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This question is a revision of this question. My theory of harmony was a bit wrong, especially for the G major chord was off. I created a new theory to re-tune the 12 tones. The just ratio of the tones are now:

    C = 1/1
    C♯ = 21/20
    D = 9/8
    E♭ = 6/5
    E = 5/4
    F = 4/3
    F♯ = 7/5
    G = 3/2
    A♭ = 8/5
    A = 5/3
    B♭ = 9/5
    B = 15/8

So this is basically just intonation with septimal C♯ and F♯.

My questions are now:

  1. Is it acceptable that ♯ and ♭ are by different ratio? (♯ is by septimal chromatic semitone. ♭ is by just chromatic semitone.)

  2. Is the phrygean mode on C in tune? What about the lydian mode on C?

  3. What is the name of A - C - E - F♯ chord?

Dannyu NDos
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  • Acceptable for what purpose? As with any just intonation scale, it works fine if you avoid certain chords. In this case, you can't use D minor or D major because the fifth from D to A has the ratio 40:27. – phoog Oct 18 '19 at 08:10
  • @phoog I mean are sharps and flats supposed to be by the same interval? – Dannyu NDos Oct 18 '19 at 08:40
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    No. Nothing needs to be the same interval, just as your whole steps are not the same interval (C to D is 9:8 while D to E is 10:9). In fact, in true just intonation, even the pitches change from one moment to the next, so sometimes C to A will be 5:3 and sometimes it will be 27:16. That's why it's not generally useful (or possible) to tune a keyboard in just intonation. – phoog Oct 18 '19 at 12:26
  • Your assumption that they are supposed to be the same probably stems from learning equal tempered tuning. They do not have to be the same. –  Oct 18 '19 at 14:12

2 Answers2

2

"Acceptable" doesn't matter in the following sense: unless you are using equal temperament, there will be an interval (or more than one) with two differing ratios. (You can prove this by the pigeonhole principle.)

Whether one finds a mode in better or worse tuning than other modes on the same set of notes doesn't matter much either; once the basic intervals are chosen, you're stuck with whatever these yield.

As written, A-C-E-F# is an F# diminished (stack by thirds) with an added minor seventh; (I think this is a half-diminished F#7).

One think to check on when using integer ratios, not all intervals will be unique. However, if the difference in these intervals is smaller than that of using equal temperament, there's no reason to prefer that tuning. All such tunings (Pythagorean, just, mean-tone, etc.) have preferred keys; whether this is desirable depends on the modulations used in music being played using this tuning.

Because no power of 2 is equal to a power of 3 (except the 0th power = 1), there is no "best" solution.

ttw
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1

To try to answer your questions:

(1) I don't know what you mean by "acceptable." Many just intonation scales use various types of modifications to generate flats and sharps. Sometimes they use the same modification everywhere; sometimes not. Sometimes they use both, creating two different accidentals, such as a different tuning for C-sharp vs. D-flat in the same scale. There are a huge variety of possibilities in JI tunings, and what a particular listener considers "acceptable" depends on the person.

(2) I don't know what you mean by "in tune."

Here are the intervals in the Phrygian subset of your scale:

        1     2     3     4     5     6     7  
 1/1  : 21/20 6/5   4/3   3/2   8/5   9/5   2/1
 21/20: 8/7   80/63 10/7  32/21 12/7  40/21 2/1
 6/5  : 10/9  5/4   4/3   3/2   5/3   7/4   2/1
 4/3  : 9/8   6/5   27/20 3/2   63/40 9/5   2/1
 3/2  : 16/15 6/5   4/3   7/5   8/5   16/9  2/1
 8/5  : 9/8   5/4   21/16 3/2   5/3   15/8  2/1
 9/5  : 10/9  7/6   4/3   40/27 5/3   16/9  2/1
 2/1

The first column has the basic notes of the Phrygian mode, as applied to your scale. The other columns show ratios from each note up a specified number of steps within the Phrygian mode.

Here is the same data in cents:

         1     2     3     4     5     6      7     
 0.0   : 84.5  315.6 498.0 702.0 813.7 1017.6 1200.0
 84.5  : 231.2 413.6 617.5 729.2 933.1 1115.5 1200.0
 315.6 : 182.4 386.3 498.0 702.0 884.4 968.8  1200.0
 498.0 : 203.9 315.6 519.6 702.0 786.4 1017.6 1200.0
 702.0 : 111.7 315.6 498.0 582.5 813.7 996.1  1200.0
 813.7 : 203.9 386.3 470.8 702.0 884.4 1088.3 1200.0
 1017.6: 182.4 266.9 498.0 680.4 884.4 996.1  1200.0
 1200.0

Whether those intervals count as "in tune" or not depends on what you mean by that phrase. Some are significantly wider or narrower than equal temperament. Many fall into standard simple just intonation ratios; others are more complex and rare.

Here's a similar table for the Lydian subset of your scale:

       1     2     3     4      5     6     7  
 1/1 : 9/8   5/4   7/5   3/2    5/3   15/8  2/1
 9/8 : 10/9  56/45 4/3   40/27  5/3   16/9  2/1
 5/4 : 28/25 6/5   4/3   3/2    8/5   9/5   2/1
 7/5 : 15/14 25/21 75/56 10/7   45/28 25/14 2/1
 3/2 : 10/9  5/4   4/3   3/2    5/3   28/15 2/1
 5/3 : 9/8   6/5   27/20 3/2    42/25 9/5   2/1
 15/8: 16/15 6/5   4/3   112/75 8/5   16/9  2/1
 2/1

And here it is in cents:

         1     2     3     4     5     6      7     
 0.0   : 203.9 386.3 582.5 702.0 884.4 1088.3 1200.0
 203.9 : 182.4 378.6 498.0 680.4 884.4 996.1  1200.0
 386.3 : 196.2 315.6 498.0 702.0 813.7 1017.6 1200.0
 582.5 : 119.4 301.8 505.8 617.5 821.4 1003.8 1200.0
 702.0 : 182.4 386.3 498.0 702.0 884.4 1080.6 1200.0
 884.4 : 203.9 315.6 519.6 702.0 898.2 1017.6 1200.0
 1088.3: 111.7 315.6 498.0 694.2 813.7 996.1  1200.0
 1200.0

Again, some scalar intervals are much wider or narrower than equal temperament, but whether they are "in-tune" is subjective. Similarly, chords produced within this subset may vary from very "smooth" with small ratios to rather complex or odd-sounding (compared to ET and JI scales close to it).

(3) As noted in the other answer, F#-A-C-E would typically be called an F# half-diminished chord (jazz theorists might say minor seventh chord with flat five instead), but it could also be an A minor chord with added sixth, depending on harmonic context. The A-C-E portion creates a nice just minor triad, and the septimal F# will color the whole chord in an interesting way in this tuning.

Athanasius
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