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tag:blogger.com,1999:blog-70030669089570860412017-08-16T01:34:16.326+01:00Clamor Vincit OmniaA place to discuss the various ways of organising noises. Some mathematics is assumed.Paul email@example.com:blogger.com,1999:blog-7003066908957086041.post-32488833189283083002017-02-10T12:33:00.000+00:002017-06-30T00:18:32.039+01:00A B♭ in my bonnet
There's no shortage of material about George Russell's "Lydian Chromatic Concept" on the 'tubes, but much of it is geometrically justified by some interpreters and – consequently - may come across as something from the Green Ink Brigade, i.e. a little cranky.
But the geometry seems sound, being as it is just a representational consequence of a harmonic 'reality' – at least no less so than the "cycle of fifths" is based on the fifth's frequency's being (classically, anyway) 3/2 times the root note's frequency (or 27/12 times, if you're well-tempered). As this is aurally (arguably, I suppose, since everything's arguable) the next most obvious interval after the octave's 2/1, its importance in music is well established.

If you allow the standard 12 hour clock (or dodecagon) as both a useful and reasonable model for talking about dodecaphonically partitioned octaves then you're already happy about (by which I mean that you are mathematically and unavoidably led to) using twelve 7 semitone jumps having as much legitimacy as a 'generator' of all 12 notes as is the more direct single 1 semitone stepping up the sequence. That's because 7 (and 5) is relatively prime to 12, just as is 11 (and, trivially, 1).

If you're comfortable (and many are not) with letting the maths repurpose its role from being merely a usefully descriptive modeller to its being a prescriptive constructor of musics, with a constructor's often concomitant value judgements, then the Lydian ends up as 'tops'. It cannot help it!

So if you decide that the (diatonic, seven note) scale you're generating is to begin on the tonic note of that scale (bearing in mind that - modally – you're quite free not to) then starting on (for example) C takes you to G then D then A then E then B then F#, at which point you stop (you've got your 7 notes) and reorder those notes into the (tada – Lydian, not Ionian) scale/mode with that telltale sharpened fourth. And its relative 'minor' is of course three semitones back to starting on the A, with its F# making it a Dorian and not an Aeolian (which would have the F).

Another way of seeing the 'distinguishedness' of the Lydian is to order all 7 diatonic modes alphabetically (which, as it happens, turns out to be numerically) with their halfstep/wholestep descriptions (not their names - that would be silly).

1221222 Locrian

1222122 Phrygian

2122122 Aeolian

2122212 Dorian

2212212 Mixolydian

2212221 Ionian

2221221 Lydian

Which is – effectively – the modes ordered by their 'majorness' starting from the most minorish. And there's the Lydian right at the end of the list, with the Ionian coming in only as the runner-up. Naturally the 'ugly duckling' Locrian brings up the rear (but personally I'm quite fond of that next 'loser', the Phrygian).

Note that these (key independent) semitone-step-determinatives of the diatonic modes are the exact same consequence of the 'generative fifthiness' – there's no new information there - but it's still interesting.

This kind of modelling will work with any sized scale built up from stacked fifths – perhaps the next most familiarly the pentatonic (with its five modes) embedded within 12-note systems.

There'll be a 'most major' ordering (the ordering with all the biggest skips at the beginning of the scale) of an octatonic scale too. It's 22122111, the dominant bebop scale (=Ionian plus an extra – functionally dominant - seventh), since you ask. As to why you'd select that particular octatonic (and its eight - permutationally cycling - modes) pattern of steps (as opposed to - say - 22221111, or 23112111) it's because we're (here) considering only scales constructed with stacked fifths:

The 'Majorest' modes built from stacked fifths, for scales of varying degree

 	Hexatonic
322122

Heptatonic
Lydian mode (diatonic)
2221221		Pentatonic
Major Mode 3
32322

Octatonic
Bebop Dominant
22122111		Tetratonic
5232

Nonatonic
221112111		Tritonic
552

Decatonic
2111211111		Duotonic
Alternating Tonic-Dominant
75

Note that - as is typical with paired n-note and 12-n note scales - the 'majorest' 12-n note scale is one of the modes of the scale constructed from the notes missing from the n-note scale. (Hexatonic scales are, naturally, their own 'anti-scales').

Microtonality

This construction principle will also work with scales embedded within the more exotic world of microtonality. Consider, for example, a scale divided into 17 'equal' (or as near as dammit) divisions. This, by the way, is a real thing. To generate the 'best' (value judgment!) scale/mode from some root note of this scale, you'd ascertain which of the 16 remaining notes was nearest in frequency to 3/2 times the root note. If it's an even-tempered microtonality (it need not be - that's a human choice, not some law of the cosmos) then 210/17 ≈ 1.5034, comes closest to 3/2 (almost as closely as does 'our' 27/12). In other words, the most consonant sounding scales within a 17 note microtonality would be generated from its 'cycle of fifths' based on 10 (as opposed to 7) semitone jumps.

Regardless of 'key', you may generate the whole set of 17 from the sequence 0, 10, 3, 13, 6, 16, 9, 2, 12, 5, 15, 8, 1, 11, 4, 14, 7 (successive remainders of successive multiples of 10 when divided by the 17 - see 'star polygon {17,10}').

Compare that with the cycle of fifths 0, 7, 2, 9, 4, 11, 6, 1, 8, 3, 10, 5, familiar to our dodecaphonic culture (a star polygon {12, 7}).

Again, you might choose to elect some equivalent to a 'diatonic' scale comprising just over half of the available tones from the beginning of that 17 note sequence, i.e. a nine note 'octave'. Then you'd arrange those 9 numbers in ascending order to generate your 'diatonic' scale. It would be 0, 2, 3, 6, 9, 10, 12, 13, 16. Analogously there'd be nine modes, and the most major of those modes would be the one with the largest internal steppings up front.

As the generated stepping is 213312131 (a scale built from three types of skip - 4 semitones, 2 tones and 3 sesquitones!), the alphabetically highest one would be 331213121 (i.e. the scale sequence 0, 3, 6, 7, 9, 10, 13, 14, 16) [see right].

Build up some analogous triads (0, 7, 10) as a 'major chord' in this scale. A 'minor' would correspondingly be (0, 6, 10). Note that both contain the fifth (the 10) and that the minor has a 'flattened third' (a 6 instead of a 7).

This 17 note scale still has room for a separate 4th, close to the fifth for that super major 'Lydian' feel. Furthermore you have two sevenths (like the two thirds) at 13 and 16 - a dominant one and a major one for a leading tone, in the same scale.

Below's a picture of an imaginary heptadecaphonic piano with 17 note scale support. You could play with both thirds (minor and major) and with both kinds of sevenths (minor/dominant and major) without ever leaving the white keys, in its "C-Major" mode. Although we - like Miles Davis - would think of the piano layout as being based on a white-noted F-Lydian, with a middle F.

Paul Sampsonhttps://email@example.com:blogger.com,1999:blog-7003066908957086041.post-1230648340422295702015-04-29T00:29:00.001+01:002015-08-10T16:03:15.515+01:00The Hungarian Major won't be Inverted
Until now we've been working in musical scales, all of which allow melodic inversion. We've looked at the standard western major and minor seven-note (heptatonic) scales, the old Greek or Church modes (Aeolian, Dorian, Mixolydian etc), a couple of pentatonic scales (major and minor) and an octatonic scale or two for jazz, and slightly more unusual heptatonic scales such as the Hungarian Minor and its relatives.

All of those scales will permit you to invert a musical phrase, i.e. a phrase using only the notes of the scale it is written in, where the inversion may be forced to stay in the same scale without having to fudge things. Such inversions are performed by a subtraction from a fixed note (or, as in the case of the two octatonic scales presented, from any of a choice of four) within that scale – and no others.

Furthermore, we've seen how if you choose the 'wrong' (mathematically, not musically – there's nothing 'wrong' in music!) note of the scale to subtract each note of the phrase from, you will fail to get a true inversion (because some notes in the inverted phrase will stray out of the original scale). But, notwithstanding such failures, you will end up in a different scale – one of a family of related scales analogous to the modes of the standard western scale.

For example, we know it's possible to properly, flawlessly, invert a phrase in the Hungarian Minor. Just subtract each note of the phrase from the quasi-supertonic of that scale, which happens to be the business-as-usual major 2nd. Or if you're still unwilling to dispense with the idea of pitch-axis then reflect your phrase along a horizontal line set on the minor 2nd of the scale (a pitch which isn't even in the scale for heaven's sake – that's how silly 'pitch axis' is) – and trudge along, following the melody line and moving your new line in the opposite direction.

We also know that if, instead, we rebelliously subtract a Hungarian Minor melody from its augmented subdominant (a Devil's Interval) our new phrase leaps out of the Hungarian Minor scale and ends up in its sibling scale, the Double Harmonic (aka Gypsy, aka, Byzantine, etc).

So, what about the Hungarian Major scale? Yet another of one of the many heptatonic scales, its 'pitch class signature' is 0, 3, 4, 6, 7, 9, 10. If we insist on using scale degree terminology, its sequence starts (as always) with the tonic, its supertonic is an augmented 2nd, its mediant a major 3rd, its subdominant an augmented 4th, its dominant and submediant are perfectly cromulent 5th and 6th, and its subtonic a minor 7th. Here it is in the key of C, and also represented more abstractly with its (key-independent) polygonal representation. (It's pink, not blue, for a reason).

As usual, to see which inversions work, we will take each of the scale pitch classes in turn and subtract the entire scale sequence from that selected pitch. Thus:

  	 0, 3, 4, 6, 7, 9, 10

 from	 subtracted	 modulo 12	 reordered

0
	 0,-3,-4,-6,-7,-9,-10	 0,9,8,6,5,3,2	 0,2,3,5,6,8,9

3
	 3,0,-1,-3,-4,-6,-7	 3,0,11,9,8,6,5	 0,3,5,6,8,9,11

4
	 4,1,0,-2,-3,-5,-6	 4,1,0,10,9,7,6	 0,1,4,6,7,9,10

6
	 6,3,2,0,-1,-3,-4	 6,3,2,0,11,9,8	 0,2,3,6,8,9,11

7
	 7,4,3,1,0,-2,-3	 7,4,3,1,0,10,9	 0,1,3,4,7,9,10

9
	 9,6,5,3,2,0,-1	 9,6,5,3,2,0,11	 0,2,3,5,6,9,11

10
	 10,7,6,4,3,1,0	 10,7,6,4,3,1,0	 0,1,3,4,6,7,10

The 'reordered' column is there only for information, to show the 'scale' of the transformed notes. It's the 'modulo 12' column which embodies the actual 'note to note transformation'.

