dominant, 2) superlocrian? Wait let's list: Phrygian dominant (1) is: 1, 2♭, 3, 4, 5, 6♭, 7♭? Actually Phrygian dominant is the 5th mode of the harmonic minor. So the intervals: 1, ♭2, 3, 4, 5, ♭6, ♭7? Wait, the Phrygian dominant is a scale with major third, minor second, major third? Let's recall: In key of E: E, F, G#, A, B, C, D, E. That is Phrygian dominant: E F G# A B C D E. So intervals: 1, ♭2, 3, 4, 5, ♭6, ♭7. It's basically the harmonic minor scale starting on the 5th degree. Its modes would be 1) Phrygian dominant (E F G# A B C D E), 2) superlocrian? Actually let's list modes of Phrygian dominant: Starting on each degree. The modes are (for scale degrees):

Phrygian dominant

Super Locrian (i.e., Locrian #2). The second mode has intervals: 1, 2, ♭3, 4, 5, ♭6, ♭7. Let's check: Starting from the second degree: F G# A B C D E F? Wait, if we take the Phrygian dominant on E: E F G# A B C D E. The second degree is F. Starting at F: F G# A B C D E F. That scale: F G# A B C D E F. That's an interesting scale. Let's examine the intervals: F to G# is major third (4 semitones). G# to A is minor second (1 semitone). A to B is major second (2 semitones). B to C is minor second (1). C to D is major second (2). D to E is major second (2). E to F is minor second (1). So the intervals: 4, 1, 2, 1, 2, 2, 1. That's the second mode. It is called the "Super Locrian" or "Locrian #2"? Actually super locrian is the 7th mode of the melodic minor? Wait, I'm mixing. Let's think of the modes of harmonic minor: The 5th mode (Phrygian dominant). The 2nd mode of Phrygian dominant is the "Super Locrian" (7th mode of harmonic minor?), but I'm not sure. Let's recall: Modes of harmonic minor:

Harmonic minor (tonic)

Locrian #6? The second mode is the "Locrian #6" or "Harmonic Locrian"? Actually the 2nd mode of harmonic minor is "Locrian #6" which has intervals: 1, b2, b3, 4, 5, b6, 7? Wait let's compute: harmonic minor: 1, 2, b3, 4, 5, b6, 7. The second mode starts on the second degree: 2, b3, 4, 5, b6, 7, 1. Intervals: 1, b2, 3, 4, 5, b6, 7? Actually let's compute precisely: Let's say harmonic minor in C: C D Eb F G Ab B C. Second mode: D Eb F G Ab B C D. Intervals: D to Eb (b2), to F (b3? Actually D-Eb is half step, to F is whole step, so total: D to Eb: 1 semitone (b2). Eb to F: 2 semitones (b3?), but we should compute in terms of scale degrees: Starting on D: D Eb F G Ab B C D. That's a scale with intervals: D-Eb: minor second; Eb-F: major second; F-G: major second; G-Ab: minor second; Ab-B: major third; B-C: minor second. So the pattern is 1, 3, 5, 6, 8, 10, 11? Actually I'm mixing.

But the second mode of Phrygian dominant is known as "Super Locrian"? Let's confirm. Actually super locrian is the 7th mode of the melodic minor (or 7th mode of the harmonic minor?), but maybe not.

We can derive the mode from the E Phrygian dominant: E F G# A B C D E. The 2nd mode starting at F: F G# A B C D E F. That's a scale with a major third from F to G#, a minor second G# to A, etc. This is basically the scale of "Phrygian dominant" but starting on its second degree. This is actually the "Lydian Dominant" scale? No, Lydian Dominant is 1, 2, 3, #4, 5, 6, b7. But our scale has 1, #2, b3, 4, 5, b6, b7? Wait let's list: F, G#, A, B, C, D, E, F. The intervals from F: F to G# is a major third (M3). G# to A is a minor second (m2). A to B is a major second (M2). B to C is a minor second (m2). C to D is a major second (M2). D to E is a major second (M2). E to F is a minor second (m2). So it's M3 + m2 + M2 + m2 + M2 + M2 + m2. In terms of scale degrees relative to F: F G# A B C D E: that's basically the "Lydian augmented" or something? But the question: "What is the second mode of Phrygian dominant?" We need to answer with the name: "Super Locrian (also called the 7th mode of the harmonic minor)"? Wait, "Super Locrian" is the 7th mode of the melodic minor? Let's check.

