in various tunings

[Joe Monzo]

John Coltrane's landmark tune Giant Steps (1960) utilizes a harmonic plan which showcases 12-edo tuning's membership in both the aristoxenean and the augmented temperament families: the former by repetition of the dominant-7th-to-tonic functions, and the latter by the arrangement of tonics as a cycle of major-3rds. The chord progressions are as follows:

LEGEND:

- In the chord symbols, "z" is used to represent the triangle which is commonly used for major; the other two symbols are the usual "x" for dominant and "m" for minor;
- it is assumed that all chords include the 7th, and they may also include the extension 9th and/or 13th;
- the analysis underneath the chord symbols shows how the V-I and II-V-I progressions circulate thru the three keys, which are indicated on the left in brackets.

Bz Dx | Gz Bbx | Ebz | Am Dx | [G]: V | I | | II V | [Eb]: | V | I | | [B]: I | | | | Gz Bbx | Ebz F#x | Bz | Fm Bbx | [G]: I | | | | [Eb]: V | I | | II V | [B]: | V | I | | Ebz | Am Dx | Gz | C#m F#x | [G]: | II V | I | | [Eb]: I | | | | [B]: | | | II V | Bz | Fm Bbx | Ebz | C#m F#x | [G]: | | | | [Eb]: | II V | I | | [B]: I | | | II V |

Below is another version of the analysis, using more familiar abbreviations for the chord symbols: "M7" = major-7th, "m7" = minor-7th, "7" = dominant-7th:

BM7 D7 | GM7 Bb7 | EbM7 | Am7 D7 | [G]: V | I | | II V | [Eb]: | V | I | | [B]: I | | | | GM7 Bb7 | EbM7 F#7 | BM7 | Fm7 Bb7 | [G]: I | | | | [Eb]: V | I | | II V | [B]: | V | I | | EbM7 | Am7 D7 | GM7 | C#m7 F#7 | [G]: | II V | I | | [Eb]: I | | | | [B]: | | | II V | BM7 | Fm7 Bb7 | EbM7 | C#m7 F#7 | [G]: | | | | [Eb]: | II V | I | | [B]: I | | | II V |

31-edo is the first non-12-edo tuning i tried for Giant Steps. I chose 31-edo because it offers low error for prime factors 5 and 7, which are featured prominently in jazz, and its error for 3 is good enough to preserve the approximation to pythagorean root-movement. The arrangement is my own, as i play it on the piano, and the instrument timbre used is acoustic guitar. The original was done in Tonescape^{®}, and all the other formats were generated automatically from that by Tonescape; below are the downloadable links:

- Tonescape Musical Piece
- MIDI
- Csound unified file (.sco + .orc) -- see also the printout of the Csound output screen
- .wav file

Below is a screenshot of the opening chord being played in Tonescape, with the Lattice using Rectangular Geometry:

Below is the same chord, with the Lattice using Closed Curved crumpled-napkin Geometry:

The chords use the same spelling as in the analysis above, and therefore their progressions do not always follow identical pitch patterns in 31-edo as they do in 12-edo. I plan to make another 31-edo version which does preserve the same pitch patterns, but this will modulate into remote keys which in 12-edo are enharmonic equivalents but in 31-edo are not.

a&b temperament [a&b are numbers]

55-edo (comma) (Mozart's tuning)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

a&b temperament [a&b are numbers]

apotome (Greek interval)

aristoxenean (temperament family)

atomic (temperament family)

augmented / diesic (temperament family)

augmented-2nd / aug-2 / #2 (interval)

augmented-4th / aug-4 / #4 (interval)

augmented-5th / aug-5 / #5 (interval)

augmented-6th / aug-6 / #6 (interval)

augmented-9th / aug-9 / #9 (interval)

blackjack (tuning)

cent / ¢ (unit of interval measurement)

centitone / iring (unit of interval measurement)

chromatic-semitone / augmented-prime (interval)

daseian (musical notation)

dekamu / 10mu (MIDI-unit)

diapason (Greek interval)

