# Microtonality in Western Art Music



## GucciManeIsTheNewWebern (Jul 29, 2020)

Given the vast amount of innovation and experimentation in 20th century and contemporary WAM, I would just go out on a limb and assume that microtones have been used by a composer somewhere and sometime. Microtonality is an attribute one usually associates with Arab or Turkish music which imparts its wonderfully unique sound, but certainly there are composers working within the parameters of WAM that have experimented with microtones? I think it opens up a whole universe of possibility. Think about how much composers have achieved to this point with just whole tones and semitones!


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## SONNET CLV (May 31, 2014)

Alois Hába


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## GucciManeIsTheNewWebern (Jul 29, 2020)

Listening that string quartet right now. Those rich and tight harmonies are a delight!


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## Bwv 1080 (Dec 31, 2018)




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## cheregi (Jul 16, 2020)

I haven't delved too deeply but I believe microtonality is a vein that has been explored extensively in 20th-century WAM. La Monte Young's Well-Tuned Piano, in 7-limit just intonation, is probably the most canonical example, and I think it's gorgeous. There's a quote somewhere that I can't find where he jokes that once he's figured out the tuning for a new composition, he's basically already done. In this case the tuning is "derived from various partials of the overtone series of an inferred low fundamental E-flat reference ten octaves below the lowest E-flat on the Bösendorfer Imperial."






Harry Partch is another name to investigate - already in the early 20th century he was building custom instruments to explore octave division into 43 unequal tones. Partch's most significant student to carry on work in this realm was probably Ben Johnston, who I only know about because his student, Kyle Gann, has a great website where he both hosts his own microtonal music and writes a lot about the advantages of different tuning systems: https://kylegann.com/index.html.

Also, you may be interested to know that Renaissance music theorist Nicola Vicentino built a harpsichord with 31 tones per octave in 1555 in order to experiment with microtonal music - https://en.wikipedia.org/wiki/Archicembalo - obviously it didn't exactly take off but it's a neat curiosity!


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## SuperTonic (Jun 3, 2010)

Ben Johnston is another composer who wrote microtonal music. He has a cycle of 10 string quartets that has been recorded in its entirety by the Kepler Quartet. They are all available on Youtube (last time I checked).


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## SONNET CLV (May 31, 2014)

GucciManeIsTheNewWebern said:


> Listening that string quartet right now. Those rich and tight harmonies are a delight!


I'm glad for your delight.
I know folks whose ears get twisted out of shape by most anything written by those "barbarian" composers who came anywhere after the time of Mozart. Hába would likely make those folks go deaf (a condition they'd likely prefer to having to hear any more music by the microtonalist).


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## Mandryka (Feb 22, 2013)

Can someone explain to me where the 12 note scale came from? Is there anything in the nature of sound which leads you to it, or is it a totally arbitrary convention?

In unfretted instruments, the musician can use microtones if he wishes easily enough. Someone said to me once that this sort of thing goes on all the time in a string quartet, that the each musician will adjust their tone slightly to create interesting harmonies with the others, even if they’re playing Mozart.


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## Mandryka (Feb 22, 2013)

This is the opening music from Stockhausen's opera Montag aus Licht, for oboe and electronics


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## Mandryka (Feb 22, 2013)

Michael Finnissy Awaz-y-Nyaz

https://www.21stcenturyoboe.com/performance-videos-and-recordings/michale-finnissy-âwâz-e-niyâz


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## Mandryka (Feb 22, 2013)

Harry Partch's Delusion of the Fury


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## cheregi (Jul 16, 2020)

Mandryka said:


> Can someone explain to me where the 12 note scale came from? Is there anything in the nature of sound which leads you to it, or is it a totally arbitrary convention?
> 
> In unfretted instruments, the musician can use microtones if he wishes easily enough. Someone said to me once that this sort of thing goes on all the time in a string quartet, that the each musician will adjust their tone slightly to create interesting harmonies with the others, even if they're playing Mozart.


I don't totally understand this but I believe it's because Pythagorean circle-of-fifths tuning gets you 12 tones per octave. And of course the circle-of-fifths method was independently discovered and used as a basis for many non-Western music theory systems, such as the Chinese.

As for the string quartet phenomenon you're describing, my understanding is that players gravitate towards Pythagorean pitch for unaccompanied passages, just intonation for chords with other string players, and equal temperament if accompanying piano etc. - so even though it's technically many more than 12 tones per octave they're conceptualized as variants of the basic 12 tones. On the other hand I'm sure you already know Indian classical music derives 24 tones per octave via sharps and flats of each of the Pythagorean 12 tones...


