Just & Equal Temperaments

[apollo33]

So, I just started attempting to learn music theory so I can compose something, and being a mathematical person I tried to go down to the very basics of frequencies and the natural harmonics of strings and everything. And the first thing I've discovered is the different ways of tuning a scale. From what I've read, it seems that for a given key the just temperament should sound the best, but almost everything is in equal temperament. Is that right?

Then my main question is: is there synth software or something where I could try out the two temperaments just to hear for myself how they sound?

[gravehill]

You can compare different tuning systems with Scala.

I admire the route you are taking, with lots of different tuning systems and all. The easy way of course, is to forget all about it and go equal temperament all the way, but I honestly believe that exploring other tuning systems enables you to see music as a whole from totally different perspective.

There are several historical texts trying to explain the differences between different keys. One can be found HERE, but bottom line is that with different tuning systems how do you compare? On the other had, with equal tempered tuning, you don't...

[JumpingJackFlash]

So, I just started attempting to learn music theory so I can compose something, and being a mathematical person I tried to go down to the very basics of frequencies and the natural harmonics of strings and everything. And the first thing I've discovered is the different ways of tuning a scale. From what I've read, it seems that for a given key the just temperament should sound the best, but almost everything is in equal temperament. Is that right?

Then my main question is: is there synth software or something where I could try out the two temperaments just to hear for myself how they sound?

Very basically, nowadays we use Equal Temperament. This means:

1) You can start on any note, go up in perfect fifths, and after a cycle of 12 perfect fifths, you will end up on the same note you stated on, exactly 7 perfect octaves higher.
2) You can start on any note, go up in major thirds, and a after 3 major thirds, you end up on the same note you started on, exactly one perfect octave higher.
3) All semitones are equidistant from each-other.

However, in order for this to work, you have to compromise the natural physical laws, as determined by Pythagoras. He worked out that the the ratio of a "pure" fifth is 3:2, and the ratio of a "pure" octave is 2:1, the ratio of a "pure" major third is 5:4, and the ratio of a "pure" minor third is 6:5. etc. ("Just intonation").

The problem with pure fifths, is that if you go up this interval twelve times, the resultant note is NOT the same as the one you started on. It is about a quarter of a semitone sharp. Therefore, in Equal Temperament, the fifths are flattened slightly by roughly 1/50th of a semitone, to enable clause 1 above to work.

Similarly, a series of 3 "pure thirds" produces a note almost half a semitone flatter than the perfect octave. - So again, the thirds are altered in Equal Temperament to compensate.

There are other tuning systems that play around with these intervals differently.

Pythagorean Tuning
Was used for vocal music in the early Renaissance. This has two different sized semitones and was designed to maintain the "pure fifths" at their correct ratio, downward to Eb and upward to G#. The resultant "wolf" interval between G# and Eb is badly out-of-tune, and therefore this system is unsuitable for any music that uses both of these notes. The sacrifice in this tuning is that major thirds are slightly sharper, and minor thirds are slightly flatter that their "pure" form.

Although this was used in early Keyboard music, by 1482 keyboards were being tuned in a different system:

Meantone Tuning
In a way, this was almost the opposite of Pythagorean tuning, in the sense that it favoured the "pure" thirds, while sacrificing the fifths, which were almost three times flatter that they are in equal temperament. This has its own "wolf" fifth, and was favoured by the English, who used more thirds in their early music.


Generally, in the 17th and 18th centuries, equal temperament was not accepted (for keyboard instruments), as it involved too much compromise. As music was only ever written with a few sharps or flats (usually flats in very early music), the other systems of tuning were better because they allowed these simple 'keys' to be in-tune (while 'keys' with more sharps/flats were out-of tune).

With later music that uses many sharps and flats though, equal temperament is essential. Equal temperament also has the advantage of strengthening the polarity between tonic and dominant, which is so important in almost all modern Western music.

Note that in equal temperament, notes such as G# and Ab are enharmonically equivalent. - That is, they sound the same. This is not so with the other methods of temperament.

There are many more systems of temperament, not only in Western music, but in music from other cultures also.

Look in Wiki:
Equal temperament
Pythagorean tuning
Meantone temperament
Pythagorean comma
Just intonation
etc.

[Toxikator]

As was said above, scala can do it for you. The only synth I know of off the top of my head, though, that works with scala files is z3ta+ (I am quite sure there are others, but oh well) but its tuning files don't include just intonation.

you might also try "Hermode" tuning; I think that's what it's called anyway. What it does is orients root notes in equal temperament so, as JJF said, you have equal semitones making using different keys possible. However, it retunes CHORDS in just tuning.

