Talk:70:90:105:126

Name

This should be the subharmonic seventh chord, not 1/(7:6:5:4). 1/(7:6:5:4) is a half-diminished chord that doesn't even include a perfect fifth, and the term subharmonic alone isn't enough to signify that as the altering on the fifth should always be explicit. Also, the subharmonic ninth chord 1/(9:7:6:5:4) should extend the subharmonic seventh chord, so if 1/(7:6:5:4) is the subharmonic seventh chord, then 1/(9:7:6:5:4) can't be the subharmonic ninth chord. Finally, there's the neat symmetry that the harmonic sixth chord 6:7:9:10 inverts to the subharmonic seventh chord 1/(7:6:5:4) and the harmonic seventh chord 4:5:6:7 inverts to the subharmonic sixth chord 1/(12:10:8:7). —FloraC (talk) 08:58, 11 March 2026 (UTC)

Please see https://en.xen.wiki/w/Kite%27s_thoughts_on_harmonic_and_subharmonic_nomenclature. I believe it addresses all your objections. --TallKite (talk) 18:55, 20 March 2026 (UTC)

I think that's a good try. It's almost the best one can make out without awareness of the basics of inverting a chord in the practice of negative harmony. Unfortunately due to the lack of awareness, your nomenclature for seventh chords and onwards, as well as sixth chords, are all unnecessarily irregular.

In negative harmony practice, a chord is inverted not w.r.t the tonic but w.r.t. the midpoint of the tonic and fifth. The major triad inverts to the minor triad, but the dominant seventh chord inverts to the minor-major sixth chord. Since the minor seventh is a minor third above the fifth, inverting it makes it a minor third below the tonic, and octave-reducing it gives the major sixth.

Becuz you paired the dominant seventh chord with the minor seventh flat-fifth chord, your nomenclature shows several asymmetries:

• Your harmonic ninth chord is an extension of your harmonic seventh chord, but your "subharmonic ninth chord" isn't an extension of your "subharmonic seventh chord" up to octave reduction.
• Your harmonic eleventh chord corresponds to a common name, but your "subharmonic eleventh chord" doesn't.
• Your sixth chords are irregular special cases.

None of them is necessary. They are all fixed if you follow the negative harmony practice I laid out above, by pairing dominant seventh chords with minor-major sixth chords. From here, we further have dominant ninth chords with minor-major sixth added-eleventh chords, dominant added-eleventh chords with minor-major sixth-ninth chords, and dominant eleventh chords with minor-major sixth-eleventh chords. Assign 5/4, 7/4, and 11/4 to the appropriate diatonic categories and you'll sort all the JI chords out.

So in conclusion, I think I've made a good case for my initial request that 1/(9:7:6:5) is the subharmonic seventh chord, not 1/(7:6:5:4).

—FloraC (talk) 09:34, 21 March 2026 (UTC)

Before we debate our disagreements, can we start with saying what we agree with? 1) I agree that the negative harmony approach of melodically inverting the chord and then moving the root down a fifth is a reasonable approach. BTW it's not true that I wasn't aware of it. I'm just not convinced it's the best approach. 2) Do you agree that "7th chord" can refer to not just the dom7 chord but also the min7, dim7, etc.? And the same for 6th chord, 9th chord, etc.? 3) Of all the chord names I have listed on my "Kite's Thoughts" page, which ones are the same as your names? --TallKite (talk) 19:48, 22 March 2026 (UTC)

Right, I think the negative harmony practice is the best practice since it minimizes asymmetries. I agree on your (2). However, I don't see a need to assign harmonic/subharmonic-based names to all common JI chords, so I simply don't try. For example, 8:10:12:15 doesn't need a harmonic-based name, since it's 5-limit. Same with 10:12:15:18. I think you agree, as you said: "Often the resulting terms are rather useless, because it duplicates a much more widespread term." The chords that do need such terms are those that mix intervals of multiple primes, like 4:5:6:7.

As for (3), my nomenclature is the same as yours for harmonic chords from seventh to thirteenth, as well as harmonic and subharmonic sixth chords. The subharmonic seventh chord is obviously different, and the subharmonic ninth chord I'm not super sure of. I don't really assign simple subharmonic eleventh or thirteenth chords as they are not very notable/useful to begin with. The useful ones are extensions of the subharmonic sixth chord: the subharmonic sixth added-eleventh chord 1/(24:20:16:14:9), the subharmonic sixth-ninth chord 1/(24:20:16:14:11), and the subharmonic sixth-eleventh chord 1/(24:20:16:14:11:9).

—FloraC (talk) 14:26, 24 March 2026 (UTC)

Yes, I agree 5-limit chords like 8:10:12:15 don't need a harmonic name. But I do sort of like "harmonic third" for 5/4, since "major 3rd" includes 81/64, 4\12, etc.

