unequaltemperament

Top 10 Temperaments from PianoScope + CyberTuner

This page applies a Fred Sturm-style simplification: many temperament names are reduced into recurring structural patterns (“buckets”) using fifth-deviation signatures.

Input temperaments: 192 · Structural buckets found: 140 · Top buckets displayed: 10

Research Series: from Top-10 buckets to Top-20 emotional profiles

A new article series reframes this page as data-science research inspired by Fred Sturm: ranking temperaments, modeling beat-rate behavior in thirds and fifths, and predicting emotional color for all 24 major/minor triads.

Series overviewRoadmap for methods, assumptions, and outcomes. Top-20 methodologyScoring framework, robustness checks, and uncertainty. Chord brightness mapHow each temperament colors the 12 major and 12 minor triads. Emotion from beat-rate spectrumFrom calm to manic brightness, and minor-mode counterparts.

Math used for selection (typeset)

For each temperament and each fifth, we measure deviation from a pure fifth and then construct a rotation-invariant fingerprint.

Fifth deviation
\(d_i = f_i - 701.955\)
Quantization
\(q_i = \mathrm{round}(d_i/0.5)\cdot 0.5\)
Rotation-invariant signature
\(\sigma = \min_{r \in \{0,\dots,11\}} \operatorname{rot}_r(\mathbf q)\)
Coverage
\(\mathrm{coverage}=\frac{\lvert\mathcal D_{bucket}\rvert}{2}\)
Cohesion
\(\mathrm{cohesion}=\frac{1}{1+\overline{\mathrm{RMS}}}\)
Bucket score
\(\mathrm{score}=n\cdot\bigl(1+0.5(\mathrm{coverage}-1)\bigr)\cdot\mathrm{cohesion}\)

The equation typography is rendered with KaTeX for readability and publication-quality mathematical notation.

Top 10 representative temperaments

Rank Representative Temperament Graph Links Dataset Bucket Size Score
1 Tuner's Guide (Marsh plan) Graph Editor Scale Explorer CyberTuner 6 4.4969
2 Vallotti-Young Graph Editor Scale Explorer PianoScope 7 3.0950
3 2/7 Syntonic Graph Editor Scale Explorer PianoScope 3 2.9635
4 Broadwood's Best - Ellis #4 Graph Editor Scale Explorer PianoScope 3 2.9595
5 1/4 Syntonic Graph Editor Scale Explorer PianoScope 3 2.9505
6 1/5 Syntonic - Holden Graph Editor Scale Explorer PianoScope 3 2.9381
7 Young Graph Editor Scale Explorer PianoScope 3 2.8735
8 1/3 Syntonic Graph Editor Scale Explorer PianoScope 2 2.0000
9 1/6 Syntonic - Silbermann Graph Editor Scale Explorer PianoScope 2 2.0000
10 Broadwood's Best Graph Editor Scale Explorer PianoScope 2 2.0000

