Reference Tables — Pitch-Class Space ℤ12
GeCo-Tool allows for generation of any scale or intervallic pattern, and is therefore very useful for exploring and practicing scales. By setting the first note to the desired one, the scale can be transposed in every key, or otherwise by using the transposition tool.
GeCo-Tool is based on cyclic interval generation within the pitch-class space ℤ12. Instead of constructing scales from traditional tonal functions, the system operates through ordered sequences of semitone intervals.
A musical scale can therefore be represented as a cyclic interval vector, where each number indicates the distance in semitones between consecutive notes.
For example, the C Major scale:
can be encoded as:
Because GeCo operates within the standard twelve-tone equal temperament system, it can represent and generate virtually all scales, modes, arpeggios, and harmonic structures commonly used in Western music. This includes major and minor scales, modal scales, pentatonic systems, blues scales, whole-tone collections, diminished scales, bebop scales, and dodecaphonic structures.
Many musical traditions outside the Western equal-tempered system can also be explored through approximate twelve-tone representations. However, GeCo does not currently model microtonal intervals such as quarter-tones, neutral intervals, or the complex tuning systems found in Arabic, Persian, Turkish, Indian, Southeast Asian, and other non-Western musical traditions. For this reason, GeCo should not be considered a complete representation of these musical cultures. Instead, it provides a twelve-tone interval framework that can approximate some of their modal structures while preserving the mathematical simplicity of pitch-class space modulo 12.
The choice of a twelve-note system is intentional. It allows scales, chords, interval structures, transformations, and cyclic processes to be explored within a common mathematical environment, making it possible to compare tonal, modal, blues, quartal, and dodecaphonic systems using the same interval-based language.
Below we present tables that allow for implementation of scales in GeCo-Tool, enabling fast transposition in any key for instrumental practice and ear training.
| Mode | Semitone Sequence | Notes |
|---|---|---|
| Ionian (C) | 2 2 1 2 2 2 1 | C D E F G A B C |
| Dorian (D) | 2 1 2 2 2 1 2 | D E F G A B C D |
| Phrygian (E) | 1 2 2 2 1 2 2 | E F G A B C D E |
| Lydian (F) | 2 2 2 1 2 2 1 | F G A B C D E F |
| Mixolydian (G) | 2 2 1 2 2 1 2 | G A B C D E F G |
| Aeolian (A) | 2 1 2 2 1 2 2 | A B C D E F G A |
| Locrian (B) | 1 2 2 1 2 2 2 | B C D E F G A B |
| Mode | Semitone Sequence | Notes |
|---|---|---|
| Harmonic Minor (A) | 2 1 2 2 1 3 1 | A B C D E F G# A |
| Locrian #6 (B) | 1 2 2 1 3 1 2 | B C D E F G# A B |
| Ionian #5 (C) | 2 2 1 3 1 2 1 | C D E F G# A B C |
| Dorian #4 (D) | 2 1 3 1 2 1 2 | D E F G# A B C D |
| Phrygian Dominant (E) | 1 3 1 2 1 2 2 | E F G# A B C D E |
| Lydian #2 (F) | 3 1 2 1 2 2 1 | F G# A B C D E F |
| Ultra Locrian (G#) | 1 2 1 2 2 1 3 | G# A B C D E F G# |
| Mode | Semitone Sequence | Notes |
|---|---|---|
| Melodic Minor (A) | 2 1 2 2 2 2 1 | A B C D E F# G# A |
| Dorian b2 (B) | 1 2 2 2 2 1 2 | B C D E F# G# A B |
| Lydian Augmented (C) | 2 2 2 2 1 2 1 | C D E F# G# A B C |
| Lydian Dominant (D) | 2 2 2 1 2 1 2 | D E F# G# A B C D |
| Mixolydian b6 (E) | 2 2 1 2 1 2 2 | E F# G# A B C D E |
| Locrian #2 (F#) | 2 1 2 1 2 2 2 | F# G# A B C D E F# |
| Altered / Super Locrian (G#) | 1 2 1 2 2 2 2 | G# A B C D E F# G# |
The Harmonic Major scale may be viewed as a major scale with a lowered sixth degree.
Parent scale| Mode | Semitone Sequence | Notes |
|---|---|---|
| Harmonic Major (C) | 2 2 1 2 1 3 1 | C D E F G Ab B C |
| Dorian b5 (D) | 2 1 2 1 3 1 2 | D E F G Ab B C D |
| Phrygian b4 (E) | 1 2 1 3 1 2 2 | E F G Ab B C D E |
| Lydian b3 (F) | 2 1 3 1 2 2 1 | F G Ab B C D E F |
| Mixolydian b2 (G) | 1 3 1 2 2 1 2 | G Ab B C D E F G |
| Lydian Augmented #2 (Ab) | 3 1 2 2 1 2 1 | Ab B C D E F G Ab |
| Locrian bb7 (B) | 1 2 2 1 2 1 3 | B C D E F G Ab B |
The Neapolitan scales are distinguished from the harmonic and ascending melodic minor scales by the lowered supertonic or second scale degree. Both systems are characterized by the presence of the lowered second degree, which gives them their distinctive color.