By looking down the column headed 'reordered' it's easy to see that none of the scales are Hungarian Major. The one in the second row, beginning "0,3" is indeed the only one which even begins with a match on the first two pitches,

This means that, however hard you try, no musical phrase in the Hungarian Major scale can be inverted along any axis so that every note in the resultant phrase remains in the Hungarian Major scale.

Musically speaking of course, this is no tragedy. A musician need not commit suicide on learning of such an impossibility, If a musical – as opposed to a scale-preserving mathematical - inversion is required, the musician will either allow the notes to go out of scale as they will, or else they will relax the requirement that every transition between successive notes in the 'inversion' exactly match (but in reverse direction) the interval transitions taken by the original phrase, A minor third up for a major third down here, a minor sixth leap for a major sixth drop there, and the like.

Nonetheless, let's just have a quick look at the plesio-inversion represented by the first working row of the above table – the row having us subtracting the scale from 0. I.e. the one which moves 0 to 0, 3 to 9, 4 to 8, 6 to 6, 7 to 5, 9 to 3 and 10 to 2 (the modulo 12 column).

If in the key of C, then C and F# keep their pitches, D# (aka an enharmonic E♭) swaps with A, E is replaced by G#, G by F and B♭ by D. Altogether, only four of the original pitch classes remain and three are swapped out of the scale entirely.

None of the other attempted inversions fare any better. But the sharp-eyed amongst you will have noticed that this second heptagon is just a lateral inversion (in the optical sense, reflected in the vertical axis) of the first one. And it should come as no surprise, therefore, that the scales turning up in the second through seventh rows are simply the other six (we always require a vertex at 0) rotations of this second polygon. So here are those seven polygons, row by row:

 2121213	 3212121	 1321212	 2132121	 1213212	 2121321	 1212132

 0,2,3,5,6,8,9	 0,3,5,6,8,9,11	 0,1,4,6,7,9,10	 0,2,3,6,8,9,11	 0,1,3,4,7,9,10	 0,2,3,5,6,9,11	 0,1,3,4,6,7,10

It would be tempting to say that these seven scales were all 'musical modes' of essentially the same scale and that they just started on a different (tonic) note. But as this author is unaware that any of these seven scales is used anywhere in the world. that might be stretching a point. Rotations of the same polygons they definitely are, Maybe you can find one that sounds pleasing enough to noodle around in? If so, then you might plesio-invert out of it and find yourself in the Hungarian Major (as it were).

It will not have escaped your attention that – corresponding to the Hungarian Major – there will be another six scales, rotations of the Hungarian Major's polygon, and that each one of these is a one for one (vertical axis) reflection of the ones above. Here they all are (the Hungarian Major is the first of them):

 3121212	 1212123	 2121231	 1212312	 2123121	 1231212	 2312121

 0,3,4,6,7,9,10	 0.1.3.4.6.7.9	 2,3,5,6,8,11	 0,1,3,4,6,9,10	 0,2,3,5,8,9,11	 0,1,3,6,7,9,10	 0,2,5,6,8,9,11

Together, these fourteen scales form a nicely complicated, closed, family in which any musical phrase written entirely within one of them can be plesio-inverted into any of the others, As far as I know, the only actual scale in use out of these fourteen is that Hungarian Major. But of course I would be delighted to hear of any others.

The especially eagle-eyed – and non-colour-blind, will note that these polygons are pink. We've been using blue ones in all the previous articles. They're pink because they're representations of uninvertible scales. All our blue polygons represent invertible scales – and they're all symmetric. All of our pink polygons are non-symmetric and occur in families with corresponding mirror image families. There are many more pink polygons than blue ones.
Paul Sampsonhttps://email@example.com:blogger.com,1999:blog-7003066908957086041.post-27601042738187661522015-04-25T21:58:00.001+01:002015-08-10T16:04:26.844+01:00The Invertible Hungarian Minor
We've seen how to invert a piece of music so that it's guaranteed to stay in the same scale (or mode) as the original melody (again assumed to consist exclusively of scale notes). The subject here is melodic, or phrase, or even horizontal inversion. It concerns neither chord nor vertical inversion.

We know how to do this for any of the traditional western classical 7 note diatonic scales in all their modes (Ionian, Aeolian, etc.). We can do it in major and minor pentatonic scales (and their further three but seldom mentioned sibling modes). We can even invert four different ways when the original melody is either of the two common octatonic (aka diminished) scales used in jazz.

But what about other scales, those off the beaten track (to use a phrase very much on the beaten track)?

Let's pretend to pick one at random. Say, the Hungarian Minor. As you may (or may not) know, this is another flavour of 7 note scale which – when rooted on the A above middle C (just so we can stick to as many 'white notes on the piano' as possible) – looks like this:

The top A is of course present just to keep you from becoming too tense with an unresolved sequence. It really is a 7 note scale.

There are a couple of sesquitonic (one and a half tone) leaps larger than whole tones in this scale. In terms of (key independent) pitch numbers, it can be represented as 0, 2, 3, 6, 7, 8, 11. It's pretty easy to see that it's quite similar in shape to the standard Western Minor except that its IV and vii have been sharpened. Compare their heptagonal clock diagrams:

 2131131	 2122122

The one on the left is the Hungarian one, the one on the right is the standard ("all the white notes" if you start on A) minor (or the Aeolian Mode). It may not be easy to spot, but both are symmetric heptagons, The Hungarian's prow points at seven o'clock (an E for an A at twelve o'clock) and the Aeolian's points at five o'clock (a D for its A). The headings are the intervals (number of semitones) between the scale notes clockwise from the top.

Now we already know that the Aeolian mode is invertible (in our strict scale-preserving sense) and that to invert an Aeolian tune, you subtract each note from the Aeolian's subtonic (i.e. the modulo 12 number 10 corresponding to vii, the minor seventh degree, in that scale). So if you're in A minor then the quickest way to an inversion of a phrase is to subtract its notes from G. Which boils down to leaving all the Ds alone, swapping all C with E, all B with F and all G with A and adjust the contour of the melodic line, as you commit it to the staff, by appropriate octave shifts where necessary. This is far quicker than trying to follow the original phrase and move correspondingly down and up the exact same interval for its every up and down, as if you were reflecting with a silly old mirror placed on an imaginary pitch-axis located on the G# line (a note which isn't even in the scale we're working in).

Aeolian Inversion, showing the pitch swaps (including the 'do-nothing' swap
at five o'clock) resulting from subtraction from ten o'clock

And like the Aeolian, the Hungarian Minor's heptagon is symmetric. So we might hope that we can invert a tune in this scale too. However, it should be immediately apparent that there's no 10 in the scale to subtract from – it got sharpened. As before, we can try subtracting each note in the Hungarian Minor (the 0, 2, 3, 6, 7, 8, 11) from each of its scale notes (modulo 12) in turn. If one (or more) of those subtractions results in the same numbers then we're done.

Let's try one. Subtract each of the seven in turn from 3 and we get 3, 1, 0, –3, –4, –5, –8, which in modulo 12 is 3, 1, 0, 9, 8, 7, 4, which rearranged into ascending order is 0, 1, 3. 4, 7, 8, 9. Clearly this is not at all the same scale (the out of scale notes are in bold red). But fear not – there is one answer (and only one – can you see why it's only one?) – and it is by subtraction from 2. Here we go:

2 – 0 = 2 → 2
2 – 2 = 0 → 0
2 – 3 = -1 → 11
2 – 6 = -4 → 8
2 – 7 = -5 → 7
2 – 8 = -6 → 6
2 – 11 = -9 → 3

Here's the heptagon again with the inversion mappings:

Hungarian Minor Inversion, showing the pitch swaps (including the 'do-nothing' swap
at seven o'clock) resulting from subtraction from two o'clock

To demonstrate, let's be slightly more adventurous and pen a quick phrase in D Hungarian Minor. The root note is of course D, the supertonic is E (which is the note we'll eventually be subtracting from), the mediant F, the (augmented) subdominant G#, the dominant A, the submediant B♭ and the (augmented) subtonic C#. As you know, the D minor scale is usually written with the single (i.e. B) flat associated with the major key of F, so we'll go along with that, and use the accidentals at C and G to bring out the phrase's Hungarianism.

To invert the phrase, certain that we're going to stay in the exact same scale, with the exact same tonic note of D, we subtract each of its notes from E (the scale's 2 pitch). This amounts to keeping any A where it is, swapping all B♭ with G#, all F with C#, and all D with E. After adjusting octaves appropriately (we can use this whilst moving from note to note left to right without needing to worry if the interval movements are exact mirrors – they will be. it's guaranteed):

So there you go. Easy, isn't it?

Recall that in the last post we deliberately did a 'wrong' inversion (by subtracting Dorian scale notes from its subtonic instead of the tonic. thus landing in the Phrygian) and showed that for each of the seven traditional western 7 note modes, subtraction from – in turn – each of each scale's scale notes (yes, that's hard to parse) will always result in seven numbers characterising another one of the modes. In other words the collection is self preserving even if the true inversion is only one of them. Is this also true of the Hungarian Minor?

Of course it is. It must be. All subtractions, modulo 12, from each of the seven notes in the scale will result in the same heptagon. It's just that it will be rotated. This is because the heptagon is symmetric. We have in fact already seen one of them by subtraction from the 3, which gave us the scale (0, 1, 3, 4, 7, 8, 9) shown here.

It's just the Hungarian Minor scale with the 'cooker knob' turned one click forward (clockwise) – where the rules of our cooker knob turning require that there's always a vertex (of whatever polygon we have) at twelve o'clock (because obviously we need the root note of the scale to be in the scale). This is a legitimate seven note scale (for who is to say it is not?) which – rooted-on-C-wise – would comprise the notes C, D♭, E♭, E, G, A♭, A. As the little white blob at the four o'clock position indicates, this scale is invertible (in our strict sense) by subtractions from its (quasi)-subdominant.

