Guitar String Tone Optimization

Summary

This experiment investigates guitar string tone optimization. Box-Behnken design to maximize brightness and sustain by tuning string gauge, action height, and pickup height.

The design varies 3 factors: gauge thou (thou), ranging from 9 to 13, action mm (mm), ranging from 1.5 to 3.0, and pickup mm (mm), ranging from 2 to 5. The goal is to optimize 2 responses: brightness (pts) (maximize) and sustain sec (sec) (maximize). Fixed conditions held constant across all runs include guitar type = solid_body, tuning = standard.

A Box-Behnken design was chosen because it efficiently fits quadratic models with 3 continuous factors while avoiding extreme corner combinations — requiring only 15 runs instead of the 8 needed for a full factorial at two levels.

Quadratic response surface models were fitted to capture potential curvature and factor interactions. The RSM contour plots below visualize how pairs of factors jointly affect each response.

Key Findings

For brightness, the most influential factors were gauge thou (45.3%), pickup mm (33.6%), action mm (21.1%). The best observed value was 8.8 (at gauge thou = 11, action mm = 2.25, pickup mm = 3.5).

For sustain sec, the most influential factors were action mm (48.8%), pickup mm (32.9%), gauge thou (18.3%). The best observed value was 5.3 (at gauge thou = 11, action mm = 3, pickup mm = 5).

Recommended Next Steps

Run confirmation experiments at the predicted optimal settings to validate the model.
Consider whether any fixed factors should be varied in a future study.

Experimental Setup

Factors

Factor	Low	High	Unit
`gauge_thou`	9	13	thou
`action_mm`	1.5	3.0	mm
`pickup_mm`	2	5	mm

Fixed: guitar_type = solid_body, tuning = standard

Responses

Response	Direction	Unit
`brightness`	↑ maximize	pts
`sustain_sec`	↑ maximize	sec

Configuration

use_cases/157_guitar_string_tone/config.json

{
  "metadata": {
    "name": "Guitar String Tone Optimization",
    "description": "Box-Behnken design to maximize brightness and sustain by tuning string gauge, action height, and pickup height"
  },
  "factors": [
    {
      "name": "gauge_thou",
      "levels": [
        "9",
        "13"
      ],
      "type": "continuous",
      "unit": "thou"
    },
    {
      "name": "action_mm",
      "levels": [
        "1.5",
        "3.0"
      ],
      "type": "continuous",
      "unit": "mm"
    },
    {
      "name": "pickup_mm",
      "levels": [
        "2",
        "5"
      ],
      "type": "continuous",
      "unit": "mm"
    }
  ],
  "fixed_factors": {
    "guitar_type": "solid_body",
    "tuning": "standard"
  },
  "responses": [
    {
      "name": "brightness",
      "optimize": "maximize",
      "unit": "pts"
    },
    {
      "name": "sustain_sec",
      "optimize": "maximize",
      "unit": "sec"
    }
  ],
  "settings": {
    "operation": "box_behnken",
    "test_script": "use_cases/157_guitar_string_tone/sim.sh"
  }
}

Experimental Matrix

The Box-Behnken Design produces 15 runs. Each row is one experiment with specific factor settings.

Run	`gauge_thou`	`action_mm`	`pickup_mm`
1	11	1.5	2
2	11	2.25	3.5
3	13	2.25	5
4	13	2.25	2
5	11	2.25	3.5
6	11	2.25	3.5
7	9	2.25	5
8	13	1.5	3.5
9	11	1.5	5
10	13	3	3.5
11	9	2.25	2
12	11	3	5
13	9	1.5	3.5
14	9	3	3.5
15	11	3	2

Step-by-Step Workflow

1

Preview the design

Terminal

$ doe info --config use_cases/157_guitar_string_tone/config.json

2

Generate the runner script

Terminal

$ doe generate --config use_cases/157_guitar_string_tone/config.json \
    --output use_cases/157_guitar_string_tone/results/run.sh --seed 42

3

Execute the experiments

Terminal

$ bash use_cases/157_guitar_string_tone/results/run.sh

4

Analyze results

Terminal

$ doe analyze --config use_cases/157_guitar_string_tone/config.json

5

Get optimization recommendations

Terminal

$ doe optimize --config use_cases/157_guitar_string_tone/config.json

6

Multi-objective optimization

With 2 competing responses, use --multi to find the best compromise via Derringer–Suich desirability.