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## calvinpv (Apr 20, 2015)

I think in both of these pieces, the composers are mixing different tuning systems. But I'm not sure how.

Georg Friedrich Haas: *in vain*, for 24 instruments, played in darkness (2000)






Enno Poppe: *Rad*, for two electric pianos and two laptops constantly feeding the pianos new tuning systems (2003)


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## julide (Jul 24, 2020)

Microtonality in art music is jarring and ugly high pitched mess whereas it's never the case with the arabic and turkish art music.


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## calvinpv (Apr 20, 2015)

Mandryka said:


> Can someone explain to me where the 12 note scale came from? Is there anything in the nature of sound which leads you to it, or is it a totally arbitrary convention?
> 
> In unfretted instruments, the musician can use microtones if he wishes easily enough. Someone said to me once that this sort of thing goes on all the time in a string quartet, that the each musician will adjust their tone slightly to create interesting harmonies with the others, even if they're playing Mozart.


I actually just started to read a book approaching music and sound from a math perspective (and the next chapter I'll be reading is about tuning systems, so this thread is good timing), and a comment was made in passing about how the the 12-tone chromatic scale was made to accommodate for transposing the diatonic scale from one key to another. Because when you transpose that scale, which has a very particular sequence of whole and half steps, it will introduce sharps and flats that aren't in the original C major diatonic scale. So you need to fill out the octave with 5 additional potential notes to prevent a transposition that is outside the theoretical range of notes. And as to why composers decided to transpose in the first place: apparently it was because transposable instruments were starting to be made in the Baroque era, like the various clarinets and horns for example, where the same fingerings that you would play on a C instrument yield different notes.

I don't know how true this is, because there was no supporting evidence for this claim. But it sounds reasonable enough.

For me, the more interesting question is why the West settled on a 7-note diatonic scale in the first place. Was there something special about the number 7 in the Pythagorean universe?


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## Xisten267 (Sep 2, 2018)

julide said:


> Microtonality in art music is jarring and ugly high pitched mess whereas it's never the case with the arabic and turkish art music.


I believe that the future of western music lies in microtonality and that it's totally possible to conciliate it with mainstream expectations - it doesn't need to sound ugly to most people.


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## chu42 (Aug 14, 2018)

Ives wrote some of the first microtonal music for the piano.


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## Roger Knox (Jul 19, 2017)

Julian Carrillo - Preludio a Colon (1925)


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## Doctor Fuse (Feb 3, 2021)

The earliest example I can think of, is in the first movement of the 1938 Violin COcnerto of Bela Bartok, after the big climax, and immediately preceding the cadenza.

In my mind, it depicts the wreckage of a typical Budapest house, after a potential air bombing, and the Victrola is still spinning a disc, albeit it is warped and unsteady, hence the warbling quarter tones. Bartok was a pioneer recording engineer (in the field), so he knew about wow and flutter and speed accuracy issues.


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## OMD (Feb 6, 2021)

https://www.mixcloud.com/deafmix3/2319-microtonal-magik/


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## John Lenin (Feb 4, 2021)

Sliding between notes on a hungarian violin is not micro tonality


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## Phil loves classical (Feb 8, 2017)

Mandryka said:


> Can someone explain to me where the 12 note scale came from? Is there anything in the nature of sound which leads you to it, or is it a totally arbitrary convention?
> 
> In unfretted instruments, the musician can use microtones if he wishes easily enough. Someone said to me once that this sort of thing goes on all the time in a string quartet, that the each musician will adjust their tone slightly to create interesting harmonies with the others, even if they're playing Mozart.


I've heard the explanation of all 12 tones being derived from a sequence of perfect fifths of the previous note, before going back to your starting note. I recall there is a small bit of error, when you come full circle but the ear can't really detect it. Very early on, many cultures noted the perfect consonance of the perfect 5th. In math it's represented by a ratio of 3/2, so that could be one explanation why it just sounds that way.


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## premont (May 7, 2015)

cheregi said:


> As for the string quartet phenomenon you're describing, my understanding is that players gravitate towards Pythagorean pitch for unaccompanied passages, just intonation for chords with other string players, and equal temperament if accompanying piano etc...


