Yahoo Tuning Groups Ultimate Backup tuning Chromatic JI Tuning

Hi everybody,

It's no big deal (and probably well known) but I just happened to notice that the harmonics 16:17:18:19:20 give good chromatics from tonic to major third. Applying these from the fifth as 3/2 and filling out the fourth and flat-fifth gives intervals from the tonic:

1/1 17/16 9/8 19/16 5/4 4/3 45/32 3/2 51/32 27/16 57/32 15/8 and of course 2/1.

PS: I chose 45/32 for flat-fifth because this is the interval between the fourth and major seventh 15/8 x 3/4.

As you see there's many familiar intervals here, while intervals between non-tonics also seem to hold together. Or we could apply 16 -20 from fourth and see what happens. I just got a new computer and am not set up yet to test chords on my microtuner yet so I don't know to what extent we can modulate from this tuning before having to reset the scale for the new key. But it seems to sound nice as is.

Cheers

-Rick

🔗Petr Parízek <p.parizek@...>

🔗Kraig Grady <kraiggrady@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> To Rick,
>
> see this webpage:
> http://tonalsoft.com/monzo/ganassi/ganassi.htm
>
> Petr
>
Thanks Petr,

A meantone tuning, just what I was looking for. (Funny, John C just sent me an email). Thought it was too obvious not to have been looked at before. I too noticed that if we start from 64 (I'd prefer scientific C rather than C 60 for calculations) we get separations by 4 and then from 96, separations by 6 (although my scale has a pure fourth 4/3 or 85.333...instead of 84). But I don't know why Ganassi said that G major wasn't perfect because 9/8 = (3/2)^2?? One interval that does stand out is the minor third from the fourth (51/32)/(4/3) = 153/128 = 1.1953125. Since this lies between 6/5 and 19/16 and has "odd/2^N" then it should still be a good minor third.

As I mentioned to Marcel, rather than testing the tuning with arbitrary major and minor triads from here and there (especially "subharmonics" which IMO is nonsense), we should try a more modern approach with root-third-seventh and then cycle in the key of C i.e. guide tones of thirds leading to sevenths and vice-versa (More "Monk" than "Mozart"). for eg, as the left hand of a piano plays C F B E A D G C, the right hand plays E-B E-A D-A D-G C-G C-F B-F B-E. Next we can modify chords along the way, for eg preceding all the V I's by a dominant 7: E-B E-Bb E-A, for C7 to F, or D-G# C#-G C-F# for E7, A7, D7 (chromatic tritones in 12 tET). These last tritones are 51/36 = 1.41666..., 24/17 = 1.411764706 and 45/32 = 1.40625, all of which seem pretty close to the tempered sqrt2 1.41421356. We can then push the limits even further by trying altered chords with these 7ths, 5+ 5- and 9+ 9- etc...If it passes these tests then perhaps we have found a meantone tuning which may be used for more modern 12 tET harmony, at least in the key of C and its satellite keys.

Rick

🔗Andreas Sparschuh <a_sparschuh@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Rick~
> there is also a subharmonic scale on page 3, 4th scale down. also a
> 12 tone scale. Here two subharmonic series switch at an 7/5
> http://anaphoria.com/tres.PDF <http://anaphoria.com/tres.PDF3>
> --
Hi Kraig,

I couldn't quite figure out what was going on with the diagram (tho it looks very interesting, as do all of them). By the term "subharmonics" does Wilson just mean the harmonic series inverted eg 4:5:6 becomes 8/5 4/3 ? If so, my question is; isn't this just a new harmonic series where the original tonic is now placed in the role of upper harmonic? IOW still just the harmonic series? But I suppose that in terms of context this term might make sense.

Where I'm coming from is that many of the stuff I read about "subharmonics" years ago (don't ask me where, maybe Helmholtz) is nothing more than the GCD of the plain old harmonic series, gone unrecognised. For eg, it was something like, if you play 5 and 6 together you can sometimes hear a "virtual tonic" or "subharmonic" 1 come out. But this is just the resultant frequency of the wave addition and hence, just the harmonic series.

On a related note, I did notice the term "Diophantine" in the Wilson doc. Do you know what this all means? It's just that I've been exploring a vague idea that the concept of GCD might be extended to complex numbers (probably Gaussian integers) which sometimes can have more than one. During my Googling travels I came across Diophantine equations which (if I recall correctly) are polynomial equations with GCD's. My idea is that if we can find a wave application for these then a whole new arena might open up. [I haven't got very far. Given two Gaussian integers of the form a + bi they can have up to four GCD's. But if we take a wave with complex exponent we get:
e^i(a + bi)t = e^(ai - b)t = (e^-bt).(e^iat) = e^-bt(cosat + isinat), which is just a damped oscillation. Since periodicity is independent from amplitude, then adding two such waves would only give the usual GCD. IOW I can't see where the complex GCD's would fit in].
It just might be that Wilson's work might already relate to this??

-Rick
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> Mesotonal Music from:
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> a momentary antenna as i turn to water
> this evaporates - an island once again
>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> >
> >...I just happened to notice that the harmonics 16:17:18:19:20 give good chromatics from tonic to major third.
>
> Even more general Rick,
> see
> http://www.xs4all.nl/~huygensf/doc/stevinsp.html
> ..."There is a continuous range of semitones with the values 14 : 15 : 16 : 17 : 18 : 19 : 20 : 21, from major semitones 14:15 to minor semitones 20:21...."
>
> bye
> A.S.
>
Thanks A.S,

Interesting that Galileo's father Vincenzo entertained and rejected these semitones 14-21. People nowadays often don't know that the young Galileo was a mathemusician.

I've since set up my microtuner and, personally, I actually don't like my tuning so far. Both 5/4 and 3/2 sound flat to me. It's probably because I used a piano sample and I'm so conditioned to 12 tET on this instrument that it sounds "honky tonk" to my ears. Maybe it will sound better on another non ET instrument?

-Rick

🔗Andreas Sparschuh <a_sparschuh@...>

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
>
> > >... harmonics 16:17:18:19:20 give good chromatics
> > > from tonic to major third.

> I've since set up my microtuner and, personally, I actually don't
>like my tuning so far. Both 5/4 and 3/2 sound flat to me.

Agreed Rick,
in order to gain all-3rds-sharp inbetween the semitones:

c# : d : eb : e == 16 : 17 : 18 : 19 == C#1 : D1 : Eb1 : E1

use that interpolation of a dozen 5ths:

C 243 := 3^5
G 91 182 364 728 (<729 :=3^6)
D 68 136 272 (<273 := 3*G)
A 203 (<204 := 3*68)
E 19 38 76 152 304 608 (<609 := 3*A) ; 3 single 'squiggles'
B 57
F# 171
C# 1...512 (<513 := 3*F#) ; the concluding 5 triply 'squiggles'
G# 3
D# 9 := 3^2
A# 27 := 3^3
F 81 := 3^4
C=end 243 := 3^5

as overtaken from:
/tuning/topicId_78659.html#78677

>It's probably because I used a piano sample
> and I'm so conditioned to
>12 tET on this instrument
>that it sounds "honky tonk" to my ears.
>Maybe it will sound better on another non ET instrument?

alike implemented on my own old harpsichord....

!S_16_17_18_19.scl
c# : d : eb : e == 16 : 17 : 18 : 19 == C#1 : D1 : Eb1 : E1
!absolute pitches from the middle_c' on:
!c'243#256 d'272#288 e'304 f'324#342 g'364#384 a'406#432 b'456
!
12
!
256/243 ! ~90.22cent C#(cents) the limma
272/243 ! ~195.81... D 17-limit tone
32/27 !_! ~294.13... Eb pythagorean minor-3rd
304/243 ! ~387.74... E (5/4)*(1216/1215)
4/3 !___! ~498.04... F just 4th
38/27 !_! ~591.65... F# 19-limit tritone
364/243 ! ~699.58... G (3/2)*(728/729)
128/81 !! ~792.18... G# ditone 81/64 downwards
406/243 ! ~888.63... A4=406cps ~ J.S.Bach's 1722 Coethen Cammer-thone
16/9 !__! ~996.09... Bb tone 9/8 downwards
152/81 !! ~1089.69.. B 19-limit 7th
2/1
!

Quest:
How does that 16:17:18:19 sequence on the keys C#:D:Eb:E
sound in yours 12ET-biased ears?

bye
A.S.

