Just Intonation Maps
• Twelve-Tones Map:
| Interval | Ratio | Cents | ET Correction | % of deviation |
|---|---|---|---|---|
| Unison | 1:1 | 0.00 | 0.00 | 0.00% |
| Minor Second | 16:15 | 111.73 | +11.73 | 11.72% |
| Major Second | 9:8 | 203.91 | +3.91 | 1.96% |
| Minor Third | 6:5 | 315.64 | +15.64 | 5.21% |
| Major Third | 5:4 | 386.31 | -13.69 | 3.42% |
| Perfect Fourth | 4:3 | 498.04 | -1.96 | 0.39% |
| Tritone | 45:32 | 590.22 | -9.78 | 1.63% |
| Perfect Fifth | 3:2 | 701.96 | +1.96 | 0.28% |
| Minor Sixth | 8:5 | 813.69 | +13.69 | 1.71% |
| Major Sixth | 5:3 | 884.36 | -15.64 | 1.74% |
| Minor Seventh | 16:9 | 996.09 | -3.91 | 0.39% |
| Major Seventh | 15:8 | 1088.27 | -11.73 | 1.06% |
| Octave | 2:1 | 1200.00 | 0.00 | 0.00% |
Strengths: Produces perfectly pure triadic harmonies with zero beating. Major thirds (5:4) and minor thirds (6:5) create the distinctive "locked-in" sound of just intonation. Simple ratios make this system intuitive and historically grounded.
Weaknesses: Severely limited modulation capabilities due to comma pump effects. Different keys sound dramatically different, making seamless key changes impossible. The Pythagorean comma creates tuning inconsistencies across the circle of fifths.
• Twenty-Four Tones Map:
| Interval | Ratio | Cents | ET Correction | % of deviation |
|---|---|---|---|---|
| Unison | 1:1 | 0.00 | 0.00 | 0.00% |
| Small Semitone | 25:24 | 70.67 | +20.67 | 41.34% |
| Large Semitone | 16:15 | 111.73 | +11.73 | 11.72% |
| Small Tone | 10:9 | 182.40 | -17.60 | 8.80% |
| Large Tone | 9:8 | 203.91 | +3.91 | 1.96% |
| Minor Third (low) | 32:27 | 294.13 | -5.87 | 1.96% |
| Minor Third | 6:5 | 315.64 | +15.64 | 5.21% |
| Neutral Third | 11:9 | 347.41 | -52.59 | 13.15% |
| Major Third | 5:4 | 386.31 | -13.69 | 3.42% |
| Perfect Fourth | 4:3 | 498.04 | -1.96 | 0.39% |
| Augmented Fourth | 11:8 | 551.32 | -48.68 | 8.11% |
| Tritone | 45:32 | 590.22 | -9.78 | 1.63% |
| Diminished Fifth | 64:45 | 609.78 | +9.78 | 1.63% |
| Perfect Fifth | 3:2 | 701.96 | +1.96 | 0.28% |
| Minor Sixth | 8:5 | 813.69 | +13.69 | 1.71% |
| Golden Ratio | φ:1 | 833.09 | -66.91 | 7.43% |
| Major Sixth | 5:3 | 884.36 | -15.64 | 1.74% |
| Minor Seventh (low) | 16:9 | 996.09 | -3.91 | 0.39% |
| Minor Seventh | 9:5 | 1017.60 | +17.60 | 1.73% |
| Major Seventh | 15:8 | 1088.27 | -11.73 | 1.06% |
| Leading Tone | 48:25 | 1129.33 | -20.67 | 1.83% |
| Octave | 2:1 | 1200.00 | 0.00 | 0.00% |
Strengths: Provides both syntonic and Pythagorean versions of intervals, allowing for more sophisticated harmonic progressions. Neutral thirds (11:9) open up Middle Eastern and North African musical possibilities. Better accommodation of different key centers than 12-tone JI.
Weaknesses: Increased complexity makes practical implementation challenging. Still suffers from comma-related modulation issues, though less severely than 12-tone. The abundance of microtonal intervals can overwhelm performers unfamiliar with extended just intonation.
