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The following table lists the first 13 harmonics of the note C
Harmonic Note Unique note
-----------------------------
1 C C
2 C
3 G G
4 C
5 E E
6 G
7 A# A#
8 C
9 D D
10 E
11 F# F#
12 G
13 G# G#
Ordering the unique notes gives:
C - D - E - F# - G - G# - A#
Does this scale have a name? Is it used? What is this scale called?
There are two different questions that could be read here, which it is not obvious (yet extremely significant) that they are different. One question is "what is the name of the scale consisting of the notes given by the first thirteen harmonics of C", and the other is "what is the name of the scale consisting of the notes C - D - E - F# - G - G# - A#", where the crucial point making these questions different is is that the pitches produced by the harmonics of C are not the same pitches as the ones we usually refer to by the letter names – for example the 'F#' given by the 11th harmonic of is in fact almost equidistant between an F and an F# in equal temperament, and so in my view it's substantially misleading to even give it a note name at all. The seventh and thirteenth harmonics are both also substantially far from the equal-tempered pitch they have been given the name of. So if you sat down at a keyboard and played the notes C - D - E - F# - G - G# - A# (or, as per the Wikipedia article referenced in the answer from Aaron (which does a good job of answering the second sense of the question as I've set out above), the Acoustic Scale C - D - E - F# - G - A - A#), you are not in any real sense playing the scale made up of the first 13 harmonics of C at all.
To do something towards answering the first possible sense of the question, to play this scale or use it in a piece we would have to make clear we are in the world of "microtonal" intonation. The composer Ben Johnston developed a notation which permits writing such notes (see the picture here https://en.wikipedia.org/wiki/Just_intonation#Staff_notation where the "F#" is notated as "up-arrow F") however I am not greatly familiar with his and others' work using these tunings so I don't know if there are any examples of this scale being used; certainly he has composed pieces in which notes tuned to these higher harmonics are used, and I would definitely be interested to know if there is an example where this scale occurs.
A final comment is that, whilst the 7th, 11th and 13th harmonics were signficantly far from any equal-temperament pitch, the D arising as the 9th harmonic is exactly a just perfect fifth above the G arising from the 3rd harmonic, since 9=3x3.
The Musical Scale Search Tool offers four scales whose notes correspond to the OP, with only one -- C Minor Lydian -- containing the pitches in the order specified. The others would be permutations/modes of that scale.
G#/Ab; A#/Bb; C; D; E; F#/Gb; G; G#/Ab;C; D; E; F#/Gb; G; G#/Ab; A#/Bb; C;D; E; F#/Gb; G; G#/Ab; A#/Bb; C; D;G; G#/Ab; A#/Bb; C; D; E; F#/Gb; G;Ian Ring's website contains an extensive analysis of the minor lydian scale.
Since the OP scale is constructed in terms of the overtone series, it is in principle the Acoustic scale. However, as given on Wikipedia, the acoustic scale has A natural rather than Ab.
Two songs are referenced as using the acoustic scale (A-natural version) in this page from the University of Iowa:
Your scale is a mode of the scale that Wikipedia calls Neapolitan major. Unfortunately, Neapolitan major is a minor scale (at least, it harmonizes out to a minor triad at the root, not a major one) and the citations for that name are not very convincing. But you can find references to this scale more easily than yours, so it might be a good search term to know.
Wikipedia duplicates the name Aaron found for your mode: "Lydian minor". Again, this seems to be a name that someone invented and that is reproduced only in a few lists of scales. It seems to be a bad name to me -- shouldn't "Lydian Minor" at least have a b3 in it? But I'm sure that doesn't bother everyone the way it does me.
Names like these usually originated in scale books marketed towards rock guitarists; they represent a kind of folk terminology that hasn't passed into common use (because the scales are rarely used) and usually doesn't have any basis outside that. But all that might not bother you too much -- after all, everything has a name because someone decided to call it that, and names don't have to make logical sense.
It might also be useful to know that your scale is the same as the Carnatic melakata Rishabhapriya. This name connects your scale with real-world music-making rather than obsessive list-making, so it may be more helpful for you. A YouTube search throws up lots of South Indian performances of pieces in Rishabhapriya.
In xenharmonic theory, this scale is called an overtone scale, specifically "Mode 8 of the Harmonic Series" (Denny Genovese's term). To be precise, this assumes Just Intonation (pure ratios, not the tempered equivalent) and extending it to 15 (B).
Writing the scale as ratios, you get 8:9:10:11:12:13:14(:15:16), which is an Over-1 scale.
Some people do use it: guitarist Dante Rosati has refretted his guitar to play it, and calls it the "Diatonic Harmonic Series Scale" (see First Five Octaves of the Harmonic Series; this is the fourth octave), since its 8 notes are similar to the 7 notes of the usual diatonic scale.
Peter Hulen has composed some music using this scale (such as his dissertation, The Madman's Diary), and written a paper about how to use it with synthesized music.
See more compositions at Otones8-16.
This is one of the most interesting and practical overtone scales, since you get both very familiar chords such as the 4:5:6 major triad (8:10:12) and its extension to the barbershop tetrad (4:5:6:7), as well as more exotic harmonies, notably the 9:11:13:15 tetrad.
