Probably the most overlooked feature of the Transmuter, let's think about what this actually works with.
Here's the key language of Empowered Transmutation (Not the full text, but it is all the text that governs what spells are affected and how): "When you use a spell slot to cast a Transmutation spell that doesn't make an attack roll or force a saving throw... you can increase the spell's effective level by 1."
There's a couple things this does. First, for all affected spells, it increases the spell level. That matters for things like Dispel Magic. This is trivial, but not irrelevant. I will flag such spells as (trivial). Second, for affected spells which have a defined up-cast, you of course get the 'At Higher Levels' benefit. I'll flag such spells as (up) and bold them.
Now, there are spells this inarguably works with, spells this inarguably doesn't work with, and... some spells that are somewhere in-between. Let's break this down.
Inarguably Works
Expeditious Retreat (Trivial)
Feather Fall (Trivial)
Jump (Up)
Longstrider (Up)
Alter Self (Trivial)
Darkvision (Trivial)
Enhance Ability (Up)
Kinetic Jaunt (Trivial)
Knock (Trivial)
Magic Weapon (Up)
Rope Trick (Trivial)
Skywrite (Trivial)
Spider Climb (Up)
Ashardalon's Stride (Up)
Blink (Trivial)
Flame Arrows (Up)
Fly (Up)
Gaseous Form (Up)
Haste (Trivial)
Tiny Servant (Up)
Water Breathing (Trivial)
Fabricate (Trivial)
Stone Shape (Trivial)
Stoneskin (Trivial)
Animate Objects (Up)
Create Spelljamming Helm (Trivial)
Passwall (Trivial)
Skill Empowerment (Trivial)
Create Homunculus (Trivial)
Move Earth (Trivial)
Tasha's Otherworldly Guise (Trivial)
Create Magen (Trivial)
Sequester (Trivial)
Simbul's Synostodweomer (Trivial)
Control Weather (Trivial)
Iron Body (Trivial)
Should work
Dragon's Breath (Up)
Songal's Elemental Infusion (Trivial)
Investiture of Flame/Ice/Stone/Wind (Trivial)
Draconic Transformation (Trivial)
But wait, you say, there are saving throws in these descriptions! Yes, but it's not the spell forcing the saving throw. The spell gives the target an ability. Using that ability forces the saving throw. Similarly, Tasha's Otherworldly Guise gives you an extra attack, but despite an attack being involved, it's not the spell making the attack, and I don't think anyone would argue Tasha's Otherworldly Guise doesn't meet the criteria. Providing an ability that allows the target to make an attack or force a save is not the same as the spell making an attack or forcing a save.
So I would say it's pretty clear these should work, Rules as Written. And as far as balance/intent goes, nothing is breaking the game here. Most of them are trivial. Only Dragon's Breath is gaining any up-cast benefit, and +1d6 damage on a 2nd level spell at level 6 is.... not amazing. There's no reason to make a ruling against RAW here.
Depends on target willingness?
Enlarge/Reduce (Trivial)
Levitate (Trivial)
Polymorph (Trivial)
Each of these spells only have a save if the target is unwilling. There's two ways you could understand this. Option 1: The spell doesn't force the save at all, so Empowered Transmutation always applies. Choosing to be unwilling is a choice, so the save isn't forced. Option 2: The spell only forces a save when the target is unwilling, so Empowered Transmutation applies in all cases except when the target is unwilling. There's no reason to feel strongly about these options, since it only affects what spell level the spell is cast at, so DM's call for unwilling target uses is fine.
It's complicated
Pyrotechnics (Trivial)
Control Water (Trivial)
Control Winds (Trivial)
Telekinesis (Trivial)
Transmute Rock (Trivial)
Tenser's Transmutation (Trivial)
Leaving aside Tenser's for a moment, these are complicated for a couple reasons. Many of them have multiple things they can do, not all of which involve saving throws. But some of them allow you to change what you're doing during the spell duration, which means it could be cast without a saving throw being involved in what's going on, but then switched to something that involves a saving throw in a later round.
Let's set that aside for a moment. I would argue that none of these spells directly force a saving throw. Rather, most of them change the environment, and then the environment can force a saving throw. ie, Transmute Rock just makes a bunch of rock into mud (or vice versa). The saving throws are incidental to what the magic is doing.
Telekinesis is a bit of a special case, because like the Should Work category, Telekinesis is probably best understood as giving the caster an ability that the caster can then use to cause a saving throw, or do other things. However, it gives the range to thing affected, rather than 'self', so RAW it's not clear cut. That makes it a harder call than the environmental affects.
Which brings us to the weirdest of these: Tenser's, which is weird for an entirely different reason. You may remember we placed Tasha's Otherworldly Guise in Inarguably Works. How is Tenser's different? Well, the thing of it is, Tenser's forces a saving throw on the caster when the spell expires. But the counter-point here is that it's not the spell which forces the saving throw, it's the spell ending. I honestly have no opinion.
The bright side is, all of these would be trivial beneficiaries if Empowered Transmutation worked. As such, I would be inclined to just say it does, since the feature isn't exactly jumping the power level here, and reasonable arguments can be made for all of them qualifying.
Inarguably Doesn't Work
Blindness/Deafness
Earthbind
Maximillian's Earthen Grasp
Erupting Earth
Slow
Elemental Bane
Disintegrate
Flesh to Stone
Reverse Gravity
Mass Polymorph
Shapechange
Timestop
True Polymorph
The last 4 on the list would go in other categories, except you can't cast a spell at higher than 9th level, so there's no effect, even if they would otherwise qualify.
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Now, you may have noticed there's only 12 spells which benefit from up-casting from this feature. That's.. not a large number.