I believe it operates on the principle that the human ear can't hear differences of greater than about 1 cent (maybe even more) so melodically we don't notice the differences, but by retuning chords we can eliminate the beating that does indeed occur.

Give it a shot, I know z3ta+ does hermode, too.

[apollo33]

gravehill:
Thanks a lot for the reply and link to Scala. It looks like a really cool (and confusing) program. I didn't try it much yet but I'm looking forward to it.

JackFlash:
Thanks for the long and informative reply. I really don't know how you have the time to write so much--especially all the music theory lessons--but I definitely appreciate it.

And thanks for killing my dream of finding the perfect scale where all notes would vibrate happily in natural harmony and drink beers with one another instead of fighting. But yeah, the inevitable tuning imperfections you present does kind of make me want to give up on exploring alternate tunings, since equal temperament sounds like the best compromise. Maybe I should concentrate on learning all about the 12 notes I already have on my keyboard first?

Toxikator:
Thanks for the suggestions, I'll check into them.

[bernhardtjeff]

for synths, albino can load .tun files. native instruments pro53 loads its own format (which you can save as in scala). i like lounge lizard's handling best, though: it loads native scala files and you can choose the key right there without having to make a special key map like with a .tun file.

there are many, many others synths that load .tun or scala files, too (z3ta, rhino, etc.), but the above are the ones i have. oh, and the korg legacy collection can do alternate tunings, but i've never figured out how to save the scales i make internally in the synths, so it kinda sucks.

anyway, get scala. it does lots of stuff you'll never need (or even understand), but it's pretty easy once you it down.

[Dreamw]

So, I just started attempting to learn music theory so I can compose something, and being a mathematical person I tried to go down to the very basics of frequencies and the natural harmonics of strings and everything. And the first thing I've discovered is the different ways of tuning a scale. From what I've read, it seems that for a given key the just temperament should sound the best, but almost everything is in equal temperament. Is that right?

Then my main question is: is there synth software or something where I could try out the two temperaments just to hear for myself how they sound?

Aha !
Glad you asked. I have designed a MaxMSP patch that lets you do just that. I have experimented with dividing the octave up into various types of whole number ratios including half/quarter/eighth/sixteenth/32th etc and various other non pythagorian experiments. Actually the scales sound quite good if I dont say so myself (well at least not the mad cacophony of noise you might expect)
I wanted to see if the natural mathematics in nature could form the basis of a pleasant sounding scale. The other problem to me was HZ - HZ are a completely aribitary unit based on seconds whos only relationship to the natural world is how many digits we have on out hands (thats why we have 60 seconds, count the sections of your fingers with your thumb and multiply by five) . So I invented a new HZ based on the amount of time it took the Earth to go round the sun and soon one based on the atomic clock. Then started making just scales using a starting frequency of 2,4,8,16,32,64 ,128,256 etc of these units .
To answer your question Max MSP will allow you to do ANYTHING in this regard including wxriting your own scales/ratios.
Another thing - we are used from birth to hear the equal tempered scale - just tunings take a while to get used to, at first the may sound a bit "wrong" but later you may find equal tempered scales sound "wrong" !
Check out the music of Harry Parch for music in traditional just tunings.

[gravehill]

there are many, many others synths that load .tun or scala files, too (z3ta, rhino, etc.), but the above are the ones i have. oh, and the korg legacy collection can do alternate tunings, but i've never figured out how to save the scales i make internally in the synths, so it kinda sucks.

It's worth mentioning, that even if your synths couldn't directly import .tun files, Scala also has an option to export tuning files in formats understood by individual synths.

Exports tuning data to a variety of synthesizers with an internal tuning table. You can save tuning data in a MIDI file or send it directly to your instrument or sound card (not available on all platforms). Supported tuning dump formats include:

