Sounds Like F#

Last time, I thought that I was going to have to do some special research to account for multiple sources of sound. Reading this short physics snippet I can see that multiple sources is really as simplistic as my initial thinking was inclined to. You can ignore all of the decibel stuff and focus on the statements about amplitude, because, of course, I am already working in terms of amplitude. What can I say, I let the mass of voices on the internet trouble my certainty unduly? Sort of.

There's two important issues to consider:

1. Offsetting of the signals.
2. Psychological issues regarding the perception of sound.

Regarding number 2, here's a quick hit I found: http://hyperphysics.phy-astr.gsu.edu/hbase/Sound/loud.html. It seems credible, but I'm not sufficiently interested to pursue it further. And so, my dear Perception, I salute you in passing, Good day, Sir!

Offsetting of the signals is something that is implicitly accounted for by naively adding the amplitudes, but there is a question of what the sineChord() function is trying to achieve. Do I want:

1. The result of the signals beginning at exactly the same time (the piano player hits all his notes exactly[?] at the same time—lucky dog, guitar player not as much[?]. Hmm...)?
2. The average case of sounds which may or may not be in phase?

Regarding case 2, Loring Chien suggests in a comment on Quora that the average case would be a 41% increase in volume for identical sounds not necessarily in phase. At first glance, I'm inclined to think it is a plausible statement. Seeming to confirm this suggestion is the summary about Adding amplitudes (and levels) found here, which also explains where the the 41% increase comes from, namely, from \(\sqrt{2}\approx 1.414\dots\). Good enough for me.

All such being said, I will pass it by as I am approaching this from a mechanical perspective. I'm neither interested in the psychological side of things, nor am I interested in getting average case volumes. I prefer the waves to interact freely. If would be neat, for example, to be able to manipulate the offset on purpose and get different volumes. The naive approach is little more subject to experimentation.

I would like to go at least one step further though and produce a sequence of notes and chords and play them without concern of fuzz or pops caused by cutting the tops of the amplitudes off. We need two pieces: a) a limiter and b) a means of producing sequences of sounds in a musically helpful way.

A limiter is pretty straight-forward and is the stuff of basic queries. I have the advantage of being able to determine the greatest amplitude prior to playing any of the sound. The function limiter() finds the largest absolute amplitude and scales the sound to fit.

let limiter volume wave = let amp = volume / 2.0 let max = wave |> Seq.map (fun u -> abs(u)) |> Seq.max if max <= amp then wave else let compression = amp / (float max) Seq.map (fun u -> u*compression) wave
Here is an example usage:

let ChordAOpen = seq[69; 76; 81; 85; 88] notes.sineChord 65535.0 ChordAOpen sample_rate 1.0 |> notes.limiter 65535.0 |> Seq.map (fun u -> int(u)) |> notes.PlayWaveAsync sample_rate

One concern here is that we may make some sounds that we care about too quiet by this approach. To address such concerns we would need a compressor that affects sounds from the bottom and raises them as well. The general idea of a sound compressor is to take sounds within a certain amplitude range and compress them into a smaller range. So, sounds outside your range (above or below) get eliminated and sounds within your range get squeezed into a tighter dynamic (volume) range. This might be worth exploring later, but I do not intend to go there in this post.

Last post I had a function called sineChord(), which I realize could be generalized. Any instrument (or group of identical instruments) could be combined using a single chord function that would take as a parameter a function that converts a frequency into a specific instrument sound. This would apply to any instruments where we are considering the notes of the chord as starting at exactly the same time (guitar would need to be handled differently). So, instead of sineChord(), let's define instrumentChord():

// expects: // volume (2 x amplitude) // frequency (Hertz) // sample_rate (Hertz) // duration (seconds) let instrumentChord (instrument: (float->float->int->float->seq<float>)) (volume:float) (sample_rate:int) (duration:float) notes = notes |> Seq.map frequency |> Seq.map (fun f -> instrument volume f sample_rate duration) |> Seq.reduce (fun a b -> Seq.map2 (fun sa sb -> sa + sb) a b)

Next we will create a simple class to put a sequence of notes/chords and durations into which we can then play our sounds from.