Note the use of the quasi- there by the way. That four o'clock position is usually occupied by the mediant, but we've already 'used up' the mediant's (albeit minor) slot at three o'clock (because the supertonic looks like it turned up early at one o'clock). These scale degree terms aren't going to scale well (no pun intended) in more general cases. The Pentatonic Major may appear to be simply deficient in the subdominant and subtonic departments, but what are we to do with only seven scale degree terms in an octatonic (or nonatonic) scale? But that's by the way.

As it happens, subtracting Hungarian Minor notes from its (augmented) subdominant (i.e subtracting each of 0, 2, 3, 6, 7, 8, 11 from 6) yields the following rotation - not one, but three big clicks - clockwise:

This scale (0, 1, 4, 5, 7, 8, 11) which, rooted on C, would comprise C, D♭, E, F, G, A♭, B is invertible by subtraction from its tonic. All strictly invertible scales which subtract from their tonics (such as the Dorian mode) have the rather visibly obvious axis of symmetry vertically down the middle of the (perforce) symmetric polygon they live on. They are self-negative (which is subtractions of their integer members from 0) scales. You might think of this scale and the one above as, respectively, the quasi-Dorian and the quasi-Mixolydian (in addition to the quasi-Aeolian orientation of the Hungarian Minor) modes of an alternative system of seven modes.

As it happens, this scale has a name, the Double Harmonic. In fact it has other names. It's also called Gypsy, the Byzantine, the Chahargah, and the Arabian. That's quite a few genres you can now accurately invert in! And use it to try out those double backflip modulations by first inverting (say) out of the Hungarian Minor into (say) the Double Harmonic and once more by further inversion from Double Harmonic back into the Hungarian Minor.
 Minor Hungarian Minor Waltz by LemoUtan Paul Sampsonhttps://email@example.com:blogger.com,1999:blog-7003066908957086041.post-15933582545368294792015-04-21T02:05:00.001+01:002015-08-10T16:04:50.125+01:00Phrygian = subtonic – Dorian
What do we mean by Phrygian = subtonic - Dorian, which you might be tempted to rewrite as Phrygian + Dorian = subtonic? Both look nonsensical. The units are all wrong for a start. It may mean something to 'add' a mode to a mode because they're at least from the same 'object' space. But the term 'subtonic' is in another realm, that of relationships between the notes of a scale, and independent of scale to boot (e.g. the mediant of the A minor scale is C, a minor third above the tonic, whereas the same term is commonly used for the major third above the tonic, as E is to C in the C major scale). It's like adding a green apple to a red one and a kangaroo shows up.

Furthermore, just because things are the same type of object, 'adding' them doesn't necessarily make sense. You can add 10 minutes (a duration object) to 14:34 hrs (a time of day object) to produce a new time (14:44 hrs), or you can subtract a time from a time (e.g. 13:10 hrs from 13:58 hrs) to produce a duration (48 minutes) but what would we mean by adding 17:14 hrs to 11:08 hrs? Note that we're not attempting to add a duration (of 11 hours and 8 minutes) to a time of day – that could make sense (it would take you to 04:02 hrs of the 'next day', if we're pretending we are on Earth and not, say, Mars with its shorter day), we really meant trying to add 'just after 11AM' to 'teatime'. It's nonsensical. (You might temporarily add them, and divide by two, to get a time of day midway between them, but that's part of a bigger operation).

Bear in mind that things that look like numbers might not be numbers. It makes no sense to expect any meaning to result from the addition (or subtraction, multiplication or division) of two credit card 'numbers'. Not even on the way to (say) an average credit card number.

But enough of this. That was all to get you used to the idea that 'addition' or 'subtraction' may – or may not – legitimately be applied sometimes to like-things and sometimes to unlike-things, i.e. not just numbers.

So, let's see what we might mean by our article's title. We know, from our earlier post on reliable inversions, that to phrase-invert a piece of music in the Dorian mode so that it stays Dorian – and thus doesn't need any further adjustments (such as sharpening a non-Dorian note here, flattening another there), we must subtract every note from the tonic.

For example, here's a rather familiar bleeding chunk of Dorianistry (we'll base it in C-Dorian, which looks like the key of B♭ on paper – a two-flats key signature – but its tonic note is C, not B♭):

<audio controls><source src="http://sonic.storistry.com/drunk1.mp3" />unsupported audio element</audio>

The octave tone pattern of the 7 note Dorian mode scale, recall, is 0, 2, 3, 5, 7, 9, 10 where – in this case – those note values are attached to the 'real' notes (in the above) C, D, E♭, F, G, A, B♭ respectively. Subtraction from the tonic means that an inversion will require that every note value in the piece be subtracted from 0 (always the scale representation of the tonic note of any scale whatever). 0-0 is 0 – which means the tonic note will stay put. In the above, that means that every occurrence of the note C goes on being a C. It may shift octave, as the movement of the line dictates, but it's the only note of the scale which remains as it started.

Now the 2 note is subtracted from 0 to become a –2 note, which, when 12 is added to keep it positive (remember we're in the modulo 12 number land of what is sometimes called clock arithmetic) becomes a 10. The 3 is subtracted from 0 to become –3, which is 9. The 5 is subtracted from 0 (or 12) to produce –5 (or 7) etc.

So 0, 2, 3, 5, 7, 9, 10 is inverted to the set 0, 10, 9, 7, 5, 3, 2. Thus wherever you have a D in the original you'll have a B♭ in the inversion – and vice versa. All E♭ and A swap places, as do all F and G. C, as we know, stays put. Recall our dorian knob from our heptagonal representations of the ancient greek (aka church) modes.

The horizontal white lines on the blue heptagon show, for each one of the seven notes in the scale, what its inverted note number is (mentally add outward-pointing arrowheads on the lines, if it helps you see this as a transformation showing how 7 maps to 5 and 5 maps to 7, for example). Each pair connected to the white lines sums to 12 (even the 0 to 0 at the top, because 0 is 12, right?).

The fully inverted piece is shown below. You should maybe check everything about it (the ups and downs corresponding to the inversion's downs and ups, the note mappings as described above, etc). When played, it will probably remind you of "What shall we do with the drunken sailor", but – being an inversion of a piece originally ending on the tonic chord of C-Dorian (which sounds rather like Cm – C E♭ G), it'll now end on C A F. Which is F-major, a slightly weird major-sounding subdominant of a minor-sounding tonic key. But that's the Dorian mode for you:

<audio controls><source src="http://sonic.storistry.com/drunk2.mp3" />unsupported audio element</audio>

Note the complete absence of accidentals. And we didn't need to tweak anything to get this pure, flawless inversion. It happened automatically because we subtracted from the tonic, we didn't reflect in any silly old pitch axis.

Now let's do it wrong

As the title of this article suggests, let's subtract the original Dorian piece instead from its subtonic. The subtonic note of a scale is the one before (below) the tonic, often used in minor keys as an alternative lead in, instead of the dominant, to the final tonic. It's the minor seventh (Roman vii) in such keys (and in the minor-sounding church/greek modes). It's note numbered 10 in our Dorian clock (but it would be VII, sometimes notated as ♮VII, or note number 11 in other, more major sounding modes).

Subtracting each of our Dorian notes 0, 2, 3, 5, 7, 9, 10 from 10 is a lot easier (no pesky 12s to add to keep things positive!) and we get 10, 8, 7, 5, 3, 1, 0.

As expected, the inversion is (technical term) buggered up. The 0, 3, 5, 7 and 10 are OK – they're Dorian notes – but we've got 8 and 1 where we had 9 and 2. Our (Roman) II (or supertonic) has been flattened, as has our (Roman) VI (or submediant).

Let's look at the top line of our original four bars of Dorian. We can usefully get away with considering only this one line because it just happens to contain every one of the Dorian scale notes. Which means that every scale note gets to be 'exercised' by the inversion. Compare it with its (faulty) inversion (top – original 'Drunken Sailor' melody line, bottom – not-quite-completely inverted 'Drunken Sailor'):

<audio controls><source src="http://sonic.storistry.com/drunk3.mp3" />unsupported audio element</audio>

The three points where it has 'gone wrong' are pretty obvious – they're where the accidentals turn up in bar 2, bar 3 and bar 4.

The D♭ in bar 3 came from (in the 'traditional' way of performing pitch inversions) the original step up of a whole tone (from G to A) which required us to move correspondingly a whole tone down from E♭. Thus we've been taken out of the scale.

Similarly in bar 2 of the original we moved up a major third (two whole tones) from B♭ to D, where – in the inversion – we've reached a C (corresponding to the B♭) so requiring a major third step down from that C to the (out-of-scale) A♭.

You may work out for yourself why a traditional reflection gifts us with the accidental A♭ in bar 4.

In 'pitch axis' terms it looks as if the inversion is being done along (i.e. reflected in) the F between the two bottom E and G lines of the treble staff. Indeed F inverts to F (because, of course, 10-5=5).

At this point we may care to examine what actually happened when we subtracted every one of the 'Dorian note values' from the 'Dorian subtonic value' of 10. We ended up with notes which, when reassembled into ascending order, looked like this:

0, 1, 3, 5, 7, 8, 10

Which is itself a seven note scale. Is it one we know? Well, yes it is. We've seen it before as one of the seven possible orientations of our single heptagon, specifically the one which 'points' at the 10 o'clock position – the Phrygian mode:

And that is what we mean by Phrygian = subtonic – Dorian.

We've already had another such statement right under our nose all this time. It is Dorian = tonic – Dorian. In fact this last is but one of the seven from our original list in our attack on the concept of pitch axis, viz

Ionian = mediant – Ionian
Dorian = tonic – Dorian
Phrygian = submediant – Phrygian
Lydian = subdominant – Lydian
Mixolydian = supertonic – Mixolydian
Aeolian = subtonic – Aeolian
Locrian = dominant – Locrian

Of course this kind of 'arithmetic' must be carefully interpreted. The above statements should be more accurately expressed as

Ionian.pitch = Ionian.mediant.pitch – Ionian.pitch

Etc. And our 'contentious arithmetic' example is intended to be read as

Phrygian.pitch = Dorian.subtonic.pitch – Dorian.pitch

On the right hand side of the equals sign we have 'subtonic'. But, scalewise (or modewise), subtonic is a relative term. Although note-like, or pitch-class-like, it does depend upon which key you're in, or more generally within which mode. Sometimes its note value is 10, sometimes 11. Sometimes the mediant note is 4 (e.g. in Ionian) and sometimes it's 3 (as in the Dorian). But once you've chosen your mode, the scale degree is effectively a constant.

