Not all instruments in gamelan feature paired tunings; “solo” instruments like the rebab can be played alone; at the opposite extreme, while individual pairs are tuned together, gamelan builders generally think of the tuning of the whole ensemble, so just doing pairing out of context wouldn’t really work. That just covers Java and Bali; other countries have various issues of their own. Talking about tuning and computer music could easily fill a book. At the very least, I hope this software will support Scala microtunings, since it’s the only real standard for implementing this on computers. Even in the Western world, 12-TET tuning doesn’t cover the full range of what people actually play. Sorry to get off on this topic, but I imagined someone would bring this up if I didn’t. It does illustrate why the challenges of creating this software are fascinating in themselves. -PK

The goal is to have multiple levels of complexity. A kid can start making music very easily by clicking some buttons that will generate music based on algorithms. Eventually he can start sequencing himself note by note. He will also be able to write python scripts to generate music based on rules of his own. Same goes for synthesis, a sound bank based around samples will
be included but it will be possible for the kid to dig into csound to create his own sounds. The OLPC being quite modest regarding performance. We won’t be able to do things like physical modeling, but simple synthesis should be possible.

The resulting software will be completely open source and the goal is to have 2 versions. A version targeted at the OLPC which will deal with the fact the machine is not powerful. Another version will be made available to other platforms so we can play with it ourselves. Even though the concept is simple, we “wasted” lots of hours with the max demo generating music and, it can be used as a composition tool or just for the fun of it. The coolest feature we did is the ability to record short samples and load the samples
as instruments. It will be interesting to see also if the community can come up with custom algorithms generated using our scripting interface. These could eventually be added to the OLPC version. It would be fairly easy for instance to write an algorithm that would generate different Gamelan music each time you run it.

Thanks for posting the story and I will keep you updated as it evolves!

Tuning pianos involves some heavy science and art. In other words, Reyburn Pocket RCT has absolutely no relation to that simple guitar tuner you’ve got in your gig bag. It’s a US$900 powerhouse of visual tuning:

Reyburn Cyber Tuner / Pocket RCT, for PocketPC (Windows Mobile)

This is probably old news if you’re a piano tuner (either this or the Mac/Windows laptop version), but I saw it this weekend while I was staying at my parents’ house and a tuner came over to adjust our Baldwin grand. The tuner was more than happy to show it to me. You can’t tell in this screenshot, but the UI pulses like some sort of alien eye as you near the pitch. The software was able to guess that the piano was a grand of more than six feet just by listening to the harmonic content of the sound (already impressed); it can compensate tuning for the size of the piano. The system uses aural tuning, meaning it looks not only at the fundamental but directly samples and matches partials, which is the way tuners are trained to work.

The tuner was especially pleased by two features: first, that you can keep records of tunings of different pianos, giving the tuner virtual “medical records” of the way a piano has held its tuning over time. (That helps diagnose how the piano itself behaves, and how it responds to the environment.) She was also happy that she could perform extremely accurate overtuning that would anticipate how the tuning would settle over time; because of the enormous sensitivity of pianos, they don’t hold their initial tuning exactly.

I know a couple of musicologist friends who would love playing around with this, particularly the 57 historical tunings from Owen Jorgensen. Now you can finally play the Well-Tempered Clavier on a piano that’s actually well-tempered. For those learning to tune professionally, the software even includes exams, but it sounded as though pros could comfortably use the technology to augment rather than replace their existing craft and experience.

We have at least a couple of pro tuners reading CDM, so I’d love to hear what you think of this. Maybe some of you think this new-fangled tech is useless; I don’t know.

Updated: Via comments, Veritune is a formidable competitor to this product. The concept is the same, but Veritune has a real-time spectral display, multiple simultaneous partials, far more notes measured (76 vs. 6), note switching for all notes, no required measurement step, and other features. It’s also available in an integrated, rugged hardware unit as well as for your existing PocketPC, and Veritune claims it’s easier to use. Anyone who’s used one or the other, let us know what you think. Thanks to Carl Lumma, formerly of Keyboard Magazine.