Terminal

$ doe optimize --config use_cases/157_guitar_string_tone/config.json --multi

7

Generate the HTML report

Terminal

$ doe report --config use_cases/157_guitar_string_tone/config.json \
    --output use_cases/157_guitar_string_tone/results/report.html

Features Exercised

Feature	Value
Design type	`box_behnken`
Factor types	`continuous` (all 3)
Arg style	`double-dash`
Responses	2 (brightness ↑, sustain_sec ↑)
Total runs	15

Analysis Results

Generated from actual experiment runs using the DOE Helper Tool.

Response: brightness

Top factors: gauge_thou (45.3%), pickup_mm (33.6%), action_mm (21.1%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
gauge_thou	2	1.3087	0.6543	2.930	0.1110
action_mm	2	0.3498	0.1749	0.783	0.4891
pickup_mm	2	0.8658	0.4329	1.938	0.2059
Lack	of	Fit	6	17.9224	2.9871
Pure	Error	2	0.4467
Error	8	18.3690	0.2233
Total	14	20.8933	1.4924

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Half-Normal Plot of Effects

Model Diagnostics

Response: sustain_sec

Top factors: action_mm (48.8%), pickup_mm (32.9%), gauge_thou (18.3%).

ANOVA

Source	DF	SS	MS	F	p-value
Source	DF	SS	MS	F	p-value
gauge_thou	2	0.2894	0.1447
action_mm	2	2.4954	1.2477
pickup_mm	2	0.8894	0.4447
Lack	of	Fit	6	6.5899	1.0983
Pure	Error	2	0.0000
Error	8	6.5899	0.0000
Total	14	10.2640	0.7331

Pareto Chart

Main Effects Plot

Normal Probability Plot of Effects

Half-Normal Plot of Effects

Model Diagnostics

Response Surface Plots

3D surfaces fitted with quadratic RSM. Red dots are observed data points.

brightness action mm vs pickup mm

brightness gauge thou vs action mm

brightness gauge thou vs pickup mm

sustain sec action mm vs pickup mm

sustain sec gauge thou vs action mm

sustain sec gauge thou vs pickup mm

Multi-Objective Optimization

When responses compete, Derringer–Suich desirability finds the best compromise. Each response is scaled to a 0–1 desirability, then combined via a weighted geometric mean.

Overall Desirability

D = 0.7293

Per-Response Desirability

Response	Weight	Desirability	Predicted	Dir
`brightness`	1.5	0.8012	8.09 0.8012 8.09 pts	↑
`sustain_sec`	1.0	0.6334	4.21 0.6334 4.21 sec	↑

Recommended Settings

Factor	Value
`gauge_thou`	9 thou
`action_mm`	1.5 mm
`pickup_mm`	5 mm

Source: from RSM model prediction

Trade-off Summary

Sacrifice = how much worse than single-objective best.

Response	Predicted	Best Observed	Sacrifice
`sustain_sec`	4.21	5.30	+1.09

Top 3 Runs by Desirability

Run	D	Factor Settings
#12	0.5567	gauge_thou=11, action_mm=2.25, pickup_mm=3.5
#14	0.5438	gauge_thou=11, action_mm=3, pickup_mm=2

Model Quality

Response	R²	Type
`sustain_sec`	0.1603	linear

Full Multi-Objective Output

doe optimize --multi
============================================================
MULTI-OBJECTIVE OPTIMIZATION
Method: Derringer-Suich Desirability Function
============================================================