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> >
> > > >... harmonics 16:17:18:19:20 give good chromatics
> > > > from tonic to major third.
>
> > I've since set up my microtuner and, personally, I actually don't
> >like my tuning so far. Both 5/4 and 3/2 sound flat to me.
>
> Agreed Rick,
> in order to gain all-3rds-sharp inbetween the semitones:
>
> c# : d : eb : e == 16 : 17 : 18 : 19 == C#1 : D1 : Eb1 : E1
>
> use that interpolation of a dozen 5ths:
>
> C 243 := 3^5
> G 91 182 364 728 (<729 :=3^6)
> D 68 136 272 (<273 := 3*G)
> A 203 (<204 := 3*68)
> E 19 38 76 152 304 608 (<609 := 3*A) ; 3 single 'squiggles'
> B 57
> F# 171
> C# 1...512 (<513 := 3*F#) ; the concluding 5 triply 'squiggles'
> G# 3
> D# 9 := 3^2
> A# 27 := 3^3
> F 81 := 3^4
> C=end 243 := 3^5
>
> as overtaken from:
> /tuning/topicId_78659.html#78677
>
> >It's probably because I used a piano sample
> > and I'm so conditioned to
> >12 tET on this instrument
> >that it sounds "honky tonk" to my ears.
> >Maybe it will sound better on another non ET instrument?
>
> alike implemented on my own old harpsichord....
>
> !S_16_17_18_19.scl
> c# : d : eb : e == 16 : 17 : 18 : 19 == C#1 : D1 : Eb1 : E1
> !absolute pitches from the middle_c' on:
> !c'243#256 d'272#288 e'304 f'324#342 g'364#384 a'406#432 b'456
> !
> 12
> !
> 256/243 ! ~90.22cent C#(cents) the limma
> 272/243 ! ~195.81... D 17-limit tone
> 32/27 !_! ~294.13... Eb pythagorean minor-3rd
> 304/243 ! ~387.74... E (5/4)*(1216/1215)
> 4/3 !___! ~498.04... F just 4th
> 38/27 !_! ~591.65... F# 19-limit tritone
> 364/243 ! ~699.58... G (3/2)*(728/729)
> 128/81 !! ~792.18... G# ditone 81/64 downwards
> 406/243 ! ~888.63... A4=406cps ~ J.S.Bach's 1722 Coethen Cammer-thone
> 16/9 !__! ~996.09... Bb tone 9/8 downwards
> 152/81 !! ~1089.69.. B 19-limit 7th
> 2/1
> !
>
> Quest:
> How does that 16:17:18:19 sequence on the keys C#:D:Eb:E
> sound in yours 12ET-biased ears?
>
> bye
> A.S.
>
Hi A.S.

In answer to your question, 16:17:18:19 seem to sound fine to my 12tET biased ears. In fact I've been working so long on the maths side without any means of testing the results that I am surprised to learn that 5/4 and 3/2 do not sound as "pure" as the maths seemed to suggest. (OTOH Carl's low-limit "Barbershop" tunings sounded very harmonic to me so it might be a context or timbre thing?). What's more, my original theoretical calculation that the tempered minor third would more likely be approximating 19/16 rather than 6/5 seems confirmed, which is ironic since I was basing it on the fact that 5/4 and 3/2 have "prime/2^N". For the major third and fifth, the intervals I prefer are 161/128 and 191/128 respectively (NB: I think 3/2 might have sounded sharp, not flat).

As far as your suggested tunings go, I am a rank amateur at both the microtuning software and some of the historical lingo. I'm used to thinking of 256Hz as "scientific C", not C#, so please forgive me if it takes some time for me to figure out and test. What's ! and "squiggles"?

-Rick

🔗Kraig Grady <kraiggrady@...>

Hi Rick~
The subharmonics series is the inverse of the harmonic. It's interest lies in that it is found of flutes of equidistant holes. There is a certain 'depth ' that subharmonic material doesn't and one can move the material around much easier in many cases. Harmonic based stuff can just sit there sometimes. The Diophantine equations leads to the epimoric interval. I cannot tell what these formula relate to. It is beyond me.
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "Andreas Sparschuh" <a_sparschuh@...> wrote:
>
Hi Andreas,

I gave it a second look and took the time to put the notes in (with harpsichord sample). You're right, it sounds much better. Like real Bach. Thanks for that. Even other chords like Eb and F# major sound fine, as do diminished and augmented. It's a meantone tuning right? Has it got a specific name?

Just as an experiment I replaced the Pythagorean minor third with 19/16 which is very slightly sharper. It sounds fine too. And while I was there I substituted F# with (19/16)^2 and A with (19/16)^3, my logic being that (19/16)^4 is closer to 2 than the other b5 squared. Again these modifications seem entirely workable.

Wow its fun being able to test things finally

Rick

> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> >
> > > >... harmonics 16:17:18:19:20 give good chromatics
> > > > from tonic to major third.
>
> > I've since set up my microtuner and, personally, I actually don't
> >like my tuning so far. Both 5/4 and 3/2 sound flat to me.
>
> Agreed Rick,
> in order to gain all-3rds-sharp inbetween the semitones:
>
> c# : d : eb : e == 16 : 17 : 18 : 19 == C#1 : D1 : Eb1 : E1
>
> use that interpolation of a dozen 5ths:
>
> C 243 := 3^5
> G 91 182 364 728 (<729 :=3^6)
> D 68 136 272 (<273 := 3*G)
> A 203 (<204 := 3*68)
> E 19 38 76 152 304 608 (<609 := 3*A) ; 3 single 'squiggles'
> B 57
> F# 171
> C# 1...512 (<513 := 3*F#) ; the concluding 5 triply 'squiggles'
> G# 3
> D# 9 := 3^2
> A# 27 := 3^3
> F 81 := 3^4
> C=end 243 := 3^5
>
> as overtaken from:
> /tuning/topicId_78659.html#78677
>
> >It's probably because I used a piano sample
> > and I'm so conditioned to
> >12 tET on this instrument
> >that it sounds "honky tonk" to my ears.
> >Maybe it will sound better on another non ET instrument?
>
> alike implemented on my own old harpsichord....
>
> !S_16_17_18_19.scl
> c# : d : eb : e == 16 : 17 : 18 : 19 == C#1 : D1 : Eb1 : E1
> !absolute pitches from the middle_c' on:
> !c'243#256 d'272#288 e'304 f'324#342 g'364#384 a'406#432 b'456
> !
> 12
> !
> 256/243 ! ~90.22cent C#(cents) the limma
> 272/243 ! ~195.81... D 17-limit tone
> 32/27 !_! ~294.13... Eb pythagorean minor-3rd
> 304/243 ! ~387.74... E (5/4)*(1216/1215)
> 4/3 !___! ~498.04... F just 4th
> 38/27 !_! ~591.65... F# 19-limit tritone
> 364/243 ! ~699.58... G (3/2)*(728/729)
> 128/81 !! ~792.18... G# ditone 81/64 downwards
> 406/243 ! ~888.63... A4=406cps ~ J.S.Bach's 1722 Coethen Cammer-thone
> 16/9 !__! ~996.09... Bb tone 9/8 downwards
> 152/81 !! ~1089.69.. B 19-limit 7th
> 2/1
> !
>
> Quest:
> How does that 16:17:18:19 sequence on the keys C#:D:Eb:E
> sound in yours 12ET-biased ears?
>
> bye
> A.S.
>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> Hi Rick~
> The subharmonics series is the inverse of the harmonic. It's interest
> lies in that it is found of flutes of equidistant holes. There is a
> certain 'depth ' that subharmonic material doesn't and one can move the
> material around much easier in many cases. Harmonic based stuff can just
> sit there sometimes. The Diophantine equations leads to the epimoric
> interval. I cannot tell what these formula relate to. It is beyond me.
> --
>
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> Mesotonal Music from:
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> a momentary antenna as i turn to water
> this evaporates - an island once again
>
Thanks Kraig,

I'm not saying that it isn't real or musically useful but that it is just the usual harmonic series seen from the inverted point of view. For eg, F minor triad as inverted major (i.e. (2/3):(4/5):1 from C=1) places the C in the role of maj 3 to Ab and fifth to F. So from a certain point of view we now have two harmonic series in play. However, we can also get to an harmonic version of the minor triad, for eg as 16:19:24, where 19/16 is the minor 3 and 24/26 = 3/2 the fifth. If we call the 16 here "F", then 19/16 is Ab and 3/2 is C, both upper harmonics to tonic F. In fact, rewriting the F minor (2/3):(4/5):1 as 10:12:15, this is also an harmonic series where the 1 is actually a Db. So we see that either way there is no need to invoke a mysterious "subharmonic series".

There was however one other thing that finally clinched it for me. That if we look at the actual waves being produced, then the overtone series can be explained by the existence of the GCD frequency. For eg, 9/6 = 3/2 have GCD 3 so that 3 is the tonic, 6 the 2nd harmonic and 9 the 3rd. The wave that is produced will actually be 3Hz for this will be the number of cycles per second. But I've never found anything comparable to justify the presence of a subharmonic series.

