Transcendental numbers evade all attempts at capture by finite hands but only reveal themselves at the limits of intuition, as wonderful geometrical projections from infinity. 

This piece unfolds of one of these limiting sequences of Euler’s number e utilizing the method of Diophantine Approximation, i.e. it demonstrates the arithmetic construction of a sequence of rationals converging towards the transcendental number e, including semi-convergents. Each convergent is compared with the digital approximation of e in both frequency and time domains.

...11/4, 30/11, 49/18, 68/25, 87/32...

त्रिपुरा सुन्दरी {p/q: |e - p/q| < |e - p'/q'|} (Tripurā Sundarī)

भैरवी (Bhairavi)
Nirantar Yakthumba
[voice, clarinet, cello, piano, tuned metal bars]
2023

Purnima 
June 2023
Orgelpark, Amsterdam

How can the repetition of a simple and singular relation produce an unprecedented difference? The differential line stretches out in a fixed direction but eventually returns to its origin, almost intersecting itself and repeating the same path. Difference, however, accumulates along this path. As it circles back, the line is catapulted at another angle, like a precessing orbit, periodically repeating but never the same.

1/1
4/3
8/9
32/27
64/81
256/243
1024/729
2048/2187
8192/6561
16384/19683
65538/59049
131072/177147
524288/531441

Performed by Kali:
Kristia Michael - voice
Giuseppe Sapienza - clarinet
Beste Yıldız - cello
Joel Gester Suárez - piano
Nirantar Yakthumba - tuned metal bars

भैरवी (Bhairavi) {3ⁿ} [live excerpt]

बगलामुखी (Bagalāmukhi)
nirantar
[baroque traverso, live electronics]
2024

performed by Kali:
Eduardo Gaspar Polo Baader – baroque traverso
nirantar – live electronics

Resonances
May 2024
Maranatha Church, The Hague

Navigating the harmonic field exponentially generated by the 31st root of 2. Discovering simple connections within remote relations and articulating latent potentials of melody.

बगलामुखी (Bagalāmukhi) {xⁿ: x³¹ - 2 = 0} [live excerpt]

कमला (Kamalā)
nirantar
[harmonium, live electronics]

Crawl Space
October 2024
The Hague

20:22:24:26:27:28:30:32:33:36:39

how many paths are there across a discrete field?

performed by nirantar

कमला (Kamalā) {p/q: 32, 33, 36, 39, 40, 42, 44, 48, 52, 54, 56, 60} [live excerpt]

बगलामुखी (Bagalāmukhi)
nirantar
[voice, clavichord, zither]
2024

performed by:
Kristia Michael – voice, zither
Alex Mastichiadis – clavichord

In this work, we find methods to organize a gamut of twenty-four tones generated with an exponential unit: namely, the fourth-root of the fifth harmonic. Beyond generating the gamut, if we take this unit as an organizational structure that circumscribes various tonal configurations within it, one finds a vast tapestry of tonal figures, interconnected by this unit, that resonate with various traditional tonal organizations—most notably, Arabic and Turkish maqam, Persian dastgāh, Byzantine octoechos, Greek harmoniai, and Hindustani rāga—without direct referentiality to any particular tradition. 

This work is an ongoing charting of a fractal roadmap within this torus-like tonal field, where multiple paths circle back to the same harmonic regions. Kristia and Alexis walk together through this field, marking landmarks with melodic fragments, and repeating and varying the phrases that the other performs as a way to assure each other as to where they are at any given moment. 

In this way, listening becomes a relation of space and movement: taking small steps together and determining the size of these steps; and listening also becomes a relation of time and memory: taking the next step together when it is clear where one is and where one should to go towards.

nirantar

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