School me on... tuning temperaments...

Discussion in 'Music Theory, Lessons & Techniques' started by odibrom, Aug 17, 2018.

  1. The Omega Cluster

    The Omega Cluster n00b

    Messages:
    1,359
    Likes Received:
    198
    Joined:
    Jun 12, 2011
    Location:
    QC, Canada
    Yeah it's such a shame. Whenever I see a musician I like and respect with one of these instruments, there's that one part of respect that flies off, chipped away because of their credulousness...
     
    Winspear likes this.
  2. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,169
    Likes Received:
    844
    Joined:
    Mar 5, 2007
    Location:
    UK
    odibrom,
    A 'temperament' is a tonal system created from 'tempered intervals', it's just one type of tonal system.
    A 'tempered interval' ('tempered' as in 'changed') is a 'just intonation' ('just' meaning 'correct') interval that has been altered in size.

    For example, many tonal systems have been created by stacking equally-sized fifths on top of each other but altering the size of the fifths away from the 'Just Intonation' fifth (which is 702 cents or 7.02 12TET semitones). These systems are called 'regular temperaments' and 'Meantone' is one example.
    "Let's start with Europe's most successful tuning, if endurance can be equated with success. Meantone tuning appeared sometime around the late 15th century, and was used widely through the early 18th century. In fact, it survived in pockets of resistance, especially in the tuning of English organs, all the way through the 19th century. No other tuning has survived in the west for 400 years."
    From https://www.kylegann.com/histune.html

    The modern and popular 12TET is a 'temperament' because it is created by stacking 700 cent fifths (slightly flat of 'just intonation'). The result is 12 equally-spaced pitches in an octave, so it is called '12 tone equal temperament'.

    The most important subject to study is 'just intonation' (also known as 'natural intonation', 'pure' or 'perfect' intervals), as all other tonal systems either arise from this, are derived from this, or are compared against this. 'Just intonation' also explains what 'harmony' actually is.
     
    ElRay, Winspear and odibrom like this.
  3. Winspear

    Winspear Tom Winspear Vendor

    Messages:
    10,906
    Likes Received:
    1,492
    Joined:
    Oct 23, 2009
    Location:
    Southend-on-Sea, Essex, U.K
    ^ What a brilliant webpage, hadn't stumbled across that before! Useful charts
     
  4. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,169
    Likes Received:
    844
    Joined:
    Mar 5, 2007
    Location:
    UK
    The page for an introduction to Just Intonation is https://www.kylegann.com/tuning.html
    A list of Just Intonation intervals showing their frequency ratio and interval expressed as cents https://www.kylegann.com/Octave.html

    I feel Just Intonation can be explained more clearly though.

    ////////////////////

    What is 'harmony'? What is it about 2 pitches that makes the interval sound 'harmonius' and 'in tune'?

    The 'octave' is a clue. This is the most harmonius sounding interval (after the 'unison') and most know that an octave is an exact doubling of frequency, so the ratio of the frequencies is exactly 2:1.
    Therefore, in the time that the lower pitch completes 1 cycle of vibration, the higher pitch completes exactly 2 cycles of vibration, after which they have both come back to their starting points again and this behaviour repeats.
    See the top 2 waves in the image, they form an octave interval:

    harmony-waves-7IND.png

    If you have 2 pitches, keep the lower one constant and smoothly sweep the higher one, the audible harmonicity (or consonance) of the interval goes through many peaks and valleys. This is usually shown as a dissonance curve:

    7535820520_f9e8afe7c5_b.jpg

    The most consonant intervals correspond exactly to simple frequency ratios, that is, 'x:y' where 'x' and 'y' are fairly small numbers. These are 'Just Intonation' intervals.

    On the dissonance curve the next most consonant interval is a frequency ratio of 3:2. This is an interval of 7.02 semitones (702 cents) which is of course extremely close to the very consonant 12TET interval called the 'fifth' at exactly 7 semitones (700 cents).

    3:2 or 702 cents is the 'Just Intonation fifth', and 3:2 is the next most complex ratio after 2:1.
    In the time that the lower pitch completes 2 cycles of vibration, the higher pitch completes exactly 3 cycles of vibration, after which they have both come back to their starting points again and this behaviour repeats.
    See the lower 2 waves in the first image, they form a 'Just Intonation fifth' interval.

    Consonance is determined by how simple the frequency ratio is.
     