List of modes of the harmonic minor in C: 1) Harmonic minor (C D Eb F G Ab B C). 2) Locrian #6? 3) Phrygian dominant? Wait, the Phrygian dominant is the 5th mode of the harmonic minor. The 5th mode is Phrygian dominant. The 2nd mode of Phrygian dominant is the 6th mode of the harmonic minor? Wait, the 2nd mode of the Phrygian dominant is actually the 7th mode of the harmonic minor? Let's check.

The harmonic minor in C: C D Eb F G Ab B C. Its 5th mode: Starting on G: G Ab B C D Eb F G. That's Phrygian dominant: G Ab B C D Eb F G. Intervals: 1, b2, 3, 4, 5, b6, b7. Now the 2nd mode of that scale would start on Ab: Ab B C D Eb F G Ab. That scale: Ab B C D Eb F G. Intervals: Ab to B is major third; B to C is minor second; etc. So that would be the 6th mode of the harmonic minor? Let's compute: The harmonic minor 6th mode is starting on the 6th degree: Ab. Let's list all modes of the harmonic minor in C:

Harmonic minor: C D Eb F G Ab B C
Locrian #6: D Eb F G Ab B C D? Wait, second mode: D E♭ F G A♭ B C D? Actually second mode: starting on D: D Eb F G Ab B C D. Intervals: 1, b2, 3, 4, 5, b6, 7.
Phrygian #4? Actually third mode: Eb F G A♭ B C D Eb? Wait, but let's list systematically:
Let's get the harmonic minor scale: C D Eb F G Ab B C. Represent degrees: 1 2 b3 4 5 b6 7 8. Then modes are:

(C) Harmonic minor
(D) Locrian #6? The 2nd mode: D Eb F G Ab B C D. Intervals: 1 b2 b3 4 5 b6 7. This is "Locrian #6" or "Lydian #6"? Wait, the pattern is 1 b2 3 4 5 6 7? Actually let's compute. Starting on D: D to Eb = half step: b2. D to F = whole step: major 2? Wait we need relative to D: D-Eb (b2). D-F (M2). D-G (M3). D-Ab (b4). D-B (M5). D-C (b6). That's not right. We should systematically compute by transposing the scale. Let's do easier: The harmonic minor modes correspond to certain exotic scales. According to sources:
Harmonic Minor
Locrian #6
Phrygian Dominant (?), no, that's the 5th mode.
Dorian #4
Lydian #2
Mixolydian #11? Hmm.
Better to search memory: The modes of harmonic minor are:

Harmonic minor
Locrian #6
Phrygian Dominant
Dorian #4
Lydian #2
Mixolydian #11
Altered scale (also called superlocrian) is the 7th mode of melodic minor? Wait, no.
But I recall that the 7th mode of harmonic minor is the "Altered scale"? Let's confirm: The Altered scale (superlocrian) is the 7th mode of the melodic minor. Actually it's the 7th mode of melodic minor. But it's also the 7th mode of harmonic minor? Let's check: The melodic minor scale: C D Eb F G A B C. 7th mode: B C D Eb F G A? Wait, not.

Alternatively, the superlocrian (or altered) is the 7th mode of the melodic minor, not the harmonic minor. So maybe the 7th mode of harmonic minor is "Locrian #2"? Not sure.

Let's think again. The question: "What is the second mode of Phrygian dominant?" It's likely they want the name of that scale. The second mode of Phrygian dominant is sometimes called "Lydian dominant"? But no, Lydian dominant is a different scale. Wait, the second mode of Phrygian dominant has a major third from root. That may be "Super Locrian" because it has a major third? No, superlocrian has a minor third. Let's derive again.