The Diophantine equation is just a condition for finding coprimes which is helpful for finding the gcd. The 2 and 3 above are coprimes. What are epimoric intervals?

PS: Your concert looks good. Enjoyed the last one so I'll check if I can make it.

Cheers

Rick

🔗Petr Parízek <p.parizek@...>

Rick wrote:

> I'm not saying that it isn't real or musically useful but that it is just
> the usual harmonic series seen from the inverted point of view. For eg,
> F minor triad as inverted major (i.e. (2/3):(4/5):1 from C=1) places the C
> in the role of maj 3 to Ab and fifth to F. So from a certain point of view
> we now have two harmonic series in play. However, we can also get to an harmonic
> version of the minor triad, for eg as 16:19:24, where 19/16 is the minor 3
> and 24/26 = 3/2 the fifth. If we call the 16 here "F", then 19/16 is Ab and 3/2 is C,
> both upper harmonics to tonic F. In fact, rewriting the F minor (2/3):(4/5):1
> as 10:12:15, this is also an harmonic series where the 1 is actually a Db. So we see
> that either way there is no need to invoke a mysterious "subharmonic series".

Except that the 5-limit subharmonic F minor is, in most cases, more easily recognizable than the 19-limit harmonic F minor. The reason is simple, 5/4 is clearer than 24/19 and 6/5 is clearer than 19/16. The other thing is that if in one case we decide to make major triad by stacking 6/5 above 5/4, then the easiest and most consistent way to get a minor triad is to do it the other way round.

> There was however one other thing that finally clinched it for me. That if we look
> at the actual waves being produced, then the overtone series can be explained
> by the existence of the GCD frequency. For eg, 9/6 = 3/2 have GCD 3 so that 3 is the tonic,
> 6 the 2nd harmonic and 9 the 3rd. The wave that is produced will actually be 3Hz
> for this will be the number of cycles per second. But I've never found anything comparable
> to justify the presence of a subharmonic series.

Haven't you really?It can be explained in several ways. First of all, very often, ancient theories about intervals were based on ratios of lengths, not frequencies. These can be either relative string lengths or wavelengths or whatever lengths you choose. This means that a tone with a 1 ms period is one octave higher than 2 ms, and that, for example, 3 ms is a perfect fifth lower than 2 ms, and so on. So if you have periods of 1 ms, 2 ms, 3 ms, 4 ms, etc., you get the subharmonic series whose "guide tone" (the opposite of the fundamental) has the frequency equal to the LCM of all the sounding frequencies, which is, in this case, 1000Hz or a period of 1 ms. -- For another thing, I'm not sure how much you know about overtone singing, but let's say you want to get an A6 out of your voice's overtones. After you shape your tongue accordingly to get the desired resonance, then you have to sing (with your actual voice) a tone which is "subharmonic" to this A6 (in other words, to which the A6 is harmonic), which can be an A3 (the 8th subharmonic), a G3 (the 9th subharmonic), an F3 (the 10th subharmonic), or any other subharmonic of the A6 that you choose. -- If you don't know about overtone singing and have some versatile effect processing tools, you can try something similar by applying a very very narrow bandpass filter (let's say, from 1740Hz to 1780Hz), running a clear sawtooth wave with decreasing pitch through that, and hearing what happens.

> The Diophantine equation is just a condition for finding coprimes which is helpful for
> finding the gcd. The 2 and 3 above are coprimes. What are epimoric intervals?

Integer ratios whose numerator is 1 higher or lower than the denominator, like 5/4, 2/1 or 6/7. A few years ago, I also used the term "semi-epimoric intervals" for the cases where it's 2 higher or lower (like 5/3 or 7/9) and, at the same time, I started using what I called "ED" (meaning "epimoric denominator") as a unit for telling how far an interval is from an epimoric ratio. For example, the ED for 3/2 is 2, the ED for 2/1 is 1, and therefore the ED for 5/3 is 1.5. Scala also has the possibility to convert between EDs and other units -- you can find it in the "epiden" section of the documentation for "Set attribute".

Petr

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Rick wrote:
>
> > I'm not saying that it isn't real or musically useful but that it is just
> > the usual harmonic series seen from the inverted point of view. For eg,
> > F minor triad as inverted major (i.e. (2/3):(4/5):1 from C=1) places the C
> > in the role of maj 3 to Ab and fifth to F. So from a certain point of view
> > we now have two harmonic series in play. However, we can also get to an harmonic
> > version of the minor triad, for eg as 16:19:24, where 19/16 is the minor 3
> > and 24/26 = 3/2 the fifth. If we call the 16 here "F", then 19/16 is Ab and 3/2 is C,
> > both upper harmonics to tonic F. In fact, rewriting the F minor (2/3):(4/5):1
> > as 10:12:15, this is also an harmonic series where the 1 is actually a Db. So we see
> > that either way there is no need to invoke a mysterious "subharmonic series".
>
> Except that the 5-limit subharmonic F minor is, in most cases, more easily recognizable than the 19-limit harmonic F minor. The reason is simple, 5/4 is clearer than 24/19 and 6/5 is clearer than 19/16. The other thing is that if in one case we decide to make major triad by stacking 6/5 above 5/4, then the easiest and most consistent way to get a minor triad is to do it the other way round.
>
> > There was however one other thing that finally clinched it for me. That if we look
> > at the actual waves being produced, then the overtone series can be explained
> > by the existence of the GCD frequency. For eg, 9/6 = 3/2 have GCD 3 so that 3 is the tonic,
> > 6 the 2nd harmonic and 9 the 3rd. The wave that is produced will actually be 3Hz
> > for this will be the number of cycles per second. But I've never found anything comparable
> > to justify the presence of a subharmonic series.
>
> Haven't you really?It can be explained in several ways. First of all, very often, ancient theories about intervals were based on ratios of lengths, not frequencies. These can be either relative string lengths or wavelengths or whatever lengths you choose. This means that a tone with a 1 ms period is one octave higher than 2 ms, and that, for example, 3 ms is a perfect fifth lower than 2 ms, and so on. So if you have periods of 1 ms, 2 ms, 3 ms, 4 ms, etc., you get the subharmonic series whose "guide tone" (the opposite of the fundamental) has the frequency equal to the LCM of all the sounding frequencies, which is, in this case, 1000Hz or a period of 1 ms. -- For another thing, I'm not sure how much you know about overtone singing, but let's say you want to get an A6 out of your voice's overtones. After you shape your tongue accordingly to get the desired resonance, then you have to sing (with your actual voice) a tone which is "subharmonic" to this A6 (in other words, to which the A6 is harmonic), which can be an A3 (the 8th subharmonic), a G3 (the 9th subharmonic), an F3 (the 10th subharmonic), or any other subharmonic of the A6 that you choose. -- If you don't know about overtone singing and have some versatile effect processing tools, you can try something similar by applying a very very narrow bandpass filter (let's say, from 1740Hz to 1780Hz), running a clear sawtooth wave with decreasing pitch through that, and hearing what happens.
>
> > The Diophantine equation is just a condition for finding coprimes which is helpful for
> > finding the gcd. The 2 and 3 above are coprimes. What are epimoric intervals?
>
> Integer ratios whose numerator is 1 higher or lower than the denominator, like 5/4, 2/1 or 6/7. A few years ago, I also used the term "semi-epimoric intervals" for the cases where it's 2 higher or lower (like 5/3 or 7/9) and, at the same time, I started using what I called "ED" (meaning "epimoric denominator") as a unit for telling how far an interval is from an epimoric ratio. For example, the ED for 3/2 is 2, the ED for 2/1 is 1, and therefore the ED for 5/3 is 1.5. Scala also has the possibility to convert between EDs and other units -- you can find it in the "epiden" section of the documentation for "Set attribute".
>
> Petr
>
Hi Petr,

First of all thanks for explaining epimoric intervals. Are ED's "integer ratios whose denominator is 1 higher or lower than the numerator"? Excuse my stupidity but I'm not exactly sure what you mean or how they can be applied.

Concerning the subharmonic series, however, one needs to keep track of inversions between periods and frequencies on the one hand and inversions between intervals on the other. Choosing convenient units where 1 ms is unity, the periods 1 ms, 2 ms, 3 ms give frequencies 1, (1/2), (1/3), which have a GCD of (1/6). That is, (1/2)/(1/3) = 3/2, gcd = (1/2)/3 = (1/3)/2 = (1/6). IOW, 2 x (1/6) = (1/3), 3 x (1/6) = (1/2), and these will be the 2nd and 3rd harmonics of tonic (1/6), respectively. So it doesn't matter how a particular wave is produced (by singing lower harmonics for eg), the resultant wave will always give the gcd provided the component frequencies bear an integral relation.