* Alphakanal Automat
* AnaMark softsynth
* Big Tick Angelina, Rainbow and Rhino softsynths
* Bitheadz Unity softsynth
* Cakewalk Dimension Pro
* Cakewalk Rapture
* Camel Audio Cameleon5000 softsynth
* DashSignature EVE one (not two)
* E-mu Morpheus
* E-mu Proteus series
* Ensoniq EPS/EPS16/ASR10
* Ensoniq VFX
* Fluidsynth (iiwusynth) software synthesizer
* HERCs series, Abakos Pro softsynths
* Korg M1, M1R octave tuning dump
* Korg X5DR octave tuning dump
* Korg OASYS PCI soundcard (and softsynths supporting its .tun tuning textfile)
* LinPlug Albino 2, Alpha 2, CronoX, Octopus, Organ 3 and Sophistry softsynths
* Max Magic Microtuner for Max/MSP and Pluggo softsynths
* MIDI Tuning Standard (both bulk tuning dump and single-note tuning change, 3 byte), supported in Timidity and Audio Compositor, E-mu: Proteus 3, UltraProteus, Audity/Proteus 1000 and 2000 series, Virtuoso 2000, Proteus FX, Orbit, Planet Phatt, B3, Carnaval, Ensoniq: ASR-X, MR Rack, MR-61, MR-76, ZR-76, Turtle Beach: Multisound, Monterey, Maui, Tropez, Rio
* MIDI Tuning Standard 2-byte octave tuning dump
* MIDI Tuning Standard 1-byte octave tuning dump
* MIDI to CSound
* Native Instruments Absynth 2 (via .gly file)
* Native Instruments FM7 and Pro-52/Pro-53
* Native Instruments Kontakt 2 (via script file)
* Native Instruments Reaktor (via semitones file, frequency file or NTF file)
* Pure Data
* rgc:audio z3ta+ softsynth
* Roland GS & JV/XP families
* Smart Electronix Foorius
* Synapse Audio Orion Pro softsynth
* Timidity and Audio Compositor MIDI to audio renderers
* Tobybear Helios softsynth
* VAZ Plus, 2001 and Modular softsynths VirSyn Cube, Cantor and TERA 2 softsynths
* Yamaha DX7II/TX802
* Yamaha SY77/TG77/SY99/VL-1/VL-7
* Yamaha TX81Z/DX11/DX27/DX100/V50 (both octave and full keyboard bulk data)
* Yamaha XG family
* Yamaha VL70m
* Wusik Wusikstation v2
* Zebra 2.0 softsynth

Several software synthesizers like AlsaModularSynth, Tobybear Helios, MAZ Sound VSampler, Orion Pro, VirSyn Cube, Cantor, TERA 2, rgc:audio z3ta+, Cakewalk Dimension Pro and Rapture, Oblivion VSTi and ZynAddSubFX have adopted the Scala scale file format as a means to tune them instead of via a special tuning dump file. More are following all the time.
Other instruments can be supported through modification of an external data file, if the system exclusive data format is straightforward.
Not all synthesizers have microtuning support in the form of a tuning table, or one with sufficient resolution, and therefore cannot be directly tuned by Scala. Hopefully more future synthesizers will be equipped with a full keyboard variable tuning capability. Be careful to check this before you buy.

[JumpingJackFlash]

The other problem to me was HZ - HZ are a completely aribitary unit based on seconds whos only relationship to the natural world is how many digits we have on out hands (thats why we have 60 seconds, count the sections of your fingers with your thumb and multiply by five) . So I invented a new HZ based on the amount of time it took the Earth to go round the sun and soon one based on the atomic clock. Then started making just scales using a starting frequency of 2,4,8,16,32,64 ,128,256 etc of these units

Hertz is a measuring system, like seconds, centimetres, gallons, miles etc.

The pitch of a note depends on its frequency. Frequency is defined as the number of occurrences of something in a set time interval. - It is scientific, and not arbitrary.

Because we measure time in seconds, it makes sense to deal with seconds as the time. Hertz therefore measures cycles per second. This then gives a numerical value which you can compare with other numerical values.

You can measure the frequency of sound however you like, but however you do it, it will always be possible to convert it to Hertz. - Just as seconds can be converted into minutes, miles can be converted into kilometres etc. All frequencies (and therefore, all fundamental pitches) can be measured in Hertz.

You cannot invent a "new" Hertz system anymore than you can change the laws of physics.

To demonstrate, simply play your "new" notes into a Tuner, and it will tell you their frequency in Hertz. - It might not correspond to the values used in equal temperament, but you can still measure them in Hertz.

BTW, it is also necessary to differentiate between 'tuning' and 'temperament'. - Temperament was standardised long before tuning was. - In other words, you can play consistently sharp or flat, by any amount, and still be in equal temperament (or not).

[Dreamw]

The other problem to me was HZ - HZ are a completely aribitary unit based on seconds whos only relationship to the natural world is how many digits we have on out hands (thats why we have 60 seconds, count the sections of your fingers with your thumb and multiply by five) . So I invented a new HZ based on the amount of time it took the Earth to go round the sun and soon one based on the atomic clock. Then started making just scales using a starting frequency of 2,4,8,16,32,64 ,128,256 etc of these units

Hertz is a measuring system, like seconds, centimetres, gallons, miles etc.