Many of these up-casts are an extra target. That's actually pretty useful. Getting a free extra person to Fly or even Spider Climb helps save higher level spell slots. It's a noticeable reduction of your resource expenditure.
Magic Weapon actually goes straight from +1 to +2 at its base 2nd level casting. With 1h duration and no concentration, that actually sounds like a playable spell with this feature.
Animate Object adds a damage die to the attacks of the animated objects. Useful.
Tiny Servant creates two more servants for free. Strongest right at level 6. 8h duration and no concentration for some tiny minions, especially at levels where they might still hit stuff.
Dragon's Breath adds 1d6 damage. At level 6, 3d6 -> 4d6 isn't setting the world on fire (pun intended), but you can use it every turn with a magic action, and it is a cone. It's okay. I don't think anyone is playing a transmuter for direct damage though. Unless your DM just drops a scroll of this in your lap, it's probably not worth one of your limited learned spells at level-up.
Ashardalon's Stride does a little extra damage, but the big thing is it increases your movement speed and allows you to avoid opportunity attacks, and +1 spell level means another 5' of extra speed for free. This is actually a reasonable buff from Empowered Transmutation, especially if you have a good reason you want that speed boost. (And don't ignore the damage. Ashardalon's is guaranteed damage: no attack roll, no saving throw.)
And if you do happen to be in a campaign where enemies are actually trying to dispel your magic, don't ignore the trivial benefit. Sure, the spell doesn't do anything extra, but casting at one higher level does make it harder to dispel. If that makes the difference between your critical spell staying up or not, that's significant. The Transmuter's Transmutation buff spells are going to be harder to dispel than other wizards.
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Probably the most overlooked feature of the Transmuter, let's think about what this actually works with.
Here's the key language of Empowered Transmutation (Not the full text, but it is all the text that governs what spells are affected and how):
"When you use a spell slot to cast a Transmutation spell that doesn't make an attack roll or force a saving throw... you can increase the spell's effective level by 1."
There's a couple things this does. First, for all affected spells, it increases the spell level. That matters for things like Dispel Magic. This is trivial, but not irrelevant. I will flag such spells as (trivial). Second, for affected spells which have a defined up-cast, you of course get the 'At Higher Levels' benefit. I'll flag such spells as (up) and bold them.
Now, there are spells this inarguably works with, spells this inarguably doesn't work with, and... some spells that are somewhere in-between. Let's break this down.
Inarguably Works
Should work
But wait, you say, there are saving throws in these descriptions! Yes, but it's not the spell forcing the saving throw. The spell gives the target an ability. Using that ability forces the saving throw. Similarly, Tasha's Otherworldly Guise gives you an extra attack, but despite an attack being involved, it's not the spell making the attack, and I don't think anyone would argue Tasha's Otherworldly Guise doesn't meet the criteria. Providing an ability that allows the target to make an attack or force a save is not the same as the spell making an attack or forcing a save.
So I would say it's pretty clear these should work, Rules as Written. And as far as balance/intent goes, nothing is breaking the game here. Most of them are trivial. Only Dragon's Breath is gaining any up-cast benefit, and +1d6 damage on a 2nd level spell at level 6 is.... not amazing. There's no reason to make a ruling against RAW here.
Depends on target willingness?
Each of these spells only have a save if the target is unwilling. There's two ways you could understand this. Option 1: The spell doesn't force the save at all, so Empowered Transmutation always applies. Choosing to be unwilling is a choice, so the save isn't forced. Option 2: The spell only forces a save when the target is unwilling, so Empowered Transmutation applies in all cases except when the target is unwilling. There's no reason to feel strongly about these options, since it only affects what spell level the spell is cast at, so DM's call for unwilling target uses is fine.
It's complicated
Leaving aside Tenser's for a moment, these are complicated for a couple reasons. Many of them have multiple things they can do, not all of which involve saving throws. But some of them allow you to change what you're doing during the spell duration, which means it could be cast without a saving throw being involved in what's going on, but then switched to something that involves a saving throw in a later round.
Let's set that aside for a moment. I would argue that none of these spells directly force a saving throw. Rather, most of them change the environment, and then the environment can force a saving throw. ie, Transmute Rock just makes a bunch of rock into mud (or vice versa). The saving throws are incidental to what the magic is doing.
Telekinesis is a bit of a special case, because like the Should Work category, Telekinesis is probably best understood as giving the caster an ability that the caster can then use to cause a saving throw, or do other things. However, it gives the range to thing affected, rather than 'self', so RAW it's not clear cut. That makes it a harder call than the environmental affects.
Which brings us to the weirdest of these: Tenser's, which is weird for an entirely different reason. You may remember we placed Tasha's Otherworldly Guise in Inarguably Works. How is Tenser's different? Well, the thing of it is, Tenser's forces a saving throw on the caster when the spell expires. But the counter-point here is that it's not the spell which forces the saving throw, it's the spell ending. I honestly have no opinion.
The bright side is, all of these would be trivial beneficiaries if Empowered Transmutation worked. As such, I would be inclined to just say it does, since the feature isn't exactly jumping the power level here, and reasonable arguments can be made for all of them qualifying.
Inarguably Doesn't Work
The last 4 on the list would go in other categories, except you can't cast a spell at higher than 9th level, so there's no effect, even if they would otherwise qualify.
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Now, you may have noticed there's only 12 spells which benefit from up-casting from this feature. That's.. not a large number.
And if you do happen to be in a campaign where enemies are actually trying to dispel your magic, don't ignore the trivial benefit. Sure, the spell doesn't do anything extra, but casting at one higher level does make it harder to dispel. If that makes the difference between your critical spell staying up or not, that's significant. The Transmuter's Transmutation buff spells are going to be harder to dispel than other wizards.