The seven modes, by rotation
Each mode starts the family on the next degree up. Names are the source’s; degrees, neighbour counts and mirrors are worked out here. | Mode | Name, as printed | Degrees | One degree away | Reflected |
| Mode 1 | Double harmonic | 1 b2 3 4 5 b6 7 | 3 | itself |
| Mode 2 | Lydian #2 #6 | 1 #2 3 #4 5 #6 7 | 2 | Locrian bb3 bb7 |
| Mode 3 | Phrygian bb7 b4 (or Altered Diminished #5) | 1 b2 b3 b4 5 b6 bb7 | 2 | Ionian #5 #2 |
| Mode 4 | Hungarian minor or Lydian Diminished b6 | 1 2 b3 #4 5 b6 7 | 3 | Locrian natural 6 natural 3 |
| Mode 5 | Locrian natural 6 natural 3 or Mixolydian b5 b2 | 1 b2 3 4 b5 6 b7 | 3 | Hungarian minor |
| Mode 6 | Ionian #5 #2 | 1 #2 3 4 #5 6 7 | 3 | Phrygian bb7 b4 |
| Mode 7 | Locrian bb3 bb7 | 1 b2 bb3 4 b5 b6 bb7 | 2 | Lydian #2 #6 |
Names as printed in Mode (music), “Other heptatonic scales” (Wikipedia, read 2026-09-24).
Interval content
How many of the twenty-one pairs of notes in the family lie each distance apart, the shorter way round: the interval vector <4 2 4 5 4 2>. | Semitones apart | 1 | 2 | 3 | 4 | 5 | 6 |
| Pairs | 4 | 2 | 4 | 5 | 4 | 2 |
Mirror and symmetry
Reflected around its root, every mode of the double harmonic family lands on another mode of the same family, and Double harmonic lands on itself. No shift short of the octave maps the family onto itself: seven notes cannot divide twelve evenly.
One degree away
What the sources say about it
Every relation a source states that involves this family, tested against the atlas. | The claim | In the atlas | The source’s words |
| Ionian (diatonic, mode 1): lower the 2nd and lower the 6th, and it is Double harmonic (double harmonic, mode 1). | Borne out in two altered degrees, stopping at Harmonic major (harmonic major, mode 1) on the way. | lowering both the second and sixth of the Ionian mode by a semitone. — Double harmonic scale, “Details” (Wikipedia, read 2026-09-24) |
| Harmonic minor (harmonic minor, mode 1): lower the 2nd and raise the 3rd, and it is Double harmonic (double harmonic, mode 1). | Borne out in two altered degrees, stopping at Neapolitan Minor (Neapolitan minor, mode 1) and Harmonic major (harmonic major, mode 1) on the way. | lowering the second note and raising the third note of the harmonic minor scale by one semitone. — Double harmonic scale, “Details” (Wikipedia, read 2026-09-24) |
| Phrygian Dominant (harmonic minor, mode 5): raise the 7th, and it is Double harmonic (double harmonic, mode 1). | Borne out: the two are one altered degree apart. | raising the seventh of the Phrygian dominant scale (a mode of the harmonic minor scale) by a semitone. — Double harmonic scale, “Details” (Wikipedia, read 2026-09-24) |
| Neapolitan Minor (Neapolitan minor, mode 1): raise the 3rd, and it is Double harmonic (double harmonic, mode 1). | Borne out: the two are one altered degree apart. | raising the third of the neapolitan minor scale by a semitone. — Double harmonic scale, “Details” (Wikipedia, read 2026-09-24) |
| Harmonic major (harmonic major, mode 1): lower the 2nd, and it is Double harmonic (double harmonic, mode 1). | Borne out: the two are one altered degree apart. | lowering the second note of the harmonic major scale by a semitone. — Double harmonic scale, “Details” (Wikipedia, read 2026-09-24) |
| Double harmonic (double harmonic, mode 1) reflected around its root is itself. | Borne out: the reflection lands there. | The double harmonic major is its own reflection around the root, itself being the palindromic mode. — Double harmonic scale, “Symmetry and balance” (Wikipedia, read 2026-09-24) |
The family itself is read from Double harmonic scale, “Details” (Wikipedia, read 2026-09-24): lowering the second note of the harmonic major scale by a semitone.
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Two lowered degrees
The double harmonic family starts from the double harmonic, 1 b2 3 4 5 b6 7: the major scale with its second and sixth both lowered. Its rotations include Lydian #2 #6, the Hungarian minor, 1 2 b3 #4 5 b6 7, and Locrian bb3 bb7, each named as the source prints it.
Two alterations, one stop
Lowering both of those degrees at once is two alterations, so the double harmonic is not a neighbour of Ionian. Take them one at a time and the route passes through a declared set: lower the sixth first and Ionian becomes the harmonic major; lower the harmonic major’s second and it becomes the double harmonic. Lowering the second first leads out of the atlas, so that order has no stop here. From the harmonic minor the same thing happens twice over: lower its second and raise its third, in either order, and the route stops at the Neapolitan minor one way and at the harmonic major the other.
Its nearest neighbours
One altered degree connects its modes to the harmonic minor, harmonic major and Neapolitan minor families, and to nothing else. Its modes have few neighbours: none has more than three, and three have only two.
Symmetric around its root
Reflected around its root, the double harmonic is itself: the steps above its root, read upward, are the steps below it read downward. The family’s other modes pair off under the same reflection, Lydian #2 #6 with Locrian bb3 bb7 among them.