Singers can do that, but I do not think string instruments can do it consequently, the limit being the way the strings are tuned. I mean if the strings are tuned in pure Pythagorean fifths, you can't suddenly play in equal tuning. Or do they tune the strings differently for the different situations?


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## BabyGiraffe (Feb 24, 2017)

Phil loves classical said:


> I've heard the explanation of all 12 tones being derived from a sequence of perfect fifths of the previous note, before going back to your starting note. I recall there is a small bit of error, when you come full circle but the ear can't really detect it.


It's bigger than 1/5 of a semitone, so it's not a small bit of error at all.
Still, stacking pure fifths doesn't give plenty of good major and minor thirds in reasonable number of notes, that's why Baroque, Renaissance and Classical music are based on meantone tuning (and are not really in 12 tones, because most keyboard instruments had split black keys). Irregular temperaments like these employed by Bach and some others are worse than meantone in some keys, but better than pythagorean or meantone in some others.
I have searched today the best equal tunings that are better than all the smaller ones in approximating the harmonic series with certain additional criteria that will take too long to explain, but here are some of the smallest ones:
5,7,12,19,22,27,31,41,46,58,72,87,94, 111 etc.
Out of these 5,7,12,19, 31 are meantone systems (even if the first two are hardly harmonic scales at all).
72 = 6 x 12 is basically 11-limit just intonation (Harry Partch used 43 notes scale for marimba that can be translated to it), so I would say this would be the biggest utility of 12 equal in the future, if higher harmonics up to 11th are accepted in chord construction in more mainstream music.
31 is equivalent to historical 1/4 comma meantone.
https://en.wikipedia.org/wiki/Quarter-comma_meantone


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## John Lenin (Feb 4, 2021)

If you can't say what you want to say with real notes then some pretend sound system will always get the untalented some 'airtime'..


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## cheregi (Jul 16, 2020)

premont said:


> Singers can do that, but I do not think string instruments can do it consequently, the limit being the way the strings are tuned. I mean if the strings are tuned in pure Pythagorean fifths, you can't suddenly play in equal tuning. Or do they tune the strings differently for the different situations?


As I understand it, it's true that the moment-by-moment microtonal adjustments I'm describing can't be applied to open strings, but if we assume that most of the notes a string players plays are _not_ open, and of course we're talking about unfretted instruments, then _most_ of the time the capacity for microtonality is as extensive as that of the human voice.

Actually since my last post I've talked about this with someone who specializes in Western classical instrument design. In brass ensembles, or string ensembles, or vocal groups, or any other situation where there's no equal tempered instrument tying everyone down, everybody is, without even thinking about it, simply because it sounds right, gravitating towards the just-tuned / 'pure' version of every single chord or harmony that occurs in the music at any given moment, which means not just a conflict between equal temperament and _one_ just intonation, but potentially _dozens_ of different just intonations. There are experimental keyboard instruments that use A.I. to understand what each note's tuning should be at any given moment based on this principle, but it's quite unwieldy and has yet to match the nuance that singers and instrumentalists naturally bring. This is also partially why it's so hard to stay in tune singing, say, Gesualdo madrigals whose modal centers are so unclear, because you need to resist, to some extent, the voice's inclination to constantly retune itself.

Anyway as for the wider topic - my inclination is that (active, deliberate) explorations of microtonality in Western art music do tend to lack nuance, and 'show off' their microtonality as the 'point' rather than incorporate it gracefully into a coherent system like Arabic or Indian classical musics... on the other hand, it seems fair to compare the current golden age of xenharmonic music to Ars Subtilior or Mannerist musics, where an avant-garde emerged alongside or because of new 'notational' technologies (smaller and smaller rhythmic units in the former case, neomodality in the latter), and I do love both of those repertoires, in part _because_ of their seeming awkwardnesses... so maybe this xenharmonic age will be more exciting to me in retrospect than it is right now.