-Rick

🔗Petr Parízek <p.parizek@...>

Rick wrote:

> First of all thanks for explaining epimoric intervals. Are ED's "integer ratios
> whose denominator is 1 higher or lower than the numerator"? Excuse my stupidity but
> I'm not exactly sure what you mean or how they can be applied.

What you describe are epimoric intervals (or ratios or factors or whatever you want to call them). ED is the value which determines how close the interval is to an epimoric interval. And also, the larger the value, the closer the interval is to unison (you can't express an unison as an ED because you would have to use a value of 1/0). If the interval in question is epimoric, then the ED is equal to the denominator of the ratio (that's why I chose the name). So, let's say that a pure octave has a ratio of 2/1, which means that the ED for a pure octave is 1 as 2/1 is an epimoric ratio. Similarly, the ratio of a pure fifth is 3/2, which is also an epimoric ratio, and therefore the ED for a pure fifth is 2. You can convert any factor to ED with the formula "1/(f-1)" where "f" is the factor in question, which means that you get 1 for 2/1, 2 for 3/2, and so on. If the ratio is not epimoric, then it can tell you how close it is to an epimoric ratio. For example, if an ED of 1 means an octave and an ED of 2 means a fifth, then an ED of 1.5 means a pure major sixth because this is exactly the "partial" mean of the two former ratios (you can think of them as 4/2 and 6/4 so that there's 5/3 in the middle). The ED can also tell you which frequencies you would use if you wanted to have a 1Hz difference. For example, to get an octave with a 1Hz difference, you would use 1Hz and 2Hz and the ED is essentially the lower of the two. Similarly, if you wanted to find a pure major sixth with a 1Hz difference, then you would use 1.5Hz and 2.5Hz -- and, again, the ED is the lower of the two. This can be useful if you're trying to figure out how much detuned an interval is from unison. The lower the value, the larger the detuning.

> Concerning the subharmonic series, however, one needs to keep track of inversions
> between periods and frequencies on the one hand and inversions between intervals
> on the other. Choosing convenient units where 1 ms is unity, the periods 1 ms, 2 ms, 3 ms
> give frequencies 1, (1/2), (1/3), which have a GCD of (1/6).

Not sure whether you're doing this intentionally or not, but you're completely omitting the LCM. If you také frequencies of 4Hz and 6Hz, then the GCD (or fundamental) is 2 and the LCM (or guide tone) is 12. If you try to, (don't know what) let's say, tap 6Hz with one hand and 4Hz with the other, you'll realize that if you start tapping with both hands at the same time (i.e. the initial tap is with both hands), then the shortest time between two consecutive taps is 1/12 of a second. These "guide tones" are useful for tuners while counting beats in tempered intervals because tempered intervals make them beat. For example, if you mix 200Hz and 301Hz, then the 3rd harmonic of the lower tone is 600Hz and the 2nd harmonic of the higher tone is 602Hz, which results in 2Hz beating (as long as the overtones are audible, of course). Similarly, if I shaped my mouth in such a way that it had some strong resonance at 600Hz, this resonance would be most audible if I was singing tones like 300Hz, 200Hz, 150Hz, and so on.

Petr

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Rick wrote:
>
> > First of all thanks for explaining epimoric intervals. Are ED's "integer ratios
> > whose denominator is 1 higher or lower than the numerator"? Excuse my stupidity but
> > I'm not exactly sure what you mean or how they can be applied.
>
> What you describe are epimoric intervals (or ratios or factors or whatever you want to call them). ED is the value which determines how close the interval is to an epimoric interval. And also, the larger the value, the closer the interval is to unison (you can't express an unison as an ED because you would have to use a value of 1/0). If the interval in question is epimoric, then the ED is equal to the denominator of the ratio (that's why I chose the name). So, let's say that a pure octave has a ratio of 2/1, which means that the ED for a pure octave is 1 as 2/1 is an epimoric ratio. Similarly, the ratio of a pure fifth is 3/2, which is also an epimoric ratio, and therefore the ED for a pure fifth is 2. You can convert any factor to ED with the formula "1/(f-1)" where "f" is the factor in question, which means that you get 1 for 2/1, 2 for 3/2, and so on. If the ratio is not epimoric, then it can tell you how close it is to an epimoric ratio. For example, if an ED of 1 means an octave and an ED of 2 means a fifth, then an ED of 1.5 means a pure major sixth because this is exactly the "partial" mean of the two former ratios (you can think of them as 4/2 and 6/4 so that there's 5/3 in the middle). The ED can also tell you which frequencies you would use if you wanted to have a 1Hz difference. For example, to get an octave with a 1Hz difference, you would use 1Hz and 2Hz and the ED is essentially the lower of the two. Similarly, if you wanted to find a pure major sixth with a 1Hz difference, then you would use 1.5Hz and 2.5Hz -- and, again, the ED is the lower of the two. This can be useful if you're trying to figure out how much detuned an interval is from unison. The lower the value, the larger the detuning.
>
> > Concerning the subharmonic series, however, one needs to keep track of inversions
> > between periods and frequencies on the one hand and inversions between intervals
> > on the other. Choosing convenient units where 1 ms is unity, the periods 1 ms, 2 ms, 3 ms
> > give frequencies 1, (1/2), (1/3), which have a GCD of (1/6).
>
> Not sure whether you're doing this intentionally or not, but you're completely omitting the LCM. If you také frequencies of 4Hz and 6Hz, then the GCD (or fundamental) is 2 and the LCM (or guide tone) is 12. If you try to, (don't know what) let's say, tap 6Hz with one hand and 4Hz with the other, you'll realize that if you start tapping with both hands at the same time (i.e. the initial tap is with both hands), then the shortest time between two consecutive taps is 1/12 of a second. These "guide tones" are useful for tuners while counting beats in tempered intervals because tempered intervals make them beat. For example, if you mix 200Hz and 301Hz, then the 3rd harmonic of the lower tone is 600Hz and the 2nd harmonic of the higher tone is 602Hz, which results in 2Hz beating (as long as the overtones are audible, of course). Similarly, if I shaped my mouth in such a way that it had some strong resonance at 600Hz, this resonance would be most audible if I was singing tones like 300Hz, 200Hz, 150Hz, and so on.
>
> Petr
>
Hi Petr,

I'll get back to you about epimoric intervals. I'm not omitting the LCM intentionally because I was talking about the GCD. So what you're driving at is that the LCM frequency is used for tuning? Interesting. So for a simpler example, given 2Hz and 3Hz, their LCM is 6Hz which gives the count required to beat them out. i.e., dividing 1 second into 6 parts, then the 3 bps is on the numbers 1 3 5 while the 2 bps is on 1 and 4. Now some time ago I spoke to Bill about this and he said that studies have shown that our perception of harmonies are not the same as sped up rhythms, so I left it there. But concerning the GCD, everything I said in the last post was intentional. Gotta run but I'll write again tomorrow because I find this interesting.

-Rick

🔗Petr Parízek <p.parizek@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Rick wrote:
>
> > I'm not omitting the LCM intentionally because I was talking about the GCD.
>
> Right, but wasn't it you who wanted to find something characteristic for the subharmonic series in a similar way as the GCD is for the harmonic series?
>
> > Now some time ago I spoke to Bill about this and he said that studies have shown
> > that our perception of harmonies are not the same as sped up rhythms,
>
> They're not, but if you can, listen carefully to this example and tell me what you can hear then: www.sendspace.com/file/g4zho8
>
> Petr
>
Hi Petr,

The link is not there any more so I couldn't hear it. My original point was not to show that the LCM of the subharmonic series was the inverse of the GCD of the harmonic series but that this is an unfounded assumption. The so-called proofs for the existence of this LCM invariably involve an unconscious confusion between inverting intervals and inverting frequencies to obtain the period. If you reread the original posts on this thread you'll see that this is precisely what happens i.e you took 1ms, 2ms, 3ms which are periods, and said that their LCM was 1000Hz which it isn't. What I was saying was that the period of these waves is 6000Hz, the GCD, and that the same harmonic (overtonal) laws apply regardless of the fact that it might seem that the series 1, 1/2, 1/3,...has a whole new "inverted" logic going on. And if it turns out that the LCM has meaning too, then this will apply to both the series 1,2,3, and 1,1/2,1/3...IOW, there are no special laws for the "undertone series". The real meaning lies in the fact that inversion creates harmonic bitonalities.

Thanks and by all means refresh and send me the link again.

-Rick

🔗Petr Parízek <p.parizek@...>

Rick wrote:

> The link is not there any more so I couldn't hear it.

