Triangle / Lambdoma
This instrument uses simple fractions to make a wide palette of just intonation intervals available all at once.
The rows display the overtone series (1/1, 2/1, 3/1 ...). The columns multiply out the undertone series (1/1, 1/2, 1/3, ...). The resulting intervals are instantly musically meaningful, though they arise from simple ratios.
Color indicates position in the octave, with red being the root or unison interval 1/1. Brightness indicates octave, with white and black tending toward the extremes of human hearing.
- The default instrument is a Hokema sansula, a type of kalimba.
- Right-click notes to turn on sine waves matching the interval.
- Drag-and-drop samples into the window to play with your own sounds.
keyboard shortcuts
| ESC | Stop all sound |
| ESC ESC | Return to home position |
| ? | Show this help |
| + - | Change scale |
| up down left right |
Scroll the grid |
| 0-9 a-z | Keyboard mapped to the top-left 8x8 grid, sorted by pitch |
| ? | Toggle this help |
| \ | Detect MIDI device (listening on channel 1) |
| ⇧ + ⇧ - |
Change scale root by +/- 1 hz |
| ⌘⇧ + ⌘⇧ - |
Change scale root by +/- 10 hz |
| ⌘ up ⌘⇧ up ⌘⇧⌃ up |
Change pitch of sampler by +10 / +1 / -0.1 hz |
| ⌘ down ⌘⇧ down ⌘⇧⌃ down |
Change pitch of sampler by -10 / -1 / -0.1 hz |
scales
Alternate scales are accessed by pressing the +/- keys:
| natural | Natural numbers: 1, 2, 3 ... |
| undertone | Subharmonic intervals under the line 1/1 |
| overtone | Harmonic intervals above the line 1/1 |
| primes | Prime numbers only (most dissonant) |
| arithmetic | Multiply all cells by an interval rather than scrolling |
| Collatz | Hailstone numbers of Lothar Collatz |
| Pythagorean |
Pythagorean intervals
where each ratio is a power of 2 |
about this page
This webpage was inspired by Peter Neubäcker, inventor of the Melodyne software. In the short biographical documentary Wie klingt ein Stein? (What does a stone sound like?), Neubäcker describes the basic principles of harmonic intervals. He first demonstrates how one plays harmonics on a monochord. He then shows it next to a grid of whole-number ratios, and demonstrates how one can use these intervals to find specific ratios. I had never seen just intonation demonstrated so elegantly, so I made this page to explore the concept interactively.
I later learned that I had rediscovered the Lambdoma, so called by the ancient Greeks for its resemblance to the letter Lambda. The synergy of color and sound in the Lambdoma, linking the octave to the color wheel, had been studied in depth by artist and sound practitioner Barbara Hero. Hero made the Lambdoma her life's work, and built an 8x8 electronic Lambdoma instrument for sound healing purposes. Hero herself learned of the Lambdoma from Tone: A Study in Musical Acoustics (1968) by Levarie and Levy, which traces the Lambdoma back to Pythagoras (ca. 500 BCE) via the Introduction to Arithmetic by Nichomachus of Gerasa (ca. 100 BCE). They suggest it has been rediscovered several times, including in the 19th century by Albert von Thimus, who depicts it in Die harmonikale Symbolik des Alterthums (1876). More can be gleaned from Hero's paper, The Lambdoma Matrix and Harmonic Intervals (1999).
mathematics and perception
With the root, fifth, and fourth in the top-left corner, the Lambdoma shows how the 3:2 proportion is essential to the perception of consonance. The musical circle of fifths, derived from these simple proportions, can be studied in more detail in this program's Pythagorean scale mode. Similar notes can be found by color and compared. One can easily hear how stacked fifths overshoot the octave by finding two far-apart red notes and playing both at once, which makes them beat against each other. This is the "comma" which is averaged out in various keyboard tuning systems.
12-tone equal temperament is based not in harmonics, but in irrational
numbers. Equal-tempered semitones are separated by a ratio of the 12th
root of 2 (1:
In the Lambdoma, Barbara Hero also sees the image of Georg Cantor's transfinite set of rational numbers ℚ, which Cantor proved countably infinite by arranging fractions along two axes by numerator and denominator. One may easily grasp this countable infinity of rationals by considering that, while there are infinitely many fractions, in between any two there lies an uncountable continuity of real numbers in ℝ.
thank you!
Sansula samples by Freesound user cabled_mess. Thanks to Dave Noyze for telling me about Barbara Hero. Thanks to Hems for the support!
Jules LaPlace / asdf.us / 2018-2025