• Zigzag Thirds (Minor) tuning •



A ‘stack of thirds’, alternating between minor and major (i.e. semitone gaps of 3, 4, 3, 4, 3). Proposed by Greek mathematician Dr. Costas Kyritsis as an ideal landscape for simplifying the shapes of all common diatonic chords – a logical point, given that much of Western harmony is built from ‘towers of thirds’ (n.b. also see its ‘flipped’ counterpart: the ‘maj-min’ zigzag).


Outlines a min9(11) arpeggio from low to high, with each string having its own unique tone. The odd-jumping regularity is surprisingly easy to settle into – in fact, the narrowed range is probably its most constraining element (17 semitones: 7 fewer than Standard). Top tip: try playing melodies using only natural harmonics, which scatter in mellifluous fashion – e.g. the positions above <5fr>, <7fr>, and <12fr>.

Pattern: 3>4>3>4>3
Harmony: Fmin9(11) | 1-b3-5-b7-2-4




I first encountered this ‘alternating thirds’ concept via the blog of Dr. Costas Kyritsis, an Associate Professor in mathematics at the University of Ioannina. In a lengthy 2016 post entitled Melodic-optimal and chord-optimal tunings: Alternative tunings and comparison with violin, mandolin, and other instruments, he analyses questions around which tunings might be best suited to different types of guitar playing – concluding that:


When chord-playing is the main target, and not so much solo playing, [try] alternating minor and major 3rds [4>3>4>3>4 or 3>4>3>4>3]…This may be called the ‘Harmonic Tuning’…as it is based on the harmonic 2-octave 7-note scale [i.e. all its tones come from the 2-octave natural minor scale]. The latter [i.e. this one] is the most natural open tuning.” His reasoning process – as far as I can decipher it – runs as follows:

  • Triads: The tuning provides a highly regular canvas for reaching all major and minor chords in all octaves. The same basic ‘line’ shape can even be used for both – just hop it up or down a string to switch between maj and min (e.g. x-3-3-3-x-x is major, so 3-3-3-x-x-x & x-x-3-3-3-x are minor).
  • 7ths: The geometry of jazzier voicings is also simplified (and who doesn’t want that?) – with handy major 7ths [x-4-4-4-4-x], minor 7ths [4-4-4-4-x-x], dominant 7ths [x-4-4-4-3-x], diminished 7ths [4-4-3-3-x-x], and augmented 7ths [x-4-4-5-5-x]. Impressively, all diatonic 7th chords are playable in uninverted form right up to the 3rd octave, even in low fret positions (impossible in Standard without some crafty inversions).
  • Modulations: Due to the tuning’s multiple symmetries, “the relations of relative chords [e.g. Emin & Gmaj] and also chords in the wheel of 4ths [e.g. Em7 > Am7 > D7 > Gmaj7] is immediate to grasp”. This also eases the general jenga-block panic of moving melodic phrases across strings on a Standard-tuned guitar – instead offering simple diatonic scale modulations via such things as “very symmetric zig-zag patterns“.

What can you make of the zigs & zags?


Kyritsis adds that this ‘alternating intervals‘ concept has historical precedent too: “One similar example is the keyboards [and key buttons] of some accordions, concertinas, or bandoneons: where 4 notes increase in pitch by one semitone, [on] a ‘skew line’…”.


He also expanded on his thinking in a 2018 Music.StackExchange thread: “It can be proven mathematically that the…tuning [with] the largest number of [uninverted triads]…[is] the alternating minor 3rd-major 3rd: exactly as music theory gives that major and minor triad chords are created…Sometimes it is used in therapeutic harps…[It] is also very instructive [for learning] harmony – as it is represented directly [and] geometrically…[However] it is not designed for…strumming on all 6 strings: its simple magic occurs only when we play 3 or 4 [string] chords”.


I commend the breadth and odd analytic clarity of Dr. Kyritsis’ investigations (…and honestly, I’m always just happy to stumble across another hardcore tuning nerd). Inevitably, it’s hard to know what to make of all this frantic abstract reasoning without diving in to see how it fits on the fretboard – however I can’t track down any examples of recorded music in this tuning (although Kyritsis mentions using it to “record improvisations, before I pass [them] to the [EADGBE] guitar”).


Thus, it provides you with a vast expanse of largely uncharted guitaristic territory – so go try it out! Does it really simplify chordal movements as advertised? And if it does, what compromises are made elsewhere? (Long live these gonzo avenues of DIY musicology!)


More musings from Dr. Costas Kyritsis


6str 5str 4str 3str 2str 1str
Note F Ab C Eb G Bb
Alteration 1 -1 -2 -4 -4 -6
Tension (%) +12 -11 -21 -37 -37 -50
Freq. (Hz) 87 104 131 156 196 233
Pattern (>) 3 4 3 4 3
Semitones 0 3 7 10 14 17
Intervals 1 b3 5 b7 2 4
  • 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!


—Associated tunings: proximities of shape, concept, context, etc…


—Further learnings: sources, readings, lessons, other onward links…

  • The imaginarium of Dr. Kyritsis: I highly recommend browsing his other music writings too – a vast collection of dense, fiendishly detailed, topic-hopping tracts which combine abstract musical geometry with ancient rhythmic poetry and much, much more. This madcap corner of the online musicology world has certainly provided me with a few fruitful sparks in my overall tuning quest – and while I’m not quite sure the blog’s title of ‘Simpler Guitar Learning‘ is necessarily the most accurate, I certainly hail the Doc’s mission (“This blog is for making…guitar learning, song composition, and improvisation easier – based on more abstract mathematics of the music…rhythm, and the guitar…It is a new awareness and method, to link mental perception-images, the creation of feelings, and finger actions. The true goal of composition and improvisation is the existential process of creating and listening…We concentrate most of all on the feeling of the sound and new math-musical concepts”)
  • ‘Neutral’ thirds: what would it sound like if the two constituent intervals of this tuning – the equally-tempered major and minor 3rds – declared a truce, and joined forces? Well, I guess they could coexist harmonically as part of the ‘Hendrix‘ chord (a dom.7#9 voiced as 1-3-b7-#9) – or melodically in the Hindustani Raag Jog (1-3-4-5-b7-8 <> 8-b7-5-4-3-4-b3-1 = SGmPnS <> SnPmGmgS). Although the ultimate compromise would surely would be a ‘neutral third‘: the microtone lying halfway between them (i.e. 350 cents above the root) – which, in looser form, has been a mainstay of blues vocalism from its earliest Delta days (“Cluster analysis was performed on these note collections…The ‘neutral’ third was confirmed to occur in this sample. A similar blending of the perfect fourth and tritone was demonstrated…”)


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