You might want to make the case that a scale degree is not a pitch, but an interval – specifically understood to be implicitly from the tonic. But things get a tad recursive if you regard the tonic degree as an interval from the tonic, umm, degree. This may be why musicological terminology is not systematic enough to be scientific.

Whenever one of these context dependent 'scale degrees' (such as mediant, tonic etc) occurs, the context is supplied by the mode you're in. So when we say Ionian = mediant – Ionian, we mean the Ionian mode's mediant note, which is always 4 semitones up from its (always) 0 tonic note. No matter what actual key you're in.

Right at the beginning we gave an example where subtracting a pair of 'time-of-day' objects yielded a 'duration' object. What we are doing here is something similar in that we are subtracting a note from another note within the context of a specific musical mode, i.e. Dorian.subtonic.note – Dorian.note.

Now every musician knows that note subtraction (i.e. the difference between a pair of notes) does not give you a note but an 'interval'. So this arithmetic should seem a bit suspicious, or at least peculiar.

It's only because we're working with numbers, specifically those integers found in modulo 12 arithmetic which are applicable to both pitch classes and interval classes, that we're able to get away with such 'arithmetic'. It's as if we're coercing an object type of note out of an object type of interval. We can get away with this because musical intervals themselves are commonly regarded as interval classes (like pitch classes) insofar as intervals spanning distances larger than an octave are in some way equivalent to intervals within an octave (e.g. a major 9th is 'the same' as a major 2nd, a 13th is the same as a 6th, etc – just keep subtracting those 12s – they're always executed as the same note; they're just in different octaves).

But - regardless of whether or not you're happy with the equivalence of intervals, or the legitimacy of type-overloading from interval to note, it remains an undeniable fact that the bunch of seven numbers you get out of those subtractions of each of your original mode's pitches from your original mode's single distinguished degree note is another bunch of seven numbers which, modulo 12, is indistinguishable from a bunch of seven numbers characterising a mode - which may be the original mode, but more often is a sister mode.

The point of the inversion list above is to invert into the exact same mode where the degree you're subtracting from remains in exactly the same place. But when you use one of these scale degrees in non-inverting transformations the mode you end up in is different, as is perforce its degree.

All traditional modes can in fact be transformed into each of the seven modes (including themselves) by their notes' subtraction from each of their seven degrees. We'll use our (already established) three letter abbreviations for the modes, and the following two letter labels for the degrees

tn tonic
ut supertonic
md mediant
bd subdominant
dm dominant
bm submediant
bt subtonic

The following table can be used to transform pieces from mode to mode. To use it, select your starting mode from the seven in the top row, then look at the left hand column to pick out your desired target mode. Where the row and column intersect you will find the abbreviation for the degree (always relative to your original mode, here your column heading) from which you must subtract each note value of your original piece to produce the corresponding note value in your target mode.

 to\from	 ion	 dor	 phr	 lyd	 mix	 aeo	 loc

 ion	 md	 ut	 tn	 bt	 bm	 dm	 bd

 dor	 ut	 tn	 bt	 bm	 dm	 bd	 md

 phr	 tn	 bt	 bm	 dm	 bd	 md	 ut

 lyd	 bt	 bm	 dm	 bd	 md	 ut	 tn

 mix	 bm	 dm	 bd	 md	 ut	 tn	 bt

 aeo	 dm	 bd	 md	 ut	 tn	 bt	 bm

 loc	 bd	 md	 ut	 tn	 bt	 bm	 dm

more usefully, perhaps, we can use the note numbers of the two letter degree names corresponding to their column heading mode:

 to\from	 ion	 dor	 phr	 lyd	 mix	 aeo	 loc

 ion	 4	 2	 0	 11	 9	 7	 5

 dor	 2	 0	 10	 9	 7	 5	 3

 phr	 0	 10	 8	 7	 5	 3	 1

 lyd	 11	 9	 7	 6	 4	 2	 0

 mix	 9	 7	 5	 4	 2	 0	 10

 aeo	 7	 5	 3	 2	 0	 10	 8

 loc	 5	 3	 1	 0	 10	 8	 6

The bold red diagonals of these tables represent, of course, the mode preserving pure inversion transformations. It may look a bit peculiar (considering we have all numbers 0 to 11 available to us) that only even numbers appear down that diagonal – and that, furthermore, 6 turns up twice. The 6 (the tritone note, yer actual Devil's Interval, in any scale) turns up twice because it's the dominant of the Locrian and the subdominant of the Lydian. So all seven degrees do turn up for inversions (as the first table shows) despite the second table's (misleading) suggestion that only six of them are used.

It should go without saying that these so-called 'failures' of inversion (the ones that are not in bold red, i.e. 42 of them, i.e. 72%, i.e. most) by no means result in bad music. Or that 'real' musicians must justify themselves (if they happen to want to stay in scale – nobody's forcing them to!) when they find they must tweak notes as they invert. Inversion as a musical process has been around much longer than any formalisations introduced by maths.

You might want to investigate these transformations as useful (or at least novel) ways of modulating from one mode to another. By useful, I mean 'no thought required' – you don't have to tweak, you just do the appropriate subtractions.

For example, to move (the threepenny word for modulate) from (say) Lydian to Phrygian, just subtract all of your Lydian notes (turning up in various octaves, as they do, as 0, 2, 4, 6, 7, 9, 11) from 7 (to get 7, 5, 3, 1, 0, –2, –4 which yield 7, 5, 3, 1, 0, 10, 8) which, reordered, are notes in your desired Lydian scale of 0, 1, 3, 5, 7, 8, 10.

Inversions, even if they're not quite 'right' are still close enough to a theme (a sixpenny word for a musical motif) you may have developed as a useful transitioning mechanism to get you into another mode on your way to another key.

Try switching from an Ionian D mode (the traditional 'D major') into some intermediate non-major mode such as Mixolydian – but in another key, say D♭-with a recognisable 'pseudo-inverted' phrase still reminiscent of your motif. From there, do a second 'pseudo inversion' (thus recovering your original motif, or something very close to it) by switching from Mixolydian back to Ionian (use the table to show you this is a 'subtract from 9' this time), but this time in C. Thus performing a double semitone dropping modulation from D major to C major. Try it and see!
 Paul Sampsonhttps://firstname.lastname@example.com:blogger.com,1999:blog-7003066908957086041.post-23380968566499717092015-04-21T00:23:00.001+01:002015-08-10T16:05:17.250+01:00Major minor teatime diner
We've seen in a previous post that a musical phrase inversion in the major scale is guaranteed to remain within the major scale if and only if each note in the phrase is numerically subtracted from the mediant note of the scale. And we've seen that to achieve the same guarantee in the minor scale, the subtraction must be done from its subtonic.

Thus, for example, to invert the following phrase in the key of F#m:

The (blue) numbers underneath the notes are not figured-bass – they're too big and too colourful for that – but are the numerical values of the notes within the scale they are embedded. Thus F# is note number zero (aka the tonic note) in the F#m key, and you will find the subtonic E of that scale (marked in red) ten semitones up.

Since the subtonic of the minor scale acts as the 'inversion rail' (we're deprecating both the phrase and idea of 'pitch axis' and are replacing it with something of utility) from which all notes of the phrase are to be subtracted, we can see that – from left to right – the inverted phrase will begin on the subtonic (10 – 0 = 10), will drop three semitones down to the dominant (10 – 3 = 7), will then rise a semitone to the submediant (10 – 2 = 8) … etc and will end on the mediant (10 – 7 = 3):

Subtraction is simpler than reflection because you need no longer traverse the whole phrase from beginning to end, tracking the changes in direction in your first phrase and making, correspondingly, the exact opposite moves in your new phrase. Instead, you take each note on its own (start anywhere you like, jump around) and simply subtract its value from 10. There's no need to remember where you were, so that you add or subtract some number dependent on that - you simply need to know where you are, and always subtract from 10. If your notes were in a minor key, they'll stay there. Guaranteed.

If you use a clock face to represent all twelve chromatic notes, within which the seven notes of the minor scale are embedded, it's relatively straightforward to see why subtractions from 10 yield inversions which remain in the same minor key.

minor inversions: { 0 ↔ 10, 2 ↔ 8, 3 ↔ 7, 5 ↔ 5 }

The thick white lines represent the inversions. The tonic note 0 is taken to the subtonic 10, and vice versa (10–0=10 and 10–10=0). The supertonic 2 is taken to the submediant 8 (10-2=8, 10-8=2), the mediant 3 to the dominant 7 (and vice versa) and the subdominant 5 (necessarily, shown with a very short white line) remains unaltered.

Now, grasp the big blue heptagonal knob (consider the white lines as being gouged into the polygon) and turn it 3 clicks anticlockwise. After that, you're here:

major inversions: { 0 ↔ 4, 2 ↔ 2, 5 ↔ 11, 7 ↔ 9 }

Congratulations. We have just switched into the major scale.

The tonic note of the scale, 0, is always - necessarily - at the top. The knob is engineered so that one of the heptagonal vertices is pointing at zero, so there are only seven distinct positions and not the twelve you might think.

The subdominant (5) and dominant (7) happen to be in the same place, but the mediant (3), submediant (8) and subtonic (10) have each moved up a notch, to 4, 9 and 11, respectively. Which is of course exactly what you'd expect when switching from minor to major. What were the minor third, minor sixth and minor seventh become the (major) third, (major) sixth and (major) seventh (except you don't usually bother to say 'major'). The three 'flattenings' required to take us from any major key to the minor of the same name are evident here.

It should also be clear (from whichever gouged white line connects the tonic to its inversion rail) why inverting a phrase written in a major scale requires that you subtract each note of the original phrase from 4.

