Chords, a Plomp-Levelt analysis

Published

June 12, 2022

Status: draft.

We’re hoping to analyze chord consonance/dissonance using the ideas behind Plomp-Levelt curves.

Plomp-Levelt analysis shows sounds are perceived as dissonant when tones, including harmonics, are “near-misses” of each other. If many tones land right on top of one another, then the overall sound is “consonant”. If many tones land near each other, then the overall sound is “dissonant”.

So we can analyze chords as if they were “polyrhythms in frequency”, because each harmonic note “beats” an evenly-spaced (in frequency domain) of tones. Since Plomp-Levelt have shown that dissonance and consonance arise from clashing near-misses of tones, we should expect more consonant chords have more “notes” hitting at the same time.

Basics: intervals

Octaves: (“2:1 polyrhythm”)

Fifths (3:2 polyrhythm)

Fourth, 4:3:

Major third, 5:4:

Minor third, 6:5:

Chords

Triads

Major triad. A 5:4:z and x:6:5. That’s 15:12:10 overall:

Major triad, first inversion. A 4:3:z and x:5:4. That’s 20:15:12 overall:

Major chord, second inversion. A 6:5:z and 5:y:3. That’s 30:25:18 overall:

Major chord as root-fifth-tenth. 3:2:z and x:5:3, overall 15:10:6. Fewer “notes” than basic major triad. Less rich, but more consonant?

Minor chord. A 6:5:z and a 3:y:2. That’s 6:5:4 overall.

Diminished chord. Messy, depends on choice of tritone! Here, 6:5:z and 36:y:25, overall 36:30:25. We don’t show that far.

Sevenths

Dominant seventh, 5:4:z:w, 3:y:2:w, 9:y:z:5. That’s 45:36:30:25, (matching the dimished chord and adding a 45:36 = 5:4 on top). We don’t show that far.

Major seventh, 5:4:z:w, 3:y:2:w, 15:y:z:8. That’s 15:12:10:8. Compare this to the dominant seventh, and the difference is quite clear.

Minor seventh, 6:5:z:w, 3:y:2:w, 9:y:z:5. That’s 18:15:12:10. Again a much simpler ratio than the dominant seventh. This seems a decent explanation for why people think the major and minor sevenths are “stable” and the dominant seventh wants to be resolved somewhere.

Resolutions

V7 -> I in first inversion:

Appendix

Code

function makeDiagram(chords, xMax)
{
    xMax = xMax || 1500;
    const svg = html`<svg width=100% viewBox="0 0 990 ${chords.length * 30 + 30}"></svg>`;
    
    d3.select(svg).append("rect")
        .attr("width", 990).attr("height", 30 * chords.length).attr("fill", "#eee");

    const chordGroups = d3.select(svg)
      .selectAll("g")
      .data(chords)
      .enter()
      .append("g");
    
    const x = d3.scaleLinear().domain([50, xMax + 50]).range([0, 990]);
    const y = d3.scaleLinear().domain([-0.5, chords.length - 0.5]).range([30 * chords.length, 0]);

    chordGroups
      .append("line")
      .attr("x1", 0).attr("x2", 990)
      .attr("y1", (d, i) => y(i)).attr("y2", (d, i) => y(i))
      .attr("stroke", "gray");

    const markGroups = chordGroups
      .selectAll("g")
      .data((d, i) => d3.range(1, 50).map(x => ({ x: x * d, y: i }))) // 30 harmonics
      .enter()
      .append("g");

    markGroups
      .append("circle")
      .attr("r", 2)
      .attr("fill", "black")
      .attr("cx", d => x(d.x))
      .attr("cy", d => y(d.y));
    return svg;
}

Notes

Discouraging: most of this ends up just being “simple ratios sound consonant”. We now have an explanation for why, but the thing itself was already known.

So maybe we need to include even temperament here since they’re not simple ratios but they can get close enough to just that our ears can’t tell. This will require taking the actual Plomp-Levelt curve into account for each note.

We can also take the Pythagorean tuning and predict how chords using wolf intervals will sound dissonant.

Maybe we can do this analysis with closed-tube wind instruments, since they produce a different overtone series? And then we compare that to fully-harmonic instruments?

TODO: compute clashes among tone pairs.

Ideas

What happens if instead of a harmonic sound, we play a super “fat” sound by including a large amount of non-harmonic tones very near one another, so that a dissonance is “lost”? Hmmm…

