Today is Monday the 14th of October 2002
I've been enquiring further into this whole 12 equal/Just temperament business. I figured that if there's any truth in the theory that Just Intonation
is the 'correct' tuning for a music's harmony - then there should be a host of comparisons on the web all waiting to prove me wrong! ;-)
Well... I asked this very question at the eGroups Tuning Forum
. I simply asked if there were any tunes on the web to compare 12 eT against various Just/Mean temperaments. There was a response - yay! :-) This took me to the page located at http://bellsouthpwp.net/j/d/jdelaub/jstudio.htm
This looks very interesting. There are tunes comparing stuff like Chopin's "Fantaisie Impromptu" and Gershwin's "Rhapsody in Blue" in various temperaments. I'm going to dedicate myself now to hearing a certain kind of adaptive Meantone temperament called 5-limit adaptive tuning
. It aims to keep as close as possible to 'perfect' Just tuning - while also allowing for flexible melodic 'jumping'.
As expected, I found it 'off-tune' at first (chords also have a slightly 'metallic' timbre - thanks to the Just intervals), but what I plan to do is hear this (and only this) 'initially alien' tuning for the next few days/weeks - and see if I can really get to prefer it. I somehow still suspect that I won't, but I'm going to keep an open mind about this - since obviously quite a few people seem to think otherwise. I'll be posting the results here in a few days/weeks time.
For the 'other side of the argument', also see http://bellsouthpwp.net/j/d/jdelaub/bend_inf.htm
. I don't (currently) agree of course, but it provides useful information should I actually be wrong... =)
Monday the 25th of October 2002
Well I tried. I really did. For more than a week, I've heard the same tunes with as little 'exposure' to 12-eT as possible (quite hard in this day and age one might find ;-). Almost like a diet really, but at least the tunes were quite good to begin with!
And the outcome:
Well, to be honest, I can't really say I like the new temperament any more than I did to begin with. Maybe if I stuck at it longer, I'd eventually get used to it a bit more, but whether I would (or even should) is doubtful in my opinion.
I haven't 'given up' though. After a request from the Tuning group, I'm going to learn more about something to do with 'combinational tones', and try out even /more/ music in Mean temperament (slightly different to "5-limit adaptive" tuning - although its ultimate aim (i.e. getting close to JI chords) appears to be similar).
I've also recently been having flame wars with the tuning group again ;-) (that's a joke, they're quite friendly!). At least one person in the group has always preferred Mean/Just intonation, even after getting used to the same tune/s in 12-eT initially! This means my earlier claim on this page about a tolerance level (i.e. getting 'used' to a new tuning) would seem to be at least partially wrong (or at least not the whole truth).
Anyway, all this could mean one of a few things:
A 'false' preference to JI might be due to a timbre phenomenon with traditional instruments - such as the harmonic partials clashing with the fundamental tones and/or each other (which could be rectified if an instruments' tones /and/ overtones were tuned to 12-eT (probably need a synthesizer to do that though ;-).
Either I am 'wrong' to prefer music in 12-eT, or he is 'wrong' not to.
Unlikely, but there might be an interesting 'pseudo' mathematical effect with sounds exclusive to Meantone/JI tuning. I haven't particularly 'noticed' this, but if true, this would mean both tunings have fundamental advantages and disadvantages.
Also unlikely, but maybe some people hear tones differently from others. However, this is almost like saying some people see the colour brown as blue ;-) I certainly haven't dismissed this notion though, as the ear itself is a very complex piece of hardware.
Maybe tuning 'preference' is determined the day you're born, or perhaps picked up from an early age. From what I've gathered, this seems to be half the story at least. This doesn't invalidate my belief that there is
a perfect tuning, but well... ...
things have become a whole lot more unclear...
Late December to January 2003
I was again trying to 'get used to' the sound of various Just Intonation temperaments over 12-eT - without success.
