Preset group · Fibonacci · ratios
Fibonacci between two tones. Closer and closer to φ.
If you divide each Fibonacci number by the one before it, the result gets closer to 1.618…, the golden number, φ. Kepler already noticed: 5 is to 8 almost as 8 is to 13. Sacred geometry calls φ the divine proportion.
This group turns that idea into an interval: two tones playing at once, one in each ear, closer and closer to φ.
The formulas and the figures in the code.
The low tone is always 220 Hz (A3) in the left ear. The high one, in the right, is 220 × F(n+1)/F(n): 1:1, 2:1, 3:2, 5:3, 8:5, 13:8, 21:13, 34:21, 55:34 and 89:55. The last preset is exact φ, 1.618034, with the high tone at 356 Hz.
F(n+1)/F(n) heads toward φ without ever reaching it: 21:13 sits 2.84 cents away, 34:21 1.08, 55:34 0.41 and 89:55 0.16. The ear tells apart about 5 cents: from 21:13 on, they sound the same as φ.
One line per preset.
- 1:1 (1.000)
220 Hz
1:1, unison: both ears with the same tone, a single line in XY.
Physical figure (6, 2) · No layer
- 2:1 (2.000)
220 Hz + 440 Hz
2:1, the octave: the high tone at double.
Physical figure (6, 2) · No layer
- 3:2 (1.500)
220 Hz + 330 Hz
3:2, the just fifth: the curve closes in three lobes by two.
Physical figure (6, 2) · No layer
- 5:3 (1.667)
220 Hz + 366.7 Hz
5:3, the just major sixth.
Physical figure (6, 2) · No layer
- 8:5 (1.600)
220 Hz + 352 Hz
8:5, the just minor sixth, already close to φ.
Physical figure (6, 2) · No layer
- 13:8 (1.625)
220 Hz + 357.5 Hz
13:8, you can still hear the distance to φ.
Physical figure (6, 2) · No layer
- 21:13 (1.615)
220 Hz + 355.4 Hz
21:13, from here on the ear can't separate it from φ.
Physical figure (6, 2) · No layer
- 34:21 (1.619)
220 Hz + 356.2 Hz
34:21, only XY shows the difference.
Physical figure (6, 2) · No layer
- 55:34 (1.618)
220 Hz + 355.9 Hz
55:34, the curve takes a long time to close.
Physical figure (6, 2) · No layer
- 89:55 (1.618)
220 Hz + 356 Hz
89:55, the closest to φ in the sequence.
Physical figure (6, 2) · No layer
- φ 1.618034
220 Hz + 356 Hz
Exact φ: the curve that never closes.
Physical figure (6, 2) · Symbolic layer: Golden Spiral
Tab. 01 · Plate figure calculated by the app's core (cym.core.js) for each frequency: a square plate driven at the center, f = 5.5·(n² + m²) Hz. Two modes = the plate answers with both.
The physical figure and the symbolic layer.
In Chladni the plate follows both tones: the low one, 220 Hz, lands on mode (6, 2), and the high one changes with each ratio (with φ, mode (8, 0)). The real difference shows in XY: one tone against the other draws a Lissajous curve. With 3:2 it closes and stays still; with φ it never closes.
Only φ carries a layer, the Golden Spiral: symbolic, not calculated from the sound.
What people ask about this group.
Q1Why do Fibonacci ratios get closer to φ?
Because each number is the sum of the two before it, and that rule pushes the division between neighbors toward 1.618…, the golden number. The further along the sequence, the closer.
Q2Why do 21:13 and φ sound the same?
Because the difference is a few cents and the ear tells apart about 5. The app tells you under the preset's name, and in XY you can still see the difference.
Q3What is a Lissajous curve?
The figure you get by drawing one tone against another, one horizontal and one vertical. If the ratio is simple, it closes; if it's φ, it turns without closing.