Just to show that the 'knob' view works, we can re-present our original phrase (at the beginning of this post) as one in the key of A major. The only difference is our blue annotations numbers under each note, which will now be the note numbers found in a major key.

phrase now interpreted as being in A

This time we've reddened the 'inversion rail' required for the major scale, which is the scale's mediant (which in the key of A major is the C#). As we know, the mediant in a major key is the III (in roman numeral notation), the major third, and is a 4 semitone interval above the tonic.

These are exactly the same notes as before. Nothing has changed. The phrase is simply being re-viewed as one in a major key (necessarily the one corresponding to its earlier, relative minor, re-shown here for comparison)

phrase originally interpreted as being in F#m

Inverting it, by subtracting every note from the value 4 (and adding 12, wherever necessary to re-notate it as a positive note number) gives us the new phrase:

Compare it with the earlier inversion, where the original was considered as being in a minor key:

Again, exactly the same notes as the inversion before.

At this point, we remember our old Greek modes, and that the modern major and minor scales used to be known as, respectively, the Ionian and Aeolian modes. The following diagram shows all seven (where each vertex of the heptagon gets a turn at pointing to the zero) knobby orientations, each of which corresponds to one of the modes.

 dor	 ion	 loc	 aeo	 mix	 lyd	 phr

Abbreviations should be obvious, but here they are anyway – ion=Ionian, dor=Dorian, phr=Phrygian, lyd=Lydian, mix=Mixolydian, aeo=Aeolian, loc=Locrian.

If you traverse the edges of each of these polygonal orientations clockwise from 0, and write down a 2 for a whole tone skip and a 1 for a semitone skip (almost, but not quite, the lengths of the polygonal edges clockwise from the top), you may represent the seven orientations/modes as dor=2122212, ion=2212221, loc=1221222, aeo=2122122, mix=2212212, lyd=2221221 and phr=1222122.

These are the seven possible permutations of a permutation group, typically notated with parentheses as, for example, (2212221) – where the parentheses contain a string of digits allowed to 'rotate' inside the parentheses by moving the last digit from the end to the beginning, or vice versa. E.g. take the final 1 of the 2212221 inside the parentheses and move it to the first position and you get (1221222). Which represents a rotation of the knob (clockwise) from an ionian to a locrian orientation.

It may be slightly surprising that it's the Dorian mode which most clearly shows the vertical symmetry of the modern seven note diatonic scale, but it must be that way because it's only the inversions of dorian phrases (intended to remain dorian) which subtract from the tonic (or 0). The short white line has to be there.

It's an accident of symmetry, first in that the heptagon is symmetric at all (it need not have been) and second in that there's no law of the universe which says that the 0 direction is 'up'.

So it can surely be no accident that the Ionian (the modern major) and the Aeolian (the modern minor) appear to be set at two o'clock and five o'clock. Representing, as they do, the twin pinnacles of civilised society of maximal satisfaction and self-assuredness just after lunch and the more sober contemplation of the approaching teatime.
 Paul Sampsonhttps://email@example.com:blogger.com,1999:blog-7003066908957086041.post-88872117330086664182015-04-21T00:13:00.000+01:002015-08-10T16:05:43.583+01:00Pitch Axis Considered Harmful
Musical Inversion is not a reflection, it's a subtraction. You may hear about a 'pitch axis' when talking about inversion. This is to justify the (mistaken) idea that – to get a musical (phrase) inversion – you reflect the notes of the phrase along a horizontal line, the so-called pitch axis.

To illustrate, let's start with the following

and 'reflect' it about a supposed pitch axis, let's pick the D of the first note. We get

As it happens, all the notes are in the same key as the original (C major or A minor). The 'essence' of the phrase is preserved.

But suppose instead we had reflected that first phrase along the middle line of the stave, i.e. suppose the pitch axis to be the B. We'd get

There's a B♭ in the second bar because where the original rose a perfect fifth (i.e. seven semitones) from E to B, our inversion must perforce fall a perfect fifth from F to (the out-of-scale) B♭. We've broken something - we've lost our C-majority. But let's not worry about it. Fix (or kludge) it by going to the nearest in-scale note (a B). Or maybe we've changed key. Perhaps to F major? A musician isn't all that bothered.

In fact, in this particular case, the 'pitch axis' of D is the only one that works, to the extent that it keeps you in the original scale. I.e. Inversion may be a scale preserving operation, but in general (in actuality, 6/7ths of the time, i.e. mostly) it's not.

Clearly, if you want to stay in C major when you invert then your starting note must be one of seven C, D, E, F, G, A or B. Mutatis – of course – mutandis for other keys.

Starting on the original D (where the pitch axis is the very line you're already on), or the F above it (where the pitch axis would be the E gap between them), or the B below it (where the pitch axis is the C between them), or the G below (where our first mid-line pitch axis of B was) the first D etc all seem reasonable.

But starting on the C below the D? That would mean the pitch axis is a C#. We know that C# exists - but shouldn't it bother us just a bit that we must allow a pitch axis to be on a note not even in the scale we're trying to preserve? Well, OK, let it go. After all, nobody said a pitch axis must be a note in the scale.

So let's proceed on that basis. As it happens, an inversion starting on C would take us down a minor third to A (reflecting the first minor third transition from D up to F), down again a tone to G (reflecting the original's tone up from F to G), to our two quaver rest, then up a minor third (reflecting the minor third drop from G to E) to – oops, again a B♭.

But hang on a minute. Before we go any further, once we've allowed that a C# exists then there's another problem. Starting on the B below the D meant that we were assuming a pitch axis of C. But in actual fact the pitch axis should surely be (since there are three semitones between that B and D) one and a half semitones up from B, i.e. a quarter-tone lying between C and C#. It's not even in the chromatic scale we grudgingly allowed a pitch axis to inhabit!

This is getting silly. The idea of pitch axis has been stretched beyond reason. It's broken. Throw it away. It just doesn't work.

So what, if anything, does work?

Subtraction does. Always, and reliably and consistently. But you have to know your modes and your moods. To see this, we consider an abstraction of a major scale – in any key whatsoever – lying embedded within a standard western twelve tone musical octave.

Any major scale is represented – in modulo 12 arithmetic – by the seven note sequence of integers 0, 2, 4, 5, 7, 9, 11. Corresponding, for example, to the notes C, D, E, F, G, A, B of the C major scale. Or to the notes E, F#, G#, A, B, C#, D of the E major scale, etc.

The 0 is the tonic note, the 2 represents the two semitones above it supertonic, the 4 the mediant (to continue to use these old musical terminologies) two whole tones or four semitones above the tonic (aka the major third), the 5 the five-semitonally distant subdominant (or fourth), the 7 the dominant (aka the fifth). the 9 the submediant and the 11 the subtonic last semitone below the octave.

So any musical phrase within any chosen major scale may be written (in pitch terms alone, we're not concerned with duration, or with rests) as a sequence of 0s, 2s, 4s, 5s, 7s, 9s and 11s. Note that if you do wish to consider the octaves important just add (or subtract) the relevant 12 – e.g. 14 is still a '2', –13 is still an '11', etc (it's all just 'clock arithmetic'). If you're bothered that chords aren't covered here, that's just a notational problem – write the chord sequence C, Em, Dm7 as [0,4,7] [4,7,11] [2,5,9,12 (or 0)] if you like, or come up with your own scheme – it really doesn't affect anything discussed here.

Inversion as subtraction, not reflection, simply means that you need a note – represented by one of the seven values in your scale – to subtract your sequence from.

For the major scale, the subtraction happens to be from 4 (or – when convenient, after adding 12 - 16). I.e subtracting the seven values 0, 2, 4, 5, 7, 9, 11 respectively from 4 exactly reproduces the original scale – but in a different order (viz. 4, 2, 0, 11, 9, 7, 5).

4 – 0 = 4; 4 – 2 = 2; 4 – 4 = 0; 16 – 5 = 11; 16 – 7 = 9; 16 – 9 = 7; 16 – 11 = 5

That's the only number (modulo 12) that works with the major scale. Try any other, for example 7, and you get 7–0=7 (ok); 7-2=5 (ok); 7-4=3 (not ok!); 7-5=2 (ok); 7-7=0 (ok); 7-9=-2 (equivalent to 19-9=10) (not ok!); 7-11=-4 (equivalent to 19-11=8) (not ok). I.e. subtracting from 7 pulls in the out-of-scale number 10, 8 and 3 (i.e. B♭, A♭ and E♭ in C major) - we've upset the applecart even as far as turning something written in a major key into its own minor. That's a pretty significant change there!.

So our first trial inversion worked – not because we happened to accidentally pick the correct pitch axis of D – but because we happened to subtract each note of the original phrase from the mediant of the major scale.

What about the minor scale? This is represented by the seven member set (0, 2, 3, 5, 7, 8, 10). If you want to invert in the minor scale, you subtract from its subtonic. Again, it's the only one that works, keeping you in your minor scale. For example, to invert an A minor phrase, you subtract each note in the phrase from its subtonic G.

Using our first example phrase considered as one in A minor rather than C major, the sequence D, F, G, E, B, A, C is represented by 5, 8, 10, 7, 2, 0, 3. Subtracting each of those numbers from 10 gives you, respectively, 5, 2, 0, 3, 8, 10, 7 – i.e. all of them within the (abstraction of the) A minor scale.

As with 4 for the major scale, 10 is the only one which works. Subtracting from any of the other six possible numbers will introduce a number (modulo 12) which lies outside the minor scale. E.g. 2–3 brings in a –1, i.e. an 11, which is a 'wrong note'. As does 4–5, or 5–8 = –3, aka 9 – wrong again. Or 7–8, another –1; or 8–2 = 6 (nope, missing in the minor scale).

So to invert in the major, subtract from its mediant (or its major third, if you will). To invert in the minor, subtract from its subtonic - its (minor) seventh.

