Electronic music - sync
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Sync in general
Devices can keep pretty damn regular rhythm with an internal clock, much better than people can ever hear.
However small the imprecision (could be order of .01%), multiple devices with their own time would eventually still go out of sync - or be so unless you start both at precisely the same time.
We could instead do what modular synths do: not actually synchronize up, they just make a sound in direct response when they see an event on their input. But that basically means one thing must be in control of everything, and only one thing can keep time.
What we end up wanting instead of either of those is devices regularly matching up, so that each device can do its own thing to a shared beat that will never go off.
Electrically, sync is typically a square impulse, because that is best defined electrically, and easy to produce and consume.
Functionally, sync just tells another device when to move on to the next step in a sequence.
This means there is only one clock, and everything else will keep synchronized to it. Each device will do the same thing, no matter how (ir)regular that one clock is.
But this raises some questions.
Say, you can tell another device to progress on every tick, then you can only share four-to-the-floor sort of beat.
You couldn't have one thing play faster (e.g. an eighth-note melody over a fourth-note rhythm, a slow kickdrum and faster hihats), things like syncopation, off-beats, swing, triplets, slugghishly late beats, polyrhythm.
Assuming the ticks are entirely regular, then doing those things comes down to how a device consumes them.
The eighth-note melody on top of quarter note beat would be served by having one device progress only every second tick.
That kickdrum-and-hihat example by making the kickdrum only react every fourth or eighth tick.
PPQN (a.k.a. PPQ and TPQN) is fairly literal: the amount of electronic-level pulses (/ticks) before a listening device moves on to the next quarter note.
Defining this with a quarter notes at the center seems purely because it's music theory's default. It needn't actually be a quarter note - you can play things faster or slower - but when combining more than one device, that tends to become the common denominator.
PPQN is often seen in the form 24PPQN, e.g. in MIDI beat clock (and the earlier DIN sync).
'Why 24? Why not fewer? Why not more?
More exists. Fewer exists. 24PPQN just became a convention.
Consider our needs.
If 1PPQN is a quarter note, then 2PPQN lets you talk eights (and do basic syncopation/offbeats), 4PPQN sixteenths (and basic swing). And maybe 32th notes, but since those are already hard to use or hear, 8PPQN is a soft limit to basic beat sequencing needs.
If you want to do triplets, or clave-style rhythm, you want a factor 3 in there, as well as 2 or 4. So yes, 6 or 12PPQN would also have worked.
If you want swing, you want extra steps in the entirely regular pulse whre you can place it, so you quickly want to place at sixteenth-note positions (4PPQN for 16 things per bar)
If you want polymeter, you can use the same musical pulse but have one thing loop earlier. This doesn't really change anything (and was simple to do even in early electronic music).
The math says you need the lowest common multiple, and it turns out that 24PPQN covers all the just-mentioned cases.
Remembering that 24PPQN means 24 input ticks move onto the next quarter note, this allows
- typical ones:
- whole note (96 counts each)
- half note (48 counts each)
- quarter notes (24 counts each)
- eight notes (12 counts each),
- sixteenth notes (6 counts each)
- 32nd notes (3 counts each)
- and triplets
- quarter note triplets (16 each) (note this aligns only every second beat, at 48 ticks, 2*24 and 3*16)
- eight-note triplets (8 each; more common for musically practical reasons)
- sixteenth-note triplets (4 each),
- thirtysecondth-note triplets (2 each),
- sixtyfourth-note triplets (1 each)
Note that while 24PPQN (e.g. in MIDI) was probably aiming for 32nd notes and triplets, the way the least common multiple of 3 and 4 interact means it technically allows 64th-note triplets, but a lot of 24PPQN don't expose this, because you don't really perceive this.
Some practical details
One practical limitation is that such a pulse alone can't communicate when a measure should start over, so it's not very hard to get two devices playing at the same speed, but start/restart their measures at different times.
One workaround is to, if possible, prime each listening device at the start, and only then start the sync pulses.
Sync in the real world
MIDI beat clock, a.k.a. MIDI timing clock, MIDI clock, is using regular MIDI connection to transmit a (single-byte) message at 24PPQN.
- see also MIDI#Beat_clock
MIDI Timecode, a.k.a. MTC, is a variant of SMPTE
- Which isn't really a beat, so it turns out to be somewhere between finicky and useless to use for rhythm.
- It's useful to mention, to point out that this is not the same as beat clock.
- see also MIDI#MTC
 is a PPQN style pulse on a dedicated pin of a DIN socket. (A second pin set whether the drum machine should be running or paused. There are a few pinout variations.)
DIN sync is typically 24PPQN, and also known as Sync24.
Some old drum machines (e.g. some from Linn, Korg) used 48PPQN, many others 24PPQN (e.g. Roland). If DIN-synced between those, they'd go at twice/half speed. Which some people used intentionally, others has a converter box for.
Modular synth sync - there's usually just one thing keeping time, and all other things make a sound as soon as they see an event (trigger or gate), and don't run their own timer at all.
- Even deriving varied clock signals using modules tend not to use what you'd classically call sync.
- As such, it's usually only the clock generator you use that needs to know about rhythm, everything else just needs to react.