Top buckets and why they ranked highly

# Signature (quantized deviation in cents) Datasets Mean RMS Cohesion Score Sample Members
1 [-2.5, -2.0, -2.5, -1.5, -2.0, -1.5, -2.0, -2.5, -1.5, -2.0, -1.5, -2.0] CyberTuner PianoScope 0.3343 0.7495 4.4969 Graupner (PianoScope); Hummel Equal-Beating (PianoScope); Hummel Viennese (PianoScope); Jousse Quasi Equal (PianoScope)
2 [-4.0, -4.0, -4.0, -4.0, -4.0, -4.0, +0.0, +0.0, +0.0, +0.0, +0.0, +0.0] CyberTuner PianoScope 1.2617 0.4421 3.0950 Anton Bemetzrieder (CyberTuner); Francesco Antonio Vallotti (CyberTuner); Sifted Vallotti or Young No.2 (CyberTuner); Vallotti (PianoScope)
3 [-6.0, -6.0, -6.0, -6.0, -6.0, -6.0, -6.0, -6.0, -6.0, -6.0, -6.0, +44.0] CyberTuner PianoScope 0.0123 0.9878 2.9635 2/7 S.c. Extended - meantone (CyberTuner); 2/7 Syntonic (PianoScope); Gioseffo Zarlino 2/7 syntonic (CyberTuner)
4 [-5.0, -4.0, -1.0, -1.0, -1.0, -1.0, -1.0, -1.0, -1.0, -2.0, -2.0, -4.0] CyberTuner PianoScope 0.0137 0.9865 2.9595 Boadwood Best (Jorgensen) (CyberTuner); Broadwood's Best - Ellis #4 (PianoScope); Ellis tuner#4 Broadwood Best (CyberTuner)
5 [-5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, -5.5, +35.5] CyberTuner PianoScope 0.0168 0.9835 2.9505 1/4 S.c. Standard - meantone (CyberTuner); 1/4 Syntonic (PianoScope); Pietro Aaron (1/4 syntonic) (CyberTuner)
6 [-4.5, -4.5, -4.5, -4.5, -4.5, -4.5, -4.5, -4.5, -4.5, -4.5, -4.5, +24.0] CyberTuner PianoScope 0.0211 0.9794 2.9381 1/5 S.c. Homogeneous - meantone (CyberTuner); 1/5 Syntonic - Holden (PianoScope); John Holden 1/5 syntonic 1&2 (CyberTuner)
7 [-4.0, -4.0, -4.0, -4.0, -2.0, -2.0, +0.0, +0.0, +0.0, +0.0, -2.0, -2.0] CyberTuner PianoScope 0.0440 0.9578 2.8735 Representative 18th-century (CyberTuner); Young (PianoScope); Young No. 1 (CyberTuner)
8 [-7.0, -7.0, -7.0, -7.0, -7.0, -7.0, -7.0, -7.0, -7.0, -7.0, -7.0, +55.5] CyberTuner PianoScope 0.0000 1.0000 2.0000 1/3 Syntonic (PianoScope); 13 S.C. Minor 3rds - meantone (CyberTuner)
9 [-3.5, -3.5, -3.5, -3.5, -3.5, -3.5, -3.5, -3.5, -3.5, -3.5, -3.5, +16.0] CyberTuner PianoScope 0.0000 1.0000 2.0000 1/6 S.C. Attenuated - meantone (CyberTuner); 1/6 Syntonic - Silbermann (PianoScope)
10 [-3.0, -3.0, -2.0, -1.0, -2.0, -2.5, -2.5, -2.0, -2.0, -1.0, -1.5, -1.5] CyberTuner PianoScope 0.0000 1.0000 2.0000 Broadwood's Best (PianoScope); Ellis tuner #5 Broadwood Best (CyberTuner)

Embedded graphics for the top temperaments

These embeds mirror the graph/editor pages you referenced, with direct links for each temperament when an SCL ID could be resolved.

1. Tuner's Guide (Marsh plan) (ID 0)

wgm_graphs.php
temperament_editor.php

2. Vallotti-Young (ID 43)

wgm_graphs.php
temperament_editor.php

3. 2/7 Syntonic (ID 10)

wgm_graphs.php
temperament_editor.php

4. Broadwood's Best - Ellis #4 (ID 54)

wgm_graphs.php
temperament_editor.php

5. 1/4 Syntonic (ID 9)

wgm_graphs.php
temperament_editor.php

6. 1/5 Syntonic - Holden (ID 17)

wgm_graphs.php
temperament_editor.php

7. Young (ID 44)

wgm_graphs.php
temperament_editor.php

8. 1/3 Syntonic (ID 0)

wgm_graphs.php
temperament_editor.php

9. 1/6 Syntonic - Silbermann (ID 0)

wgm_graphs.php
temperament_editor.php

10. Broadwood's Best (ID 80)

wgm_graphs.php
temperament_editor.php