I guess the reason I'm still trying at all - is the way that simple JI ratios (such as 3/2 and 4/3) form the basis for most instrument harmonics (orchestral or synthesizer timbres). These will then (very, very slightly) clash with any 12-equal pitches. As I have said earlier on, you could tune all harmonics to 12-equal partials, but perhaps one would miss out on the 'different' instrument timbres that the natural harmonics of JI produces.
I'm still as stubborn as ever about how it sounds though - I'll pick the sound of 12-eT over JI any day ;-)
Actually, I did some further research, and as far as I know, this is the only site on the whole net
that argues the theory that the 12 equal temperament might not be a compromise in any way. This came as a real surprise to me - considering the dominating use of 12-equal for most of the world's music. In email exchanges with people, a few have agreed with me how there might be a chance that 12-eT isn't a compromise, but I really expected a few such pages on the web too.
If there is one thing I have noticed about the sound of JI (apart from it sounding off tune), I would say it has a more 'metallic' sound than 12-equal. This will in turn give the timbre a distinctive flavour, but that still doesn't get away from the fact that it also sounds off to me.
From what I've heard, JI advocates tend to look at 12-eT as 'plain', 'bland' or 'colourless', (though I guess they find 12-eT simply off-tune too). I would love to hear some opinions about this. If you prefer JI over 12-eT, why do you prefer it? Does 12-eT sound boring in comparison, or does it sound 'off-tune' as well?
March 24th 2003
I've been experimenting with harmonics on my computer more thoroughly lately, and have confirmed to myself that hundreds and thousands of partials (in both 12-eT and Just Intonation) unavoidably clash very slightly with the fundamental tones from other notes present in the music. And so I tried (in vain) to look for a way around it.
The first thing I tried was to alter the (usually) harmonic partials to unorthodox 12-eT partials and 'mold' them with the accompanying chord (even if it meant limiting the number of partials and/or raising or dropping a certain partial by a semi-tone or two). But all too often, doing this will make the partials sound like an extra 'layer' of melody, as opposed to 'thickening' and 'camouflaging' with the fundamental harmonic to produce a unique timbre. In other words, it doesn't sound like a single instrument anymore - more like a bunch of sine wave pitches. Even ignoring this problem, if the 'partial molding' is happening throughout the music, then you'll never get a instrument with a constant timbre, but instead get a constantly 'evolving' timbre (not that that would necessarily be a bad thing, but it's nice to have the choice!).
Similar to the above, I didn't remove or drastically change any partials, but instead altered the partials (overtones) to their nearest 12-eT equivalents (so instead of 1, 2, 3, 4, 5 etc., you get overtones of 1, 2, 2.9966, 4, 5.0396 etc.). Apart from being mathematically inelegant though, there's the problem I've mentioned in the previous paragraph - 'whole' numbers work better to form a proper timbre - while 12-eT partials are closer to forming 'separate' notes. Somehow, the whole number overtones 'camouflage' in with the timbre better to make a 'new' instrument.
I suppose the only way to make music in 12-eT without it being a compromise in this way is to use only the fundamental partial and any 'octave' partials (e.g. 2nd, 4th, 8th etc. harmonics). But of course, music would sound quite boring restricted to these 'vanilla' timbres. Then you've also got the 'problem' of the 'virtual pitch' phenomenon, and combinational tones - such as 'summation' tones and 'difference' tones to add even more chaos ;-)
Of course, music based around a Just Intonation tuning scale would be a compromise for similar reasons. For example, if you play a C minor chord, the 5th harmonic (E) would very slightly clash with the Eb note in the C minor chord. It would be mostly inaudible, but nevertheless there.
So, perhaps the right thing to say is that /all/ scales and tunings have this 'problem' or 'compromise', in that the harmonic series behind the timbre will most likely clash with the fundamental notes in certain chords. (Note - this is a different kind of 'compromise' that I am using as the main argument for this site).