The pitch of a note depends on its frequency. Frequency is defined as the number of occurrences of something in a set time interval. - It is scientific, and not arbitrary.

Because we measure time in seconds, it makes sense to deal with seconds as the time. Hertz therefore measures cycles per second. This then gives a numerical value which you can compare with other numerical values.

You can measure the frequency of sound however you like, but however you do it, it will always be possible to convert it to Hertz. - Just as seconds can be converted into minutes, miles can be converted into kilometres etc. All frequencies (and therefore, all fundamental pitches) can be measured in Hertz.

You cannot invent a "new" Hertz system anymore than you can change the laws of physics.

To demonstrate, simply play your "new" notes into a Tuner, and it will tell you their frequency in Hertz. - It might not correspond to the values used in equal temperament, but you can still measure them in Hertz.

BTW, it is also necessary to differentiate between 'tuning' and 'temperament'. - Temperament was standardised long before tuning was. - In other words, you can play consistently sharp or flat, by any amount, and still be in equal temperament (or not).

I feel you havent really understood what I was saying no doubt due to my poor explaining skills so I shall try again :
Of course you can measure any frequency in Hertz. As you say frequency is a measure of cycles per set time interval, this is not arbitrary, what is arbitrary is how long this interval is, as seconds are not based on any scientific principle or cycle in nature (unlike days and years) but on the number of segments in our fingers as counted by our thumb , I have merely made that time interval something based on a naturally occurring cycle not the number of digits on our hands.
IE Hertz are based on cycles per second and seconds are a completely arbitrary unit of time in that they are not based on any cycle found in nature such as atomic clock or the cycle of the Earth around the sun . Of course this new interval of time are not Seconds neither are the amount of cycles that occur in them Hertz (as Hertz is defined as cycles per second) so I'll call them something else call them something else - Cesonds for example - it doesn't really matter and the amount of cycles per Cesond shall be called "Shirtz".
Now you may ask why ? seeing as "Shirtz" can be converted into "Hertz" . Well as I mentioned I was seeking to find some sort of music based on the natural mathematics inherent in nature (as many have done before me) it made sense to extrapolate these mathematically based scales from something based on maths or at least a naturally occurring cycle. I wanted to have frequencies/whole number ratios based on basic mathematical principles. So using a starting frequency of 256 which is a multiple of 1, (2,4,8 etc ) (each octave was divided into 16 in various ways) for example. 256 hertz is 256 cycles per unit of time based on some random value related to the amount of fingers we have. 256 "Shirtz" is 256 cycles per precise division of the Earth around the sun, or extrapolation of the atomic clock etc etc Of course you could argue that the length of the Earth around the sun or the Atomic clock are just as arbitrary as the number of digits in your hand but at least they are naturally occurring cycles and that is how we are accustomed to measure time.
Dont know if that makes anymore sense - of course music based on these things often sounds pretty awful (like the fibonaci sequence which I also tried but wasn't very impressed by) , but experimenting with new scales and divisions of the octave has led to some pretty interesting results so far , and experiments can go wrong but sometimes the come up with things.

[frequency_algorithm]

seconds are based on the rotation of the earth around the sun and are mandated by the swiss.

[frequency_algorithm]

if you cut off a wave in the middle of a cycle you just clip it? And all hertz vibrates at set fundamentals. Isn't that the theory of an atom and quantum mechanics? Meaning you can't reach another phase until you have the exact amount of energy required? Trying to follow you here, bro.

[Dreamw]

seconds are based on the rotation of the earth around the sun and are mandated by the swiss.

Theyre just a 60th of a 60th of a 24th of a 365th of a year (cycle of Earth round sun) which I believe happens to be " 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom."

[Kingston]

Under the International System of Units, the second is currently defined as the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium-133 atom.[1] This definition refers to a caesium atom at rest at a temperature of 0 K (absolute zero). The ground state is defined at zero magnetic field.[1] The second thus defined is equivalent to the ephemeris second.

That's not exactly arbitrary.

...although 89th century scientists disagree.

[Toxikator]

Not that it matters at all, since whether you call a frequency 440Hz or 421.25 Shirtz its octave is still double that value and it's perfect fifth is still 1.5x it. Pitch is relative, so the absolute measure is purely academic and has no real musical bearing.