Phrygian dominant on G: G Ab B C D Eb F G. Starting on the second degree Ab: Ab B C D Eb F G. What scale is that? Let's identify by pattern: Ab B C D Eb F G. Intervals from Ab: Ab to B = major third (M3). B to C = minor second (m2). C to D = major second (M2). D to Eb = minor second (m2). Eb to F = major second (M2). F to G = major second (M2). G to Ab = minor second (m2). So pattern: M3, m2, M2, m2, M2, M2, m2. That's like an augmented scale? But maybe it's called "Lydian #2"? Let's check Lydian #2: pattern: 1, 2, 3, #4, 5, 6, 7. Not that. "Locrian #2"? That would be 1, 2, b3, 4, 5, b6, b7. But our pattern: 1, #2? Actually we have 1, #2 (major third is a major third away, but that's two steps). But let's map intervals in terms of scale degrees: Root is 1. The second degree is a major third away, so it's a "third" not a "second". So the pattern is: 1, 3, 4, 5, 6, 7? Wait, we need to express the scale using the diatonic scale degrees: Starting at Ab: the scale notes are Ab, B, C, D, Eb, F, G. Let's compare to the major scale in Ab: Ab Bb C Db Eb F G Ab. Our scale has Ab, B, C, D, Eb, F, G. So we replaced Bb with B (raised 2nd). We replaced Db with D (raised 4th). So we have Ab (1), B (raised 2nd, so major 2), C (3), D (raised 4th, so major 4 or #4?), but D is major 4 from Ab? Wait, major 4 from Ab is Db. So D is #4. So we have major 2, major 3? Actually we have 1, 2 (raised), 3, #4, 5, 6, 7. That is the "Lydian #2" scale: 1, #2, 3, #4, 5, 6, 7. But we also have B which is major 2, but it's also the major 3? Wait, B is a major 2 from Ab? Let's count: Ab to Bb is 2; Ab to B is 3 semitones? Actually Ab to B: Ab to A (m2?), but let's do semitone: Ab to A: 1 semitone (m2). A to B: 2 semitones (M2). So Ab to B is 3 semitones, so that's a minor third (b3) from Ab? Wait, the Ab to A is 1 semitone, A to B is 2 semitones, so Ab to B is 3 semitones which is a minor third (b3). Actually major third is 4 semitones. So B from Ab is a minor third (b3). So the pattern is 1, b3, 4, #4, 5, 6, 7? Wait, we need to re-evaluate: The scale is Ab, B, C, D, Eb, F, G. Let's compute intervals:

Ab to B: 3 semitones = minor third (b3). B to C: 1 semitone = minor second (b2). C to D: 2 semitones = major second (M2) but that's 4th from Ab? Actually C to D is 2 semitones but relative to Ab: C is 5th? Wait.

Let's recalc: Use semitone from Ab as root:

Ab: 0 B: +3 (b3) C: +5 (4th) because 5 semitones is perfect 4th? Actually 5 semitones is a perfect fourth. So C is the 4th degree. Good. D: +7 (b5?) 7 semitones is perfect 5th? Wait, 7 semitones is a perfect fifth. So D is 5th. But we have D at +7? Wait, check: Ab to D: Ab-A (1), A-B (2), B-C (3), C-D (4). That's 4 steps, each step maybe? Actually let's use formula: Each semitone is 1 step: Ab to A: 1; A to A# (Bb): 2; A# to B: 3; B to C: 4; C to C#: 5; C# to D: 6. So Ab to D is 6 semitones? Wait, we might have miscalculated. Let's use standard: Ab to A: +1; A to A#/Bb: +1; A#/Bb to B: +1; B to C: +1; C to C#: +1; C# to D: +1. So total: 1+1+1+1+1+1 = 6 semitones. So Ab to D is 6 semitones, which is a tritone (augmented 4th). So D is an augmented 4th (#4). Good. So the pattern is: Ab (1), B (b3), C (4), D (#4), Eb (b5? Actually Ab to Eb is 7 semitones? Let's compute: Ab to Eb: Ab-A:1; A-A#:2; A#-B:3; B-C:4; C-C#:5; C#-D:6; D-D#:7. So yes 7 semitones is perfect 5th. So Eb is 5th. So we have 1, b3, 4, #4, 5. Next F: Ab to F: count: Ab-A:1; A-A#:2; A#-B:3; B-C:4; C-C#:5; C#-D:6; D-D#:7; D#-E:8; E-F:9. So 9 semitones = major 6th. So F is 6. G: Ab to G: 10 semitones? Ab-A:1; A-A#:2; A#-B:3; B-C:4; C-C#:5; C#-D:6; D-D#:7; D#-E:8; E-F:9; F-F#:10; F#-G:11. So 11 semitones = major 7th. So the scale is 1, b3, 4, #4, 5, 6, 7. That is the "Lydian augmented" or "Lydian sharp 4"? Wait, Lydian augmented is 1, 2, 3, #4, 5, 6, 7? That would have a major second, not a minor third. But we have a minor third. So it's not Lydian augmented. It's "Phrygian augmented"? Hmm.

But I think the second mode of Phrygian dominant is "Lydian augmented"? Let's find references: The second mode of the Phrygian dominant is also called the "Lydian dominant" if you shift? Wait, no. Lydian dominant is the fourth mode of the melodic minor. I'm mixing.

Alternatively, maybe the second mode of Phrygian dominant is called the "superlocrian" or "altered scale"? Wait, the altered scale is the 7th mode of melodic minor. The second mode of Phrygian dominant may be "Locrian #2"? Let's see: Locrian #2 is 1, #2, 3, 4, 5, b6, b7. That has a major third? Actually 1 to #2 is a major second, so it's 1, 2? No. Wait, Locrian #2 would have a major third, so 1, #2, 3? That seems weird.

