Calculate track's key at current tempo? How-to ideas?

[muthafunka]

Pretty much stopped using Live for djing and using Traktor again, no real problems, just a matter of taste. Anyways, I have key and tempo info for all my tracks and was wondering if it'd be possible to make an Excel doc. or something that would allow you to calculate a track's key at the current tempo ie I know track A is in E at its original 124bpm, but what would the closest key be if synced to the currently playing track's 126.3bpm?
I know the math roughly but some kind of on-the-spot calculator would be filthy good. even better were it built into Traktor but don't see this happening anytime soon. Posted this as a feature request at NI but seeing as there's no shortage of smart 'uns here and this would be pretty useful generally, figured I'd ask here too.

[xherv]

The pitch shift applied by playing something in a known key at X bpm, at Y bpm, can be expressed in terms of ratios:

key/x bpm = newkey/y bpm

and can be rewritten:

(key * y bpm) / x bpm = newkey

Since we don't need a numeric value for 'key', we can consider it to equal 1 for the purpose of this calculation.

126.3/124 ~ 1.018.

Western music has very clearly defined semitones* that are neat ratios of two fairly low integers (http://en.wikipedia.org/wiki/Interval_(music) has a good table, but doesn't want to sit inside a url tag here). For example a full octave is 2:1 (2, which results when y = 2x in the above context); a fifth is 3:2 (1.5 - try putting an Operator osc @ 500 to hear this). Here is where it gets tricky though; the semitone scale is logarithmic, not too easy to calculate in your head (well, at least not my head). The difference between the first and second semitones is 16:15 versus 9:8, not something I could tell without a calculator.

So if you're doing an Excel sheet I guess I'd suggest calculating the 11 semitones by solving:

+1 semitone :: 16/15 = z / 124(bpm, just an example) :: (z ~ 132.3)
+2 semitone :: 9/8 = z / 124 :: (z = 139.5)
...

My guess is this would give you the information you want, I think at this point guesstimates would be reasonably accurate.

* - There are a few tricks to the ratios underlying semitones as well, that result in some concrete audible effects resulting from how notes in a polyphonic context can modulate a little against each other, but culturally we live in one that has semitones.

I have a question here - how closely do keymatching DJs follow the rules? It seems to me like you're likely to find some harmonic / modal structures that might not look to be in 'key' with each other but can mesh in interesting ways, but also possible to have some accidents doing it this way.

[8O]

For a single excel formula, would this work:

Semitones Difference to New Key = LOG((NewBPM/CurrentBPM),2)*12

Examples:
CurrentBPM = 124
NewBPM = 132
Formula result = 1.08
So you're just over one semitone up compared to current key e.g. A -> Asharp

CurrentBPM = 124
New BPM = 104
Formula result = -3.04
So you're just over three semitones down compared to current key e.g. A -> Fsharp

xherv - you want to confirm whether I'm just talking rubbish or not?

[chapelier fou]

, it seems right... I guess.
xherv : it is almost right, but in the pythagorician system, which is not the temperated system. But it can do an easy approximation.

[xherv]

That's pretty clever and more accurate than using the ratios listed in the Wiki . . . I haven't visited this topic in quite a while, since I was in high school tuning a guitar a couple times a day, Wiki is really fascinating.

I'm tempted to demand Live 8 include Just, Even, Well Temperament as well as all other mathematical and cross-cultural tuning maths. It's kinda like MPC swing, right? edit: Not even kidding, in the case of something like Tension modeling a guitar, I wonder if this is accounted for, I think you can definitely feel and hear some of those micro-tuning issues on the 'real' thing.

[muthafunka]

Wow, thankyou for all that, now to figure out how to apply all that in some kind of in-the-mix kind of format/process that my brain can deal with. For sure there are many uses for key-matching and not simply like-with-like. The Mixed In Key software uses an interesting non-musical notation which suggests 4 mix-friendly keys at the most basic level. In a software such as Traktor which will displays tempo and key info. in the browser this kind of functionality would seem to be a very logical step.

[blastique]

A 6% change in tempo results in approximately a semitone shift in key.

When mixing, it's just a matter of doing a quick calculation in your head. Eventually when you've been doing it often enough, you kinda tend to get a rough idea as to which situations will and won't work

i.e. a track in B minor is going to work with a track in C minor, if say Bn is 125 and Cn is 130, or a F Minor at 132 may work with a B minor at 126. If you work out the bpm difference between the two however, you'll find that they are 4% and 4.7% respectively. As long as the difference is a bit more than 3%, there will be a discernible shift, and likewise, if it is leaning towards something less than 3%, there won't be a majorly discernible shift in pitch during your mix. However, I've noticed that some tracks are in fact keyed/detuned by as much as 50cents which puts the tune in a lil bit of a limbo zone, but it's always bound to work with something