View source for Kite's thoughts on harmonic and subharmonic nomenclature

One can prefix any conventional theory term with the term ''harmonic'' or ''subharmonic'', and ideally it would refer to a specific JI interval, chord or scale. This is my (Kite's) attempt to systematically explore this. Often the resulting terms are rather useless, because it duplicates a much more widespread term. There's not much point in calling 3/2 a harmonic fifth when we can call it a perfect fifth!

== Intervals ==
7/4 is often called the harmonic 7th. 9/5 is not a harmonic 7th because the denominator isn't a power of 2. 15/8 is <u>''a''</u> harmonic 7th, but it's not ''<u>the</u>'' harmonic 7th, because 7/4 is a simpler (lower odd-limit) ratio than 15/8.

The octave complement of a harmonic interval is a subharmonic interval. In such ratios, the numerator is always a power of 2. Thus the subharmonic 2nd is 8/7.

We can generalize this to other degrees:
{| class="wikitable center-all"
|+Harmonic intervals (the more useful terms are bolded)
!interval
![[Color notation|color name]]
!ratio
| rowspan="8" |
!interval
!color name
!ratio
|-
|harmonic 2nd
|wa 2nd
|9/8
!subharmonic 2nd
!ru 2nd
!8/7
|-
!harmonic 3rd
!yo 3rd
!5/4
!subharmonic 3rd
!thu 3rd
!16/13
|-
!harmonic 4th
!ilo 4th
!11/8
|subharmonic 4th
|wa 4th
|4/3
|-
|harmonic 5th
|wa 5th
|3/2
!subharmonic 5th
!lu 5th
!16/11
|-
!harmonic 6th
!tho 6th
!13/8
!subharmonic 6th
!gu 6th
!8/5
|-
!harmonic 7th
!zo 7th
!7/4
|subharmonic 7th
|wa 7th
|16/9
|-
|harmonic 8ve
|wa 8ve
|2/1
|subharmonic 8ve
|wa 8ve
|2/1
|}
One could go further and call 17/16 a harmonic minor 2nd, 19/16 a harmonic minor 3rd, etc. But beyond that, it breaks down fairly quickly because a higher harmonic's quality and even degree can be debatable, especially for higher prime limits. And how precisely are the qualities named? Certainly 11/8 is a neutral 4th, but is 21/16 a harmonic perfect 4th or a harmonic subperfect 4th? If the latter, what would the harmonic perfect 4th be? 43/42 = 512¢? 85/64 = 491¢?

== Chords ==
4:5:6:7 is often called the harmonic seventh chord. What about 9th chords and 11th chords? What about triads?

What exactly is a harmonic chord? A narrow definition requires that the first number in the extended ratio be a power of 2, e.g. 16:19:24. But that would exclude most common JI chords. For our purposes here, every chord is a harmonic chord. The question is, what is ''the'' harmonic  (or subharmonic) chord of a given type (minor, maj7, etc.)? In general, ''the'' harmonic chord is the simplest one, i.e., the one that occurs the lowest in the harmonic series. So while 16:19:24 is ''a'' harmonic minor chord, it's not ''the'' harmonic minor chord.

=== Triads ===
These names are somewhat arbitrary. Arguably, 7:9:11 isn't an augmented chord, and the simplest augmented chord should be 8:10:13 or 16:20:25.
{| class="wikitable center-all"
|+Harmonic and subharmonic triads (the more useful terms are bolded)
!chord type
!color name
![[EFR]]
| rowspan="7" |
!chord type
!color name
!SEFR
|-
|harmonic major chord
|yo
|4:5:6
|subharmonic minor chord
|gu
|6:5:4
|-
!harmonic minor chord
!zo
!6:7:9
!subharmonic major chord
!ru
!9:7:6
|-
!harmonic diminished chord
!gu(zogu5)
!5:6:7
!subharmonic diminished chord
!zo(zogu5)
!7:6:5
|-
!harmonic augmented chord
!ru(loru5)
!7:9:11
!subharmonic augmented chord
!lo(loru5)
!11:9:7
|-
|harmonic sus4 chord
|sus4
|6:8:9
|subharmonic sus4 chord
|sus4
|12:9:8 = 6:8:9
|-
|harmonic sus2 chord
|sus2
|8:9:12
|subharmonic sus2 chord
|sus2
|9:8:6 = 8:9:12
|}
Each subharmonic chord is the melodic inversion of the corresponding harmonic chord, and vice versa.

=== 7th, 9th, 11th and 13th chords ===
The term ''7th chord'' properly refers broadly to multiple chord types (maj7, dom7, min7, min7(b5), dim7, etc.), but sometimes refers narrowly to only the dom7 chord type. Here it is used in the broad sense, as are 9th, 11th and 13th chords.

The simplest possible harmonic JI chord of N notes is a run of N odd harmonics, e.g. 1:3:5:7:9. (In practice, harmonics 1 and 3 are written as 4 and 6, for a closer voicing.) Thus in this section, because we're using the broader definition of ''chord type'', we're using the narrower definition of ''harmonic chord''.