But wait. Another (but other worldly) name for the major scale is the Ionian Mode. Another name for the minor scale is the Aeolian Mode. These are two of the seven 'ancient' modes corresponding to the seven notes of a diatonic scale, considered as starting on the 'wrong' (for six of them, anyway) note. Each mode has its own distinct subtraction point from which mode/scale preserving inversions can be made. Here's the list

Ionian (0,2,4,5,7,9,11) inverts from its mediant (4)
Dorian (0,2,3,5,7,9,10) inverts from its tonic (0)
Phrygian (0,1,3,5,7,8,10) inverts from its submediant (8)
Lydian (0,2,4,6,7,9,11) inverts from its subdominant (6)
Mixolydian (0,2,4,5,7,9,10) inverts from its supertonic (2)
Aeolian (0,2,3,5,7,8,10) inverts from its subtonic (10)
Locrian (0,1,3,5,6,8,10) inverts from its dominant (6)

 ionian
(major)	 dorian	 phrygian	 lydian	 mixolydian	 aeolian
(minor)	 locrian

 Seven modal scales as rotations of a heptagon

Again, we emphasise that the modal numbers that you subtract from (add as many 12s as you require) are the only ones – out of the seven possibilities for each – which will preserve the mode. The 'pitch axis' – or reflection – idea is quite unnecessary. We'll see more of these polygons in the next article.

Inversion as subtraction carries over into other scales too. For example, Jazz musicians are rather partial to the octatonic (aka diminished) scale which – as its name suggests, is an eight note scale (but still embedded within the chromatic 12-tone universe). Its alternating wholetone/semitone stepping nature can be represented as the eight note set 0, 2, 3, 5, 6, 8, 9, 11 – or alternatively 0, 1, 3, 4, 6, 7, 9, 10. To invert a phrase written in this scale, you have a bit more freedom than that which the traditional heptatonic modes permit you. You may subtract from any one of four of its scale notes, i.e. from 2 or 5 or 8 or 11 (or 1, 4, 7, 10 in the alternate version). The term tonic may still apply to such a scale, but the notions of dominant, supertonic etc won't fit any more – we're one short.

 octatonic	 diminished

 Octatonic Scales with quadruple inversion points

And pentatonic scales also have inversions. The (five note) major pentatonic scale is represented by 0, 2, 4, 7, 9. To invert a phrase in this scale, subtract its notes from 4 (just like the ordinary major scale). 4-0=4, 4-2=2, 4-4=0, 4(=16)-7=9, 4(=16)-9=7. Subtraction from the other notes (0, 2, 7 or 9) won't work. The minor pentatonic – i.e. 0, 3, 5, 7, 10 – like the ordinary 7 note minor scale – is invertible from 10. I.e. 10-0=10, 10-3=7, 10-5=5, 10-7=3, 10-10=0. Like the ordinary heptatonic/diatonic/modal scales, these inversion points are unique.

And distinct – corresponding to the seven modes of the diatonic scales are the five 'modes' of pentatonic ones. Being unaware of any formal names for these, we'll just go by analogy:

io-pentatonic (aka major) (0,2,4,7,9) inverts from (4)
do-pentatonic (0,2,5,7,10) inverts from (0)
ph-pentatonic (0,3,5,8,10) inverts from (8)
mi-pentatonic (0,2,5,7,9) inverts from (2)
ae-pentatonic (aka minor) (0,3,5,7,10) inverts from (10)

 io
(major)	 do	 ph	 mi	 ae
(minor)

 Five Invertible Pentatonic Scales as rotated pentagons

These are just a few of the scales which can be fashioned from the 12 notes available to us. One may consider scales consisting of all 12 notes – there's only the one, the full chromatic scale. Inversion in that scale is of course possible from any of its notes since subtraction from anything is bound to produce one of the 12 notes you have. Scales of 11 notes – of which there are of course 12 actual (each with one of the 12 notes missing) – but only one in practice (they're all just transposed versions of each other, i.e. all represented by the same set 0,1,2,3,4,5,6,7,8,9,10) must have limited inversion (you will be unable to subtract from 11, because 11-0=11, which isn't in the scale). Symmetrical with 11 note scales are the 12 actual (but only one really) single note scales which can – pretty clearly – invert only by subtraction from their only note – 'One note Samba' considered trivial. There's also only the one zero-note scale. but the only piece written in that may be Cage's 4'33" assuming it's written in any scale at all.

There are – technically – 60 scales comprising 10 notes (and, symmetrically, 2 notes) but – again in practice – many of them are merely transpositions of each other and there are really only six distinct patterns of 10 (or 2) note scales (each may be considered to have 10 – or 2 - 'modes' starting on different notes).

Of the 171 possible 9 note scales there are 19 distinct patterns – each with something akin to 9 modes, and this is also (symmetrically) the case for the 19 possible patterns for 3 note scales (with each of their 3 'modes').

The 'jazz octatonic' isn't the only way to arrange an 8 note scale. There are 43 different patterns available. Likewise 4 note scales.

There are in fact 66 different ways to have 7 note scales. The two (or seven) we are familiar with represent just one of these 66. Being - as they are - only transpositions of each other in their various modes. Most of them are rather pathological. For example a run of seven consecutive semitones might not provide a very interesting scale. Similarly there are 66 pentatonic patterns.

Finally there are 80 distinct hexatonic scale patterns, only one of them being the - possibly familiar – whole tone scale. The whole tone scale in particular is – like the full 12 tone scale – invertible from any of its 6 constituent scale notes.

By no means do all of these (350) scales permit in-scale-phrasal-inversion. And in those that do, reflection – or pitch axis – is of no real use in any of them except by accident.

Here's a table of all possible N note scales, and how many of them are properly invertible

 #Notes	 #Scales	 #Invertible

 12	 1	 1

 11	 1	 1

 10	 6	 6

 9	 19	 5

 8	 43	 15

 7	 66	 10

 6	 80	 20

 5	 66	 10

 4	 43	 15

 3	 19	 5

 2	 6	 6

 1	 1	 1

That there are exactly as many, say, 4 note scales as there are 8 note scales should be reasonably obvious. A scale with N notes missing from it can be considered the anti-scale of the corresponding N note one and although musically it will sound rather different (except possibly when N is 6), there'll be exactly one such scale to correspond to its N note scale. We've omitted the no-note scale (the anti-12-tone) as being too musically challenging (Cage again).

Diagram showing that invertible scales (blue) are in the minority
 Paul Sampsonhttps://firstname.lastname@example.com


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The exodus story is America's story. Moses is our real founding father. In this groundbreaking book, New York Times best-selling author Bruce Feiler travels through touchstones in American history and traces the biblical prophet's influence from the Mayflower through today.Feiler visits the island where the pilgrims spent their first Sabbath, climbs the bell tower where the Liberty Bell was inscribed with a quote from Moses, retraces the Underground Railroad where "Go Down, Moses" was the national anthem of slaves, and dons the robe Charlton Heston wore in The Ten Commandments.One part adventure story, one part literary detective story, one part exploration of faith in contemporary life, America's Prophet takes readers from Gettysburg to Selma, the Silver Screen to the Oval Office, to understand how Moses shaped the nation's character. America's Prophet is a thrilling, original work of history that will forever change how we view America, our faith, and our future.More
The author narrates his "narrative of hope" with clarity and authority. Using well-documented research, he illustrates why African-American slaves clung to the Bible, in particular the Exodus narrative and Moses's story, as examples of hope. More journalist than historian, Feiler brings to life four hundred years of American history with what he calls "touchstone" narratives. He describes visiting the island where America's Founding Fathers experienced their first Sabbath, then the Liberty Bell, inscribed with the words of Moses. His depictions of slave trade stories, the Underground Railroad, and the inspirational singing that was part of slave culture - in particular the song "Go Down, Moses" - are emotional and believable.
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Give NOCTURN's fun style of contemporary dance a whirl this autumn. Classes starting back this September.
Tuesday night's...
New Greenham Arts, Newbury 7.30 - 9.00pm - starts 13 Sep, 10 wk course, drop in rate available.
Book online at Corn Exchange, Newbury, by phone 0845 5218 218 or in person at the Corn Exchange Box Office.
Thursday night's....
South Hill Park, Bracknell - starts 22 Sep, 10 wk course, booking in advance only.
▪ Beginners: 6 - 7.15pm
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Joint Class Deal: Join a Contemporary Dance course and the ILLUME Choreographic Group at SHP and get one course half price. You can only book this offer by phone on 01344 484123 or in person at South Hill Park.
Book online at South Hill Park, Bracknell, by phone (0)1344 484 123 or at the Box Office.
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John Darvell's Berkshire based company NOCTURN creates & delivers experimental contemporary narrative performances based on
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Using a mixture of dance & accessible technology, they engage audiences in interactive artistic events that they may experience accidentally, either in an installation setting, or online.
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G20 focuses on cooperation with Africa
On February 16 and 17, the G20 Foreign Ministers will convene for a meeting in Bonn organised by Germany's G20 Presidency. The meeting will focus on the 2030 Agenda for Sustainable Development, crisis prevention and peace policy, and cooperation with Africa.
Enlarge image German Foreign Minister Sigmar (© Thomas Koehler/photothek.net) First major event hosted by Germany's new Foreign Minister
Besides 20 Foreign Ministers international organisations will be represented at the meeting, too. "Germany will be a forum for global politics in the coming days," German Foreign Minister Sigmar Gabriel said about the upcoming meeting, which will also be his first major event as German Foreign Minister. Many of the participants will travel directly from Bonn to the annual Munich Security Conference.
"Shaping the global order – foreign policy beyond crisis management" will be the overarching theme of the meeting in Bonn. Recently, foreign policy has been preoccupied with addressing acute crises and conflicts. This informal meeting, however, is intended to examine long-term objectives and the ways in which foreign policy can tackle key global challenges. "Foreign policy is, and must be, more than crisis management," Gabriel said prior to the meeting.
Cooperation within the framework of the G20
Enlarge image The city of Bonn will host the G20 Foreign Ministers Meeting (© dpa) The German Foreign Minister pointed out that no country in the world can tackle the major international problems of our time alone. He went on to emphasise that terrorism, water scarcity, climate change, forced migration and humanitarian emergencies cannot be resolved by isolation, but only through cooperation and joint strategies. That is, he said, what makes the G20 Summit so important. Together, the G20 nations account for almost two-thirds of the global population, more than four-fifths of gross world product and three-quarters of global trade. The world's largest economies that are represented in the G20 bear special responsibility and must jointly address the root causes of conflicts, as well as the options for peaceful crisis prevention and conflict resolution, Gabriel went on to say.
Meeting will focus on the 2030 Agenda, peace and cooperation with Africa
The meeting will therefore address implementation of the UN's 2030 Agenda and Sustainable Development Goals. The Agenda was adopted at a UN summit in September 2015 by all UN Member States. Civil society, too, played a key role in drawing up the Agenda. With its 17 goals, the Agenda promotes sustainability in the social, environmental and economic spheres. The G20 want to become a key driving force for implementation of this global project.
Enlarge image AU soldiers in Sudan (2005) (© picture-alliance/dpa) Another focus is the long-term sustainment of peace in the world today. Prevention and forward-looking foreign policy will be discussed in this regard.
A third focus of the meeting is cooperation with Africa. Together with an African Union representative, the G20 will discuss what general conditions are needed to combat poverty, strengthen state institutions and ensure that more is done to harness the potential of many African states.
Shaping an interconnected world
The meeting of Foreign Ministers is being organised by Germany in connection with its G20 Presidency. The major event of this year's Presidency will be the summit meeting of G20 Heads of State and Government in Hamburg on July 7 and 8.
The motto of Germany's G20 Presidency is "shaping an interconnected world", and discussions centre around "building resilience", "improving sustainability" and "assuming responsibility".