Let's recall the names: The 5th mode of harmonic minor (Phrygian dominant) has known other names: "Spanish gypsy", "Harmonic minor dominant". The 2nd mode of Phrygian dominant (starting on the 2nd degree) is called the "Lydian dominant"? No.

Wait, I recall reading that the second mode of Phrygian dominant is "Lydian augmented" or "Lydian augmented"? Actually Lydian augmented is the 4th mode of harmonic minor? Let's check: Harmonic minor modes:

Harmonic minor: 1, 2, b3, 4, 5, b6, 7
Locrian #6: 2, b3, 4, 5, b6, 7, 1
Phrygian dominant: 3, 4, 5, b6, 7, 1, 2
Dorian #4: 4, 5, b6, 7, 1, 2, b3
Lydian #2: 5, b6, 7, 1, 2, 3, 4
Mixolydian #11: 6, 7, 1, 2, 3, 4, b5
Altered (Super Locrian): 7, 1, 2, 3, 4, 5, b6
So the 5th mode (Phrygian dominant) is as above. Then the 2nd mode of Phrygian dominant would be the 6th mode of harmonic minor? Wait, the 6th mode of harmonic minor is the "Mixolydian #11" or "Lydian #2"? Wait, let's index:

Mode 1: C harmonic minor: C D Eb F G Ab B C
Mode 2: Locrian #6: D Eb F G Ab B C D
Mode 3: Phrygian dominant: Eb F G Ab B C D Eb
Mode 4: Dorian #4: F G Ab B C D Eb F
Mode 5: Lydian #2: G Ab B C D Eb F G
Mode 6: Mixolydian #11: A B C D Eb F G A
Mode 7: Altered: B C D Eb F G A B
Thus the 2nd mode of Phrygian dominant (starting on the second degree of Phrygian dominant) would correspond to Mode 6 of harmonic minor: Mixolydian #11. Let's test: Phrygian dominant on G: G Ab B C D Eb F G. The second mode: starting on Ab: Ab B C D Eb F G. That should be Mixolydian #11? Wait, Mixolydian #11 is starting on the 6th degree of harmonic minor: That would be A? Actually Mode 6 of harmonic minor in C: A B C D Eb F G. So the 2nd mode of Phrygian dominant (starting on Ab) would be A? Wait, our second mode is Ab B C D Eb F G. That is not A. So I'm mixing.

Let's step back. The second mode of Phrygian dominant is simply the second mode of that scale. Let's define Phrygian dominant: [1, b2, 3, 4, 5, b6, b7]. Starting on the root 1. Now the second mode starts on the 2nd degree, so its intervals relative to the new root: The 2nd degree of Phrygian dominant is b2? Wait, no, it's the second degree relative to the original key. But we want to treat the second mode as a separate scale. Let's write the notes of Phrygian dominant: [1, 2b, 3, 4, 5, 6b, 7b] relative to root. In pitch class, e.g., on G: [G, Ab, B, C, D, Eb, F]. Now the second mode will be starting on Ab. The notes will be [Ab, B, C, D, Eb, F, G]. We can name this scale by its interval pattern relative to Ab. Let's derive pattern: Ab to B is major third? Wait, earlier we found Ab to B is a minor third? Let's double-check: The interval between Ab and B is: Ab to A is 1 semitone, A to Bb is 1 (so total 2), Bb to B is 1 (so 3). So Ab to B is 3 semitones: minor third (b3). So the pattern: 1, b3, 4, #4, 5, 6, 7. That pattern matches "Lydian Augmented" (1, 2, 3, #4, 5, 6, 7). But we have 1, b3, not 1, 2. So it's not Lydian. Maybe it's "Locrian #2"? Let's examine Locrian #2: 1, 2, b3, 4, 5, b6, b7? Wait, Locrian has 1, b2, 3, 4, b5, b6, b7. Locrian #2 would raise the b2 to a 2. So 1, 2, 3, 4, b5, b6, b7. That has 1, 2, 3, 4, b5. Our scale has 1, b3, 4, #4, 5. So not that.

Let's think of "Super Locrian" is 1, b2, b3, b4, b5, b6, b7. Not that.

Wait, perhaps I'm misinterpreting the second mode. Maybe the second mode of Phrygian dominant is "Lydian dominant"? Let's check: Lydian dominant has intervals 1, 2, 3, #4, 5, 6, b7. That would be from root: 1, 2, 3, #4, 5, 6, b7. That includes major third and augmented 4th. But our second mode has a minor third and augmented 4th. So not that.

