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 ...), while the undertone series (1/1, 1/2, 1/3, ...) is shown by the columns. The resulting intervals are instantly musically meaningful, though they arise from simple ratios.
Brightness indicates octave. Color indicates position in the octave, with red being the root or unison interval 1/1.
- 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 understand the concept more deeply.
I later learned that I had recreated the Lambdoma as described by Barbara Hero. Hero made an electronic lambdoma instrument for sound healing purposes. She traces the Lambdoma back to the Introduction to Arithmetic by Nichomachus of Gerasa (ca. 100 CE), and suggests it was rediscovered by Albert von Thimus who depicts it in Die harmonikale Symbolik des Alterthums (1876). More can be read in Hero's article, The Lambdoma Matrix and Harmonic Intervals (1999). In the Lambdoma, Hero also sees the image of Georg Cantor's transfinite set of rational numbers ℚ, which Cantor showed to be countably infinite through use of a Cartesian plot (versus the uncountable continuity of real numbers ℝ).
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 "Pythagorean" mode. Similar notes can be found by color and compared, and one can easily hear how stacked fifths overshoot the octave by finding the next red note.
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