    ElRay and odibrom like this.
  5. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,169
    Likes Received:
    844
    Joined:
    Mar 5, 2007
    Location:
    UK
    On the dissonance curve, one of the next most consonant intervals is a frequency ratio of 4:3. This is an interval of 4.98 semitones (498 cents) which is of course extremely close to the very consonant 12TET interval called the 'fourth' at exactly 5 semitones (500 cents).

    4:3 or 498 cents is the 'Just Intonation fourth'.
    In the time that the lower pitch completes 3 cycles of vibration, the higher pitch completes exactly 4 cycles of vibration, after which they have both come back to their starting points again and this behaviour repeats.

    ///////////////////////

    So far these intervals have been extremely close to 12TET intervals, however when we get to 5:4 which is the Just Intonation major third at 386 semitones, we find that the 12TET major third at 400 cents is now 14 cents sharp, about 1/6th of a semitone sharp.
    Guitarists especially notice this because they often use distortion which amplifies dissonance, and why they have tuning issues when playing the very common major triad chord.
    It's in tune with 12TET but sounds out of tune. If they retune a string to tune the third by ear they will of course tune it to 386 cents which is out of tune with 12TET, then that out of tune string causes issues for other chords they play.

    6:5 is the Just Intonation minor third at 316 cents. The 12TET minor third at 300 cents is 16 cents flat, about 1/6th of a semitone flat.
    Likewise the 12TET sixths are out of tune by similar amounts.

    9:8 (not labelled on the dissonance curve but is one of the downward pointing spikes to the left of 6:5) is the Just Intonation major second at 204 cents, this is very close to the 12TET major second at 200 cents.

    In 12TET the major second, fourth and fifth are close to Just Intonation, all other intervals are up to 1/6th of a semitone away.
     
    ElRay and odibrom like this.
  6. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,169
    Likes Received:
    844
    Joined:
    Mar 5, 2007
    Location:
    UK
    The history of tuning is a shift from Just Intonation, through Meantone and Well temperament, to 12TET, in order to gain the ability to freely modulate to a larger number of keys, at the expense of perfect harmony.
    This has happened mostly due to the limitations of keyboard instruments with their limited number of fixed pitches per octave.

    Indian classical music took the other route, it sacrificed modulation to preserve Just Intonation. Each Raga is in one key with no modulation. The harmony can sound exotic but it also sounds extremely 'in tune':
     
    Bobro and odibrom like this.
  7. The Omega Cluster

    The Omega Cluster n00b

    Messages:
    1,359
    Likes Received:
    198
    Joined:
    Jun 12, 2011
    Location:
    QC, Canada
    That's interesting, but how did you (or the one who made the graph) calculate dissonance? It's not specified and I'm curious.
     
  8. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,169
    Likes Received:
    844
    Joined:
    Mar 5, 2007
    Location:
    UK
    The curve was taken from https://music.stackexchange.com/que...ss-curve-have-a-dip-for-complex-intervals-lik But it's the curve for 'all audible harmonics' that is linked to in the 'answer'. The same author answered here too https://music.stackexchange.com/que...asure-the-consonance-or-dissonance-of-a-chord
    So it's actually computed instead of a human experimental response, however it seems reasonable.
    The curve depends on how many harmonics you consider to be in the tones, and if it's just odd harmonics (for example a clarinet).
    Nice that it's somewhat 'fractal'.
     
  9. The Omega Cluster

    The Omega Cluster n00b

    Messages:
    1,359
    Likes Received:
    198
    Joined:
    Jun 12, 2011
    Location:
    QC, Canada
    This is very interesting because prior to that I had to rely on a subjective dissonance curve. I'm intrigued to see that the octave is not perfectly consonant too. The BP one is really cool, but I'd like to make such curves for some other temperaments. It's a shame that the researcher doesn't include their equations...
     
  10. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,169
    Likes Received:
    844
    Joined:
    Mar 5, 2007
    Location:
    UK
    To complete my Just Intonation explanation ...
    Polyrhythms are a very good analogy for harmony, they are essentially 'slow harmony'.

    If you have two drummers where one plays two beats in the exact same time period that the other plays one beat, it's a simple 2:1 polyrhythm and is an analogy for the octave interval. The simplicity of the polyrhythm is an analogy for how consonant an octave interval sounds.

    If you have two drummers where one plays five beats in the exact same time period that the other plays four beats, it's a 5:4 polyrhythm and is an analogy for the Just Intonation major third interval. The polyrhythm is more complex, which is an analogy for how a J.I. major third interval sounds consonant, but less consonant than an octave interval.