There must be something weird happening on your side. I've checked it a few minutes ago and the link is still valid. Anyway, if it doesn't work for you, I've tried to make a different link on another server: www.yousendit.com/download/cmcyQmtld0FiV3dLSkE9PQ

> My original point was not to show that the LCM of the subharmonic series was
> the inverse of the GCD of the harmonic series but that this is an unfounded assumption.
> The so-called proofs for the existence of this LCM invariably involve
> an unconscious confusion between inverting intervals and inverting frequencies
> to obtain the period. If you reread the original posts on this thread you'll see that this is
> precisely what happens i.e you took 1ms, 2ms, 3ms which are periods, and said that their
> LCM was 1000Hz which it isn't.

Okay, let's make order in it. It's important to note that the acoustical inversion of the "fundamental" is the "guide tone". So the frequency of the fundamental is the GCD of all the sounding frequencies, while the frequency for the guide tone is the LCM of the sounding frequencies. Similarly, the period length of the fundamental is the LCM of the sounding "lengths", while the period length for the guide tone is the GCD of the sounding lengths. So if the sounding tones are, for example, 250Hz and 400Hz, then the period lengths are 4 ms and 2.5 ms. This means that the fundamental frequency is 50Hz, the guide tone frequency is 2000Hz, the length for the fundamental is 20 ms, and the length for the guide tone is 0.5 ms. You seem to be saying that the 200Hz frequency (or 0.5 ms length) is an unimportant thing that doesn't need to be taken into account. That's why I was trying to explain that it is and why.

Petr

🔗Petr Parízek <p.parizek@...>

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Rick wrote:
>
> > The link is not there any more so I couldn't hear it.
>
> There must be something weird happening on your side. I've checked it a few minutes ago and the link is still valid. Anyway, if it doesn't work for you, I've tried to make a different link on another server: www.yousendit.com/download/cmcyQmtld0FiV3dLSkE9PQ
>
> > My original point was not to show that the LCM of the subharmonic series was
> > the inverse of the GCD of the harmonic series but that this is an unfounded assumption.
> > The so-called proofs for the existence of this LCM invariably involve
> > an unconscious confusion between inverting intervals and inverting frequencies
> > to obtain the period. If you reread the original posts on this thread you'll see that this is
> > precisely what happens i.e you took 1ms, 2ms, 3ms which are periods, and said that their
> > LCM was 1000Hz which it isn't.
>
> Okay, let's make order in it. It's important to note that the acoustical inversion of the "fundamental" is the "guide tone". So the frequency of the fundamental is the GCD of all the sounding frequencies, while the frequency for the guide tone is the LCM of the sounding frequencies. Similarly, the period length of the fundamental is the LCM of the sounding "lengths", while the period length for the guide tone is the GCD of the sounding lengths. So if the sounding tones are, for example, 250Hz and 400Hz, then the period lengths are 4 ms and 2.5 ms. This means that the fundamental frequency is 50Hz, the guide tone frequency is 2000Hz, the length for the fundamental is 20 ms, and the length for the guide tone is 0.5 ms. You seem to be saying that the 200Hz frequency (or 0.5 ms length) is an unimportant thing that doesn't need to be taken into account. That's why I was trying to explain that it is and why.
>
> Petr
>
Hi Petr,

First up, my apologies. Of course 1000Hz is the LCM of the inverse frequencies. What you get when doing maths at 4pm after a long gig! Secondly, yes everything you say above checks out exactly from my end too. Just to simplify even further (I'll choose simplest units possible for reasons that will become clear):

harmonic series: 2Hz and 3Hz have GCD 1Hz, LCM 6Hz, and in period "lengths" (1/6) seconds is the common "count" while 1 sec is the time it repeats.

Sub-harmonic series: (1/2)Hz and (1/3)Hz give GCD (1/6)Hz, LCM 1Hz, and the periods give 1 sec as the count and 6 secs as the time it repeats.

So we see that the GCD of the first series is the LCM of the second and vice-versa, which has historically been a point of confusion. Now, while I knew of its existence, I didn't know that the LCM could be important for purposes of tuning (only for finding the common count) and am glad to learn otherwise since it widens the playing field. Thanks for that. However, the original debate was that the LCM acts as a type of fundamental of a "subharmonic" series, just as the GCD does for the harmonic series, thereby justifying its existence. My point was that whenever a GCD exists then the result comes from the harmonic series.

To see this once and for all, let us take a trivial change of units such that 6 seconds becomes "1" in our new units. Then 2 sec must be renamed "1/3", 3 sec becomes "1/2", and the frequencies, defined either as inverse period or (equivalently) as number of cycles per our new "unit" time, must be renamed "1", "3" and "2" respectively. Calculating the GCD of "2" and "3" in these new units gives "1" (i.e. original (1/6)Hz) while the LCM is "6" (1Hz in original). Therefore, we see that the "subharmonic series" is purely relative. Unless it appears as an inversion of a given interval then it is still just the harmonic series.

-Rick

🔗Petr Parízek <p.parizek@...>

Rick wrote:

> the original debate was that the LCM acts as a type of fundamental
> of a "subharmonic" series, just as the GCD does for the harmonic series, thereby justifying
> its existence. My point was that whenever a GCD exists then the result comes
> from the harmonic series.

I think I'm slowly beginning to understand what you mean. I believe the main source of confusion here is new unintroduced terminology. I think I've even heard someone who made a distinction between a "fundamental" and an "inverse fundamental", the latter of which meant the guide tone. However, this doesn't mean that they both have the same roles. I would just say that every chord made up of integer ratios has two characteristic properties, one being the fundamental and the other being the guide tone, each of which has its own role in the actual sound of this chord. And even though the role of the guide tone is completely different from the role of the fundamental, I consider both being essentially equally important. The only situation I can think of where the guide tone may be almost meaningless could be a set of sine waves. But in nearly all other cases, the fact that a JI chord has not only its own general "frequency" but also its own general "length count" can be made audible and can be useful even while composing music. So, to put it straight, while I find the harmonic series characteristic for its fundamental, I find the subharmonic series similarly characteristic for its guide tone. And as I can make the fundamental clearly audible in the case of the harmonic series, I could make the guide tone similarly audible in the case of the subharmonic series, if I wanted to.

Petr

🔗Kraig Grady <kraiggrady@...>

On Subharmonic series:
Guide tones of the harmonic series i have heard in context where they have been cued. Erv Wilson for fun once played me a series of dyads then adding the guide tone (combination tone) and after only two i was able to guess what tones i was going to hear next. Possibly this just might be some quick pattern recognition on my part.
One 'practical use of the subharmonic series is when uses a drone higher than other tones. By oral communications it has been passed down that at least certain Native American tribes would use a high drone kept up my a female choir and the men would sing the melodies below this. Most melodies start high and descend, starting on what looks like the 'fifth' and proceeds downward on what too often appears to be subharmonic intervals being what one gets when one uses just intervals to the high drone.
--

/^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere: North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_ ^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Rick wrote:
>
> > the original debate was that the LCM acts as a type of fundamental
> > of a "subharmonic" series, just as the GCD does for the harmonic series, thereby justifying
> > its existence. My point was that whenever a GCD exists then the result comes
> > from the harmonic series.
>
> I think I'm slowly beginning to understand what you mean. I believe the main source of confusion here is new unintroduced terminology. I think I've even heard someone who made a distinction between a "fundamental" and an "inverse fundamental", the latter of which meant the guide tone. However, this doesn't mean that they both have the same roles. I would just say that every chord made up of integer ratios has two characteristic properties, one being the fundamental and the other being the guide tone, each of which has its own role in the actual sound of this chord. And even though the role of the guide tone is completely different from the role of the fundamental, I consider both being essentially equally important. The only situation I can think of where the guide tone may be almost meaningless could be a set of sine waves. But in nearly all other cases, the fact that a JI chord has not only its own general "frequency" but also its own general "length count" can be made audible and can be useful even while composing music. So, to put it straight, while I find the harmonic series characteristic for its fundamental, I find the subharmonic series similarly characteristic for its guide tone. And as I can make the fundamental clearly audible in the case of the harmonic series, I could make the guide tone similarly audible in the case of the subharmonic series, if I wanted to.
>
> Petr
>
Hi Petr,

Okay, you said "So, to put it straight, while I find the harmonic series characteristic for its fundamental, I find the subharmonic series similarly characteristic for its guide tone". But this isn't what I'm saying at all. It is precisely the point of historical confusion that I was trying to avoid. Even at face value both the "upper" and "lower" harmonics of the examples in the last post gave both GCD and LCM.