Let's derive the scale using a known pattern: Starting on Ab, the scale: Ab, B, C, D, Eb, F, G. That pattern is 1, #2? Wait, B is a major third away, so 1, b3? Actually we had 3 semitones, that's b3. So it's 1, b3, 4, #4, 5, 6, 7. That pattern corresponds to the "Lydian Augmented" of the harmonic minor? Wait, "Lydian Augmented" is 1, 2, 3, #4, 5, 6, 7. That's different. But "Lydian Dominant" is 1, 2, 3, #4, 5, 6, b7. That also differs. "Dorian Augmented" maybe? Let's check Dorian Augmented: 1, 2, b3, #4, 5, 6, b7. That also differs. "Phrygian #3"? But maybe this is called "Lydian Augmented" of the harmonic minor? Wait, maybe the second mode of Phrygian dominant is the "Lydian augmented" but I'm not certain.

However, I recall reading that the second mode of the Phrygian dominant is the "Lydian augmented" because it has the same set of intervals as the Lydian augmented. But check: Lydian augmented from C: C D E F# G A B C. That is 1, 2, 3, #4, 5, 6, 7. Our scale from Ab: Ab B C D Eb F G. Compare: Lydian augmented from F? Let's try C: C D E F# G A B C. The pattern: 1, 2, 3, #4, 5, 6, 7. Not our pattern. So not that.

Alternatively, maybe the second mode of Phrygian dominant is simply called the "Mixolydian #11" scale. Let's examine Mixolydian #11: 1, 2, 3, 4, 5, b6, b7. That doesn't match our pattern either.

Let's check "Altered scale" or "Super Locrian": 1, b2, b3, b4, b5, b6, b7. Not that.

Maybe it's "Locrian #6" or "Locrian #2"? Let's list known exotic modes: The 5th mode of harmonic minor (Phrygian dominant) has second mode called "Lydian augmented"? I need a reliable source.

Let's approach systematically: We have the scale Ab B C D Eb F G. We can determine its interval set: 0, 3, 5, 7, 8, 10, 11 semitones relative to Ab. Let's compare with known modes. Standard 7-note modes:

Ionian (major): 0,2,4,5,7,9,11
Dorian: 0,2,3,5,7,9,10
Phrygian: 0,1,3,5,7,8,10
Lydian: 0,2,4,6,7,9,11
Mixolydian: 0,2,4,5,7,9,10
Aeolian: 0,2,3,5,7,8,10
Locrian: 0,1,3,5,6,8,10
Now exotic modes: e.g., Lydian augmented: 0,2,4,6,7,9,11? Wait, that's Lydian augmented? Actually Lydian augmented: 0,2,4,6,7,9,11? No, Lydian augmented adds a sharp 4 and a major 6? But I'm not sure.

Let's compute our pattern: 0,3,5,7,8,10,11.

We can reorder as: 0,3,5,7,8,10,11. Let's see if that matches any known mode. Compare to "Phrygian dominant": 0,1,3,5,7,8,10? Actually Phrygian dominant: 0,1,3,5,7,8,10. Our pattern has 0,3,5,7,8,10,11. So it's like Phrygian dominant but the 2 is missing and the 11 is added (the 7 is shifted). That is indeed the 2nd mode of Phrygian dominant. What is that called? It might be called "Super Locrian" but no. Let's try to find in music theory texts: The modes of the Phrygian dominant are named: 1) Phrygian dominant (1), 2) Lydian augmented (2), 3) Mixolydian b2? Hmm.

Let's recall the modes of the Phrygian dominant (5th mode of harmonic minor). According to some references:

Mode 1: Phrygian dominant
Mode 2: Lydian augmented
Mode 3: Mixolydian b2
Mode 4: Dorian b2?
Mode 5: Aeolian augmented?
Mode 6: Locrian b3?
Mode 7: Ionian b3?
But this is speculation.

Wait, maybe we should derive the pattern of mode 2 from the root Ab: 0,3,5,7,8,10,11. We can try to find which common mode matches that pattern. Let's try the "Phrygian dominant" pattern relative to Ab: That would be 0,1,3,5,7,8,10. Our pattern differs by having 3 instead of 1 as the second step, and 11 instead of 10 as the final step. So it's like a phrygian dominant with the 2 raised to 4 and the 7 raised to 8? Actually the 2 in Phrygian dominant is a minor second (1 semitone). In our pattern it's a minor third (3 semitones). That is like taking the 3rd of Phrygian dominant as the 2nd degree? Wait, Phrygian dominant has 3rd as a major third. So 3rd of Phrygian dominant is 4 semitones from root. But we have 3 semitones. So it's not that.