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|October 17 - October 23, 1996
Grassroots GoliathThe Pima County Interfaith Council, a champion of the poor and middle classes, takes its fight for a better Tucson to a whole new level.
By Dan Huff and Dave Devine
MailbagHowls of outrage and squeals of delight from our astonished fans.
SmithRelax, Little Susie, have some Midol, and I'll show you mine if you show me yours.
By Jeff Smith
The SkinnyFife's latest tax fantasy...Republican candidates are friends of Barney Fife?...Why County Recorder F. Anne Rodriguez is now the Lady of the Lake...And more!
Money MattersLooks like Vicki Cox-Golder's Growth Lobby supporters are hauling out their wallets for a last-minute assault.
By Jim Nintzel
The KamikazesThree Republicans launch a doomed campaign in Legislative District 11.
By J.E. Relly
Currents ForumHere's your place to vent, Tucson webheads.
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Gap is having a three day sale. Perfect time to stock up on some shorts and summer tees. Starting today, 5/21-5/22, shop Gap.com and get 25% off your purchase. Shop 5/23 and get 20% off your purchase. Use code GAPLOVE at checkout. Gap.com offers everyday free shipping on orders over $50.
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This past February, I had the opportunity to produce an event in Los Angeles for a client. I was pretty excited because I'd never been to LA before, but I quickly found out it's quite a terrible city and I maybe shouldn't have been so excited.
Well, that's a bit of an exaggeration. Flying in, I was instantly disgusted by the sprawl of the city. Upon landing, I was concerned by the color of the air. Upon making my way through LAX, I was stressed out by the frenzy of the airport. While driving to my hotel in an Uber, the traffic was nightmarish. My attempts to see my friends were thwarted by conflicting schedules and the fact that Beverly Hills is apparently really far away from everything else and I didn't have my own vehicle. In the end, I was confined to staying in 90210 for four days at which point I started to feel out of touch with reality. But in my attempt to make the best of it, here's a list of the best parts of Beverly Hills:
The Beverly Wilshire
The event was taking place in the Beverly Wilshire Hotel (where Pretty Woman was filmed!), and I'd been sent down a few days early so that I could understand the lay of the land and establish in-person relationships with the hotel staff.
Now here's the thing: I've stayed in nice hotels before! Or so I thought, because as it turns out, none of my hotel experiences have even come close to the level of customer service provided by the Beverly Wilshire staff. From my second that my Uber driver opened my car door, I was treated like royalty. I'd arrived before check-in so I had an hour or so to kill before my room was ready. The concierge stored by bag, handed me a bottle of Evian and a wifi passcode and sent me on my way to the pool, where I basked in the sun and worked until my room was ready – my room keys were brought to me, poolside, in a black velvet pouch.
It turns out that I had been assigned a suite (event coordinators always get the best rooms!) so I had a giant marble bathroom, a living room and a giant bedroom with a king-sized bed and a double-balcony that looked directly onto Rodeo Drive. A teeny tiny view of the Hollywood sign could be seen in the distance.
Every single time I left my hotel room – even if it was just for an hour – somehow the housekeeping staff would know it and go into my room to tidy up after me. A casually discarded piece of clothing would be folded up. My makeup would be neatly arranged on a little mat on the bathroom counter. The toilet paper would be folded into a triangle. At night, my bed would be turned down and a towel would be placed next to the bed to step onto before getting into bed. Or maybe it was where you were supposed to kneel before you said your prayers. That's still unclear.
My only celebrity sightings the entire time I was there were some of the Houston Rockets who had lunch in the room next to the luncheon we hosted for our group, but I didn't know who any of them wore so it doesn't really count.
Rodeo Drive
Rodeo Drive is only like, three blocks long. Does everyone know that? I didn't. I DID know that it was full of stores that I will likely never be able to afford to set foot in, let alone shop at, so I was content to stroll along and window-shop and people-watch. I didn't see any celebrities but I was extremely confused by the people who I DID see shopping. Who were they!? How do they have so much money? What kind of life do they lead that allows them to wake up in the morning and think, "I'm going to go shop on Rodeo Drive on a Monday morning?" So strange.
Olive & June
I hadn't done much research about Beverly Hills prior to my arrival, but I'd seen a photo of the nail polish wall at Olive & June on Cupcakes & Cashmere. I needed a manicure and was excited to see that it was within four blocks of the hotel.
I would have preferred a less timid manicurist, but I loved the ambiance, and I liked chatting with the people around me. I also loved the intake process – where you write down all of your preferences – nail shape, amount of pressure preferred for hand massages, etc.
My mani turned out fantastic but next time I'd be sure to get a nail wrap with a gel polish on top. Wish I'd seen those sets earlier.
I cannot resist a good candy shop, and I found Sugarfina when I browsing things to go see that were within the 5 block bubble that I was up stuck in because everything in LA is too expensive to get to if you're relying on Uber. Sugarfina is a gourmet candy shop with all SORTS of delightful things like Champagne Gummy Bears, Peach Bellini Gummies, Tahitian Vanilla Malt Balls, White Chocolate almonds disguised to look like martini olives, etc. And because I also can't resist good packaging, I came home with a Sugarfina Faves Bento Box which I've been grazing from for the last three weeks.
Chaumont Bakery & Cafe
Unable to comprehend the astronomical prices associated with breakfast items at the hotel, I set out to find a more affordable option and fortunately, Chaumont Bakery & Cafe exists just around the corner from the Beverly Wilshire. I breakfasted three out of four mornings at this charming French bakery, each time spending about 1/3 of what it would have cost for room service or to eat at the hotel restaurant.
I ordered the smoked salmon scrambled eggs every time and even though I couldn't really justify ordering one of the mouthwatering desserts in the early hours of the morning, I definitely drooled over them.
I'd passed Verameat a few times on my walks through the neighborhood but they were never open when I needed them to be, so I mustered up what was left of my energy after the event wrapped and headed over to actually SHOP instead of just window shop. This creepy thing was what made my head turn in the first place:
The sales team was so sweet and took me around to try on just about every single piece that I wanted to – encouraging me to try some of the more delicate midi-rings and to experiment with stacking (neither of which I can pull off, as it turns out).
I'd hoped that there would be a lion-themed piece of jewelry but instead I walked out with this sweet vampire ring instead:
So that just about wraps up the highlights. I didn't really get to experience Los Angeles outside of Beverly Hills so part of me thinks that I missed out on it – and the other part of me thinks, "why would you ever live somewhere where most of your time is spent in your car, trying to get somewhere else?". So while there's plenty more to explore, there are plenty of other places I'd rather go first!


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Day 2: domestic flight to Axum. Axum is been capital city of Abyssinia former name of Ethiopia so this ancient town has several historical attractions such as obelisks of Axum they are a lot of obelisks but the dominants are three those implies sample of multi story building those obelisks are build in 12 century UNESCO also gives recognition ,also the church of st. marry or Zion where the original Ark of the convenient is kept secretly. This Kingdom is dominant, also its territory was wide to Middle East and its own coin.
Day 3: domestic flight to Lalibela. Lalibela is magnificent rock-hewn church of 14 century, which are carved out of living volcanic tuff with amazing architectural skill, this churches erected by king Lalibela his ambition was to make this place holly land of Christianity so there are about seven surrounding churches both of them have their own specialties by architectural but they are same been erected from one single bedrock it is amazing place that you should watch in your life that the world has never seen yet elsewhere.
Day 4: Domestic flight to Gonder. This town is founded by King Fasil also this town is been used as capital of Ethiopia the dominant attraction of this town is The Fasil palace founded by king Fasil in 16 century, this palace have amazing architectural design it is two story battlemented structure its inner walls were decorated with ivory, mirrors and paintings of palm trees and its ceiling was covered with gold-leaf and precious stones it surrounded by 900-m-long palisade inside this garden there are a lot of palaces including king Fasil children's palaces and other kings palace both palaces have there attractions also in this garden there is rectangular swimming pool which was supplied by canal from the near by river.
Days 5-6: drive to Semen Mountain National Park. This park is the world heritage in 1978 due to the park amazing land escape going up to 4,500 meter mountain called Ras Dashen the fourth highest in Africa also endemic animals such as gelada baboons and walia Ibex and endemic birds Thick-billed raven,white-collared pigeon, Black-headed siskin, wattled Ibis ,white-billed starling and so much more this place is vantage if you like to camp and trekking.
Day 7: drive to Bahirdar. Bahirdar is one of Ethiopia tourist destination many tourists didn't want to leave this town because the town has beautiful well designed ,friendly people,the weather is good and have the wide lake called "Tana " this lake depth is 14 m its area is about 3673 km2 it is the hub of world's longest river blue Nile.
Day 8: watching the Islands monasteries and churches. This trip made by boat trip there are 30 ancient churches and monasteries in the lake "Tana"islands this all are made before 14 century re painted by 17 century both of them fund in different island the journey is more interesting when you watch different topography of the islands.
Day 9: flight back to Addis Ababa. After having rest and eating lunch we go to small town called "Debrezit" this town is found only 40 km from Addis it is a small town but got have potential of natural attraction there are seven lakes this lakes formed due to volcanic this area is one part of the great East African riftvally area you can swim in the beautiful lakes watching amazing nature and bird watching you will stay the night at beautiful beach resort watching the sun set.
Day 10: back to Addis. Late in the morning having sun shower with birds singing and eating break fast we will back to Addis in 45 min Drive then you will watch the rest of Addis Ababa then shopping of cultural dresses for the memo and taking you to Bole international air port.
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If you're looking for a bus from Ebenthal in Kaernten to Kronach, you can pick between several options because the route of your choice can be found in the timetables of . Usually there's tickets for just a few pounds available. However, if you book at short notice or in times of unusually high demand, tickets might be slightly more expensive. It's up to you: Just click the "Find Now" button and find all available buses!
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- Deep-carry pocketclip
- SpeedSafe assisted opening
- Protective black oxide coating
With its blade length of just 2 inches, the new Kershaw Spoke is easy to take along with you just about anywhere. But even with such a compact blade, this knife is plenty useful. It slices through tasks quickly and easily because the Spoke's big belly provides plenty of cutting surface—even though the overall blade length is shorter.
The knife opens quickly and easily with SpeedSafe assisted opening and the built-in flipper. Slightly oversized, the flipper doubles as a finger guard when the blade is opened. The steel handle features an unusual textured pattern that also provides a solid grip on this smaller knife. Black-oxide coating gives it a great look and a sturdy liner lock secures the blade open during use.
The Spoke has a reversible, deep-carry pocketclip and a black-oxide coated blade stop.
Technical Information
Folding or Fixed Blade: Folding
Blade Steel: 4Cr14
Handle Material: Stainless steel
Blade Shape: Drop point
Serrated: No
Pocket Clip: Yes
Carry Options: Reversible; tip-up
Locking Mechanism: Liner lock
Blade Finish: Black Oxide
Rockwell Hardness: .1"
Opening Method: Flipper and Dual Thumbstud
Blade length: 2"
Blade Thickness: Not provided by manufacturer
Closed Length: 3.5"
Overall Length: 4.75"
Weight: 2.8 oz