To sum up, complex timbres (trumpet, piano, violin - most any instrument you could name) will introduce extra tones - from a single note - that may not be desired for melodical reasons. Just as well they camouflage so well though.
Late March 2003
Remember that 'virtual pitch' phenomenon I mentioned earlier? Well I tried something to get rid of that extra 'unwanted' tone. It involved playing C in the left ear, and Eb in the right ear. Interestingly, the 'problem' still occurred - I still heard a faint G# (hmm... perhaps /slightly/ quieter), which signifies to me that it's not the ear mechanism at 'fault' in particular, but rather the brain's innate response.
Early May 2003
I remembered my experiment I created some time ago which first convinced me about 12-eT.
For people who prefer the pitch 1.25 for the major third, then 1.25992 should sound as 'off-tune' as the 'inverse' of 1.25 - which is 1.24016 (logarithmically, 1.24016 to 1.25, is what 1.25 is to 1.25992).
So my question then is, what major chord sounds better:-
1.0 - 1.25992 - 1.49831
...Or this one:
1.0 - 1.24016 - 1.50169
(with the perfect fifth, I did the equivalent 'inverse' thing).
Now in my opinion, the 1st one sounds /miles/ better, but to people who think the Just Intonation Major triad is 'sweetest', then theoretically, they would expect both of the above two triads to sound /equally as bad/ or 'off-tune'. But to be honest, I think practically everyone would prefer the first one.
I've created these sounds in WAV format, so you can listen:
Now this is fairly conclusive for me, but I also considered how the subtle secondary and tertiary 'beating' might be different for 1.24016 than 1.25992. But then I tried the test with /broken/ chords (i.e. as a melody - C E G E C E G). This way, beating is taken out of the equation. As you might imagine, the 12-eT third was again - a lot better than the 'inversed' third. Secondly, could the perfect fifth be complicating matters? Maybe if I just played C and E, there would be some kind of difference...? Nope, the results from the two-tone diad was no different from the triad.
Lastly, could it be because I've 'got used to' the 12-eT interval? Maybe over time, I would like the 1.24 interval just as much. Again, I really doubt this, but hear for yourself and see what you think...
Late May 2003
links I found quoting the same thing from Chicago Reader
"Moreover, professional musicians appear to prefer equally tempered intervals to their just counterparts"
This is the direct opposite of what you'll hear from other sites! :)
- Consonance and Dissonance - Huron's Numerosity Conjecture
- Critical comparison of acoustical and perceptual theories of the origin of musical scales.
I've been researching 'World Music' lately. It turns out that practically all cultures (even remote island tribes who have had no previous contact with the rest of the world) have some form of scale based on the 12 note chromatic scale (often the diatonic or pentatonic subset).
Yes, there are occasionally cultures that base their scale on something like say... 7-eT (Siamese), or on 5 notes (Gamelan) or even a 'quarter-tone' Indian scale*
, but either the music is monophonic in nature, and/or it relies on unusual instrument timbres to create an atonal-esque 'textural' sound, rather than harmonious melodies, chord patterns and structures with a proper tonal key. You'll also find cultures where people don't have one 'standard' scale system, and instead pick notes seemingly at random, with no relation to any equal tempered system (5, 12, 19 etc.) /or/
pure ratios. Again, this type of music is either fully or at least to some extent 'pseudo-atonal', where the emphasis is on the 'texture'. That's not to say that music using such scales is invalid, but I believe it's certainly missing the important 'chromatic' dimension. The best music combines both aspects.
* It's worth mentioning that although there are technically 22 notes in the traditional Indian octave, some musicologists have found that practically all of these approximate to the 12 standard semitones. In my opinion, the subtle 'quartertone' differences are used as 'interpolating' notes - perhaps used as 'ornaments', or in place of a slide. It's true that 'inflexions' (gamakas) are outside the normal 12-et range, but according to the graphs in the paper below these are fleeting, and are not time-steady pitches (despite characterizing the works to a large degree).