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## BabyGiraffe (Feb 24, 2017)

cheregi said:


> As I understand it, it's true that the moment-by-moment microtonal adjustments I'm describing can't be applied to open strings, but if we assume that most of the notes a string players plays are _not_ open, and of course we're talking about unfretted instruments, then _most_ of the time the capacity for microtonality is as extensive as that of the human voice.
> 
> Actually since my last post I've talked about this with someone who specializes in Western classical instrument design. In brass ensembles, or string ensembles, or vocal groups, or any other situation where there's no equal tempered instrument tying everyone down, everybody is, without even thinking about it, simply because it sounds right, gravitating towards the just-tuned / 'pure' version of every single chord or harmony that occurs in the music at any given moment, which means not just a conflict between equal temperament and _one_ just intonation, but potentially _dozens_ of different just intonations. There are experimental keyboard instruments that use A.I. to understand what each note's tuning should be at any given moment based on this principle, but it's quite unwieldy and has yet to match the nuance that singers and instrumentalists naturally bring. This is also partially why it's so hard to stay in tune singing, say, Gesualdo madrigals whose modal centers are so unclear, because you need to resist, to some extent, the voice's inclination to constantly retune itself.
> 
> Anyway as for the wider topic - my inclination is that (active, deliberate) explorations of microtonality in Western art music do tend to lack nuance, and 'show off' their microtonality as the 'point' rather than incorporate it gracefully into a coherent system like Arabic or Indian classical musics... on the other hand, it seems fair to compare the current golden age of xenharmonic music to Ars Subtilior or Mannerist musics, where an avant-garde emerged alongside or because of new 'notational' technologies (smaller and smaller rhythmic units in the former case, neomodality in the latter), and I do love both of those repertoires, in part _because_ of their seeming awkwardnesses... so maybe this xenharmonic age will be more exciting to me in retrospect than it is right now.


In my opinion, adjusting intonation is not a real form of microtonality and is pointless, because music is usually based on some form of temperament or else you run into too many performance problems or reduced potential in harmony (pentatonic or diatonic scales basically need one additional tone to function like in meantone temperament; if someone wants pure harmony with only 7 notes, the maximum number of triads is 5, not 6). There is also no "canonical" tuning, so the tuning algorithm in your example will have to make random decisions every time there is a tempered comma in just tuning -> for example "D" in meantone tuning (standard Western notation) represents both 10/9(minor whole tone) and 9/8(major whole tone), and 729/640 (whole tone + syntonic comma) etc (any linear combination of 10/9 or 9/8 and syntonic comma).

I don't think that exploration of microtonality in Western music is "showing off". It is about exploring new musical resources (while not knowing what are you actually doing). Arabic and Indian music are coherent, but quite simple systems and focused on melody only. Western music is mostly about harmony: even most small microtonal scales that support harmony will be: a) big b) most of them not a consistent pattern that someone can memorize (for example microtonal diatonic scale in just intonation is not a single scale, but a whole family of scale, depending on where you want to place the "wolf" fifth) c)very hard to perform on 1d and hard - on 2d layout (keyboard type is 1d, guitar is 2d); they should be easier in 3d etc layouts, this is probably impossible in practice.
I haven't done quite extensive search, but "Arabic pseudo-diatonic harmonic" scale should be something like 22,24,27,31 etc notes (considering these equal divisions are somewhat close to being close to have decent tuning for such sonorities; like 5 and 7 equal temperaments are close to 5-limit just intonation).
Here are some of the first few harmonics (ignoring octaves), this can be thought as generalized major chord that can be voiced in different ways (leaving gaps, so it won't sound dissonant, the actual optimal voicing will be like in harmonic series - CCGCEGBb etc, many orchestrators actually recommend such voicings that are close to it).
0: 1/1 0.000000 unison, perfect prime
1: 9/8 203.910002 major whole tone
2: 5/4 386.313714 major third
3: 11/8 551.317942 undecimal semi-augmented fourth
4: 3/2 701.955001 perfect fifth
5: 13/8 840.527662 tridecimal neutral sixth
6: 7/4 968.825906 harmonic seventh
7: 15/8 1088.268715 classic major seventh
8: 2/1 1200.000000 octave

So, triadic major chord has 2 permutations (major and minor). If want to extend the pattern -> septimal (added 7th harmonic) major chord has 6 permutations; added 7th and 9th major chord has 24; added 7th, 9th, 11th => 24 permutations, added 7th,9th,11th,13th=> 120; added 7th,9th,11th,13th,15th-> 720. (so, 720 x 8= 5760 chord inversions, good luck composing with this; this will be a real task for computer assistance).

Here are the main consonances in 11-limit (Arabic systems uses neutral thirds and 11-limit is the smallest just system with such neutral thirds). (13 limit adds 16/13, 359.472338 tridecimal neutral third, which I don't think has much potential, sounding like mistuned major third or sharper neutral third, so maybe 11-limit is a good place to stop.)