Alternatively, maybe we can find the name "Lydian augmented" from other references: The "Lydian augmented" from C is C D E F# G A B. That's 0,2,4,6,7,9,11. Not our pattern.

What about "Lydian b7" or "Lydian b7"? That would be 0,2,4,6,7,9,10. Not ours.

What about "Mixolydian #4"? That would be 0,2,4,6,7,9,10. Not ours.

What about "Lydian dominant"? That is 0,2,4,6,7,9,10? Wait, Lydian dominant is 0,2,4,#4? No. Lydian dominant is 0,2,4,#4,5,7,10? No.

Perhaps we can look at the 2nd mode of Phrygian dominant: This is also the 7th mode of the melodic minor? Let's check: Melodic minor ascending: C D Eb F G A B C. 7th mode of melodic minor is B C D Eb F G A B. That's the altered scale. Not our pattern.

Let's compute the 2nd mode of Phrygian dominant by transposing to C? Let's use G Phrygian dominant: G Ab B C D Eb F G. The second mode: starting on Ab: Ab B C D Eb F G. If we transpose that down by a perfect fourth (or up a perfect fifth) to C? Let's map: Ab is D? Wait, we want the name of the mode independent of key. But we can see pattern: 0,3,5,7,8,10,11. That pattern maybe corresponds to "Lydian augmented" if we use a different root? Let's check if any known mode has pattern 0,3,5,7,8,10,11.

Let's check "Locrian #2": 0,2,3,5,7,8,10. Not ours.

Check "Lydian #2": 0,2,3,5,7,9,10. Not ours.

Check "Phrygian #4": 0,1,3,5,6,8,10? Not ours.

Check "Super Locrian": 0,1,3,4,6,8,10. Not ours.

Check "Altered": 0,1,3,4,6,8,10. Not ours.

Check "Mixolydian b2": 0,1,3,5,7,9,10? Not ours.

Check "Dorian b2": 0,1,3,5,7,9,10? Not ours.

Check "Lydian augmented": 0,2,4,6,7,9,11. Not ours.

Check "Lydian dominant": 0,2,4,6,7,9,10. Not ours.

Check "Mixolydian b5": 0,2,4,5,7,9,10? Not ours.

Check "Dorian b5": 0,2,3,5,7,9,10? Not ours.

Check "Aeolian b5": 0,2,3,5,7,8,10? Not ours.

Check "Phrygian augmented": 0,1,3,5,7,8,10? Wait, that's Phrygian augmented? Actually Phrygian augmented might be 0,1,3,5,7,9,10? Not.

Let's consider "Harmonic minor" pattern: 0,2,3,5,7,8,10. Not ours.

The pattern 0,3,5,7,8,10,11 maybe is called "Lydian augmented b2"? Let's search memory: I think the second mode of Phrygian dominant is also known as the "Lydian augmented" but we need to confirm. Wait, the second mode of Phrygian dominant is also called "Lydian dominant"? Let's check a reference: On Wikipedia, the "Phrygian dominant" has a "second mode" called "Lydian augmented"? But I'm not sure.

Alternatively, maybe the second mode is simply called "Lydian augmented" because the major third and major sixth are replaced? Let's find an authoritative source. Let's search memory: The "Phrygian dominant" is also known as "Harmonic minor dominant." The second mode is "Lydian dominant"? Wait, "Lydian dominant" is the 4th mode of the melodic minor. The 4th mode of melodic minor is Lydian dominant: C D E F# G A Bb C. That's 1,2,3,#4,5,6,b7. Not our pattern.

Let's compute the second mode of Phrygian dominant in terms of the "Spanish gypsy" scale. For the Spanish gypsy (Phrygian dominant) on G: G Ab B C D Eb F. The second mode: Ab B C D Eb F G. If we compare that to the "Lydian augmented" on G: G A B C# D E F#? That would be 0,2,4,#4,5,7,9? Not.

Wait, maybe we should compute the mode by re-evaluating intervals correctly. Let's write the interval pattern of Phrygian dominant in semitones: 0,1,3,5,7,8,10. That's the set for Phrygian dominant relative to G. The second mode would start on Ab, which is the 2nd degree of Phrygian dominant. So relative to Ab, we shift the pattern by -1 semitone? Actually if we shift the pattern such that Ab becomes the new root (0). The intervals from Ab to B: we earlier found 3 semitones (b3). That seems wrong. Let's recompute the pattern of the second mode relative to the new root. We have notes: Ab (0), B (3), C (5), D (7), Eb (8), F (10), G (11). So the differences from Ab: 3,5,7,8,10,11. That matches 0,3,5,7,8,10,11. This is the interval set.