Overall desirability: D = 0.7293

Response                  Weight Desirability    Predicted  Direction
---------------------------------------------------------------------
brightness                   1.5       0.8012        8.09 pts   ↑
sustain_sec                  1.0       0.6334        4.21 sec   ↑

Recommended settings:
  gauge_thou = 9 thou
  action_mm = 1.5 mm
  pickup_mm = 5 mm
  (from RSM model prediction)

Trade-off summary:
  brightness: 8.09 (best observed: 8.80, sacrifice: +0.71)
  sustain_sec: 4.21 (best observed: 5.30, sacrifice: +1.09)

Model quality:
  brightness: R² = 0.7684 (quadratic)
  sustain_sec: R² = 0.1603 (linear)

Top 3 observed runs by overall desirability:
  1. Run #15 (D=0.6190): gauge_thou=11, action_mm=1.5, pickup_mm=5
  2. Run #12 (D=0.5567): gauge_thou=11, action_mm=2.25, pickup_mm=3.5
  3. Run #14 (D=0.5438): gauge_thou=11, action_mm=3, pickup_mm=2

Full Analysis Output

doe analyze
=== Main Effects: brightness ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
gauge_thou               0.7750       0.3154            45.3%
pickup_mm                0.5750       0.3154            33.6%
action_mm                0.3607       0.3154            21.1%

=== ANOVA Table: brightness ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
gauge_thou                   2       1.3087       0.6543      2.930     0.1110
action_mm                    2       0.3498       0.1749      0.783     0.4891
pickup_mm                    2       0.8658       0.4329      1.938     0.2059
Lack of Fit                  6      17.9224       2.9871     13.375     0.0712
Pure Error                   2       0.4467       0.2233
Error                        8      18.3690       0.2233
Total                       14      20.8933       1.4924

=== Summary Statistics: brightness ===

gauge_thou:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  11                  7     6.7571     1.2095     4.6000     8.6000
  13                  4     6.2000     1.7720     4.9000     8.8000
  9                   4     6.9750     0.6801     6.3000     7.7000

action_mm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1.5                 4     6.8750     1.8062     4.6000     8.8000
  2.25                7     6.5143     0.7734     5.3000     7.4000
  3                   4     6.7250     1.5457     4.9000     8.6000

pickup_mm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2                   4     6.3250     0.7500     5.3000     7.1000
  3.5                 7     6.9000     1.2288     4.9000     8.8000
  5                   4     6.6000     1.7588     4.6000     8.6000

=== Main Effects: sustain_sec ===
Factor                   Effect    Std Error   % Contribution
--------------------------------------------------------------
action_mm                0.8643       0.2211            48.8%
pickup_mm                0.5821       0.2211            32.9%
gauge_thou               0.3250       0.2211            18.3%

=== ANOVA Table: sustain_sec ===
Source                      DF           SS           MS          F    p-value
-----------------------------------------------------------------------------
gauge_thou                   2       0.2894       0.1447                      
action_mm                    2       2.4954       1.2477                      
pickup_mm                    2       0.8894       0.4447                      
Lack of Fit                  6       6.5899       1.0983                      
Pure Error                   2       0.0000       0.0000
Error                        8       6.5899       0.0000
Total                       14      10.2640       0.7331

=== Summary Statistics: sustain_sec ===

gauge_thou:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  11                  7     3.7571     0.6973     2.8000     4.8000
  13                  4     3.7750     1.1325     2.6000     5.3000
  9                   4     3.4500     1.0344     2.2000     4.6000

action_mm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  1.5                 4     3.3500     1.1818     2.2000     4.8000
  2.25                7     4.1143     0.6842     3.1000     5.3000
  3                   4     3.2500     0.5066     2.8000     3.9000

pickup_mm:
  Level               N       Mean        Std        Min        Max
  ------------------------------------------------------------
  2                   4     3.7500     0.7371     2.8000     4.6000
  3.5                 7     3.4429     0.7525     2.2000     4.0000
  5                   4     4.0250     1.2038     2.9000     5.3000