0: 1/1 0.000000 unison, perfect prime
1: 12/11 150.637059 undecimal neutral second, 3/4-tone
2: 11/10 165.004228 Ptolemy's second, 4/5-tone
3: 10/9 182.403712 minor whole tone
4: 9/8 203.910002 major whole tone
5: 8/7 231.174094 septimal whole tone
6: 7/6 266.870906 septimal minor third
7: 6/5 315.641287 minor third
8: 11/9 347.407941 undecimal neutral third
9: 5/4 386.313714 major third
10: 14/11 417.507964 undecimal diminished fourth or major third
11: 9/7 435.084095 septimal major third, BP third
12: 4/3 498.044999 perfect fourth
13: 11/8 551.317942 undecimal semi-augmented fourth
14: 7/5 582.512193 septimal or Huygens' tritone, BP fourth
15: 10/7 617.487807 Euler's tritone
16: 16/11 648.682058 undecimal semi-diminished fifth
17: 3/2 701.955001 perfect fifth
18: 14/9 764.915905 septimal minor sixth
19: 11/7 782.492036 undecimal augmented fifth
20: 8/5 813.686286 minor sixth
21: 18/11 852.592059 undecimal neutral sixth
22: 5/3 884.358713 major sixth, BP sixth
23: 12/7 933.129094 septimal major sixth
24: 7/4 968.825906 harmonic seventh
25: 16/9 996.089998 Pythagorean minor seventh
26: 9/5 1017.596288 just minor seventh, BP seventh
27: 20/11 1034.995772 large minor seventh
28: 11/6 1049.362941 undecimal neutral seventh, 21/4-tone
29: 2/1 1200.000000 octave


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## cheregi (Jul 16, 2020)

This is really really interesting! I'm excited to learn more about xenharmonic music.



BabyGiraffe said:


> In my opinion, adjusting intonation is not a real form of microtonality and is pointless...


I really don't know enough to contest this, but... as far as I know, non-discrete-pitch instrumentalists' moment-by-moment intonation adjustments A) constitute microtonality in the most strict literal sense (unless microtonality specifically describes when there are more than 12 _explicitly-named_ pitches per octave in a pitch class system?), and B) constitute a practice that is basically foundational to the experience of harmony in western classical music on a very basic level, and it would be actively hard to _stop_ doing it.

As for the algorithm example, I haven't done much of my own research but my understanding is it's based on 12TET, i.e. 12TET pitches are what happens if you just play a single key, but then it makes constant adjustments based on what other notes are played simultaneously. Though I guess that does still run into a certain level of arbitrariness... I don't even know enough to google this, I just know what I've heard from someone closely involved with new instrument R&D.

I think you're probably right that it's unfair to characterize the current exploration of microtonality in western art music as 'showing off'... I think your summary is better suited. However it does seem to me that Arabic and Indian etc. musics are only 'simple' from the point of view of... having less sheer mathematical complexity? and the western approach is often to turn it into an engineering problem, almost, and bring to bear as much math as possible to see what happens... which isn't a bad thing, to be sure, but for me there's a lot of attraction to the sheer elegance and capacity for nuance in what you might describe as a 'simpler' system, not to mention the fact that it can be fully internalized in a human ear/mind rather than being dependent on computerization.


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## BabyGiraffe (Feb 24, 2017)

cheregi said:


> This is really really interesting! I'm excited to learn more about xenharmonic music.
> 
> I really don't know enough to contest this, but... as far as I know, non-discrete-pitch instrumentalists' moment-by-moment intonation adjustments A) constitute microtonality in the most strict literal sense (unless microtonality specifically describes when there are more than 12 _explicitly-named_ pitches per octave in a pitch class system?), and B) constitute a practice that is basically foundational to the experience of harmony in western classical music on a very basic level, and it would be actively hard to _stop_ doing it.
> 
> ...


The first one to invent an algorithm to retune music to pure was Vicentino. Most of his music is lost.






Anyway, the basic idea is to use super big division of octave with several flat fifth chains, so there are less problems with melodic gaps and pitch drifting, but while this can work for early music in meantone, retuning anything composed with equal temperament in mind will require even bigger divisions where exist several chains for each comma tempered. That's what I think your algorithm is doing, it doesn't tune to just chord, because you will modulate after just several chords unintentionally or just sound bad, it tunes some of the notes to different chains of temperaments.