    If they have the timing precise you would describe them as drumming 'in perfect harmony'. Similarly, tuning a major third interval so that the frequencies are in the ratio 5:4 is a perfectly tuned (Justly Intonated) major third.
    If one drummer is slightly slow they will go out of phase, they are no longer in harmony, this is the 12ET major third we are used to.

    If you record a 5:4 polyrhythm and speed it up until it turns into musical pitches, you will actually hear the Just Intoantion major third musical interval. A note is a fast rhythm, harmony is fast polyrhythms.
     
    Bobro, ElRay and Winspear like this.
  11. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,169
    Likes Received:
    844
    Joined:
    Mar 5, 2007
    Location:
    UK
    Here's a big list of Just Intonation intervals and their size in cents (1/100th of a modern semitone) https://www.kylegann.com/Octave.html
    There are an infinite number of them because there are an infinite number of mathematically possible frequency ratios, however only the simpler ratios sound consonant, as the ratio gets more complex the interval gets more dissonant.

    If we list JI intervals in order of consonance it would go something like:
    1/1 Unison.
    2/1 Octave.
    3/2 Fifth.
    5/3
    5/4
    6/5
    The above intervals are roughly represented in modern 12 tone equal temperament.
    7/4
    7/5
    7/6
    The above three are the 'blue notes' used in jazz and blues using note bending.
    ...
    ...

    Here's a good intro to JI with many music examples https://www.kylegann.com/tuning.html
    Here's a page with information about how tuning has changed through history https://www.kylegann.com/histune.html

    Here's a video that does a fairly good job of explaining the movement from Just Intonation, through Pythagorean, Meantone, Well temperament, to Equal temperament:

     
    ElRay likes this.
  12. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,169
    Likes Received:
    844
    Joined:
    Mar 5, 2007
    Location:
    UK
    My list of JI intervals in order of decreasing consonance missed out the fourth 4/3, can't edit it now.
    I found a webpage with a large list in this order http://www.huygens-fokker.org/docs/intervals.html
    Here's an edited version:

    1/1 unison
    2/1 octave
    3/2 fifth
    4/3 fourth
    5/3 major sixth
    5/4 major third
    6/5 minor third
    7/4 harmonic seventh
    7/5 septimal tritone
    7/6 septimal minor third
    8/5 minor sixth
    8/7 septimal tone
    9/5 minor seventh
    9/7 septimal major third
    9/8 major tone

    All these intervals are roughly represented in 12TET except the 'septimal' intervals that contain the number 7 in the frequency ratio. 12TET essentially approximates JI intervals that are combinations of the prime numbers 2, 3 and 5. Primes 7, 11, 13 are not present.

    About the word 'temperament': It comes from 'tempering a JI interval', where 'tempering' means altering a JI interval slightly in the process of creating a tuning system.
    So any JI tonal system is not a temperament.
    Any abstract mathmatical tonal system where the octave is simply divided into 'n' equal divisions is not a temperament, it is an 'EDO'.

    So modern 12TET is a temperament because it uses a tempered fifth of 700 cents (2 cents flat from the perfectly tuned JI fifth that is a 3:2 frequency ratio). Then 11 of those intervals are stacked to get all 12 tones. The tones are equally spaced so we call it '12 tone equal temperament':

    Cents
    0
    700
    1400 = 200
    900
    1600 = 400
    1100
    1800 = 600
    1300 = 100
    800
    1500 = 300
    1000
    1700 = 500
     
    Last edited: Nov 15, 2018
    Winspear and ElRay like this.
  13. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,169
    Likes Received:
    844
    Joined:
    Mar 5, 2007
    Location:
    UK
    Writing about Just Intonation intervals, i have not yet explained the fundamental aspect that these arise from the harmonic series of a vibrating string.

    The Harmonic Series
    ---------------------------
    When a string vibrates it contains many frequencies, the lowest frequency is called the 'fundamental' or '1st harmonic' and is the frequency we usually refer to when we talk about the 'frequency of a note', for example A4 being 440Hz (Hz = Hertz = vibrational cycles per second).
    Then there are the 'harmonics' or more correctly, the 'higher harmonics' above the 1st harmonic. Also known as 'overtones'. These can be isolated and heard by lightly touching a string at certain points.
    There are hundreds of these, the higher the frequency of the harmonic the quieter they are, which is why we tend to hear the frequency of the string as being the 1st or 2nd harmonic.