In summary, I believe most great Indian classical music is effectively and predominantly based on 12-equal. See these sites for further details:
Application of pitch tracking to south Indian classical music or download the (PDF version)
Musical Nirvana - Indian Scales
13th August 2003
It turns out that the 12 note system can be represented quite nicely using a honeycomb or hexagonal lattice. Here's how to build one from scratch. First off, here are the twelve notes we'll be using. I've assigned each note a tertiary colour based on the scale of fifths (see right diagram).
Okay, first we start off with the major triad, one of the basic chords in the chromatic scale (see diagram to the left). But how about the minor triad? Well, we can add the minor third (D#) to the left (see right diagram).
There are six major/minor chords that contain the note C in them (C major, C minor, A minor, G# Major, F minor and F major). Now we can surround the hexagon with 3 other notes - making the shape you can see on the left.
Can we build out further than this? It turns out we can. In this next example, I've added the notes B and C# at the upper right region. Once again, these notes are 'forced', and intriguingly, the whole pattern interrelates perfectly.
We can build indefinitely, and any 'triangle' of hexagons you care to pick will be a major or a minor chord. Furthermore, all twelve notes are used. Click here for a larger view of the below diagram.
I tried to look for further patterns. Maybe something interesting happens with the diatonic scale? Well, sort of; it turns out that each diatonic tone is always a 'leap' away (so the C hexagon is never touching the D hexagon, and the D is not in contact with E etc. etc.).
Is this honeycomb system unique to the 12 note scale? Unfortunately, not really as far as I know, as a similar pattern can be achieved with 16 notes and beyond. See this page for a 16 note lattice.
An update at last! It's been over 3 years! I'm going to spend more time researching this, especially since I'm using this topic for my dissertation at university. First off, here's a new page
with lots of listening tests based on the Major 3rd.
Secondly, I'm beginning to wonder again if two people who hear the same interval may perceive it differently in their 'mind's eye'. If this is the case, then it can take one of two forms:
They literally hear the interval differently. For example, someone who hears an interval of 390 cents could hear it as 400 cents (or vice versa).
They hear the same 'pitch height difference', so in one sense, they are still hearing a '390 cents' interval. However, if we assume that the brain abstractly maps to 12 intervals, and that these intervals are being represented by different neuron clusters in the brain, then it may be the case that they are receiving no minor 3rd neuron pollution, and only major third neurons are being excited. This is in contrast to people like myself, where the 390 cents interval would excite not many, but a few neurons for the minor third as well as the major third, and thus cause interval ambiguity.
In both of the cases above, the timbre (or feeling of the beats created by an interval) would remain the same for all people. However, I count this is a separate type of consonance to the consonance type created by the 12 intervals.
Just for the record, that doesn't necessarily mean there's something special about the number 1.25992 (24/12
) for the major third. The thing that's special is the Major third *sensation*
in the mind's eye. It's just that different people would need different input from the outside world to activate this interval as sweetly as possible.
26th April 2007
Having read a lot of material on pyschoacoustics, and discovered the difference between first and second order beats, the one factor I think is taken is overlooked in most experiments on intonation comparison in general (including on this site - will fix soon) is how certain partials will create the all-important first order beats with other partials from a different pitch in the same chord. This can more easily influence the perception of an interval if the listener is not careful.
In a nutshell, first order beats are created when two partials are close together, such as the pitches 1 and 1.01 (see Helmholtz and Plomp/Levelt), and second order beats are created when two tones form 'not-quite periodical beats', for example 1 & 1.51, or 1 & 1.249. The first type will happen because of the limitations of the basilar membrane in the ear, and sounds like a fluctuating tremelo (volume) effect, rather than the perception of two slightly different pitches as one might expect. It can be 'solved' either by improving the design of our ear (!), or by pitching each tone separately in the left ear and right ear, in which case the tremelo effect completely disappears. It can also be solved by retuning or omitting certain partials in the timbre, but that's obviously a compromise.