We can try to find a known mode. Let's search memory of "Modes of the Phrygian dominant" in a cheat sheet:

Mode 1: Phrygian dominant
Mode 2: Lydian augmented
Mode 3: Mixolydian b2
Mode 4: Dorian b2
Mode 5: Aeolian augmented
Mode 6: Locrian b3
Mode 7: Ionian b3
But we need to verify. Let's check if our pattern matches Aeolian augmented: Aeolian augmented might be 0,2,3,5,7,8,10? Wait, no.

Maybe we should think of the 2nd mode of Phrygian dominant as a subset of "Harmonic minor"? Actually if you consider the Phrygian dominant on G: G Ab B C D Eb F. The 2nd mode starting on Ab: Ab B C D Eb F G. This is basically the "Harmonic minor" on G? Let's check G harmonic minor: G A Bb C D E F#. Not ours.

Wait, maybe the second mode is the "Mixolydian b11"? Let's check: Mixolydian b11 pattern: 0,2,4,5,7,9,10? Not ours.

Alternatively, maybe the second mode of Phrygian dominant is "Mixolydian augmented"? That would be 0,2,4,5,7,9,11? Not ours.

Let's search for "Ab B C D Eb F G" maybe it's known as "Lydian augmented"? Let's check known patterns: The "Locrian 7" maybe? Eh.

Wait, perhaps we should simply answer: "The second mode of Phrygian dominant is the Lydian augmented scale." But I'm not comfortable. Let's verify: The "Lydian augmented" pattern might be 1,2,3,#4,5,6,7. Wait, that is the "Lydian augmented" pattern, not our pattern. But maybe some definitions differ. Let's check if the second mode of Phrygian dominant is known as "Lydian augmented b3"? That would be 1,2,3,#4,5,6,b7? Not ours.

Let's do a search in my mind: The second mode of the Spanish Gypsy scale (Phrygian dominant) is the "Lydian dominant" if you consider the Spanish Gypsy scale as the 5th mode of the harmonic minor, the second mode would be the 6th mode of the harmonic minor, which is Mixolydian #11. Wait, let's confirm: The 6th mode of harmonic minor is Mixolydian #11. Starting on the 6th degree of harmonic minor. For C harmonic minor, the 6th degree is A. So Mode 6 is A B C D Eb F G. That's A, B, C, D, Eb, F, G. That is exactly our second mode if we start on G Phrygian dominant and then shift to Ab? Wait, our second mode Ab B C D Eb F G. That's not A. But if we start on G and shift the root, maybe the second mode of Phrygian dominant starting on the second degree (Ab) is just Mixolydian #11. But the key is off. Wait, maybe we can map: Our second mode notes are Ab B C D Eb F G. The 6th mode of harmonic minor starting on A: A B C D Eb F G. If we transposed that down by a major third? Wait, from A to Ab is a minor second down. So the intervals differ. But the interval set is same, because both have pattern 0,2,3,5,7,8,10? Wait, A B C D Eb F G gives pattern 0,2,3,5,7,8,10. Our pattern is 0,3,5,7,8,10,11. So it's different.

We need a clear answer. Perhaps the answer is simply "the second mode of the Phrygian dominant is known as the Lydian augmented" or "the second mode of the Phrygian dominant is a Lydian augmented." Actually I think this is correct: The second mode of the Phrygian dominant is called the "Lydian augmented" (also called "Lydian augmented #3"). But I need to confirm.

Let's think of a quick cheat: The Phrygian dominant is used in Spanish classical music. The second mode might be called "Lydian augmented" because of the augmented fourth. Let me check the intervals: The Phrygian dominant's second mode would have an augmented fourth relative to its root, a minor third, and a minor seventh? Actually it has a minor third (b3) as the second degree, a major fourth, a perfect fifth, a minor sixth, a major seventh, and an augmented 7th? Wait, not exactly.

Let's re-evaluate the intervals for mode 2: relative to Ab, we have 0,3,5,7,8,10,11. That is: major third? Wait, 3 semitones is minor third, 5 semitones is perfect fourth, 7 semitones is perfect fifth, 8 semitones is minor sixth, 10 semitones is major seventh, 11 semitones is major seventh? Actually 11 semitones from root is major seventh? Wait, 11 semitones is major seventh? No, major seventh is 11 semitones. So our scale has major seventh at 11, but we also have 10 as 10 semitones (major seventh). Wait, I'm confused.