diapente (Greek interval)

diatessaron (Greek interval)

diatonic semitone (minor-2nd) (interval)

diesic (temperament family)

diezeugmenon (Greek tetrachord)

diminished-5th / dim5 / -5 / b5 (interval)

diminished-7th / dim7 / o7 (interval)

doamu / 2mu (MIDI-unit)

dodekamu / 12mu (MIDI-unit)

dominant-7th (dom-7, x7) (chord)

dorian (mode)

eleventh / 11th (interval)

enamu / 1mu (MIDI-unit)

endekamu / 11mu (MIDI-unit)

enharmonic semitone (interval)

ennealimmal (temperament family)

enneamu / 9mu (MIDI-unit)

farab (unit of interval measurement)

fifth / 5th (interval)

flu (unit of interval measurement)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

fourth / 4th (interval)

Gentle introduction to Fokker periodicity-blocks (by Paul Erlich)

grad (unit of interval measurement)

hexamu / 6mu (MIDI-unit)

Hurrian Hymn (Monzo reconstruction)

hypate (Greek note)

hypaton (Greek tetrachord)

hyperbolaion / hyperboleon (Greek tetrachord)

hypophrygian (Greek mode)

imperfect (interval quality)

iring / centitone (unit of interval measurement)

1/1 or 1:1 (ratio)

2/1 or 2:1 (ratio)

jot (unit of interval measurement)

JustMusic: A New Harmony [Monzo's book]

JustMusic prime-factor notation [Monzo essay]

kwazy (temperament family)

leimma / limma (Greek interval)

lichanos (Greek note)

limma / leimma (Greek interval)

locrian (mode)

lydian (mode)

magic (temperament family)

Mahler 7th/1 [Monzo score and analysis]

marvel (temperament family)

meantone (temperament family)

mem (unit of interval measurement)

meride (unit of interval measurement)

mese (Greek note)

meson (Greek tetrachord)

millioctave / m8ve (unit of interval measurement)

mina (unit of interval measurement)

minerva (temperament family)

miracle (temperament family)

mixolydian (mode)

monzo (prime-exponent vector)

Monzo, Joe (music-theorist)

morion / moria (unit of interval measurement)

mutt (temperament family)

mystery (temperament family)

octamu / oktamu / 8mu (MIDI-unit)

octave (interval)

oktamu / octamu / 8mu (MIDI-unit)

orwell (temperament family)

p4, perfect 4th, perfect fourth (interval)

p5, perfect 5th, perfect fifth (interval)

pantonality of Schoenberg [Monzo essay]

paramese (Greek note)

paranete (Greek note)

parhypate (Greek note)

pentamu / 5mu (MIDI-unit)

prime-factor notation (JustMusic) [Monzo essay]

proslambanomenos (Greek note)

savart (unit of interval measurement)

schismic / skhismic (temperament family)

Schoenberg's pantonality [Monzo essay]

second / 2nd (interval)

semisixths (temperament family)

semitone (unit of interval measurement)

seventh / 7th (interval)

sixth / 6th (interval)

sk (unit of interval measurement)

skhismic / schismic (temperament family)

sruti tuning [Monzo essay]

studloco (tuning)

subminor 3rd (interval)

Sumerian tuning [speculations by Monzo]

synemmenon (Greek tetrachord)

temperament-unit / tu (unit of interval measurement)

tenth / 10th (interval)

tetrachord-theory tutorial [by Monzo]

tetradekamu / 14mu (MIDI-unit)

tetramu / 4mu (MIDI-unit)

third / 3rd (interval)

thirteenth / 13th (interval)

tina (unit of interval measurement)

tone (interval, and other definitions)

triamu / 3mu (MIDI-unit)

tridekamu / 13mu (MIDI-unit)

trihemitone (Greek interval)

trite (Greek note)

tu / temperament-unit (unit of interval measurement)

Türk sent (unit of interval measurement)

twelfth / 12th (interval)

whole-tone (interval)

woolhouse-unit (unit of interval measurement)