| Mode | Semitone Sequence | Notes |
|---|---|---|
| Neapolitan Major (C) | 1 2 2 2 2 2 1 | C Db Eb F G A B |
| Lydian Augmented #6 (Db) | 2 2 2 2 2 1 1 | Db Eb F G A B C |
| Lydian Augmented (Eb) | 2 2 2 2 1 1 2 | Eb F G A B C Db |
| Lydian Dominant b6 (F) | 2 2 2 1 1 2 2 | F G A B C Db Eb |
| Major Locrian (G) | 2 2 1 1 2 2 2 | G A B C Db Eb F |
| Half-Diminished b4 (A) | 2 1 1 2 2 2 2 | A B C Db Eb F G |
| Altered Dominant bb3 (B) | 1 1 2 2 2 2 2 | B C Db Eb F G A |
| Mode | Semitone Sequence | Notes |
|---|---|---|
| Neapolitan Minor (C) | 1 2 2 2 1 3 1 | C Db Eb F G Ab B |
| Lydian #6 (Db) | 2 2 2 1 3 1 1 | Db Eb F G Ab B C |
| Mixolydian Augmented (Eb) | 2 2 1 3 1 1 2 | Eb F G Ab B C Db |
| Romanian Minor (F) | 2 1 3 1 1 2 2 | F G Ab B C Db Eb |
| Locrian Dominant (G) | 1 3 1 1 2 2 2 | G Ab B C Db Eb F |
| Ionian #2 (Ab) | 3 1 1 2 2 2 1 | Ab B C Db Eb F G |
| Ultra Locrian bb3 (B) | 1 1 2 2 2 1 3 | B C Db Eb F G Ab |
Bebop scales are extended tonal scales containing an additional chromatic passing tone. They are widely used in jazz improvisation because the added note allows chord tones to fall on metrically strong beats during continuous eighth-note lines.
In the following table, all scales are shown with C as the tonic. The note shown in parentheses represents the chromatic passing tone that creates the characteristic bebop alignment between chord tones and strong beats. For instance, in a 4/4 tune with eight notes, in the major bebop scale, the Ab is on the upbeat and allows for having the A on the downbeat, the B on the upbeat, and the C in the next bar on the downbeat.
| Bebop Scale | Semitone Sequence | Notes |
|---|---|---|
| Major Bebop | 2 2 1 2 1 1 2 1 | C D E F G (Ab) A B |
| Dominant Bebop | 2 2 1 2 2 1 1 1 | C D E F G A Bb (B) |
| Minor Bebop (Dorian Bebop) | 2 1 2 2 1 1 1 2 | C D Eb F G Ab (A) Bb |
| Melodic Minor Bebop | 2 1 2 1 1 2 2 1 | C D Eb F (Gb) G A B |
| Harmonic Minor Bebop | 2 1 2 2 1 1 2 1 | C D Eb F G Ab (A) B |
| Bebop Locrian | 1 2 2 1 1 1 2 2 | C Db Eb F Gb (G) Ab Bb |
| Bebop Altered Scale | 1 2 1 2 2 2 1 1 | C Db Eb E F# G# (Bb) B |
| Half-Whole Bebop Diminished | 1 2 1 2 1 2 1 2 | C Db Eb E F# G A (Bb) |
The following scales are simplified pitch-class approximations of modal systems used in Arabic, Turkish, Persian, and Middle Eastern musical traditions. Since many of these traditions employ microtonal intervals and non-equal temperaments, the sequences below should be interpreted as 12-tone approximations suitable for computational exploration within GeCo-Tool.
| Scale / Maqam / Makam | Semitone Sequence | Notes |
|---|---|---|
| Hijaz | 1 3 1 2 1 2 2 | C Db E F G Ab Bb |
| Hijazkar | 1 3 1 2 1 3 1 | C Db E F G Ab B |
| Bayati Approximation | 1 2 2 2 1 2 2 | C Db Eb F G Ab Bb |
| Nahawand | 2 1 2 2 1 2 2 | C D Eb F G Ab Bb |
| Rast Approximation | 2 2 1 2 2 2 1 | C D E F G A B |
| Saba Approximation | 1 3 1 1 3 1 2 | C Db E F Gb A Bb |
| Kurd | 1 2 2 2 1 2 2 | C Db Eb F G Ab Bb |
| Nikriz | 1 3 1 2 1 3 1 | C Db E F G Ab B |
| Huzzam Approximation | 1 3 1 1 2 3 1 | C Db E F Gb Ab B |
| Suznak | 2 2 1 2 1 3 1 | C D E F G Ab B |
| Ussak Approximation | 1 2 2 2 1 2 2 | C Db Eb F G Ab Bb |
| Segah Approximation | 1 3 1 2 2 1 2 | C Db E F G A Bb |
| Persian Scale | 1 3 1 1 2 3 1 | C Db E F Gb Ab B |
| Byzantine / Double Harmonic | 1 3 1 2 1 3 1 | C Db E F G Ab B |
| Oriental Scale | 1 3 1 1 3 1 2 | C Db E F Gb A Bb |
| Arabic Pentatonic Approximation | 2 1 4 2 3 | C D Eb G A |
| Turkish Hüseyni Approximation | 2 1 2 2 1 2 2 | C D Eb F G Ab Bb |
| Turkish Nihavent | 2 1 2 2 1 2 2 | C D Eb F G Ab Bb |
| Turkish Karcığar Approximation | 1 3 1 2 1 2 2 | C Db E F G Ab Bb |
The user can take any semitone sequence from the tables above, choose the multidimensional option in GeCo-Tool, set the correct K value (for instance 7 for heptatonic scales), and then generate the selected scale.
However, considering the variety of scales, a zipped file with ready-made parameter files (.txt) is provided separately. The user can import the file and the selected sequence will appear directly in the interface, ready to be generated.