Optimization Recommendations

doe optimize
=== Optimization: brightness ===
Direction: maximize

Best observed run: #11
  gauge_thou = 11
  action_mm = 2.25
  pickup_mm = 3.5
  Value: 8.8

RSM Model (linear, R² = 0.3967, Adj R² = 0.2321):
  Coefficients:
    intercept                      +6.6667
    gauge_thou                     -0.9125
    action_mm                      -0.3250
    pickup_mm                      -0.3125

RSM Model (quadratic, R² = 0.8012, Adj R² = 0.4433):
  Coefficients:
    intercept                      +7.5667
    gauge_thou                     -0.9125
    action_mm                      -0.3250
    pickup_mm                      -0.3125
    gauge_thou*action_mm           -0.8500
    gauge_thou*pickup_mm           -0.2250
    action_mm*pickup_mm            +0.1500
    gauge_thou^2                   -0.0708
    action_mm^2                    -0.4958
    pickup_mm^2                    -1.1208

  Curvature analysis:
    pickup_mm                      coef=-1.1208  concave (has a maximum)
    action_mm                      coef=-0.4958  concave (has a maximum)
    gauge_thou                     coef=-0.0708  negligible curvature

  Notable interactions:
    gauge_thou*action_mm           coef=-0.8500  (antagonistic)

  Predicted optimum (from quadratic model, at observed points):
    gauge_thou = 9
    action_mm = 3
    pickup_mm = 3.5
    Predicted value: 8.4375

  Surface optimum (via L-BFGS-B, quadratic model):
    gauge_thou = 9
    action_mm = 2.64665
    pickup_mm = 3.49453
    Predicted value: 8.5473

  Model quality: Good fit — general trends are captured, some noise remains.

Factor importance:
  1. gauge_thou  (effect: 1.8, contribution: 46.2%)
  2. pickup_mm  (effect: 1.4, contribution: 35.2%)
  3. action_mm  (effect: 0.7, contribution: 18.6%)

=== Optimization: sustain_sec ===
Direction: maximize

Best observed run: #10
  gauge_thou = 11
  action_mm = 3
  pickup_mm = 5
  Value: 5.3

RSM Model (linear, R² = 0.2687, Adj R² = 0.0692):
  Coefficients:
    intercept                      +3.6800
    gauge_thou                     +0.2375
    action_mm                      +0.1125
    pickup_mm                      +0.5250

RSM Model (quadratic, R² = 0.6001, Adj R² = -0.1196):
  Coefficients:
    intercept                      +3.4333
    gauge_thou                     +0.2375
    action_mm                      +0.1125
    pickup_mm                      +0.5250
    gauge_thou*action_mm           +0.4250
    gauge_thou*pickup_mm           -0.2500
    action_mm*pickup_mm            +0.3500
    gauge_thou^2                   -0.3042
    action_mm^2                    +0.1458
    pickup_mm^2                    +0.6208

  Curvature analysis:
    pickup_mm                      coef=+0.6208  convex (has a minimum)
    gauge_thou                     coef=-0.3042  concave (has a maximum)
    action_mm                      coef=+0.1458  convex (has a minimum)

  Notable interactions:
    gauge_thou*action_mm           coef=+0.4250  (synergistic)
    action_mm*pickup_mm            coef=+0.3500  (synergistic)

  Predicted optimum (from linear model, at observed points):
    gauge_thou = 13
    action_mm = 2.25
    pickup_mm = 5
    Predicted value: 4.4425

  Surface optimum (via L-BFGS-B, linear model):
    gauge_thou = 13
    action_mm = 3
    pickup_mm = 5
    Predicted value: 4.5550

  Model quality: Weak fit — consider adding center points or using a different design.

Factor importance:
  1. pickup_mm  (effect: 1.2, contribution: 58.2%)
  2. gauge_thou  (effect: 0.6, contribution: 30.0%)
  3. action_mm  (effect: 0.2, contribution: 11.8%)