Traditional Arabic, Indian etc types of music are simpler, because if there is no real harmony, you can pretty much use arbitrary melodic intervals without sounding bad. Systems evolve from simplicity to complexity - our most ancient ancestors were unicellular organisms for example and I personally find simple music for very boring (electronic techno styles is a good trivial music example ). Usually, ethnic types of music compensate the lack of harmony with some form of melismatic/ornamental virtuosity, odd rhythms etc.


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## Guest002 (Feb 19, 2020)

Mandryka said:


> Can someone explain to me where the 12 note scale came from? Is there anything in the nature of sound which leads you to it, or is it a totally arbitrary convention?
> 
> In unfretted instruments, the musician can use microtones if he wishes easily enough. Someone said to me once that this sort of thing goes on all the time in a string quartet, that the each musician will adjust their tone slightly to create interesting harmonies with the others, even if they're playing Mozart.


Have a watch of this now-ancient TV program. It made sense to me about 20 years ago, anyway!






Especially in the 7 - 11 minute mark, he explains that the octave is "natural"; that division by 2/3rds is also natural, and that if you keep on dividing your metal bars by 2/3rds you get a complete spiral of notes. Almost none of which are then perfectly tuned to any other!

Anyway: it was a good series in its time, and that one a particularly good episode, I thought.


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## premont (May 7, 2015)

Interesting and informative thread. Thanks particularly to BabyGiraffe and cheregi.


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## cheregi (Jul 16, 2020)

AbsolutelyBaching said:


> Especially in the 7 - 11 minute mark, he explains that the octave is "natural"; that division by 2/3rds is also natural, and that if you keep on dividing your metal bars by 2/3rds you get a complete spiral of notes. Almost none of which are then perfectly tuned to any other!
> 
> Anyway: it was a good series in its time, and that one a particularly good episode, I thought.


I watched the section you highlighted - it's probably the clearest explanation I've yet seen of the circle-of-fifth Pythagorean-tuning idea, which is indeed natural - so natural, in fact, that Pythagoras was not the only, or even the first, person to have discovered it...



BabyGiraffe said:


> The first one to invent an algorithm to retune music to pure was Vicentino. Most of his music is lost.
> 
> Anyway, the basic idea is to use super big division of octave with several flat fifth chains, so there are less problems with melodic gaps and pitch drifting, but while this can work for early music in meantone, retuning anything composed with equal temperament in mind will require even bigger divisions where exist several chains for each comma tempered. That's what I think your algorithm is doing, it doesn't tune to just chord, because you will modulate after just several chords unintentionally or just sound bad, it tunes some of the notes to different chains of temperaments.
> 
> Traditional Arabic, Indian etc types of music are simpler, because if there is no real harmony, you can pretty much use arbitrary melodic intervals without sounding bad. Systems evolve from simplicity to complexity - our most ancient ancestors were unicellular organisms for example and I personally find simple music for very boring (electronic techno styles is a good trivial music example ). Usually, ethnic types of music compensate the lack of harmony with some form of melismatic/ornamental virtuosity, odd rhythms etc.


I knew about Vicentino's 31-pitch-per-octave system, but not about his algorithm for retuning music to pure... unless the former is just an application of the latter? In any case, he's a really interesting figure, and I love that during the 20th century his work has been picked up almost exactly where it left off 400 years prior.

As for the other part of your post... I'm sorry but I just can't understand what you're trying to say. It's been demonstrated time and time again that 'horizontal harmony' i.e. the memory-based harmonic feeling established by multiple notes in sequence is absolutely real and very significant; melodic intervals are far from arbitrary; a trained Arabic or Indian classical musician has an extreme sensitivity to whether pitch intervals are in tune or not; the idea that 'ethnic types of music' are 'compensating' for lack of harmony with other stuff completely ignores the incredible amount of thought and development that goes into those rhythmic and melodic ideas...

I think most of all I'm shocked to hear such opinions from someone so invested in xenharmonic music, and here's why: you're arguing for a very limited understanding of where musical subtlety/complexity/nuance is capable of existing - i.e. that complexity/nuance/subtlety exist only in harmony, that they can't exist in melody or rhythm or texture - but, meanwhile, _your_ position, as a 'xenharmonics person', with respect to the average Common-Practice-only listener/composer/performer, is _exactly_ the position of defending the idea that complexity etc. _can_ exist in places outside where it's traditionally been found - that is, the CP-only person would argue that _you_ are just compensating for your lack of - melodicism, good counterpoint, insert whatever you want here - by fiddling with irrelevant tuning systems.