Considering the full gamut from tetrad to heptad, here are the simplest chords:
{| class="wikitable center-all"
|+
!chord type
!color name
!EFR
!12edo name
| rowspan="5" |
!chord type
!color name
!SEFR
!12edo name
|-
!harmonic 7th chord
!har7
!4:5:6:7
!dom7
!subharmonic 7th chord
!sub7
!7:6:5:4
!min7(b5)
|-
!harmonic 9th chord
!har9
!4:5:6:7:9
!dom9
!subharmonic 9th chord
!sub9
!9:7:6:5:4
!dom9
|-
!harmonic 11th chord
!har11
!4:5:6:7:9:11
!dom11
!subharmonic 11th chord
!sub11
!11:9:7:6:5:4
!???
|-
!harmonic 13th chord
!har13
!4:5:6:7:9:11:13
!dom13
!subharmonic 13th chord
!sub13
!13:11:9:7:6:5:4
!???
|}
Again, all sub chords are melodic inversions of har chords.

Harmonic chords extend upwards, so the har7 chord 4:5:6:7 is contained in the lower 4 notes of the har9 chord 4:5:6:7:9. But subharmonic chords extend ''downwards'', just like the subharmonic series extends downwards. Thus the sub7 chord 7:6:5:4 is contained in the ''upper'' 4 notes of the sub9 chord 9:7:6:5:4.

As a result, while all the har chords are extensions of a major triad, the sub chords don't share the same 3rd or 5th. In particular, the 5ths of the sub chords are 6/4, 7/5, 9/6, 11/7 and 13/9, two perfect ones, two diminished ones and one augmented one.

=== 6th chords ===
The har-N and sub-N nomenclature generally requires that N is odd, but an exception is made for har6 and sub6. Here the term ''6th chord'' is used in the broad sense.

The simplest subharmonic 6th chord is 12:10:8:7. The simplest harmonic 6th chord is technically 5:6:7:8, because 6/5 is a 3rd, 7/5 is a 5th and 8/5 is a 6th. But the 4:5:6:7 chord is so elemental and basic that 5:6:7:8 will always be perceived as 4:5:6:7 in 1st inversion. In other words, 5:6:7:8 an extremely implausible chord homonym. Calling 5:6:7:8 a har6 chord would be like calling a C major chord Emin6no5. So ''the'' har6 chord is the simplest ''plausible'' har6 chord, 6:7:9:10.
{| class="wikitable center-all"
|+
!chord type
!color name
!EFR
!12edo name
| rowspan="2" |
!chord type
!color name
!SEFR
!12edo name
|-
!harmonic 6th chord
!har6
!6:7:9:10
!min6
!subharmonic 6th chord
!sub6
!12:10:8:7
!min6
|}
Because the har6 chord "breaks the rules", it doesn't follow the same neat patterns as the other chords. The har6 chord is not the melodic inverse of the sub6 chord. And whereas the sub6 and sub7 chords are [[Chord homonym|homonyms]], the har6 and har7 chords are not.

=== Other tetrads ===
{| class="wikitable center-all"
|+Harmonic and subharmonic tetrads (the more useful terms are bolded)
!chord type
!color name
!EFR
| rowspan="8" |
!chord type
!color name
!SEFR
|-
|harmonic maj7 chord
|yo7
|8:10:12:15
|subharmonic maj7 chord
|yo7
|15:12:10:8 = 8:10:12:15
|-
|harmonic min7 chord
|gu7
|10:12:15:18
|subharmonic min7 chord
|gu7
|18:15:12:10 = 10:12:15:18
|-
!harmonic dom7 chord
!har7
!4:5:6:7
!subharmonic min7(b5) chord
!sub7
!7:6:5:4
|-
!harmonic min7(b5) chord
!gu7(zogu5)
!5:6:7:9
!subharmonic dom7 chord
!ru,gu7
!9:7:6:5
|-
!harmonic dim7 chord
!gu,sogu7
!10:12:14:17
!subharmonic dim7 chord
!soru,sogu7(so5)
!17:14:12:10
|-
|harmonic maj6 chord
|yo6
|12:15:18:20
|subharmonic maj6 chord 2nd inv.
|yo6
|15:12:10:9 = 12:15:18:20
|-
!harmonic min6 chord
!har6
!6:7:9:10
!subharmonic min6 chord
!sub6
!12:10:8:7
|}

== Scales ==

=== Harmonic and subharmonic scales ===
The obvious N-tone scale is N::2N or 2N::N, or some mode of that. Only modes with a stable tonic triad (i.e. a 3rd and a perfect 5th) are listed.
{| class="wikitable center-all"
|+
!scale type
!as a chord
!EFR
!ratios
|-
!harmonic major pentatonic
!har9
!8:9:10:12:14:16
!1/1 9/8 5/4 3/2 7/4 2/1
|-
!harmonic minor pentatonic
!har6,11
!6:7:8:9:10:12
!1/1 7/6 4/3 3/2 5/3 2/1
|-
!subharmonic major pentatonic
!sub9
!18:16:14:12:10:9
!1/1 9/8 9/7 3/2 9/5 2/1
|-
!subharmonic minor pentatonic
!sub6,11
!12:10:9:8:7:6
!1/1 6/5 4/3 3/2 12/7 2/1
|-
!harmonic octotonic
!har15
!8::16
!1/1 9/8 5/4 11/8 3/2 13/8 7/4 15/8 2/1
|-
!subharmonic octotonic
!sub15
!16::8
!1/1 16/15 8/7 16/13 4/3 16/11 8/5 16/9 2/1
|}
The terms ''major'' and ''minor'' refer to the 3rd of the scale. "Harmonic major pentatonic" is quite a mouthful. It can be abbreviated harmajor pentatonic. Likewise we have harminor, subharmajor, etc.

Note that harmajor pentatonic and subharmajor pentatonic are not melodic inverses.

I left out heptatonic scales because the 7::14 scale always feels to me like an 8::16 scale with a missing note.