Triangle / Lambdoma
This instrument uses basic 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.
- Resonator mode puts white noise through tuned bandpass filters.
- 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 sine/bandpass mode |
| \ | 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 |
| hyperbolic | Change stride rather than scrolling. Denominator magnifies along the 1/1 line, numerator emphasizes the hyperbolic extremes. |
| 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 fractions, and demonstrates how one can use these ratios to find specific intervals. I had never seen just intonation demonstrated so elegantly, so I made this page to explore the concept.
I later learned that I had constructed the "lambdoid diagram" or Lambdoma, named for its resemblance to the Greek letter Lambda Λ. The synergy of color and tone, linking the octave to the color wheel, seemed intuitive, and revealed a beautiful pattern, both visual and musical. This pattern had previously been uncovered by artist and sound practitioner Barbara Hero, who built an 8x8 electronic Lambdoma instrument for sound healing purposes, using the same pattern of colors.
Hero learned of the Lambdoma from Tone: A Study in Musical Acoustics (1968) by Levarie and Levy, who trace the Lambdoma back to Pythagoras (ca. 500 BCE) by way of the Introduction to Arithmetic by Nicomachus of Gerasa (ca. 100 BCE) and the Theologumena arithmeticae of Iamblichus (ca. 300 CE). The Lambdoma is also mentioned by Plutarch (ca. 100 CE) in his commentary on Plato's Timaeus. It was depicted in the 19th century by Albert von Thimus in the neo-Pythagorean treatise Die harmonikale Symbolik des Alterthums (1876) which connects musical intervals to other harmonic relationships in nature. The Lambdoma was also used by mathematician Georg Cantor in his theory of transfinite sets (see below). More information can be gleaned from Hero's paper, The Lambdoma Matrix and Harmonic Intervals (1999).
the mathematics of perception
With the root, fifth, fourth, and octave in the top-left corner, the Lambdoma shows how the 3:2 proportion is basic to musical perception.
The musical Circle of Fifths, created by repeatedly multiplying a frequency by 3:2, can be studied in more detail in this program's Pythagorean mode, where each ratio is a power of 2 or 3. Similar notes can be found by color and compared. One can easily hear how stacking fifths does not bring you back to the starting note. Find two far-apart red notes and play both at once. These two frequencies are not quite the same, and they will audibly vibrate or "beat" against each other. The interval between these notes is known as the "syntonic comma", and tuning systems try to correct for it in various ways.
Within a Pythagorean tuning system, each scale can sound highly
distinctive, since the notes will not be distributed evenly within the
octave. In a sense, 12-tone equal temperament "bends" all of the notes
so they are evenly spaced. Fifths in equal temperament are not quite
pure but close enough, making it easy to modulate between keys. By
comparison, thirds are quite out of tune compared to a pure ratio
(5:4). Equal temperament invokes irrational numbers, creating
in-between intervals which do not exist on the Lambdoma. (Notes in
equal temperament are separated by an irrational ratio of 1 to 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 into a grid by numerator and denominator. One may easily grasp this countable infinity of rationals by considering that, though there are infinitely many rational numbers, 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. Gradient algorithm via Inigo Quizeles. Thanks to Hems for the support!
Jules LaPlace / asdf.us / 2018-2025