Am I misunderstanding something?

EDIT: Even if I were to grant your arbitrary commitment to such a narrow type of musical complexity - I really think you should look into what the Indian tanpura drone is actually doing with respect to overtones and microtonality, especially within dhrupad singing, and remember that this instrument has been in common use for ~400 years in a totally nonwestern context. Actually, on top of that, I recommend looking into the early 20th century history of new tunings within Western art music, and you may find just how much of that history begins with Western composers encountering and being inspired by nonwestern pitch systems...


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## BrahmsWasAGreatMelodist (Jan 13, 2019)

calvinpv said:


> I actually just started to read a book approaching music and sound from a math perspective (and the next chapter I'll be reading is about tuning systems, so this thread is good timing), and a comment was made in passing about how the the 12-tone chromatic scale was made to accommodate for transposing the diatonic scale from one key to another. Because when you transpose that scale, which has a very particular sequence of whole and half steps, it will introduce sharps and flats that aren't in the original C major diatonic scale. So you need to fill out the octave with 5 additional potential notes to prevent a transposition that is outside the theoretical range of notes. And as to why composers decided to transpose in the first place: apparently it was because transposable instruments were starting to be made in the Baroque era, like the various clarinets and horns for example, where the same fingerings that you would play on a C instrument yield different notes.
> 
> I don't know how true this is, because there was no supporting evidence for this claim. But it sounds reasonable enough.
> 
> For me, the more interesting question is why the West settled on a 7-note diatonic scale in the first place. Was there something special about the number 7 in the Pythagorean universe?


Is this really true? I was under the impression that the 12-tone scale came from being directly generated by a Pythagorean temperament, rather than emerging to "fill in the gaps" as you hypothesize. But, seeing as the diatonic scale appears to be older than the 12-tone scale (though I can't seem to find too much information on its history), it may be a plausible hypothesis. It would also help to explain the arrangement of keys on keyboard instruments. It seems like there isn't necessarily a general consensus about these issues.

The emergence of equal and meantone temperaments, on the other hand, are of course far better documented.

Here's one Reddit thread I came across. What do you make of this thread, especially top comment and OP's reply?


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https://www.reddit.com/r/musictheory/comments/3jx9iq


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## tortkis (Jul 13, 2013)

calvinpv said:


> For me, the more interesting question is why the West settled on a 7-note diatonic scale in the first place. Was there something special about the number 7 in the Pythagorean universe?


The 4th octave of overtones contains 8 notes (8th to 15th partials), including 7 notes which are similar to the ones in a diatonic scale. If the base note is C, the 4th octave has C, D, E, F, G, A, Bb and B. (Some notes are a bit sharp or flat compared with 12ET scale.) I guess that, when someone played an instrument like natural trumpet, decent melodies could be played using the notes on the 4th octave? The 3rd octave has too few notes (C, E, G, Bb) and the 5th has too many notes and may be too high.


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## erki (Feb 17, 2020)

I have explored scale-less music extensively - built my own instruments and so on. I find that there is so much more outside of the tonal music than there is within. Like the world is full of it and tonal just takes a fraction out it based on some matrix. Also you do not have to practice to master in that matrix while you can just go and explore and discover.
*Hans Reichel*


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## BabyGiraffe (Feb 24, 2017)

BrahmsWasAGreatMelodist said:


> Is this really true? I was under the impression that the 12-tone scale came from being directly generated by a Pythagorean temperament, rather than emerging to "fill in the gaps" as you hypothesize. But, seeing as the diatonic scale appears to be older than the 12-tone scale (though I can't seem to find too much information on its history), it may be a plausible hypothesis. It would also help to explain the arrangement of keys on keyboard instruments. It seems like there isn't necessarily a general consensus about these issues.
> 
> The emergence of equal and meantone temperaments, on the other hand, are of course far better documented.
> 
> ...


Some of the oldest found flutes are almost equal pentatonic or heptatonic.
Still, Mesopotamian and Egyptian civilizations knew about "Pythagorean" tuning thousands of years before Pythagoras was born (or even 12 equal, if some of the math found on ancient Sumerian tablets is really right).
I don't doubt that the usage of quarter tones in Middle East is also very old, but they were derived from division of tetrachords or string lengths. Ptolemy in his "Harmonics" gives many scales as division of string lengths in a similar way.
Orthodox church/Byzantian music theory is based on 72 parts division of octave.


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