Anyway, the second order beating type occurs even if the tones are pitched in each ear separately! In other words, it's the mind that form the 'secondary beats'. They are much more subtle than first order beats, and so it's my guess the first type bother the JI enthusiasts much more than the second type. Proof? Listen to these:
Major third in Just intonation (f1 and f1.25) - first 5 harmonics
Major third in Equal temperament (f1 and f1.25992) - first 5 harmonics
As you can hear, compared to the JI version, the equal tempered version has a slight tremelo effect (quiet-loud-quiet-loud).
However, if we exclude the fourth harmonic from the major third in both cases:
Major third in just intonation (f1 and f1.25) - first 5 harmonics, excluding 4th harmonic in major third
Major third in equal temperament (f1 and f1.25992) - first 5 harmonics, excluding 4th harmonic in major third
As you can hear, these two sound much more similar to each other than before! Therefore, it should be much easier to compare the raw pitch of the two diads if we exclude the 4th harmonic of the M3rd in this way.
I prefer the ET major third in one way (melodically, tonally, more major-3rd-ally), but it goes without saying that the Just third sounds more like a fused sound (therefore being more consonant in another way). These two major types of consonance form my 'own' theory, and I really think it could be true. See January 2007 to see how the first consonance type could vary from person to person though.
7th May 2007
My dissertation is now online, and is very thorough with full references etc., though the spirit of this article is still intact ;-)
Potential Mathematical Models for the Western Musical Scale - A Historical and Empirical Comparison
13th June 2007
I've now updated this whole page to take into account some of the information present in my dissertation, along with links throughout. Although there are few references on this page, my dissertation
has almost 200 of them, so visit there if you need sources, quotes, and verification.
Despite the entry in January 2007, I still think there's a chance that there's a most 'correct' tuning, whether it be JI, ET or Pythagoras. Here is an excerpt from near the end of my dissertation
A more 'natural' tuning?
This addresses Argument 3 stated near the beginning of the dissertation. Despite the idea that two different physical intervals could sound the same to the mind's eye for two different people, there's still the chance that one tuning may be easier to get used to, and perhaps therefore in a sense provides the 'truer' intervals for chromatic consonance. An analogy could be likened to riding a bike with the handle bars turned by 30 degrees. Over time, the mind compensates, and then returning to 'normal' positioning of the handle bars would feel strange (for a short while). However, riding a bike with the handle bars twisted by 30 degrees may include costs, such as taking longer to get used to, and may ultimately provide slightly less accurate control of the bike.
As an extreme example, one may try to recondition oneself to prefer the perfect fifth at 40 cents flatter than the ET or JI tuned fifth. As one might imagine, this would take a long time to condition anyone to! It therefore follows that even if it's by fractional amounts, perhaps the just intoned interval is quicker to condition people to than the other models on average. Or maybe it's the ET or Pythagorean models that would work out quicker - who knows. In summary, there may still be ideal pitch values for the twelve intervals, whether they are JI, ET, Pythagorean or whatever else.
16th May 2008
Added video comparison created by Bradley Lehman to this page. I prefer the Well tempered version, but this isn't equal temperament's fault, but rather how the harpsichord treats equal temperament.
10th April 2019
After a long hiatus, I revisited this page, mostly to tie up loose ends with the survey. Wow it has been a while. Thank you to everyone for all your comments, thoughts, insights (and even the occasional insult, though most have been positive!). Looks like Equal Temperament won out overall, though Just Intonation won by a whisker for the harpsichord tune which I thought was a bit odd, but there we go. If I were to redo the poll, I might make all three blind tests, instead of just one of them. Using more real-sounding instruments would also be an idea, though in some ways, it is easier to distinguish between tunings if there are harsher synthetic instruments that have a greater number of harmonics.
Just read the whole of this site again too. It was kinda fun. I still don't think we're much closer to solving the mystery of the twelve notes though!