[Excuse the capitals but I don't have italics). The GCD will ALWAYS represent the fundamental/tonic of a harmonic series and the ONLY condition for it to exist is that the frequencies involved form a ratio. This applies as much to frequencies smaller than 1 as to those larger than 1. For example, take frequencies 1/N and 1/(N+1) where N is whole (i.e. consecutive values from the "Sub-series"). Their ratio is (1/N)/(1/(N+1)) = (N+1)/N and their GCD frequency is (1/N)/(N+1) = (1/(N+1))/N = 1/(N^2 + N). Therefore, both frequencies are now upper harmonics to the tonic-fundamental frequency 1/(N^2 + N). [Or to view it the other way around, 1/(N^2 + N) x (N + 1) = (1/N) and 1/(N^2 + N) x N = 1/(N + 1), so that they are the N'th and (N+1)th upper harmonics of the fundamental 1/(N^2 + N).] IOW, -and this is most important- since the ratio (N+1)/N has both (N+1)>1 and N>1, it follows that they come from the harmonic series; the idea that frequencies smaller than 1 form a unique "subharmonic" series is an illusion. In fact I already proved this in another way with the trivial change of units proof in the last post i.e. by choosing new units the smaller than 1 values became larger than 1.

All in all, since the sub harmonic series does not exist, then BOTH the GCD and LCM, or "fundamental" and "guide tone", are characteristic of the harmonic series only and apply whenever two frequencies form a ratio, even for two sine waves. The fact that they are in a sense inverse of one another has nothing to do with the notion that there exists a subharmonic series which is the "inverse" of the true harmonic series. It is intuitive guesswork. I know the maths might be a bit off putting but it is really worth it in the end. I suggest plugging in some numbers and writing it down.

-Rick

🔗Petr Parízek <p.parizek@...>

Rick wrote:

> But this isn't what I'm saying at all. It is precisely the point of historical confusion that I was
> trying to avoid. Even at face value both the "upper" and "lower" harmonics of the examples
> in the last post gave both GCD and LCM.

I didn't say they both had the same meaning, I said I found them both very important fenomena which can occur in the actual sound even though they do so in very different ways.

> [Excuse the capitals but I don't have italics). The GCD will ALWAYS represent
> the fundamental/tonic of a harmonic series and the ONLY condition for it to exist is that
> the frequencies involved form a ratio. This applies as much to frequencies smaller than 1
> as to those larger than 1.

You seem to have misunderstood my point. Of course, if 2-second and 3-second periods were tones, then the fundamental would be 6 seconds, not 1. But the 1-second length can play a part in the sound as well and I thought you were saying it didn't. The LCM frequency (or GCD length) is not the fundamental for the subharmonic series but it is simply another property of the sound which is specific for the subharmonic series -- in fact, so specific that it reflects in the sound. Don't know what else to say to make you understand what I mean.

BTW: Have you downloaded my mp3 example?

Petr

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Rick wrote:
>
> > But this isn't what I'm saying at all. It is precisely the point of historical confusion that I was
> > trying to avoid. Even at face value both the "upper" and "lower" harmonics of the examples
> > in the last post gave both GCD and LCM.
>
> I didn't say they both had the same meaning, I said I found them both very important fenomena which can occur in the actual sound even though they do so in very different ways.
>
> > [Excuse the capitals but I don't have italics). The GCD will ALWAYS represent
> > the fundamental/tonic of a harmonic series and the ONLY condition for it to exist is that
> > the frequencies involved form a ratio. This applies as much to frequencies smaller than 1
> > as to those larger than 1.
>
> You seem to have misunderstood my point. Of course, if 2-second and 3-second periods were tones, then the fundamental would be 6 seconds, not 1. But the 1-second length can play a part in the sound as well and I thought you were saying it didn't. The LCM frequency (or GCD length) is not the fundamental for the subharmonic series but it is simply another property of the sound which is specific for the subharmonic series -- in fact, so specific that it reflects in the sound. Don't know what else to say to make you understand what I mean.
>
> BTW: Have you downloaded my mp3 example?
>
> Petr
>
Hi Petr,

Yes the 1 second length can play a part in the sound as well, which I didn't know before. And sure, the "LCM frequency (or GCD length) is not the fundamental for the subharmonic series but it is simply another property of the sound". But I don't think I've misunderstood you. When you continue on to say that the LCM frequency "is specific for the subharmonic series", I honestly don't know what you mean. Since the values 1/2, 1/3 etc... also belong to the harmonic series, then I can't see how it could possibly be true. Besides, isn't it true that, say, 3Hz and 5Hz, two notes from the upper harmonic series, also have a LCM of 15Hz? Or do you have some other idea in mind that I'm just not getting?

Sorry, haven't downloaded it yet. I'll do that right now.

Cheers

Rick

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> On Subharmonic series:
> Guide tones of the harmonic series i have heard in context where
> they have been cued. Erv Wilson for fun once played me a series of dyads
> then adding the guide tone (combination tone) and after only two i was
> able to guess what tones i was going to hear next. Possibly this just
> might be some quick pattern recognition on my part.
> One 'practical use of the subharmonic series is when uses a drone higher
> than other tones. By oral communications it has been passed down that at
> least certain Native American tribes would use a high drone kept up my a
> female choir and the men would sing the melodies below this. Most
> melodies start high and descend, starting on what looks like the 'fifth'
> and proceeds downward on what too often appears to be subharmonic
> intervals being what one gets when one uses just intervals to the high
> drone.
> --
>
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> Mesotonal Music from:
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> a momentary antenna as i turn to water
> this evaporates - an island once again
>
Hi Kraig,

When you say guide tone (combination tone), do you mean the LCM or difference tone?

Sure, I can imagine that once you set up a drone it is always interesting to invert the tonality by making the drone note serve different harmonic roles. Similar thing with pedal bass. My only issue is with the term "subharmonic" rather than, say, "inverted" series, mainly because I feel that it only has meaning in relation to harmonic intervals. For eg, C:E has a clear tonic C. Ab:C gives Ab as the tonic. But Ab:C:E together, the maj 3 above and below C, creates ambiguity by setting up two tonics. In numbers we might say 4/5:1:5/4. But the lower interval 4/5:1 on its own is just 4:5, two values from the harmonic series.

Btw, I've always wanted to ask, where does the "K" come from in "kraig"?

Rick

🔗Jacques Dudon <fotosonix@...>

Subharmonic series are especially useful for diphonic singers, or any singers making use of harmonics (which is in human voice generally unescapable anyway...)

When a group sings the same vowels together, especially in resonant rooms, very likely even if unconsciously it will focus here and there on the same overtones frequencies, and the only way for harmonics to meet exactly on the same tones, which sounds wonderful, is to place your fundamentals on subharmonic series.
(that is one of the various levels of "harmonic singing" techniques described by David Hykes)

Therefore subharmonic scales, and derivatives, are vocally highly interesting.

- - -
Jak

> On Subharmonic series:
> - -
> One 'practical use of the subharmonic series is when uses a drone > higher
> than other tones. By oral communications it has been passed down > that at
> least certain Native American tribes would use a high drone kept up > my a
> female choir and the men would sing the melodies below this. Most
> melodies start high and descend, starting on what looks like the > 'fifth'
> and proceeds downward on what too often appears to be subharmonic
> intervals being what one gets when one uses just intervals to the high
> drone.
> -->
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_

🔗Petr Parízek <p.parizek@...>

Rick wrote:

> When you continue on to say that the LCM frequency "is specific for the subharmonic
> series", I honestly don't know what you mean.

It seems you haven't heard about the technique which I call "lower octave singing". If you're able to involve not only your regular vocal folds but also your "false vocal folds" (http://en.wikipedia.org/wiki/Vestibular_fold), then these will vibrate at half the speed of the proper vocal folds (interestingly, some people with high voices can alternately change this extra tone between 1/2 and 1/3 of the proper vocal fold speed). For example, in this recording, you can hear someone singing a tone close to a Bb1 but his voice is, in fact, tuned to a Bb2. The reason why you hear a Bb1 instead of a Bb2 is that he also uses his false vocal folds together with his proper vocal folds: www.khoomei.com/mp3s/kokarg.mp3

> Besides, isn't it true that, say, 3Hz and 5Hz, two notes from the upper harmonic series, also
> have a LCM of 15Hz?

Of course, they do. And if these would be in the range of our hearing, you might eventually be able to hear this relative frequency of 15 as well, even though only softly. But this effect could get a bit more "pronounced" if you added, for example, another relative frequency of 4/15 to the 3 and 5.