Reference Chords and Values

Here are a few reference chord shapes and duration values. I define the chord shape and the caller supplies the root sound to get an actual chord. The durations are relative to the whole note and the chord shapes are semi-tone counts for several guitar chord shapes. The root of the shapes is set to 0 so that by adding the midi-note value to all items you end up with a chord using that midi note as the root. This is the job of chord().

let m1st = 1.0 // whole note let m3_4 = 0.75 // dotted-half let m2nd = 0.5 // half let m4th = 0.25 // quarter let m8th = 0.125 // eighth let m16th = 0.0625 // sixteenth let m32th = 0.03125 // thirty-secondth let EShape = seq[0; 7; 12; 16; 19; 24] let AShape = seq[0; 7; 12; 16; 19] let DShape = seq[0; 7; 12; 16] let DAShape = seq[-5; 0; 7; 12; 16] let CShape = seq[0; 4; 7; 12; 16] let CGShape = seq[-5; 0; 4; 7; 12; 16] let GShape = seq[0; 4; 7; 12; 16; 24] let Gno3Shape = seq[0; 7; 12; 19; 24] let EmShape = seq[0; 7; 12; 15; 19; 24] let AmShape = seq[0; 7; 12; 15; 19] let DmShape = seq[0; 7; 12; 15] let DmAShape = seq[-5; 0; 7; 12; 15] let chord root shape = shape |> Seq.map (fun u -> root + u)

Next, InstrumentSequence is an accumulator object that knows how to output a wave form and play it. For simplicity, it is limited to the classic equal temperment western scale. Note that it takes a function has the task of turning a frequency into a wave form, which means that by creating a new function to generate more interesting wave forms, we can have them played by the same accumulator. If we only want it to produce the wave form and aggregate the wave forms into some other composition, we can access the wave form through the routine WaveForm.

type InstrumentSequence () = let chords = new ResizeArray<seq<int>>() let metrics = new ResizeArray<float>() member this.AddNote note duration = chords.Add(Seq.singleton note) metrics.Add(duration) member this.AddChord notes duration = chords.Add(notes) metrics.Add(duration) member this.Chords = chords member this.WaveForm instrument volume tempo sample_rate = let whole_note_duration = 240.0 / (float tempo) let runs = Seq.map2 (fun chord metric -> let duration = metric * whole_note_duration instrumentChord instrument volume sample_rate duration chord) chords metrics seq { for r in runs do yield! r} member this.PlayInstrument instrument volume tempo sample_rate = this.WaveForm instrument volume tempo sample_rate |> limiter volume |> Seq.map int |> PlayWave sample_rate member this.PlayInstrumentAsync instrument volume tempo sample_rate = this.WaveForm instrument volume tempo sample_rate |> limiter volume |> Seq.map int |> PlayWaveAsync sample_rate

A sample use of the class which follows a very basic E, A, B chord progression follows:

let song = notes.InstrumentSequence() // E major song.AddChord (notes.chord 40 notes.EShape) notes.m2nd // A major song.AddChord (notes.chord 45 notes.AShape) notes.m4th // B major bar chord song.AddChord (notes.chord 47 notes.DShape) notes.m4th song.AddChord (notes.chord 40 notes.EShape) notes.m2nd song.AddChord (notes.chord 45 notes.AShape) notes.m4th song.AddChord (notes.chord 47 notes.DShape) notes.m4th song.AddChord (notes.chord 40 notes.EShape) notes.m2nd song.PlayInstrumentAsync notes.sinewave 65535.0 92 44100

To get the same progression with minors, use the EmShape, etc., values. But, at the end of the day, we've still got sine waves, lots of them. And, it's mostly true that,

                           boring + boring = boring.

I will not attempt the proof. You can make some more examples to see if you can generate a counter-example.

There is also some unpleasant popping that I would like to understand a bit better—for now I'm going to guess it relates to sudden changes in the phases of the waves at the terminations of each section of sound. Not sure what the solution is but I'll guess that some kind of dampening of the amplitude at the end of a wave would help reduce this.