--- title: Chords, a Plomp-Levelt analysis format: html: default echo: false date: "2022-06-12" --- Status: _draft_. We're hoping to analyze chord consonance/dissonance using the ideas behind Plomp-Levelt curves. Plomp-Levelt analysis shows sounds are perceived as dissonant when tones, _including harmonics_, are "near-misses" of each other. If many tones land right on top of one another, then the overall sound is "consonant". If many tones land _near_ each other, then the overall sound is "dissonant". So we can analyze chords as if they were "polyrhythms in frequency", because each harmonic note "beats" an evenly-spaced (in frequency domain) of tones. Since Plomp-Levelt have shown that dissonance and consonance arise from clashing near-misses of tones, we should expect more consonant chords have more "notes" hitting at the same time. ## Basics: intervals Octaves: ("2:1 polyrhythm") ::: {.column-page} ```{ojs} makeDiagram([100, 200]) ``` ::: Fifths (3:2 polyrhythm) ::: {.column-page} ```{ojs} makeDiagram([100, 150]) ``` ::: Fourth, 4:3: ::: {.column-page} ```{ojs} makeDiagram([100, 133.333333]) ``` ::: Major third, 5:4: ::: {.column-page} ```{ojs} makeDiagram([100, 125]) ``` ::: Minor third, 6:5: ::: {.column-page} ```{ojs} makeDiagram([100, 120]) ``` ::: ## Chords ### Triads Major triad. A 5:4:z and x:6:5. That's 15:12:10 overall: ::: {.column-page} ```{ojs} makeDiagram([100, 125, 150]) ``` ::: Major triad, first inversion. A 4:3:z and x:5:4. That's 20:15:12 overall: ::: {.column-page} ```{ojs} makeDiagram([100, 100 * (4/3), 100 * (4/3) * (5/4)], 2000) ``` ::: Major chord, second inversion. A 6:5:z and 5:y:3. That's 30:25:18 overall: ::: {.column-page} ```{ojs} makeDiagram([100, 100 * (6/5), 100 * (5/3)], 3000) ``` ::: Major chord as root-fifth-tenth. 3:2:z and x:5:3, overall 15:10:6. Fewer "notes" than basic major triad. Less rich, but more consonant? ::: {.column-page} ```{ojs} makeDiagram([100, 150, 150 * 5 / 3]) ``` ::: Minor chord. A 6:5:z and a 3:y:2. That's 6:5:4 overall. ::: {.column-page} ```{ojs} makeDiagram([100, 120, 150]) ``` ::: Diminished chord. Messy, depends on choice of tritone! Here, 6:5:z and 36:y:25, overall 36:30:25. We don't show that far. ::: {.column-page} ```{ojs} makeDiagram([100, 120, 144]) ``` ::: ### Sevenths Dominant seventh, 5:4:z:w, 3:y:2:w, 9:y:z:5. That's 45:36:30:25, (matching the dimished chord and adding a 45:36 = 5:4 on top). We don't show that far. ::: {.column-page} ```{ojs} makeDiagram([100, 125, 150, 180]) ``` ::: Major seventh, 5:4:z:w, 3:y:2:w, 15:y:z:8. That's 15:12:10:8. Compare this to the dominant seventh, and the difference is quite clear. ::: {.column-page} ```{ojs} makeDiagram([100, 125, 150, 100 * 15 / 8]) ``` ::: Minor seventh, 6:5:z:w, 3:y:2:w, 9:y:z:5. That's 18:15:12:10. Again a much simpler ratio than the dominant seventh. This seems a decent explanation for why people think the major and minor sevenths are "stable" and the dominant seventh wants to be resolved somewhere. ::: {.column-page} ```{ojs} makeDiagram([100, 120, 150, 180], 1800) ``` ::: ## Resolutions V7 -> I in first inversion: ::: {.column-page} ```{ojs} makeDiagram([100, 125, 150, 100 * 7 / 4], 3000) makeDiagram([100, 100 * (4/3), 100 * (4/3) * (5/4)], 3000) ``` ::: ## Appendix ### Code ```{ojs} //| echo: true function makeDiagram(chords, xMax) { xMax = xMax || 1500; const svg = html`<svg width=100% viewBox="0 0 990 ${chords.length * 30 + 30}"></svg>`; d3.select(svg).append("rect") .attr("width", 990).attr("height", 30 * chords.length).attr("fill", "#eee"); const chordGroups = d3.select(svg) .selectAll("g") .data(chords) .enter() .append("g"); const x = d3.scaleLinear().domain([50, xMax + 50]).range([0, 990]); const y = d3.scaleLinear().domain([-0.5, chords.length - 0.5]).range([30 * chords.length, 0]); chordGroups .append("line") .attr("x1", 0).attr("x2", 990) .attr("y1", (d, i) => y(i)).attr("y2", (d, i) => y(i)) .attr("stroke", "gray"); const markGroups = chordGroups .selectAll("g") .data((d, i) => d3.range(1, 50).map(x => ({ x: x * d, y: i }))) // 30 harmonics .enter() .append("g"); markGroups .append("circle") .attr("r", 2) .attr("fill", "black") .attr("cx", d => x(d.x)) .attr("cy", d => y(d.y)); return svg; } ``` ### Notes Discouraging: most of this ends up just being "simple ratios sound consonant". We now have an explanation for _why_, but the thing itself was already known. So maybe we need to include even temperament here since they're _not_ simple ratios but they can get close enough to just that our ears can't tell. This will require taking the actual Plomp-Levelt curve into account for each note. We can also take the Pythagorean tuning and predict how chords using wolf intervals will sound dissonant. Maybe we can do this analysis with closed-tube wind instruments, since they produce a different overtone series? And then we compare that to fully-harmonic instruments? TODO: compute clashes among tone pairs. ### Ideas What happens if instead of a harmonic sound, we play a super "fat" sound by including a large amount of non-harmonic tones _very near one another_, so that a dissonance is "lost"? Hmmm...