Let's compute: Ab B C D Eb F G. Distances: Ab to B: 3 semitones (minor third), B to C: 1 semitone, C to D: 2 semitones, D to Eb: 1 semitone, Eb to F: 2 semitones, F to G: 2 semitones? Actually F to G is 2 semitones. So pattern of steps: 3,1,2,1,2,2. This doesn't correspond to standard modes.

But let's compute again from Ab: 0,3,5,7,8,10,11. Let's examine known modes: The "Phrygian dominant" pattern is 0,1,3,5,7,8,10. Our pattern is 0,3,5,7,8,10,11. That is like shifting the pattern by +2 for the second step? Actually the pattern of Phrygian dominant with the 2nd degree missing (b2 removed) and the 7th replaced by the major 7th? Eh.

Let's find the "Super Locrian" pattern: 0,1,3,4,6,8,10. Not ours.

The "Altered" pattern: 0,1,3,4,6,8,10. Not ours.

"Mixolydian b5" pattern: 0,2,4,5,7,9,10. Not ours.

Maybe this is called the "Lydian augmented" but we need to verify. Let's test: The "Lydian augmented" pattern from C: 0,2,4,6,7,9,11. Not ours. So no.

What about "Lydian dominant"? 0,2,4,6,7,9,10. Not ours.

What about "Lydian #2"? pattern: 0,2,3,5,7,9,10. Not ours.

Check "Lydian #6": 0,2,4,6,7,8,10? Not ours.

Check "Mixolydian b2"? 0,1,3,5,7,9,10. Not ours.

Check "Dorian #2"? 0,2,3,5,7,9,10. Not ours.

Check "Dorian #3"? 0,2,3,5,7,9,10? Not ours.

Check "Aeolian augmented"? 0,2,3,5,7,9,10? Not ours.

Check "Ionian b3"? 0,2,3,5,7,9,11? Not ours.

Let's step back: It's probably known as "Lydian augmented" but maybe I'm misremembering. Let's check an authoritative source: The "Phrygian dominant" is also called the "Spanish Gypsy" or "Dominant harmonic minor". Its modes:

Phrygian dominant
Lydian augmented
Mixolydian b2
Dorian b2
Aeolian augmented
Locrian b3
Ionian b3
But these are speculated. Let’s search memory: I think I've seen "Lydian augmented" used for the 2nd mode of Phrygian dominant. Actually "Lydian augmented" is 1,2,3,#4,5,6,7? Wait, that has a #4 but no b3. So not.

Let's check if the second mode might be "Mixolydian b2" because that has a major third? Wait, Mixolydian b2 is 1, b2, 3, 4, 5, 6, b7? That might be close? Let's compute Mixolydian b2 pattern: 0,1,3,5,7,9,10. Not ours.

Alternatively, maybe it's "Dorian b2"? 0,1,3,5,7,9,10. Not ours.

Let's find "Aeolian augmented" pattern: 0,2,3,5,7,8,10? Actually Aeolian augmented might be 0,2,3,5,7,9,10? Not ours.

Maybe "Ionian b3"? 0,2,3,5,7,9,11? Not ours.

Wait, maybe we should search for the second mode of Phrygian dominant in a source. Without external search, I can use logic: The second mode of Phrygian dominant is simply the Phrygian dominant but rotated by a step. It doesn't have a common name, so we can answer: "It doesn't have a widely used name beyond being the second mode of the Phrygian dominant; it's sometimes called the 'Lydian augmented' but not standard." Actually it's sometimes called "Lydian augmented". Let's verify: The second mode of the Phrygian dominant is the Lydian augmented because the pattern has a major third and major sixth. But we have minor third, major sixth. Wait, maybe we mis-evaluated intervals: Let's check the pattern again: 0,3,5,7,8,10,11. The 3rd is major third relative to the root? Wait, 3 semitones is minor third. But the major third would be 4. So we don't have a major third. But the Lydian augmented has a major third, so it's not that.

However, maybe the root is not Ab but something else. If we choose a different key for the second mode, maybe the pattern becomes standard. Let's try the second mode of Phrygian dominant on key G: Ab B C D Eb F G. Suppose we treat this as key D? Wait, if we transpose it up a major third (4 semitones) to C? Let's see: Ab -> C? Actually a major third up from Ab is C? Wait, Ab to C is 4 semitones up? Actually Ab to A is