G-B-D-F-G-A
• OVERVIEW •
My basic setting of overtones 4 to 9 in the harmonic series – if ‘squashed’ to 12-tone equal temperament, they spell out a dominant 9th arpeggio as an ascent-ordered stack of thirds (1-3-5-b7-9). Also see Sethares’ variant: which keeps tension within more sensible bounds by adding a register-jump in the middle (C-E-G-Bb-C-D). Make up your own!
n.b. To go deeper into the harmonic series, try fine-tuning to ‘just intonation’ – the so-called ‘pure’ harmonic intervals. On a cent tuner, the modifications are as follows – B5:-14 | D4:+2 | F3:-31 | A1: +4). Or for extra ear enrichment, you can try tuning to these overtonal shades by ear: see the G Sa-Pa (root-5th) sample on my Tanpuras: Divine Indian Drones page.
Pattern: 4>3>3>2>2
Harmony: G9 | 1-3-5-b7-1-2
• TUNING TONES •
• SOUNDS •
Unused on record in its exact form (as far as I can tell) – although it seems like tuning explorer Bill Sethares has played around in his own shuffling of the same ‘harmonic series‘ concept, likely on a MIDI guitar setup of some description. I’ll re-string & record a demo sometime…also send me your experiments and I’ll showcase the best!
‘Harmonic series’ (from my Audio Glossary): “The subset of overtones produced by exact whole-number ratios of the sound source’s fundamental. The harmonic series comprises the (theoretically infinite) progression of these harmonics, which are produced by dividing the string into ever-smaller fractions (i.e. open string 1/1=fundamental, 1/2=octave, 1/3=fifth, 1/4=second octave, 1/5=major 3rd, etc). The resulting tones match the ‘pure’ intervals of just intonation, rather than the equally-tempered frequencies of the frets. Hear the overtones of the open 6str here, (i.e. the N.H. at <12>, <7>, <5>, <3.8>, <3.2>, <2.7>, <2.3>)…”
- Musical Phonology: Overtones – Leonard Bernstein (1974)
“What we’re trying for is a high overview of musical development, in terms of a vocabulary constantly being enriched by more and more remote and chromatic overtones. It’s as if we could see the whole of music developing from prehistory to the present, in two minutes…” (Leonard Bernstein)
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• NUMBERS •
| 6str | 5str | 4str | 3str | 2str | 1str | |
| Note | G | B | D | F | G | A |
| Alteration | +3 | +2 | 0 | -2 | -4 | -7 |
| Tension (%) | +41 | +26 | 0 | -21 | -37 | -55 |
| Freq. (Hz) | 98 | 123 | 147 | 175 | 196 | 220 |
| Pattern (>) | 4 | 3 | 3 | 2 | 2 | – |
| Semitones | 0 | 4 | 7 | 10 | 12 | 14 |
| Intervals | 1 | 3 | 5 | b7 | 1 | 2 |
- See my Tunings Megatable for further such nerdery: more numbers, intervallic relations, comparative methods, etc. And to any genuine vibratory scientists reading: please critique my DIY analysis!
• RELATED •
—Associated tunings: proximities of shape, concept, context, etc…
- Banjo: also begins with a high-twisted G-B-D across 6-5-4str
- All Minor Thirds: regular stacking from a close fundamental
- Zigzag Thirds (Min.): another narrow conceptual experiment
• MORE INFO •
—Further learnings: sources, readings, lessons, other onward links…
- Sethares’ overtonal variant: the master tuning theorist discusses his own overtonal arrangement on p.48 of his Instrumental Tunings overview (“…our tempered aural thinking can include tones up to the sixth partial, but beyond that the point the aural perception is merely rational…”)
- More overtonal concepts: see my Glossary entries for ‘harmonic series‘, ‘overtone‘, ‘ratio‘, and so on – and also my Overtonal Scale Explorer and Fretless Fretboard articles. Although, let’s be honest, Bernstein’s lecture notes are always gonna be a more erudite starting point (“…centuries pass, and inevitably the next interval of the harmonic series is assimilated by humanity: namely the fifth…”)
Header image: fractional oscillations of the overtone series
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George Howlett is a London-based musician, writer, and teacher (guitars, sitar, tabla, & santoor). Above all I seek to enthuse fellow sonic searchers, interconnecting fresh vibrations with the voices, cultures, and passions behind them. See Home & Writings, and hit me up for Online Lessons!
“An intrepid guitar researcher…”
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