    The fundamental plus the higher harmonics form the 'harmonic series' of a string.
    For a vibrating string of, for example, fundamental frequency 100Hz, all the harmonics (including the fundamental) will have frequencies:
    100, 200, 300, 400, 500, 600, 700, 800, 900 ... etc.
    So for fundamental frequency 1, the harmonic series wil have frequencies 1, 2, 3, 4, 5, 6, 7, 8, 9 ... etc.
    It's easy to remember as the 1st harmonic has frequency 1, the 2nd has frequency 2, etc.
    Also, for example, the 5th harmonic has a frequency that is the frequecy of the 1st harmonic multiplied by 5.
     
  14. ixlramp

    ixlramp SS.org Regular

    Messages:
    2,169
    Likes Received:
    844
    Joined:
    Mar 5, 2007
    Location:
    UK
    Deriving intervals from the Harmonic Series
    --------------------------------------------------------
    Playing the open note of a string and the harmonics, it sounds pleasantly musical, the intervals sound harmonius.
    So what happens if we derive our muscal intervals from the harmonics?
    Remember that the frequencies of the harmonic series are 1, 2, 3, 4, 5, 6, 7, 8, 9 ... etc.

    The most harmonius interval is formed by the open string and the 2nd harmonic.
    The frequency ratio is 2:1, so it's a JI interval, this interval is what we call the 'octave'.
    It's so harmonius it sounds almost like the same note but higher in pitch.

    The next most harmonius interval is between the 2nd harmonic and the 3rd.
    The frequency ratio is 3:2, so it's a JI interval, this interval is what we call the JI 'fifth' (702 cents).

    Continuing, taking various pair combinations of harmonics we end up deriving these JI intervals:
    (Note that an interval with frequency ratio 3:2 is written as 3/2 for a reason i will explain later)
    (Note that intervals larger than an octave are ignored because they are considered to be an octave plus a smaller interval, because we consider scales repeat in every octave. For example 3/1 (the 3rd harmonic) = 2/1 plus 3/2 = an 'octave' plus a 'fifth')

    2/1
    3/2
    4/3
    5/3
    5/4
    (6/4 is an interval identical to 3/2)
    6/5
    7/4
    7/5
    7/6
    8/5
    (8/6 is an interval identical to 4/3)
    8/7
    9/5
    (9/6 is an interval identical to 3/2)
    9/7
    9/8
    ...
    etc.

    The convention for writing a JI interval is that we don't write the frequency ratio, instead we write what the frequency of the higher note would be if the lower note has frequency 1:
    Ratio
    3:2
    Divide both sides by 2:
    3/2 : 2/2
    = 3/2 : 1
    So write 3/2.

    Also, the convention for writing the unison interval is '1/1' instead of '1'.

    So for example the JI major scale is written:
    1/1 9/8 5/4 4/3 3/2 5/3 15/8 (2/1)
     
  15. The Omega Cluster

    The Omega Cluster n00b

    Messages:
    1,359
    Likes Received:
    198
    Joined:
    Jun 12, 2011
    Location:
    QC, Canada
    This is why it's interesting to study the Gamelan scales, which arose because of the overtones of the drums and bells they use (which are fundamentally different from string overtones).
     
    Bobro likes this.
  16. Bobro

    Bobro SS.org Regular

    Messages:
    101
    Likes Received:
    41
    Joined:
    Jun 26, 2017
    These are all excellent responses!

    I've been using only a 17-tone to the octave Just Intonation tuning system for many years now and studying microtonal music for even longer. I think one of the most important things is to be concerned about your own personal expressive and musical needs, and beware of any claims that some one tuning is the be all and end all of musical tuning. My own tuning is concerned with getting my own versions of all the middle eastern maqams and ancient Greek tetrachords, and I wound up reinventing the wheel of Ibn Sina's medieval Islamic tunings. By ear. It was a long difficult process, but worth the effort because I based it all on "what feels most natural for me to sing?". I highly recommend that any guitarist interested in microtonality invest in some kind of saz with moveable tied-on frets- then you can test whatever you please for playability and "earability". Then you can invest later in a custom fretting, with previous experience in the tuning and with great confidence that you will use it and not just have some weirdo curiosity guitar lying around unplayed.

    I hope it is not declassé and gauche to mention here that I have a lovely little cura saz, that I bought in Istanbul, for sale and you can just PM me! "Cura" means "little girl" and that describes it very well, but the sound relative to the size is pretty huge (middle eastern instruments mostly evolved outdoors, I believe). 3 double courses, holds a tuning very well and super easy to calculate where the frets should be moved to according to some tuning or other.
     

Share This Page