Petr

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Petr Parízek <p.parizek@...> wrote:
>
> Rick wrote:
>
> > When you continue on to say that the LCM frequency "is specific for the subharmonic
> > series", I honestly don't know what you mean.
>
> It seems you haven't heard about the technique which I call "lower octave singing". If you're able to involve not only your regular vocal folds but also your "false vocal folds" (http://en.wikipedia.org/wiki/Vestibular_fold), then these will vibrate at half the speed of the proper vocal folds (interestingly, some people with high voices can alternately change this extra tone between 1/2 and 1/3 of the proper vocal fold speed). For example, in this recording, you can hear someone singing a tone close to a Bb1 but his voice is, in fact, tuned to a Bb2. The reason why you hear a Bb1 instead of a Bb2 is that he also uses his false vocal folds together with his proper vocal folds: www.khoomei.com/mp3s/kokarg.mp3
>
> > Besides, isn't it true that, say, 3Hz and 5Hz, two notes from the upper harmonic series, also
> > have a LCM of 15Hz?
>
> Of course, they do. And if these would be in the range of our hearing, you might eventually be able to hear this relative frequency of 15 as well, even though only softly. But this effect could get a bit more "pronounced" if you added, for example, another relative frequency of 4/15 to the 3 and 5.
>
> Petr
>
Hi Petr (and Jacques),

Of course I'm not denying the truth of what you're saying concerning the use of lower harmonics in singing etc, all of which is very interesting and gives me opportunities to learn and hear things I didn't know. I'm only trying to nut out a clearer set of mathematical definitions for the phenomena of sub harmonics. And I now think I understand what you were driving at when you said "just as the GCD is characteristic of the harmonic series, the LCD is characteristic of the subharmonic series". At any rate, I figured out a good formula which seems to cover it.

If we take two coprimes larger than 1 then their GCD will be 1 and their LCD will be the product of the two. If we instead take the inverse of these coprimes then the roles of GCD and LCD will be reversed. That is, the LCM will be 1 and the GCD will be the inverse of their product. For simple eg, 3Hz and 5Hz are coprime, GCD = 1 and LCM = 3 x 5 = 15. For (1/3)Hz and (1/5)Hz, GCD = (1/15) and LCM = 1.

Thus, just as all coprimes have GCD of 1, all inverse coprimes will have LCM of 1. This is why one might say that the LCD is characteristic of the subharmonic series.

OTOH, if the frequencies are not coprime then other GCD's and LCD's will result. Given two non-coprime frequencies p and q with p > q, if p/q = a/b where a and b are whole, then we obtain the two equations:
1) pb = qa giving the LCM, and
2) p/a = q/b giving the GCD frequency.
Similarly, if our initial frequencies are the inverse values (1/q) and (1/p) (now with (1/q) > (1/p)), then (1/q)/(1/p) = p/q = a/b, and we get the two equations:
3) b/q = a/p for the LCM, and
4) 1/aq = 1/bp giving the GCD frequency.
Again for a simple eg, let p = 9 and q = 6. 9/6 = 3/2, LCM = 18, GCD = 3. Or for inverse frequencies, (1/6)/(1/9) = 3/2, LCM = (1/3) and GCD = (1/18). Again we see this switching going on.

These four equations will give a simple formula for quickly working out the GCD and LCD for any two frequencies.

Rick

🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
> Hi Andreas, I gave it a second look ... You're right, it sounds ... Like real Bach ... It's a meantone tuning right? Has it got a specific name?

Hi Rick,
This is not a meantone; I think it would be called a "well" temperament or "circulating" (or perhaps only "irregular"). Using "C" as the baseline and going around the circle of 5ths, the notes at first get progressively flatter when compared with 12-tET, because the 5ths are <700 cents, then progressively sharper when the 5ths are >700 cents (in this case, pure).

> Just as an experiment I replaced the Pythagorean minor third with 19/16 which is very slightly sharper. It sounds fine too. And while I was there I substituted F# with (19/16)^2 and A with (19/16)^3, my logic being that (19/16)^4 is closer to 2 than the other b5 squared. Again these modifications seem entirely workable.

Note that the 16:19:24 triads were on C#, G# and Eb in AS's file; some other minor 3rds (e.g. on A, E) are much larger, almost 5:6. I assume you mean you moved Eb, F# and A from AS's position, which will obviously change this, and break the "well" temperament pattern, I think.

Re-cap of AS's scala file:
> > !S_16_17_18_19.scl
> > c# : d : eb : e == 16 : 17 : 18 : 19 == C#1 : D1 : Eb1 : E1
> > !absolute pitches from the middle_c' on:
> > !c'243#256 d'272#288 e'304 f'324#342 g'364#384 a'406#432 b'456
> > !
> > 12
> > !
> > 256/243 ! ~90.22cent C#(cents) the limma
> > 272/243 ! ~195.81... D 17-limit tone
> > 32/27 !_! ~294.13... Eb pythagorean minor-3rd
> > 304/243 ! ~387.74... E (5/4)*(1216/1215)
> > 4/3 !___! ~498.04... F just 4th
> > 38/27 !_! ~591.65... F# 19-limit tritone
> > 364/243 ! ~699.58... G (3/2)*(728/729)
> > 128/81 !! ~792.18... G# ditone 81/64 downwards
> > 406/243 ! ~888.63... A4=406cps ~ J.S.Bach's 1722 Coethen Cammer-thone
> > 16/9 !__! ~996.09... Bb tone 9/8 downwards
> > 152/81 !! ~1089.69.. B 19-limit 7th
> > 2/1
> > !

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> > Hi Andreas, I gave it a second look ... You're right, it sounds ... Like real Bach ... It's a meantone tuning right? Has it got a specific name?
>
> Hi Rick,
> This is not a meantone; I think it would be called a "well" temperament or "circulating" (or perhaps only "irregular"). Using "C" as the baseline and going around the circle of 5ths, the notes at first get progressively flatter when compared with 12-tET, because the 5ths are <700 cents, then progressively sharper when the 5ths are >700 cents (in this case, pure).
>
> > Just as an experiment I replaced the Pythagorean minor third with 19/16 which is very slightly sharper. It sounds fine too. And while I was there I substituted F# with (19/16)^2 and A with (19/16)^3, my logic being that (19/16)^4 is closer to 2 than the other b5 squared. Again these modifications seem entirely workable.
>
> Note that the 16:19:24 triads were on C#, G# and Eb in AS's file; some other minor 3rds (e.g. on A, E) are much larger, almost 5:6. I assume you mean you moved Eb, F# and A from AS's position, which will obviously change this, and break the "well" temperament pattern, I think.
>
> Re-cap of AS's scala file:
> > > !S_16_17_18_19.scl
> > > c# : d : eb : e == 16 : 17 : 18 : 19 == C#1 : D1 : Eb1 : E1
> > > !absolute pitches from the middle_c' on:
> > > !c'243#256 d'272#288 e'304 f'324#342 g'364#384 a'406#432 b'456
> > > !
> > > 12
> > > !
> > > 256/243 ! ~90.22cent C#(cents) the limma
> > > 272/243 ! ~195.81... D 17-limit tone
> > > 32/27 !_! ~294.13... Eb pythagorean minor-3rd
> > > 304/243 ! ~387.74... E (5/4)*(1216/1215)
> > > 4/3 !___! ~498.04... F just 4th
> > > 38/27 !_! ~591.65... F# 19-limit tritone
> > > 364/243 ! ~699.58... G (3/2)*(728/729)
> > > 128/81 !! ~792.18... G# ditone 81/64 downwards
> > > 406/243 ! ~888.63... A4=406cps ~ J.S.Bach's 1722 Coethen Cammer-thone
> > > 16/9 !__! ~996.09... Bb tone 9/8 downwards
> > > 152/81 !! ~1089.69.. B 19-limit 7th
> > > 2/1
> > > !
>
Thanks Martinsj,

Not meantone hah, doesn't surprise me. It's been about 15 years since I read Helmholtz and I probably didn't understand it back then as much as I thought I did. All I seem to "remember" is that that meantone was a non-ET JI, and probably threw all rational "well temperaments" into one bag.

Yeah you're right. 16:19:24 triads are already there on C#, G# and Eb. I didn't notice this because I didn't check all consecutive intervals, only those from the tonic. No need to rewrite them from C. We could always take the scale from c'# to c''# or transpose down to c' etc.

Now, I was having a bit of trouble because I'm used to thinking of 256 = 2^8 as C, not C# i.e. scientific C is 1Hz, and 256Hz is its eighth 8ve. But I think I get it now. If C = 243 = 3^5, which is six 3rd harmonics from 1, then this would make the tonic of this tuning a C#. This would also make sense when c'# = 256 and 16:19:24 is from c'# because 16 = 2^4. And g'# = 384 = 3 x 2^7, 8ve to 3rd harmonic of c'#, and so on.

Rick

🔗rick_ballan <rick_ballan@...>

Just one more question. It was my understanding that A.S's scale was based on successive applications of the fifth as 3/2. I was transposing it taking C as 256 instead of 243 and noticed that the series seems to break down from F# onwards. From B as 486 we have 486 x 3/4 = 364.5 instead of 364. I thought perhaps he just rounded it off except from here we have 364 x 3/2 = 273 instead of 272. Now the fifth from 273 is 409.5 whereas A.S has 406. But even if we take 272 x 3/2 we get 408! I'm not sure what I'm missing, some comma shift I should know about etc?
For the record, A.S's scale from c'=256 is:
c'256#272 d'288#304 e'324 f'342#364 g'384#406 a'432#456 b'486 c'512.
The 16:17:18:19 is now from the tonic c'.

