I find it fascinating that the two highest ranked scales are precisely the two ones allowed by Plato for his Republic:
Quite the reverse, he replied; and if so the Dorian and the Phrygian are the only ones which you have left.I answered: Of the harmonies I know nothing, but I want to have one warlike, to sound the note or accent which a brave man utters in the hour of danger and stern resolve, or when his cause is failing, and he is going to wounds or death or is overtaken by some other evil, and at every such crisis meets the blows of fortune with firm step and a determination to endure; and another to be used by him in times of peace and freedom of action, when there is no pressure of necessity, and he is seeking to persuade God by prayer, or man by instruction and admonition, or on the other hand, when he is expressing his willingness to yield to persuasion or entreaty or admonition, and which represents him when by prudent conduct he has attained his end, not carried away by his success, but acting moderately and wisely under the circumstances, and acquiescing in the event. These two harmonies I ask you to leave; the strain of necessity and the strain of freedom, the strain of the unfortunate and the strain of the fortunate, the strain of courage, and the strain of temperance; these, I say, leave.And these, he replied, are the Dorian and Phrygian harmonies of which I was just now speaking.Then, I said, if these and these only are to be used in our songs and melodies, we shall not want multiplicity of notes or a panharmonic scale?I suppose not.
CORRECTION: A reader correctly points out in the comments that the names of musical scales are a bit ambivalent, so I looked into the paper itself. By "Dorian" they mean do-re-mib-fa-sol-la-sib-do which corresponds to the ancient Phrygian mode (the peace-like one according to Plato). The ancient Dorian mode (the war-like one according to Plato) seems to correspond to the "Phrygian" one. So, as far as I can tell, the highest ranked is the ancient Dorian and the next highest-ranked is the ancient Phrygian one, but any music experts are free to chime in and correct me.
From the paper:
The 50 heptatonic scales with the highest mean percentage similarity among the >4×107 possible scales evaluated are shown in Table 3. Three of the seven heptatonic modes (see Figure 1) emerge at the top of this list. The Phrygian mode holds the highest rank followed by the Dorian mode and the Ionian mode (the major scale). The fourth ranked scale is similar to the Phrygian mode but contains a neutral second (12:11) instead of a minor second; this collection is the Husayni scale in Arabic music . The Aeolian mode (the natural minor scale) and Lydian mode are the fifth and sixth ranked scales. The next three scales are similar to the Dorian mode but with slight alterations in one or two scale degrees. The seventh ranked scale may represent the Kafi scale in classical Indian music with an alternative sharp sixth scale degree . The eighth ranked scale is the Kardaniya scale in Arabic music [op cit.]. Although the ninth ranked scale does not represent any well-known musical tone collection, the Mixolydian mode is ranked tenth. The Locrian, which is the least used of the Western modes, is ranked fiftieth. Thus both the five-note and seven-note scales preferred in much music worldwide comprise intervals that conform optimally to a harmonic series.
In humans, vocal stimuli arise in a variety of complex ways, not all of which are harmonic. Harmonic series depend on vocal fold vibrations and are characteristic of the “voiced speech” responsible for vowel sounds and some consonants . Although the relative amplitudes of harmonics are altered by filtering effects of the supralaryngeal vocal tract resonances to produce different vowel phones, the frequencies of harmonics remain unchanged [op cit.]. In consequence, the presence of a harmonic series is a salient feature of human vocalizations and essential to human speech and language. It follows that the similarity of musical intervals to harmonic series provides a plausible biological basis for the worldwide human preference for a relatively small number of musical scales defined by their overall similarity to a harmonic series.
PLoS ONE doi:10.1371/journal.pone.0008144
A Biological Rationale for Musical Scales
Kamraan Z. Gill, Dale Purves
Scales are collections of tones that divide octaves into specific intervals used to create music. Since humans can distinguish about 240 different pitches over an octave in the mid-range of hearing , in principle a very large number of tone combinations could have been used for this purpose. Nonetheless, compositions in Western classical, folk and popular music as well as in many other musical traditions are based on a relatively small number of scales that typically comprise only five to seven tones –. Why humans employ only a few of the enormous number of possible tone combinations to create music is not known. Here we show that the component intervals of the most widely used scales throughout history and across cultures are those with the greatest overall spectral similarity to a harmonic series. These findings suggest that humans prefer tone combinations that reflect the spectral characteristics of conspecific vocalizations. The analysis also highlights the spectral similarity among the scales used by different cultures.