The "glue" that holds subharmonic chords together is the simplicity of the intervals between individual notes. This simplicity only occurs in certain voicings. 24:20:16:14:11:9 is a very dissonant voicing. 11:9:7:6:5:4 is much better. Because it has much simpler ratios. Surely we can agree that 9/4, 9/5 and 3/2 sound better than 16/9, 20/9 and 8/3? And likewise that 7/4, 7/5 and 7/6 sound better than 8/7, 10/7 and 12/7? And 11/4, 11/5 and 11/6 sound better than 16/11, 20/11 and 24/11?

So what to call 11:9:7:6:5:4? Is it a sixth-ninth-eleventh chord with the 6th, 9th and 11th voiced below the root? Or is it simply an eleventh chord in root position, in the obvious voicing, a close voicing of stacked 3rds? The latter is much more straightforward. This is why the 11th subharmonic makes much more sense as a root than the 3rd subharmonic.

Same with 24:20:16:14:11 vs. 11:7:6:5:4. Same for 24:20:16:14:9 vs. 9:7:6:5:4, but here the 9th subharmonic is the obvious root.

--TallKite (talk) 21:42, 31 March 2026 (UTC)

So if you mean 1/(24:20:16:14:11:9) and 1/(11:9:7:6:5:4), then no, I strongly disagree. 1/(24:20:16:14:11:9) is a much more consonant voicing cuz it's more regularly tertian and more notes are on the consonant categories, like the minor third, the perfect fifth, even the neutral ninth and the perfect eleventh. The perfect fifth above the bass is especially significant here, as it suits itself much better to traditional chord naming systems. I mean, if you don't mind me using a similar logic as yours, surely we can agree that 6/5, 3/2, 12/7, 24/11, and 8/3 sound better than 11/9, 11/7, 11/6, 11/5, and 11/4, on average?

From there, 1/(24:20:16:14:9) and 1/(24:20:16:14:11) are one-note omissions of 1/(24:20:16:14:11:9), which makes them likewise notable. But anyway, I call these chords sixth-eleventh, sixth added-eleventh, and sixth-ninth chords, so there's no conflict in names. The only special case is 1/(9:7:6:5:4), for which I think we can agree this could be the subharmonic ninth chord, but that again points to 1/(9:7:6:5) as the subharmonic seventh chord, cuz what's the ninth being added to?

Also, inflecting the third and sixth by 36/35 turns the subharmonic sixth chord into the harmonic sixth chord. Since we're using 36/35 to flip the o/utonality without changing the intervals' degrees or qualities, all we need to do in the names is to flip the harmonic/subharmonic part. It follows that inflecting the third and seventh by the same interval should similarly turn the harmonic seventh chord into the subharmonic seventh chord. This bond is so simple and clear. I don't see why you ignored it in your nomenclature.

—FloraC (talk) 08:17, 1 April 2026 (UTC) (last edited FloraC (talk) 12:28, 3 April 2026 (UTC))

I actually listened to the voicings of all these chords using normal harmonic timbres, and both Praveen and I agreed that the ones with the 11th subharmonic in the bass sound better.

"Surely we can agree that 6/5, 3/2, 12/7, 24/11, and 8/3 sound better than 11/9, 11/7, 11/6, 11/5, and 11/4?" I don't agree. Consider these two chords 1/1 6/5 3/2 5/3 (min6) vs. 1/1 6/5 3/2 9/5 (min7). Which is more consonant? By your logic, considering only the 3 intervals from the root, since 5/3 is more consonant than 9/5, the min6 is more consonant. But what one actually hears is not just those 3 intervals, but all 6 intervals within the chord. Min6 has 6/5 5/4 10/9, 3/2 25/18, 5/3. Min7 has 6/5 5/4 6/4, 3/2 3/2, 9/5. Ignoring the ratios they have in common, we have 10/9 vs 6/5 and 25/18 vs 3/2 and 5/3 vs 9/5. The dissonant 25/18 makes min7 the clear winner.

Can we agree that when judging a chord's consonance, one must take into account all intervals within the chord, not just the ones from the root?

Assuming your answer is yes, here are all 15 intervals in 24:20:16:14:11:9

6/5 5/4 8/7 14/11 11/9

3/2 10/7 16/11 14/9

12/7 20/11 16/9

24/11 20/9

8/3

And here are all 15 intervals in 11:9:7:6:5:4

11/9 9/7 7/6 6/5 5/4

11/7 3/2 7/5 3/2

11/6 9/5 7/4

11/5 9/4

11/4

Discarding those ratios in common, we have 11 ratios left. Let's compare them using Weil height, Bernadetti height, and Wilson height.

8/7 vs 7/6 and 14/11 vs 9/7 -- the latter wins both times

14/9 vs 3/2 and 10/7 vs 7/5 and 16/11 vs 11/7 -- the latter wins all 3 times

16/9 vs 7/4 and 12/7 vs 9/5 and 20/11 vs 11/6 -- the latter wins all 3 times

20/9 vs 9/4 and 24/11 vs 11/5 -- the latter wins both times

8/3 vs 11/4 -- the former wins

The latter wins 10 out of 11 times, no matter which height you use.

"It suits itself much better to traditional chord naming systems." What's more important, a chord's sound or its name?

"More notes are on the consonant categories." Is this still true when you consider all 15 ratios? I don't know how you define "consonant category", but surely it can't contradict all 3 of the heights I used.