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To those of you who did not attend the ceremony of the "Living Treasure Awards" on Wednesday, May 30, at Saint Francis, let me give you a recap. First of all, there probably would not have been any room for you. What a crowd! I was honored to have my all family present.
Our hostess, Gail Kuchavik, was super. She is charming and ever so gracious. She explained the huge area from where the contestants were able to apply for this award, and I was amazed to hear that it was just about all of Morris County. This award has been given for the past eight years.
Gail introduced Donna Costello our town clerk, who spoke about the four contestants who were Denvillites, and who won the award. We thank you Donna for saying some very wonderful things about us all. I thank God you don't play golf Donna! Otherwise you too would have been entered into the golf tournament, as our town officials were and then you would not have been able to attend the services. Donna announced that on June 19, on a Tuesday night, at the council meeting, we, the new "living treasures" will be honored by the mayor, council and the administration. I think that is super!
So few people have the opportunity to be alive and to hear their own eulogy. I was one of the very few who did so by listening to all the compliments that were said about me when I received the award. Gail, our hostess, said some wonderful things. She then read the letter my daughter had written to the judges when she entered me into the contest. Wow!!! And the icing on the cake was when I learned that Rena Rupple met with my daughter and gave her more information about my doings and informed her that the Denville Seniors had also endorsed me!!! My heart was pounding and the tears were trying to gush out. I tried to act composed and doubt if I fooled anyone. Then Walter Taylor spoke and said nice things too. As I said, I felt like I was eavesdropping from the hereafter. Thank you all.
I am sure that Gloria Toohey, John McConnell, and Henry Patterson, all from Denville, who also had their families present, had heard their eulogies too. Each and every one has lived his life, doing and trying to bring happiness to all. They were always there to help others, and to be a friend to all.
Gloria Toohey has brought joy and happiness to all the children as a Professional clown; to those in Nursing homes; Hospital; or at parties. For the Seniors, she has organized a monthly get together for chatter, dancing,or just talking. She manages to get food donated for this event. She has brought a place for seniors to go on a lonely Sunday afternoon. And let's not forget the task as President of the "Silver Brigade." That is one job we all that will be working on till we succeed.
And as for Joe, a charming friend, a perfect gentleman, a survivor of Pearl Harbor, lifetime volunteer with the Denville Fire Department and a sriver for "Meal on Wheels."
Then there was Henry Pattersen a WW2 Navy man; boy scout and cub scout master; a member of the civil patrol for the emergency system. He built furniture for our Denville Library and produced the film which he freely exhibits to all, called "God Bless America" which shows the beauty of our country.
Also honored was a wonderful man from Boonton. He is Alfonse Scerbo, a dedicated person, former Morris County Clerk, a past mayor of Boonton, a member and officer of all veterans organizations and a champion for all veterans needs.
Last but not least, a charming woman, Lois Nichols from Chester who is a very dedicated senior and works with and for the seniors in her town. She is always volunteering her services to her Community.
With all the good things that have occured in the past few days, a very sad event occured. Our friend, our president, Charles {Chuck} Bothner died suddenly Saturday morning, May 26. He had presided at the general meting only on the day before. He will be greatly missed by all. Chuck was our "fix it" man. He was always repairing our microphone, or making signs for all our events, taking on jobs. He was always volunteering and was a perfect gentleman. Our hearts go out to his wonderful wife and my friend Rose Bothner.
There weren't any meetings last week. So you will have to wait till next week for news.
At present I am trying to get myself together because on Monday, June 4 a group of us are taking off for Niagara Falls in Canada very early in the morning. I usually am prepared three days before, but with all the activities that have occured this past week I just could not do anything. One thing I must do is buy me some hosiery. I am truly looking forward to this trip. The last time I went, was when my gals were little and that seems like an eternity ago. Iill report on our activities when I get back.
June 28 off to Monmouth Race Track,as a guest of the establishment.
July 12, bus to Beach Haven, we take the Black Whale cruise at 10 a.m. to Trump Marina where we will spend time; then at 4:30 p.m., cruise back. Your cost is $37 and you get $15 comp. On July 24, it's our mystery trip;
Aug. 9 is Lake Wallenpaupack. We get a train ride plus dinner and a show. The cost is $39.
August 23 our monthly trip to AC.
On Sept 20, we are off to Tuxedo, N.Y. for a luncheon and show at a cost of $35.
This is the last chance to join the trip to Las Vegas. Book now or forget it. We have booked the plane and they will only be able to accommodate maybe two more people if they are booked now.


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--3B Evan Longoria (strained left oblique) went on the 15-day disabled list April 3. He might go on a brief rehab assignment in late April before returning to the Rays.
--LHP J.P. Howell (left shoulder surgery in May 2010) went on the 15-day disabled list retroactive to March 22. He might start a rehab assignment in late April, with a target of returning to the Rays in mid-May.
LHP David Price
RHP James Shields
RHP Wade Davis
RHP Jeff Niemann
RHP Jeremy Hellickson
RHP Kyle Farnsworth (closer)
RHP Joel Peralta
RHP Adam Russell
RHP Andy Sonnanstine
LHP Jake McGee
LHP Cesar Ramos
RHP Juan Cruz
John Jaso
Kelly Shoppach
1B Casey Kotchman
2B Ben Zobrist
SS Reid Brignac
3B Felipe Lopez/Sean Rodriguez
DH Dan Johnson
INF/OF Elliot Johnson
LF Johnny Damon
CF B.J. Upton
RF Matt Joyce
OF Sam Fuld
Copyright (C) 2011 The Sports Xchange. All Rights Reserved.


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Question: Discuss the difficulties involved in attempting to choose an optimal
Discuss the difficulties involved in attempting to choose an optimal channel structure.
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View Solution:
Answer to relevant QuestionsCompare and contrast the so-called judgmental heuristic approach to choosing a channel structure with any of the other approaches discussed in the chapter. How do distribution tasks viewed from the perspective of the channel manager in a producing or manufacturing firm differ from the traditional lists of marketing functions? Marketing channels should be designed to make products and services conveniently available to customers, how, when, and where they want them. This is exactly what several franchises such as Cousins Submarines Inc., Tasti ...What is a potential problem with the use of the sales force as a means for finding prospective channel members? Master Lock is perhaps the nation's best-known name in padlocks. Master Lock padlocks are sold intensively at the retail level through a wide range of stores, including hardware stores, home centers, automotive stores, ...
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Awards & Recognition
The Iowa Engineering Society presents the following awards.
Members and Chapters are encouraged to submit nominations to the IES state office at 515-284-7055 or email@example.com by February 28th of each year.
Please identify each individual along with their e-mail address & phone, and why you are nominating them for the award you specify.
ANSON MARSTON AWARD is awarded in recognition of outstanding service to the Iowa Engineering Society.
DISTINGUISHED SERVICE AWARD is awarded in recognition of outstanding service to the engineering profession.
HONORARY LIFE MEMBERSHIP AWARD is awarded in recognition of high professional attainment and exceptional service to IES.
VOICE OF THE ENGINEER AWARD is awarded in recognition of significant contributions to the enhancement of the image of the engineering profession, through involvement with the general public or targeted publics. These efforts should bring a broader recognition or understanding to the contributions or accomplishments of engineers.
YOUNG ENGINEER AWARD (previously the John Dunlap-Sherman Woodward Award) is awarded in recognition of outstanding technical or professional achievement reached before the age of 35.
CLICK HERE for Online Application
IES has a NEW website!
We are thrilled to announce that the IES website has been updated to a more dynamic system that will help with the work of keeping our membership connected. With social networking support, simple tools for staying in touch, space for sharing documents and photos and resources to support Chapters and committees, we can use it to communicate among members and highlight great work being done in our industry. In addition, www.iaengr.org is going to be the workhorse that will collect member dues and support event registrations and payments.
We hope you enjoy the new site!
Thu Aug 24, 20172017 Transportation Conference
Category: Other Relative Industry Events