• Thirty-One Equal Temperament Map:
| Interval | Ratio (approx) | Cents | JI Correction | % of deviation |
|---|---|---|---|---|
| Unison | 1:1 | 0.00 | 0.00 | 0.00% |
| Diesis | 128:125 | 38.71 | +2.33 | 6.41% |
| Small Semitone | 25:24 | 77.42 | +6.75 | 9.55% |
| Diatonic Semitone | 16:15 | 116.13 | +4.40 | 3.94% |
| Chromatic Semitone | 135:128 | 154.84 | +62.66 | 67.95% |
| Minor Tone | 10:9 | 193.55 | +11.15 | 6.11% |
| Major Tone | 9:8 | 232.26 | +28.35 | 13.91% |
| Pythagorean Minor Third | 32:27 | 270.97 | -23.16 | 7.87% |
| Augmented Second | 75:64 | 309.68 | +35.10 | 12.78% |
| Diminished Third | 256:225 | 348.39 | -31.02 | 8.20% |
| Minor Third | 6:5 | 387.10 | +0.79 | 0.25% |
| Augmented Third | 125:96 | 425.81 | +39.50 | 10.22% |
| Major Third | 5:4 | 464.52 | -78.21 | 16.84% |
| Perfect Fourth | 4:3 | 503.23 | +5.19 | 1.05% |
| Augmented Fourth | 45:32 | 541.94 | -48.28 | 8.18% |
| Tritone | 7:5 | 580.65 | -2.52 | 0.43% |
| Diminished Fifth | 10:7 | 619.35 | +2.52 | 0.41% |
| Diminished Sixth | 64:45 | 658.06 | +48.28 | 7.91% |
| Perfect Fifth | 3:2 | 696.77 | -5.19 | 0.74% |
| Augmented Fifth | 25:16 | 735.48 | -20.78 | 2.75% |
| Minor Sixth | 8:5 | 774.19 | -39.50 | 4.85% |
| Augmented Sixth | 225:128 | 812.90 | +31.02 | 3.96% |
| Major Sixth | 5:3 | 851.61 | -32.75 | 3.70% |
| Diminished Seventh | 128:75 | 890.32 | -35.10 | 3.79% |
| Pythagorean Major Sixth | 27:16 | 929.03 | +23.16 | 2.56% |
| Minor Seventh | 16:9 | 967.74 | -28.35 | 2.85% |
| Harmonic Seventh | 9:5 | 1006.45 | -11.15 | 1.09% |
| Augmented Seventh | 125:64 | 1045.16 | -2.33 | 0.22% |
| Major Seventh | 15:8 | 1083.87 | -4.40 | 0.40% |
| Leading Tone | 48:25 | 1122.58 | -6.75 | 0.60% |
| Octave | 2:1 | 1200.00 | 0.00 | 0.00% |
Strengths: Excellent approximation of 5-limit just intonation with nearly perfect major and minor thirds. Allows for extensive modulation while maintaining consistent interval sizes. The equal division provides predictable tuning and easy transposition to any key center. Works exceptionally well for Renaissance and Baroque music styles.
Weaknesses: High complexity requires specialized instruments or sophisticated electronic control. The 31-fold division can be overwhelming for traditional musicians. Still an equal temperament compromise, so intervals are not perfectly just. Limited availability of instruments capable of 31-tone performance.
Understanding the Deviations
The percentage deviation column shows how much equal temperament differs from just intonation as a proportion of each interval's size. The minor second shows the largest relative deviation at 11.72%, while the perfect fifth and fourth show the smallest deviations (under 0.4%).
The most significant absolute deviations are the major third (14 cents flat) and minor third/major sixth (16 cents sharp). These adjustments create the pure, beatless intervals that define just intonation's distinctive sound quality.
Practical Application
Use these maps when tuning acoustic instruments, programming synthesizers for microtonal work, or analyzing the harmonic content of just intonation compositions. The percentage deviations help illustrate why some intervals sound more "out of tune" in equal temperament than others. Choose the system that best fits your musical needs: 12-tone for simple triadic music, 24-tone for enhanced harmonic possibilities, or 31-tone for maximum just intonation accuracy with modulation capabilities.
