User:FloraC/Hard problems of harmony and psychoacoustically supported optimization: Difference between revisions
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Such a metric will guide us to psychoacoustically supported chord construction, which spontaneously obeys the rules of primodalism as well. In a primodal chord like 1–6/5–7/5–8/5, notice how all the notes consolidate each other through the bond to a common virtual fundamental of 1/5. It is not rooted, but its harmonic identity is evident. Now consider 1–6/5–10/7–8/5, where we put 10/7 in place of 7/5, and it immediately becomes a 25-odd-limit chord. The virtual fundamental is pushed even lower to 1/35 as the denominators "fight" each other. As a result, a chord like this has little harmonic identity to listen to, and little utility to be concerned with. | Such a metric will guide us to psychoacoustically supported chord construction, which spontaneously obeys the rules of primodalism as well. In a primodal chord like 1–6/5–7/5–8/5, notice how all the notes consolidate each other through the bond to a common virtual fundamental of 1/5. It is not rooted, but its harmonic identity is evident. Now consider 1–6/5–10/7–8/5, where we put 10/7 in place of 7/5, and it immediately becomes a 25-odd-limit chord. The virtual fundamental is pushed even lower to 1/35 as the denominators "fight" each other. As a result, a chord like this has little harmonic identity to listen to, and little utility to be concerned with. | ||
[[File:Frequency spectrum of just major triad in semisine wave.png|thumb|800px|Figure 1: Frequency spectrum of just major triad in semisine wave]] | |||
The same cannot be assumed for tuning optimization, since that is a vastly different scenario. In a just major triad, the 15th harmonic exists in three ways: as the harmonic of the root, of the 3rd harmonic, and of the 5th harmonic. Figure 1 is the frequency spectrum of the triad played in the semisine waveform, which has been proposed as the standard ear-training waveform in ''Proposed Standard Ear-Training Waveform''. | |||
If we play such a triad in a tempered tuning profile, the quality of the chord is determined by how the three components said above line up. In a tuning profile characterized by the mistuning map | If we play such a triad in a tempered tuning profile, the quality of the chord is determined by how the three components said above line up. In a tuning profile characterized by the mistuning map | ||
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We should also note the just minor triad is of equal complexity as the just major triad by the principle of invertibility. The just major triad is sometimes considered to be more important by being isodifferential and thus having a common beating rate. The just minor triad is also isodifferential, though not with respect to frequency but to its inverse, the length of a virtual vibrating string. Optimizing for the just minor triad requires us to put it in the context of negative harmony. Starting atop and step downwards, the optimization targets are first 1/3 and then 1/5, which are analytically equivalent to 3/1 and 5/1 respectively in the positive harmony. | We should also note the just minor triad is of equal complexity as the just major triad by the principle of invertibility. The just major triad is sometimes considered to be more important by being isodifferential and thus having a common beating rate. The just minor triad is also isodifferential, though not with respect to frequency but to its inverse, the length of a virtual vibrating string. Optimizing for the just minor triad requires us to put it in the context of negative harmony. Starting atop and step downwards, the optimization targets are first 1/3 and then 1/5, which are analytically equivalent to 3/1 and 5/1 respectively in the positive harmony. | ||
That all but suggests practically equal importance of divisive ratios and multiplicative ratios in tuning optimization. | That all but suggests practically equal importance of divisive ratios and multiplicative ratios in tuning optimization. | ||
== Chapter III. Power in Proportion == | == Chapter III. Power in Proportion == | ||