Rick

🔗rick_ballan <rick_ballan@...>

I forgot to mention that a friend of mine, Arne Hanna, can sing two notes at once. It sounds just like the kogarg example. I'll ask if I can record him and send you a copy if you like.

While I'm here, I reviewed some of my old math's work about the LCM. Recall that I gave interval p/q = a/b, gcd = p/a = q/b and lcm = pb = qa. If we take the harmonic series of the original two frequencies, p, 2p, 3p...Np and q, 2q, 3q...Np, since the ratio between corresponding harmonics also equals a/b (the N's cancel), then the same two rules apply and we get a HS for both the gcd and lcm. Eg, 2p/2q = a/b, gcd = 2p/a = 2q/b and lcm = 2pb = 2qa. IOW, instead of thinking of this as the HS of the two freq's p and q, we can instead look at this as a HS composed of both gcd's and lcm's.
Finally, note that the interval between the lcm and gcd is ab: pb/(p/a) = ab.

Thanks for the post

Rick

🔗Petr Parízek <p.parizek@...>

🔗rick_ballan <rick_ballan@...>

🔗Kraig Grady <kraiggrady@...>

🔗martinsj013 <martinsj@...>

--- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@...> wrote:
> > Re-cap of AS's scala file:
> > > > !S_16_17_18_19.scl
> > > > c# : d : eb : e == 16 : 17 : 18 : 19 == C#1 : D1 : Eb1 : E1
> > > > !absolute pitches from the middle_c' on:
> > > > !c'243#256 d'272#288 e'304 f'324#342 g'364#384 a'406#432 b'456
> > > > !
> > > > 12
> > > > !
> > > > 256/243 ! ~90.22cent C#(cents) the limma
> > > > 272/243 ! ~195.81... D 17-limit tone
> > > > 32/27 !_! ~294.13... Eb pythagorean minor-3rd
> > > > 304/243 ! ~387.74... E (5/4)*(1216/1215)
> > > > 4/3 !___! ~498.04... F just 4th
> > > > 38/27 !_! ~591.65... F# 19-limit tritone
> > > > 364/243 ! ~699.58... G (3/2)*(728/729)
> > > > 128/81 !! ~792.18... G# ditone 81/64 downwards
> > > > 406/243 ! ~888.63... A4=406cps ~ J.S.Bach's 1722 Coethen Cammer-thone
> > > > 16/9 !__! ~996.09... Bb tone 9/8 downwards
> > > > 152/81 !! ~1089.69.. B 19-limit 7th
> > > > 2/1
> > > > !
> Hi again,
> Just one more question. It was my understanding that A.S's scale was based on successive applications of the fifth as 3/2. I was transposing it taking C as 256 instead of 243 and noticed that the series seems to break down from F# onwards. From B as 486 we have 486 x 3/4 = 364.5 instead of 364. I thought perhaps he just rounded it off except from here we have 364 x 3/2 = 273 instead of 272. Now the fifth from 273 is 409.5 whereas A.S has 406. But even if we take 272 x 3/2 we get 408! I'm not sure what I'm missing, some comma shift I should know about etc?

Hi Rick,
re. "successive applications of the fifth as 3/2" (at the risk of telling you things you already know):

1) in a set of 12 notes for a keyboard you can never have more than 11 fifths of 3/2 (702 cents); and if you do that, the last fifth is very flat indeed (678 cents).

2) A.S., in common with most temperament builders, has chosen fewer fifths of 3/2; that way, although he has more impure fifths, at least none is as small as 678 cents. Note however that their main motivation was usually to get closer to 4:5 in the major 3rds, which is why the smaller fifths are near C, and the larger fifths near F#.

3) in A.S.'s scale the 3/2 fifths are on E, B, C#, G#, Eb, Bb, F; in your transposition Eb, Bb, C, G, D, A, E (I think). So the chain breaks at B (I think) - as noted above, it has to break somewhere! Did you notice that your fifth on F is also not pure 3/2; it has to be smaller to give the minor 3rd C:Eb the correct size you wanted - 16:19.

4) Finally, meantones are about "successive applications of the fifth as X" where X is usually <700 cents.

HTH,
Steve M.

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, "martinsj013" <martinsj@...> wrote:
>
> --- In tuning@yahoogroups.com, "rick_ballan" <rick_ballan@> wrote:
> > > Re-cap of AS's scala file:
> > > > > !S_16_17_18_19.scl
> > > > > c# : d : eb : e == 16 : 17 : 18 : 19 == C#1 : D1 : Eb1 : E1
> > > > > !absolute pitches from the middle_c' on:
> > > > > !c'243#256 d'272#288 e'304 f'324#342 g'364#384 a'406#432 b'456
> > > > > !
> > > > > 12
> > > > > !
> > > > > 256/243 ! ~90.22cent C#(cents) the limma
> > > > > 272/243 ! ~195.81... D 17-limit tone
> > > > > 32/27 !_! ~294.13... Eb pythagorean minor-3rd
> > > > > 304/243 ! ~387.74... E (5/4)*(1216/1215)
> > > > > 4/3 !___! ~498.04... F just 4th
> > > > > 38/27 !_! ~591.65... F# 19-limit tritone
> > > > > 364/243 ! ~699.58... G (3/2)*(728/729)
> > > > > 128/81 !! ~792.18... G# ditone 81/64 downwards
> > > > > 406/243 ! ~888.63... A4=406cps ~ J.S.Bach's 1722 Coethen Cammer-thone
> > > > > 16/9 !__! ~996.09... Bb tone 9/8 downwards
> > > > > 152/81 !! ~1089.69.. B 19-limit 7th
> > > > > 2/1
> > > > > !
> > Hi again,
> > Just one more question. It was my understanding that A.S's scale was based on successive applications of the fifth as 3/2. I was transposing it taking C as 256 instead of 243 and noticed that the series seems to break down from F# onwards. From B as 486 we have 486 x 3/4 = 364.5 instead of 364. I thought perhaps he just rounded it off except from here we have 364 x 3/2 = 273 instead of 272. Now the fifth from 273 is 409.5 whereas A.S has 406. But even if we take 272 x 3/2 we get 408! I'm not sure what I'm missing, some comma shift I should know about etc?
>
> Hi Rick,
> re. "successive applications of the fifth as 3/2" (at the risk of telling you things you already know):
>
> 1) in a set of 12 notes for a keyboard you can never have more than 11 fifths of 3/2 (702 cents); and if you do that, the last fifth is very flat indeed (678 cents).
>
> 2) A.S., in common with most temperament builders, has chosen fewer fifths of 3/2; that way, although he has more impure fifths, at least none is as small as 678 cents. Note however that their main motivation was usually to get closer to 4:5 in the major 3rds, which is why the smaller fifths are near C, and the larger fifths near F#.
>
> 3) in A.S.'s scale the 3/2 fifths are on E, B, C#, G#, Eb, Bb, F; in your transposition Eb, Bb, C, G, D, A, E (I think). So the chain breaks at B (I think) - as noted above, it has to break somewhere! Did you notice that your fifth on F is also not pure 3/2; it has to be smaller to give the minor 3rd C:Eb the correct size you wanted - 16:19.
>
> 4) Finally, meantones are about "successive applications of the fifth as X" where X is usually <700 cents.
>
> HTH,
> Steve M.
>
Hi Steve,

Thanks. I completely forgot that 12 fifths (as 3/2) exceeds 128, the twelfth 8ve, so that something had to give. Yes the chain breaks at B in my transposition of A.S's scale. Have to look into it more closely (and re-read Meantone).

Rick.

🔗rick_ballan <rick_ballan@...>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@...> wrote:
>
> If one has a high drone and one forms just intervals below you end up
> with a subharmonic series which outside of equally spaced holes is a
> common non mathematical way in which people would do it.
> [The K comes from my parents]
> --
>
>
> /^_,',',',_ //^ /Kraig Grady_ ^_,',',',_
> Mesotonal Music from:
> _'''''''_ ^North/Western Hemisphere:
> North American Embassy of Anaphoria Island <http://anaphoria.com/>
>
> _'''''''_ ^South/Eastern Hemisphere:
> Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>
>
> ',',',',',',',',',',',',',',',',',',',',',',',',',',',',',
>
> a momentary antenna as i turn to water
> this evaporates - an island once again
>
Hi Kraig,

Sorry for getting back to you so late. I actually missed this post of yours for some reason. Have you got an example of this undertone singing and/or music made on pipes. Its so much easier to comment if I know what we're referring to. (Where is everybody?).

Ah, the K comes from your parents. Thought it was some foreign spelling.

Rick.