Frequencies not adding up

Processing Libraries
2

ha, good find

just for reading 2 links of prior discussions
FFT frequencies ,
Not sure how to work with Minim's FFT feature ,
the link to the reference
where obviously most of your code come from
https://processing.org/reference/libraries/sound/FFT.html

and a very interesting line in the processing code
https://github.com/processing/processing-sound/blob/master/src/processing/sound/FFT.java#L39 ,

in words:
in the processing sound lib the bands are really bands,
and not samples == bands * 2
while in minim lib not.
that are the 2 codes i play with:
sound lib:

// https://discourse.processing.org/t/frequencies-not-adding-up/9170

import processing.sound.*;
SoundFile file;
FFT fft;
int bands = 1024;
float[] spectrum = new float[bands];

void setup() {
  size(1024, 512);
//  file = new SoundFile(this, "sin440.wav");
  file = new SoundFile(this, "the_a.mp3");
  file.play();

  fft = new FFT(this, bands);
  fft.input(file);
}

void draw() {
  fft.analyze(spectrum);
  for (int i = 0; i < bands; i++) {
    stroke(0, 0, 0);
/*
    if (i == 2) {
      stroke(0, 0, 255);
    }
    if (i == 5) {
      stroke(0, 255, 255);
    }
    if (i == 10) {
      stroke(0, 255, 0);
    }
    if (i == 20) {
      stroke(255, 0, 0);
    } 
*/
    line( i, height, i, height - spectrum[i]*height*5 );
    if ( keyPressed) if ( i < 30 ) println("i "+i+" amp "+nf(spectrum[i],1,1));
  }
}

minim lib

import ddf.minim.*;
import ddf.minim.analysis.*;

String soundfile = "the_a.mp3"; // "sin440.wav"

Minim minim;
AudioPlayer sound;
FFT spectrum;

float zoom =0.5;

void setup() {
  size(600, 600);
  noStroke();
  fill(255);
  minim = new Minim(this);
  sound = minim.loadFile(soundfile, 1024);
  sound.loop();
  spectrum = new FFT(sound.bufferSize(), sound.sampleRate());
  println("bufferSize() "+sound.bufferSize());
  println("sampleRate() "+sound.sampleRate());
  println("fft.specSize()"+spectrum.specSize());
}

void draw() {
  background(0);
  spectrum.forward(sound.mix);
  for (int i = 0; i < spectrum.specSize(); i++) {
    float x = map(i, 0, spectrum.specSize(), 0, width);
    rect(x, height-10, width/spectrum.specSize(), -spectrum.getBand(i)*zoom);
    if ( keyPressed) if ( i < 30) println("i "+i+" amp "+nf(spectrum.getBand(i), 1, 1));
  }
}

sound lib max amp at i 20 ( at bands = 1024 it uses 2048 samples )
minim lib max amp at i 10 ( use 1024 samples and give 513 values ( 512 bands ))

also i assume that both use DFT math,
while here
FFT of 1D array of 1024 measurements ,
i play with a complex algorithm i copy
and that does have 2 values per band.
( that was my first idea when i read your question )

i tested with my 440Hz mp3 file but
also i used your online link and did a crosscheck

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