I agree that 9:7:6:5:4 is the subharmonic ninth chord. But by the rules of negative harmony, this should be 1/1 6/5 3/2 12/7 8/3, a min6add11 chord 24:20:16:14:9. In choosing 9:7:6:5:4 over this, you're actually arguing against using negative harmony. Note that 24:20:16:14:9 is a more dissonant voicing than 9:7:6:5:4. Analyzing the ratios as before, we have the latter winning 8/7 vs 7/6 and 10/7 vs 7/5 and 16/9 vs 7/4 and 14/9 vs 3/2 and 12/7 vs 9/5 and 20/9 vs 9/4. Whereas the former wins only 8/3 vs 9/7.

"What's the ninth being added to?" Again, I view subharmonic chords as being extended downwards, not upwards. Because the subharmonic series itself extends downwards, not upwards. You may or may not agree with me on this, but it is in fact a logical and consistent approach. So the 9th is added to 7:6:5:4 to make 9:7:6:5:4. And adding the 11th makes 11:9:7:6:5:4.

"Inflecting the third and sixth by 36/35... why you ignored it in your nomenclature." I ignored it for two reasons. One, converting harmonic to subharmonic by commatic inflections holds for the 7-limit, but breaks down in the 11-limit. 11/8 and 18/11 differ by 301c. Two, IMO the significance of a subharmonic chord should be determined not by whether it's commatically close to a harmonic chord, but by its consonance. 7:6:5:4 is more consonant than 9:7:6:5, because 5/4 beats 9/7 and 7/4 beats 9/5.

--TallKite (talk) 22:32, 2 April 2026 (UTC)

> Can we agree that when judging a chord's consonance, one must take into account all intervals within the chord, not just the ones from the root?

I agree. However, complexities of individual ratios aren't the only factor for consonance. If anything, notes above the root deserve more weight. Tertian chords deserve to be favored. Chords with a perfect fifth above the bass in particular deserve a big plus.

So, to get to your example, in 1–6/5–3/2–5/3 there's 25/18, which is discordant, and not examining all the component intervals in the chord means failing to spot this, and incorrectly cataloguing it as a consonant chord. Perhaps by consonance you just mean concordance, but consonance is really more than that. Anyway, in 1/(24:20:16:14:11:9) and 1/(11:9:7:6:5:4) all the component intervals are concordant already since they're all 11-odd-limit, so the chord structure plays the major role of determining which chord is more consonant here.

> What's more important, a chord's sound or its name?

The question is out of context, and besides the point anyway. Of course sound is more important, but we're here to resolve a name conflict. So name is our focus.

> Is this still true when you consider all 15 ratios? I don't know how you define "consonant category", …

A consonant category is literally an interval category that is considered consonant in tonal harmony … like the major and minor third. Of course, the note should be at least fairly concordant to begin with, which is satisfied in our case. So, since all the notes are on the consonant categories, each component interval have clear functions. Some of them are neutral, but neutral intervals occuring between the notes is much less of a dissonance than on the bass.

> I agree that [1/(9:7:6:5:4)] is the subharmonic ninth chord. But …

I didn't choose this over 1/(24:20:16:14:9). I've consistently argued that 1/(24:20:16:14:9) is the more significant voicing and only brought up 1/(9:7:6:5:4) as an answer to "what is the subharmonic ninth chord if there is one" cuz 1/(24:20:16:14:9) is clearly not a ninth chord. And again I strongly disagree that 1/(24:20:16:14:9) is a more dissonant voicing than 1/(9:7:6:5:4) (see above).

> I view subharmonic chords as being extended downwards, not upwards.

In negative harmony practice, subharmonic chords are also extended downwards, but starts on the perfect fifth. This has many advantages, including making chord extensions intuitive. Surely one expects a dominant ninth chord to extend a dominant seventh chord, not a half-diminished seventh chord?

> Converting harmonic to subharmonic by commatic inflections holds for the 7-limit, but breaks down in the 11-limit.

I don't call it "break down". There's more to it in the 11-limit, like in the 7-limit, you have the pairs 4:5:6:7 ↔ 1/(12:10:8:7) and 6:7:9:10 ↔ 1/(9:7:6:5), whereas in the 11-limit commatic inflection gives you 4:5:6:7:9 ↔ 1/(24:20:16:14:11) and 4:5:6:7:11 ↔ 1/(24:20:16:14:9), but 1/(24:20:16:14:9) is the inverse of 4:5:6:7:9 and 1/(24:20:16:14:11) the inverse of 4:5:6:7:11. It's a little different, but there's still logic to it.

> IMO the significance of a subharmonic chord should be determined not by whether it's commatically close to a harmonic chord, but by its consonance.

As I said, inflecting the third and sixth by 36/35 flips the o/utonality without changing the intervals' degrees or qualities. This is a very useful property, very helpful for memorizing the 7-limit chords, and I think you should start to consider it, in addition to all the other things I've shown and explained to you.

—FloraC (talk) 12:28, 3 April 2026 (UTC)